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By Hongjie Yu, Mike Mooney

Transitions between geological/geotechnical soil units have a critical impact on the operation and performance of a tunnel boring machine (TBM). Geostatistics-based probabilistic modeling is used to quantify the uncertainty in soil transition locations. This paper presents two unique soft-ground tunneling case studies where the results from the geostatistics-based probabilistic modeling are compared to the TBM data and as-encountered ground conditions derived from the actual TBM data. The assessment enhances the confidence in using geostatistics to improve ground awareness for efficient tunnel construction, minimizing different site condition (DSC) claims, and improving decision-making on tunnel projects.

Dec 1, 2022

By P. H. Dudley

The piping and segregation of ingots of steel is a comprehensive subject, and must be studied in reference to the different grades of soft, medium, and hard steel, rather than to discuss it in a general manner, as though applicable to all grades. Rail steel, however, will receive the most consideration in this paper, for the wheel-loads of the motive power and rolling stock have doubled from 1890 to 1910. Steel rails and tires which were efficient under the light loads, became less serviceable under the double wheel-loads, quadrupled tonnage, and higher speeds. Bessemer Steel. Bessemer steel of from 0.10 to 0.15 per cent. of carbon, for splice-bars, spikes, and tie-plates, rises in setting, and is cast in bottle-mouthed molds, which must be capped to prevent an overflow from the top. This grade of steel rises in the molds and makes a longer ingot than the volume of molten steel when first teemed. The ingots which are allowed to cool and then cut open, show, particularly in the upper part, large occluded blow-holes, and when they are not oxidized or do not contain foreign matter, weld more or less completely when the steel is rolled or forged above 1,100° C., and it is in this way that the blow-holes are closed in the low-carbon steels. Boiler-Plate and Fire-Box Steel. These grades often contain more or less minute laminations, which are the remains of small blow-holes forming after the setting metal has reached a pasty condition. The blow-holes in the low-carbon steels have not been prevented by usiug deox-idizers, though the ingots are slightly improved so far as the soundness of the steel is concerned. This grade of steel also rises in the molds in setting.

Jan 1, 1914

By Sebastian Lobo-Guerrero

Welcome to DFI45! our first on-line annual meeting. We cannot deny the severe impacts that the COVID19 pandemic is having in our industry and our lives. We extend our support to all of our members that have been impacted in many different ways. But through the middle of this difficult and challenging times, our beloved annual conference emerges, trying to re-unite our industry and remind us why we love what we do so much. DFI45 was not originally planned to be delivered virtually, but as our world changed, we adapted. Adversity became an opportunity to re-invent our annual conference. Thanks to the vision of our Organizing Committee, we saw the amazing new possibilities and DFI45 is today a reality. Our theme for this year is People + Purpose + Passion. This theme was decided before the COVID 19 pandemic started, but today it is more relevant than ever before. We are a group of very diverse people at DFI. Our organization includes Contractors, Consultants, Suppliers, and Academics just to name a few sub-groups. But we all have a common purpose and try to find common ground. We are all extremely passionate about what we do, and this excitement is really what makes this organization what it is today. This conference celebrates this passion as you can see in our three main Keynote Sessions. On a local scale, we want to go back and remember the contribution of our industry to our beloved National Monuments in Washington D.C. On a national scale, we want to explore and recap the development of Government Guidelines and Specifications that became the world standards. And finally, we want to close the conference with a session celebrating the family legacy and share amazing stories of families with multiple generations working in our industry. Besides our Keynote Lectures, the conference is full of exciting content such as several case histories, experimental research in deep foundations, applications of new technologies, business aspects and risk management considerations for deep foundations, just to list a few. We also have our now traditional events such as the Hal Hunt Lecture, Awards Reception, DFI Educational Trust 5K Fundraiser, etc. For the first time, all this content will be delivered using different virtual formats, including on-demand presentations, plenary sessions, interactive chat rooms, etc. We are excited about these possibilities. To conclude, we need to recognize the people that against all odds made this conference a reality. First our sponsors, they adapted to the most abrupt change never thought before, and they always stayed there with us, providing unconditional support. Without them no conference is possible. Next, our Organizing Committee that re-invented the concept of an annual conference and found new and exciting ways to make this happen! Our Technical Program Committee led by our Technical Program Chairs, Jennifer Nicks and Gregg Piazza, showed us that nothing is impossible. And last, but not least, the amazing DFI staff that make dreams reality. It has been an honor to serve as the Conference Chair for DFI45. It is an experience that will stay with me for the rest of my career. Conference Chair: Sebastian Lobo Guerrero, Ph.D., P.E. American Geotechnical and Environmental Services (A.G.E.S.)

Oct 27, 2020

By J. Douglas

The past year has been a strange one for the fertilizer industry of the world and the United States. Listening to news stories, you would think the sky is falling on the world's agriculture and probably conclude that we should have had an exceptionally poor fertilizer season. In some respects, we did. Prices were low, profits were low, and in some cases nonexistent, but volume was good. We heard much in the press about the farmers' credit crunch. The truth of the matter is that only 6.6 percent of the U.S. farmers have debt-equity ratios of 70 percent or more, and a large percentage of our farmers have absolutely no debt but operate entirely on their own cash assets.

Jan 1, 1985

By W. M. Hart

The remote face loading and entry convergence monitoring systems, employed at some of the Cyprus Amax's underground coal mines, have been able to continuously record changes in face shield resistance across and entry convergence ahead of the longwall face at any moment during longwall mining. The instrumental data obtained from these systems plus in-situ observations during the mining operation can help us better understand the mechanism of overburden strata movement above a longwall face and verify the effectiveness of the current face and entry supports. This paper describes the functions of both the face and entry monitoring systems, and discusses our findings and the issues related to successful ground control for longwall mining.

Jan 1, 1996

By A. Kaczmarek

As the mining industry is enjoying a much needed and long awaited increase in demand for natural resources, consolidations in both the mining and mining service industries and a generational departure of experienced industry professionals is challenging today’s mining companies, not only in maintaining their existing operations, but also in securing qualified resources for developing new projects. These challenges are compounded for junior mining companies, oftentimes lacking in operating properties to draw from for experienced professionals and having to compete with the majors’ “mega-projects” for experienced project personnel from the reduced ranks of engineering companies. This paper discusses some strategies developed by Anatolia Minerals Development Limited in developing its Çöpler Gold Project in central Turkey.

Jan 1, 2006

By 1980, the U.S. mining community had reached a broad consensus regarding coal pillar design. The pillar load could be estimated from tributary area theory, and the pillar strength from empirical formulas and laboratory coal strength testing. Then the growth of longwall mining required new thinking. Recently, powerful design methods have emerged from analysis of large data bases of real-world pillar successes and failures. These include the Analysis of Retreat Mining Pillar Stability (ARMPS), the Analysis of Longwall Pillar Stability (ALPS), the Mark-Bieniawski rectangular pillar strength formula, and guidelines for preventing massive pillar collapses. Sophisticated numerical models have also helped transform the pillar design landscape. In the process, our understanding of pillar mechanics has been greatly enriched. A new paradigm divides pillar failure into three categories:

Jan 1, 1999

By AIME AIME

FOR many years it has been the invariable custom of the Institute, in addition to its annual meeting in February, to hold a technical meeting in the fall in some mining or metallurgical center in the West. As the activities of the Institute have increased, and especially since the absorption 'of the American Institute of Metals, there have for a number of years been two fall meetings. Never before in the history of the Institute have there been three meetings in addition to the annual meeting in February; in 1920, however, three important fall meetings are to be held. The general Institute trip will be made this fall to the Lake Superior iron and copper country. Members will leave Buffalo on August 20 by steamer, visit the copper mines in and about Houghton, the iron mines of the Marquette and Iron Ranges of Michigan and the iron mines of the Mesabi Range in Minnesota, and' return to Buffalo by steamer on September 3. Technical sessions will be held at Houghton, Minneapolis and Duluth. For a long period it was customary for the Institute to visit the Lake Superior country about every seven years. This trip, however, will be the first visit since 1904. The unprecedented development in both copper and iron mining in this section during this period will make this trip of unusual importance. In addition, a notable series of papers will be presented.

Jan 1, 1920

By M. Gebhardt, J. Johnson, S. Rosenthal

The use of presplitting applications in an underground mine setting is a method of reducing overbreak in the rib and back when blasting occurs. Presplitting performed at the Underground Mine Education Center (UMEC) on Montana Technological University’s campus in Butte, Montana showed a dramatic decrease in the amount of overbreak incurred in the blasting process. Application of the presplitting technique occurred after presplitting presentations at ISEE in San Antonio. Presplit testing consisted of drilling perimeter holes closer together than production holes and loading perimeter holes with a combination of specialized presplit explosive and packaged emulsion with electronic detonator initiation. The ideology was to shoot the perimeter row first, to create a fracture around the perimeter, so when the main production round is shot there is less shock exerted on the surrounding walls due to the pre-fractured perimeter rock. For the test performed, the back ended up with 83% of total holes resulting in half barrels representing a remarkable result. The right rib ended up with one half barrel visible before muck was removed while the left rib did not have any half barrels visible due to a clay seam being in very close vicinity to the perimeter row. The first test blast was a great start encouraging future work on presplitting applications in the UMEC. Additional testing will continue with different combinations of perimeter hole spacing, the addition of a buffer row along the ribs, and varying loading techniques to determine the most effective presplit method for the rock type of the UMEC. Detailed test results and loading configurations will be provided in the paper.

Jan 1, 2024

By Alexander N. Winchell

The cities of St. Paul and Minneapolis are located side by side on a double loop in the Mississippi River near the eastern boundary of Minnesota. Both cities lie on both sides 'of the Mississippi River, which in this region describes an S-shaped curve. The two cities together have a population of over 600,000 people, gathered in an area which has grown to be a single urban district inasmuch as the two cities have expanded until they occupy all the territory formerly between them, though the business centers are ten miles apart. St. Paul is located at the head of navigation on the Mississippi River and Minneapolis, farther upstream, at the Falls of St. Anthony. A recent water¬power development below the Falls of St. Anthony, constructed by the Federal Government, has at the same time made possible navigation to the foot of St. Anthony Falls through a lock around the dam which replaces the former rapids. Minneapolis, in 1920, had a population of 380,000 people, and St. Paul, 235,000. The two cities occupy land at an elevation of 700 to 900 feet above sea level. The average elevation is slightly less at St. Paul. Minneapolis has nearly 4,000 acres of parks and park¬ways, while St. Paul has nearly 2,000.

Jan 1, 1920

By Faisal Shalzad, Adnan Anwar Malik, Tadashi Maejima, Jiro Kuwano

Single helix screw piles carry most of the structural load through end-bearing if the helix is embedded in the dense bearing layer. The tip condition of the central shaft of the screw pile can be closed, closed with cutting blades, opened with flat cut, and opened with wedge shape cut and its selection is based on the insitu ground conditions. These different end conditions of the central shaft disturb the ground differently, leading to the change in bearing capacity. In this study, an attempt is made to investigate the effect of open and close-end screw piles on installation effort (installation load and torque) and bearing performance in the dense bearing layer. The model steel screw pile with a pile shaft diameter of 21.7 mm and helix diameter of 43 mm is used in this study. Test results show that the open-end screw pile required less installation effort than the close-end pile that penetrated the bearing layer by 1Dhwith an unplugged condition. Further, the open-end screw pile shows a higher bearing response than the close-end screw pile if it enters the bearing layer with an unplugged condition.

Oct 1, 2022

By X. Q. Cao

As the second largest phosphate resource in the world, Chinese phosphate ores generally grades with low of P2O5. The presentation of beneficiation test on the low grade of phosphate rock in this paper may be used for a reference for development and utilization of the similar reserves. The ore is a marine sedimentary phosphate rock containing low grade of P2O5 and high content of MgO and R2O3; The banded structure shows its suitability for heavy medium separation. The laboratory heavy liquid testing results indicate that as specific gravity of the heavy liquid is 2.85 g/cm3, a rougher concentrate assaying 28.71% P2O5 could be obtained while the rougher tailings contain only 4.47% P2O5. When the combined process with heavy liquid separation followed by direct & re-verse flotation is employed to treat the feed containing 21.15% P2O5, 3.79% MgO, 1.86% Fe2O3, 3.66% Al2O3, 23.75% A.I, and specific gravity of heavy liquid is 2.84 g/cm3, the optimum mesh of grind is 67.5% minus 200 mesh, a phosphate concentrate contained 32.51% P2O5, 0.48% MgO, 0.81% Fe2O3, 1.69% Al2O3 can be produced at a yield of 55.24%. The concentrate P2O5 recovery is 85% with which quality comply with the Class A specifications for production of phosphate fertilizer.

Jan 1, 2014

By D. E. Moore

The strength of a natural serpentinite gouge was measured at 400°C and an effective pressure of 100 MPa. The strength of the gouge decreased with increasing pore pressure to 50 MPa and then was nearly constant between 50 and 125 MPa. The samples at 50-125 MPa pore pressure slid stably, whereas those at 3-25 MPa pore pressure showed stick-slip behavior. Lengthening the pre-heating time to 72,000 s of samples at 100 MPa pore pressure led to increases in both the strength and the likelihood of sliding unstably. With still longer pre-heating times, the strengths decreased again and sliding once more became stable. The strength of the gouge at 3 MPa pore pressure was unaffected by changes in pre-heating time. A possible explanation for the results is that excess pore pressures are being generated in the gouge, causing the true effective pressure to be lower than the apparent value.

Jan 1, 1984

By O.W. Ellis

Recently there has been a considerable revival of interest in the effects of the various elements commonly added to brass for the purpose of increasing its strength. For many years the work of Guillet2 alone gave reliable information in this regard. Even today his determinations of the coefficients of equivalence of the various elements relative to zinc in brass are those universally employed in industry. This is so because these coefficients have stood the test of time—their substantial correctness has been proved in almost universal practice. It is not our intention to discuss here these coefficients or their use. Rather do we wish to emphasize the need for further systernatic studies of the mechanical properties of the ternary systems of copper, zinc, and other individual elements. Since Guillet published the results of his first investigations Millington, 16 Smalley33 and Thiband 45 have added somewhat to our information. In general they have confirmed the findings of Guillet, which may be stated in the following rather general terms: 1. The addition of aluminum to the 70:30 and 60:40 brasses progressively increases their hardness and strength and reduces their ductility and resistance to shock. 2. The addition of small amounts of iron to a 60:40 brass improves its strength without affecting its ductility. 3. The addition of small amounts of manganese to a 60:40 brass causes a progressive increase in its strength and reduction in its ductility. A considerable amount of information has been amassed in regard to the effect of nickel on brass (9, 22, 41) which it is unnecessary to refer to here, in view of the scope of this paper. The situation in regard to these alloys in 1926 was very clearly stated by an anonymous writer,47 in The Metallurgist of February of that year, whose remarks were so to the point that they are well worth quoting: It will he seen that there are still large fields remaining to be studied in each of these systems (those of the aluminum, iron, nickel and manganese brasses) before our knowledge is adequate. When we come to the more complicated systems the position is even less satisfactory, and on account of the very incomplete nature of the work which has been

Jan 1, 1929

By O. W. Ellis

Recently there has been a considerable revival of interest in the effects of the various elements commonly added to brass for the purpose of increasing its strength. For many years the work of Guillet2 alone gave reliable information in this regard. Even today his determinations of the coefficients of equivalence of the various elements relative to zinc in brass are those universally employed in industry. This is so because these coefficients have stood the test of time—their substantial correctness has been proved in almost universal practice. It is not our intention to discuss here these coefficients or their use. Rather do we wish to emphasize the need for further systernatic studies of the mechanical properties of the ternary systems of copper, zinc, and other individual elements. Since Guillet published the results of his first investigations Millington, 16 Smalley33 and Thiband 45 have added somewhat to our information. In general they have confirmed the findings of Guillet, which may be stated in the following rather general terms: 1. The addition of aluminum to the 70:30 and 60:40 brasses progressively increases their hardness and strength and reduces their ductility and resistance to shock. 2. The addition of small amounts of iron to a 60:40 brass improves its strength without affecting its ductility. 3. The addition of small amounts of manganese to a 60:40 brass causes a progressive increase in its strength and reduction in its ductility. A considerable amount of information has been amassed in regard to the effect of nickel on brass (9, 22, 41) which it is unnecessary to refer to here, in view of the scope of this paper. The situation in regard to these alloys in 1926 was very clearly stated by an anonymous writer,47 in The Metallurgist of February of that year, whose remarks were so to the point that they are well worth quoting: It will he seen that there are still large fields remaining to be studied in each of these systems (those of the aluminum, iron, nickel and manganese brasses) before our knowledge is adequate. When we come to the more complicated systems the position is even less satisfactory, and on account of the very incomplete nature of the work which has been

Jan 1, 1929

By Pengfei Wang, Guorui Feng

"With the depletion of easily minable coal seams, less favorable reserves under adverse conditions have to be mined out to meet the market demand. Due to several historical reasons, a large amount of remnant coal was left unrecovered. One such case history occurs with the remnant rectangular stripe coal pillars using partial extraction method at Guandi Coal Mine, Shanxi Province, China. The challenge that the coal mine is facing is that there is an ultra-close coal seam right under it, with an only 0.8 to 1.5 m sandstone dirt band in between the two. The simulation study was carried out to investigate the simultaneous recovery of upper remnant coal pillars while mining the ultra-close lower panel using longwall top coal caving (LTCC). The remnant coal pillar was induced to cave in as top coal in LTCC system. Physical modelling shows that the coal pillars are the abutments of the stress arch structure formed within the overburden strata. The stability of overhanging roof strata highly depends on the stability of the remnant coal pillars. The gob development (roof strata cave-in) is intermittent with the cave-in of these coal pillars and the sandstone dirt band. FLAC3D numerical modelling shows that the multiple seam interaction has a significant influence on mining-induced stress environment for mining of lower panel. The pattern of the stress evolution on the coal pillars with the advance of the lower working face was found. This demonstrates that the stress relief of a remnant coal pillar enhances the caveability of the pillars and sandstone dirt band below them.INTRODUCTIONChina is one of the world’s coal-rich countries. Coal has played a significant role in the growth of China’s economy. Coal reserves have been mined commercially in China for over one hundred years. The demand for coal in China is still going to grow at a relatively high rate (BP Energy Outlook, 2019). Long-term large-scale and intensive coal mining activities have created tension between increased demand and limited resources (Bai et al., 2018). Easily minable coal seams such as sub-horizontal, tabular, and laterally extensive coal seams are being depleted, and therefore the remaining reserves are becoming less favorable. However, in many situations, the coal reserves under adverse conditions have to be mined out to meet the market demand for energy.Due to sedimentary environments and geological conditions, many coal seams are interspersed with sedimentary bands, or partings both within an individual seam and between seams. These bands and partings lead to appearance of ultra-close coal seams, as well as giving rise to considerable variation in their structural and mechanical properties (Ghabraie and Ren, 2016). Difficulties are encountered in the mining of such coal seams. If ultra-close coal seams are mined sequentially, that requires the design of a mining system for each individual coal seam; also, a large number of gateroads have to be excavated, which is not a high-efficiency option. If ultra-close coal seams are mined at a great mining height at once, for instance, using longwall top coal caving—a longwall mining method that is widely used in China for the mining of ultra-thick coal seams—then the location and characteristics of the band would have a significant influence on success of their extraction."

Jan 1, 2019

Jan 1, 1947

By W. S. Holt, E. T. Tkac, R. A. Grange

QUANTITATIVE knowledge of the time element involved in austenite transformation in a particular steel provides a sound basis for understanding and planning heat-treatment. Such knowledge is conveniently obtained by measurement of austenite transformation as it occurs at each of a series of temperature levels. The results of these measurements usually are summarized in the form of the now familiar isothermal transformation diagram (I-T diagram, TTT diagram, or S-curve). Although a considerable number of such diagrams already have appeared in the literature, there remain many compositions of commercial importance for which no I-T diagram is available. The diagram for many such steels may be predicted from published data with sufficient accuracy for most practical purposes, but for others no satisfactory prediction can be made because the effect on austenite transformation of one or more of the important alloying elements present is not well enough known. Such a steel is Nitralloy, Type 135-Modified, which, containing about 1.25 per cent aluminum, differs from any whose I-T diagram has been published. This paper presents results of a study of the isothermal transformation behavior of a sample from a commercial heat of this aluminum-chromium-molybdenum steel, together with pertinent supplementary data including the equilibrium transformation temperatures (Ae3 and Ae1), the temperature range of martensite formation, and the end-quench hardenability. These data are directly applicable to the heat-treatment of this type of steel and are of interest in revealing some of the fundamental effects of aluminum as an alloying element in steel. MATERIAL All specimens were prepared from a 13-in. diam bar forged from a 5 by 5-in. billet from a commercial heat made in an electric arc furnace. The composition of the bar was; C, 0.41 per cent: Mn, 0.57: P, 0.016: S, 0.005: Si, 0.24: Ni, 0.17: Cr, i.57: MO, 0.36: Al, 1.26. The forged bar was normalized from 1750°F (955°C) and then tempered for one hour at 1200°F (650°C), this tempering treatment having been given to facilitate the machining of specimens. EQUILIBRIUM TRANSFORMATION TEMPERATURES In hypoeutectoid steel the minimum temperature at which austenite is completely stable, Ae3, and the maximum temperature at which austenite is completely unstable, Ae1, are of great significance in heat-treatment. Since these temperatures are dependent upon com-

Jan 1, 1946

By David A. Neudorf

Known resources of nickel laterites considerably exceed nickel sulfide resources and there is a growing consensus that most new nickel production will be sourced from laterites. The development of nickel laterite projects is very capital intensive due to inherently low grades, inability to concentrate ore prior to processing, complex processing technology and typically large infrastructure requirements. This has resulted in most recently proposed projects being "mega-projects," to achieve economics of scale. However, this may result in unacceptable project risk in many cases. Furthermore, many known laterite resources are not large enough to support a mega-project. This paper describes an alternative model for nickel laterite project development that allows, smaller scale and reduces capital exposure. The requirements of this model in terms of process technology, infrastructure and commercial innovation are described. Keywords: Nickel, Laterite, Project, Economics

Jan 1, 2004

By J. Szekely, Robert J. Fisher

A formulation is Presented and solutions are given for the problem of transient heat conduction in a sphere that undergoes a spatially uniform radiative heat exchange with its environment. The solutions were obtained by the numerical integration of the appropriate Partial differential equation and are Presented in a graphical form by Plotting a dimensionless average temperature against the Fourier number with the radiation number as a paranreter, for various initial sphere and environment temperatures. In addition to these numerical results asymptotic analytical solutions are also given. PROBLEMS involving the transient heating or cooling of droplets and particles frequently arise and are of importance in metallurgical operations. Spray refining, vacuum degassing, high-temperature gas-solid reaction systems, and pebble heaters are but few of the numerous examples that may be quoted. The practical interest in these problems from a design or operational point of view is to evaluate the change in the particle temperature with the time given its physical properties, initial temperature, and that of the environment. At the high temperatures encountered the principal mode of heat transfer between the particles and their environment is by thermal radiation; in case of vacuum degassing due to the very low thermal conductivity of the gas space the convec-tive components are negligible and radiation is the only heat transfer mechanism. Although much work has been done on transient heat conduction problems, the special class mentioned here, in which thermal radiation is specified at the bounding surface, received little attention to date. One contributory factor to this situation may be the nonlinearity of the radiation term in temperature, which in general precludes the use of analytical techniques and requires numerical or analog solution of the appropriate differential equations. In fact the only solutions of general nature that are available at present pertain to semi-infinite media and slabs.'-3 The work on which the paper reports forms part of a program aimed at the theoretical and experimental study of transient radiative heat exchange between particles and their environment. The purpose of the present paper is to develop a formulation and solutions to the problem of transient heat conduction in a sphere that undergoes a spatially uniform radiative heat exchange with its surroundings. The solutions, obtained by the numerical integration of the appropriate differential equations, will be presented graphically, by plotting dimensionless parame- ters covering the range of variables relevant to metallurgical interests, thus affording a ready reference for design calculations. In addition to these results, approximate analytical solutions are also given which represent asymptotes, corresponding to extreme values of the variables mentioned. FORMULATION Consider a spherical particle of radius, a, initially at uniform temperature, Ti (T measured on the absolute scale), having a thermal conductivity, k, density, , specific heat, C, and thermal diffusivity, a. At time = 0 let this particle be exposed to an environment which is at a temperature T, and subsequently let there be a purely radiative heat exchange between the particle and the environment resulting in a spatially uniform heat flux at the surface. If in addition to the above we make the assumptions that the sphere is a gray body radiator, homogeneous, isotropic, and opaque, and that the physical properties are independent of temperature, conservation of heat in the sphere may be written as follows: The boundary condition contained in Eq. [3] states that the initial temperature of the sphere is considered uniform, equalling Ti, whereas Eq. [4] is readily obtained from symmetry considerations. Eq. [5] expresses the continuity of the heat flux at the surface of the sphere, or in other words the condition that the rate of heat exchange between the sur-

Jan 1, 1969