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By A. Haghighat

A risk analysis on mine fires for all U.S. underground coal mines from 2000–2012 was completed to identify locations that have a high risk for fire incidents. The data for this analysis was extracted from the Mine Safety and Health Administration’s (MSHA) mine fire accident reports database. After identifying fire incidents from this database, a risk matrix was developed for underground locations that showed a significant propensity for fire incidents. This matrix associates the hazards and risks with the severity and frequency of their consequences at each location. These significant mine areas will be the concentration of future studies in fire behavior, suppression techniques, and simulation methodologies. INTRODUCTION Fires in underground mines pose significant hazards to personnel because of inherent restrictions to movement and limitations to egress routes. Despite improvements, mine fires remain serious hazards in underground coal mine. In order to better mitigate the risks from fire hazards in the modern underground mining environment, a better understanding of where fire events occur must be gained. The following paper presents an investigation on the number of fatal and non-fatal incidents in underground coal mines fires from 2000 to 2012. Three phases of risk assessment (i.e., risk identification, estimation, and evaluation) were applied to the recorded fire events from the MSHA incident database that occurred during this time period. Using this process, a fire hazard was first defined. This definition was then used to identify any incidents that had occurred as a result of the defined hazards. The seriousness of the injuries and damages were finally quantified for each incident though an estimate of severity and frequency and then aggregated according to underground location (ISO, 1999; Van Duijne, 2008). Using this approach, locations prone to incidents resulting from fire hazards were first identified. The risk for the occurrence a fire incident in general was then estimated for each of the identified locations. The results of this assessment will be used to improve fire simulation techniques for the identified, high risk underground locations. RISK IDENTIFICATION A risk for this assessment was defined as any single occurrence of a fire initiated by the ignition of gas, dust, or electricity. Each recorded occurrence of such an event in the MSHA incident database was counted as a single fire incident. Both non-fatal incidents that were reportable but did not result in a fatality and fatal incidents that fell into the aforementioned criteria were counted. Incidents in the MSHA database that were already classified as “generic” were not considered. The remaining incidents were then totaled and aggregated according to the underground location in which they occurred. In previous studies, a number of locations, which are listed in Table 1, were highlighted as high fire hazard areas (McDonald et al, 1980; Pomroy et al, 1995; De Rosa, 2004). These locations were used to group the fire incidents extracted from the MSHA incident database. The compiled numbers of non-fatal and fatal underground coal fire incidents from 2000 to 2012 are displayed in Table 1.

Jan 1, 2019

By R. O. Y. Breese

The term "diatomite" implies the nearly pure sedimentary accumulation of diatoms. Sedimentary and chemical purity, low density, inertness, intricate fine structure, high brightness and other properties give the diatom and diatomite unique industrial usefulness and commercial value in high value functional filler and specialty filter aid products. Although diatomite deposits occur commonly throughout the world in Miocene and younger marine and lacustrine sediments, deposits that contain significant reserves of crude suitable for these quality sensitive markets are uncommon. U.S. production is centered on exceptional deposits located in Arizona, Nevada, Oregon, Washington and California. Evaluation of deposits requires understanding of geology, mining, production and end use. As crude quality in deposits varies within and between strata, ore suitable for quality sensitive end uses must be identified through sampling and evaluation programs tailored to permit recognition of critical variations.

Jan 1, 1989

By Gordon M. Pugh, David G. Rasmussen

INTRODUCTION This method is a variation of cut-and-fill mining where either the wall rock or the vein is too unconsolidated to permit safe mining by conventional cut-and-fill methods. Basic development consists of driving a 2.4 x 2.7-m (8 x 9-ft) haulage heading laterally in the footwall side of the vein. Crosscuts are driven over to the vein at 91-m (300-ft) intervals. Then the vein, at each inter¬section, is mined out to a distance of about 9 m (30 ft ) above the track level. After the installation of an ore chute, scrams or intermediate drifts are driven in both directions along the vein. Mining then starts imme¬diately below the level bounding the upper limit of the stope. A cut varying between 3 and 4.2 m (10 and 14 ft) in depth and mined for a distance of 45.7 m (150 ft )along the strike is taken from the orepass in each direction (Fig. 1). In this mined-out void, the floor of the cut is equipped with 203 x 203-mm (8 x 8-in.) caps hitched into the wall with steel saddle hitches installed on 1.5-m (5-ft) centers. Double wood stringers, 152.4 x 152.4 mm (6 x 6 in.), are then installed on 1.2-m (4-ft) centers longitudinally down the length of the stope. Reinforc¬ing wire mesh is also placed, overlying each mat of timber. Directly above the stringers, stulls are installed from hanging wall to footwall to hold the timber in place during the sand-filling operation. Then a cement¬sand mixture in a ratio of about 1:10 weight is poured to a depth of about 2.4 m (8 ft). The entire cycle is then repeated down to the scram drift. Another variation of timbering is illustrated in Fig. 2. There is a vein width of approximately 5.5 m (18 ft) with the same unconsolidated ground conditions. The primary difference between handling this situation and the previously described undercut-and-fill method is that standard timbered sets are installed in each excavated stoping void rather than stulls. Once again the basic method is to work down the ore block from an upper timbered drift to a lower timbered drift. Cuts are taken from a timbered raise with a chute on the stope side. Each piece of ground excavated is essentially two sets high, 5.5 m (18 ft), by two sets wide, 5.5 m (18 ft), by the length of the cut along the strike of the vein. The timbered sets are 2.4 x 2.4 x 2.4 m (8 x 8 x 8 ft). The net result is, therefore, two sets of timber on a top floor separated by doubled-up 76-mm (3-in.) lagging from

Jan 1, 1982

By G. Righettini, P. Mousset-Jones

Diesel powered vehicles have made underground mines more efficient, but with the advent of DPM regulations, in most cases, extra ventilation and/or engine emissions control technology will be required. Using fuel cell powered vehicles will not need these measures to be taken. Nine US underground mines of varying production rates and mining methods were studied to determine possible ventilation reductions if their diesel fleets were replaced with identical fuel cell powered vehicles. Factors that might limit ventilation reductions, such as heat and the effects on other pollutants were determined for each mine. The ventilation cost savings and other benefits of fuel cell vehicles show that when they become commercially available they have the potential of being viable alternatives to diesel vehicles.

Jan 1, 2004

By M. D. Flavel, J. C. Motz, G. V. Jergensen

Introduction Compared to the gyratory crusher, the cone crusher is character¬ized by its higher speed and a flat crushing chamber design which is intended to give a high capacity and reduction ratio for materials suitable to this type of processing. The aim is to retain material longer in the crushing chamber to do more work on material as it is being processed. A concept of how a piece of stone might flow through a secondary crushing chamber is shown in Fig. 45. The number of times material will be nipped during crushing will depend on material size, friability, and the geometry of the crushing chamber as well as its speed and eccentric throw. Cone crushers are usually specified in terms of closed-side setting (CSS) and a given quantity of material passing a particular square-mesh screen size. A criterion of product passing closed-side setting is often available. This value will vary according to the particular crusher design and the details of its applica¬tion. Essentially, cone crushers can be classified into distinct types, depending on their duty. Crusher sizes for all duties are described by terms arising out of common industry usage, such as 3-ft, 7-ft, etc., which refer to the crushing-head (mantle) diameter. At present the range of sizes available varies from approximately 2 to 10 ft diam. Weights vary in the range of 5 to 200 tons, and connected horsepowers range from 10 through 700. There are so many variations that specifics should be obtained from machine manufacturers. The crushing chambers that are fitted to various machines are also often referred to by terms arising out of common usage, such as standard for secondary crushers, and short head for tertiary crush¬ers, which refer specifically to the Symons design. Also used are more specific but equally vague descriptions, such as coarse, intermedi¬ate, and fine chambers. Some manufacturers such as Allis-Chalmers with their Hydrocone crushers refer to the model number of the machine by the feed opening and the diameter of the mantle. For example, a Model 10-84 Hydrocone crusher has an 84-in. diam man¬tle, and is capable of accepting a 10-in. diam sphere at the feed point to the crushing chamber. Rexnord and Telsmith define crushing cavi¬ties in their capacity tables as feed openings at minimum recommended discharge settings (closed-side settings). Each practice offers guidelines which should be carefully defined for the specific crushing problem. The utilization of a cone crusher depends on how well it is controlled and the features of the circuit in which it operates. Normally, it will operate in the range of 75 to 85% of the peak capacity. It is likely that industry pressures will call for a standarized de¬ applied crushing power. Experience has shown that the crushing process is more controllable if certain reduction ratios are adhered to for each stage. Technically, this is defined as that size of feed (in. or mm) of which 80% will pass, divided by the size of the product (in. or mm) which 80% will pass. Secondary Crushers These are the cone crushers which accept suitable feed size material either prepared by a primary crusher or occurring naturally, and reduce it to a size suitable for marketing in one stage, or make feed for subsequent crushing or grinding stages. Secondary crushers have feed openings of 4 to 25 in. in the larger (7-ft) models down to a 2'/a to 4 in. in the smallest 24-in. models. Reduction ratios normally range between 3:1 and seldom more than 5:1. In the 7-ft models, the cone crusher typically makes a product all passing 2 in., but this is dependent on the machine design and properties of the material being treated. Secondary crushers most commonly operate in open circuit. They also are operated in closed circuit with a vibrating screen, depending upon product requirements and conditions where the reduction ratio does not cause excessive power draw and build-up of circulating load. Screening ahead of secondary crushers is generally recommended, especially where the feed contains more than 25% material smaller than the desired closed-side setting. Tertiary Crushers These are cone crushers that normally take secondary crusher product and reduce it to a marketable product or make it suitable for subsequent comminution steps. The reduction achieved is a func¬tion of the crusher design and the properties of the material to be treated. The reduction ratio is normally in the range between 1.5 and 2 to 1, and seldom more than 3 to 1. The tertiary crusher normally operates in closed circuit with a vibrating screen and makes a product smaller than all passing 'h or 5/8 in. Fine Crushers Sometimes these machines are called sand crushers and are called by various manufacturer brand names, such as Gyradisc, Hydrofine, V.F.C., and others. A cross section of a Symons Gyradisc crusher is shown in Fig. 46. Essentially, these crushers would be used in a fourth-stage crushing operation, but could be called on to reduce a screened feed size fraction from a secondary or primary crushing operation. This type of crusher normally receives feed no coarser than 1'h in. that is scalped of all over-sized material and operates in closed circuit with a vibrating screen. For average materials, typical product is '/e in. top size, although a -10 mesh product can be produced on those materials having suitable characteristics. The crusher is nor¬mally operated in a separate circuit from the main crushing plant because of the variations in output rates that are caused by varying physical properties in the feed. The reduction ratio is generally less than 2:1, and circulating loads are generally high. More interparticle crushing takes place in these machines than with conventional cone crushers and capacities can be more sensitive to changes in moisture content, feed size graduation, and other physical properties. Design Features For the various applications, crusher manufacturers normally have a range of crushers of different sizes and power ratings to select from.

Jan 1, 1985

By Allan M. Johnson

The Sherwood Mine is being monitored to determine the nature and extent of changes in the mine and its environs which result from the closing and the cessation of pumping. Of particular interest was the possibility of piping-induced subsidence occurring as the mine flooded. Previous experience in the district indicated that this possibility was very real. A monitoring system, including acoustic sensors, extensometers, surface subsidence monuments, and ground water observation wells was installed in July 1978, just prior to the mine closing. Late in 1979, three piping-induced subsidence events occurred. The monitored data, and related laboratory studies are being applied to a quantitative description of piping-induced subsidence.

Jan 1, 1980

By Robert Stefanko

The most perplexing problem that has confronted mining engineers since the dawn of mining has been strata control. For the purposes of this paper, strata control will be defined as control of ground in the immediate area surrounding the, mine opening. To prevent movement of the surrounding strata into underground mine openings, miners have traditionally relied upon some type of artificial supports placed within the openings and located in a manner determined by trial and error from previous experience. Without the means to measure changes of stress in the supports and within the mine structure itself, an evaluation of the efficiency of the supports and mining system was impossible. Strata control thus remained an art with very little application of science. During recent years a revolution has occurred in mining. The modern mining engineer is contented no longer-with trial and error methods but is utilizing all the technological information at his command to obtain a solution to this age old problem. However, serious problems have arisen due to the lack of certain basic knowledge in three important areas: 1, The original stress distribution within the earth strata prior to mining. 2, Redistribution and concentration of the original stresses with mining activity. 3. The properties of rock that are significant in failure and its prevention.

Jan 1, 1960

By Neil H. Bostick

This research, sponsored at Stanford Research Institute by the Grand Junction Operations Office of the U.S, Atomic Energy Commission, was an outgrowth of the concern that presently-known ore reserves may be inadequate to meet the projected demands of nuclear power reactors. The original program at SRI was to design a comprehensive geological information system and use it to store and analyze the very great quantity of data needed to estimate uranium ore reserves. Study of the problem showed that it could be solved only by use of modern electronic data processing techniques, so such a comprehensive system was designed and tested. The system test was satisfactory, but input lead time and total expense were shown to be too great for the system to be practicable for the AEC under present circumstances. Thus, based on results of the earlier research, a specialized and simplified procedure for predicting approximate uranium ore reserves was devised. The prediction is based on general geology of specific area cells of the earth, in order to provide complete volume coverage; the strata in these ores are evaluated for uranium ore favorability on the basis of a few selected criteria in order to allow rapid data input. Since the bulk of U.S. reserves are in clastic host rocks only such strata are considered in this work. This paper describes the structure and some details of the resource system and how it was tested in rn area of the western United States.

Jan 1, 1969

By Madeline Mello, Michael Low, Nick Karlin, Bryce Scofield

The LA Effluent Outfall Tunnel is the first tunnel in the United States to incorporate post- tensioning into the precast segmental lining. Several unique components were integrated to ensure that the post tensioning could be carried out successfully in the tunnel, even concurrently with the TBM excavation. This also required the development of new processes both in the casting facility and in the TBM to ensure that the almost 3,700 rings could be successfully manufactured, threaded, stressed, and grouted. This paper will discuss the material selection, fabrication, and installation of, as well as lessons learned from, this first in country tunnel lining.

Jun 23, 2024

Those involved in the mining and minerals industry really are an optimistic bunch. Faced with an economic downturn that has spread across the globe ? one, some say, that crept up on everyone ? mining professionals remain hopeful that the industry will return. An example of that optimism is the attendance at two of the industry?s key conferences and exhibitions. In February, more than 4,200 people attended the SME Annual Meeting and Exhibit in Denver, CO. And a week later, in Toronto, Ontario, more than 18,000 were on hand for the Prospectors and Developers Association of Canada?s (PDAC) meeting. Both are respectable numbers and run contrary to meeting attendance figures during previous downturns when attendance and the numbers of exhibitors fell off significantly. Like other downturns, this most recent one has resulted in massive layoffs, curtailments of production and outright closures. Exploration has been hard hit and many junior mining and exploration companies will either disappear or be swallowed up by others. Gold is its usual self The PDAC generally attracts the gold mining and exploration sectors. So that might explain some of the optimism at the meeting. Unlike base metals and construction materials, gold, not surprisingly, is doing well. And most of the companies in the Investors Exchange at PDAC were involved with gold. While the balance sheets for most base metals companies look bad, gold companies appear to be not as hard hit. Gold is doing what it usually does during a downturn ? act as a safe haven while the value of other assets collapse worldwide. Governments around the world are preparing to pour trillions of dollars in an attempt to resuscitate their economies. This will likely lead to inflation, some experts at the PDAC convention said, which will be good for gold. And gold mining companies are also benefiting from lower production costs. The result is that some gold watchers believe gold could reach $64/g ($2,000/oz) at some point.

Jan 1, 2009

By D. B. Spears

The sillimanite group includes the aluminosilicate minerals sillimanite, kyanite and and alusite, all of which have the same chemical composition (Al2SiO5). This discussion also includes the closely related mineral, mullite(about Al6Si3O15), which, although rare in nature, is produced artificially for industrial use. All of the minerals in this group have a low coefficient of thermal expansion and high corrosion resistance. This makes them desirable for refractory applications, especially in the steel and foundry industries. Other uses include electrical insulators, abrasives and household ceramic products. The minerals in the sillimanite group decompose with an irreversible expansion at temperatures between 1,300° C and 1,600° C (2,372° F and 2,912° F). Sillimanite and and alusite have relatively small expansion on decomposition (4 percent to 6 percent) and, therefore, can be used as aggregates in refractory products. Kyanite’s expansion on decomposition is large (18 percent). So it is sometimes used to balance shrinkage of other materials such as clays in refractory mixtures. For applications requiring high-temperature volume stability, mullite is more desirable. As an additive, mullite improves resistance to creep and corrosion, and imparts thermal and electrical insulating properties.

Jan 1, 2005

By D. Duvris

The main objective of this work was to quantitatively investigate exhaust emission levels relative to maintenance (service) intervals during the operation of diesel-powered equipment at an underground bauxite mine in Greece. Such emissions depend on a great number of parameters, including the type of engine (technology) and its tuning, the type of work performed (as related to the engine workload), the suggested maintenance intervals, the altitude, the temperature and humidity of the atmosphere in the underground mine, the properties of the fuel and the operator's training and experience levels. The necessary ventilation quantities were calculated based on exhaust gas-concentration measurements and site-specific relationships between pollutants and adjusted air-quality indices. Diagrams comparing the necessary and recommended airflow for several equipment groups, featuring two engine types and different maintenance intervals, are presented.

Jan 1, 2000

By Nigel Fung, J. D. Humphrey, R. H. Oats, J. H. Ingle

Production dozing of cast material is a cost effective method of improving dragline or mobile equipment overburden removal. Yet dozer productivity is reduced during the part of the process that requires the dozers to cut material away from the highwall. One solution is to use a hydraulic excavator to remove this material and throw it into a windrow for the dozers. By freeing up the dozers for more productive work the overall system productivity increases 25% and cost per yard is reduced by up to 10%. Dozer fleet sizing can be determined to match a specific production schedule or can be sized to balance with an excavator’s production.

Jan 1, 2004

By K. F. Unrug

The research work reported in in this paper deals with the monitoring of roof behavior in a drift mine located in Eastern Kentucky operating in the Coalburg Seam. The thickness of the seam at the monitoring site ranges between 9 and 10 feet with the immediate roof changing from sandstone to shale then back to sandstone. Shale thickness varies between 0 and 2.5 feet. The characteristics of the immediate shaly roof were monitored with 38 dynamometers which were set up in three survey lines to cover the entire range of shale distribution. The dynamometers are capable of providing accurate load readings in the range of +/- 5 pounds. To closely observe the contribution of moisture toward roof deterioration the dynamometers were set up in intake and neutral airways. A television borescoping camera was used in conjunction with the dynamo- metric instrumentations at pre- determined observation holes to physically observe and record the separation process of the shale and sandstone stratum with respect to time. Monitoring took place between January 1985 through January 1986 in order to cover a full cycle of dry and wet mine conditions associated with the seasonal changes. Analysis of the data show a trend of roof deterioration induced by condensation and eventual penetration of moisture within the immediate roof bedding planes, leading to rapid disintegration of the mine roof.

Jan 1, 1986

By QINGXIANG CAI, Lin Li, PENG LIU, IZHAR MITHAL JISKANI, XIANG LU, Wei Zhou, YONGLI ZHOU

Green mining is the latest technological advancement to reduce mining’s environmental impact with minimal energy and resource consumption. Green mining — aimed at protecting nature and providing socio-economic benefits — is a multifaceted network of its connotations and key indicators. China has not yet defined its connotation and formed an effective evaluation system for surface mining. We defined its connotation and developed an evaluation index system based on the theory of green grades. As the indicators are selected based on extensive research, they express the advantages and disadvantages of various aspects of a mine. The index is composed of three attributes (safety, efficiency and environment), nine criteria and 35 indicators. The indicators were quantified according to experts’ opinions using the fuzzy comprehensive evaluation method. The results of a case study show that the environment is the most critical aspect, followed by efficiency and safety. This study can provide technical support for the development of green mining.

By F. Habashi

Chile is at present the largest copper and rhenium producer and was before World War I the largest nitrate producer. She is also an important producer of lithium carbonate and molybdenum oxide. An important event in the history of the country is the Pacific War of 1879-1883 in which she acquired the northern provinces initially belonged to Bolivia and Peru. In 1888 Chile annexed Easter Island. The Chilean mineral and cultural history is briefly reviewed together with the people who contributed to her development. The present report is the result of a number of academic and industrial visits to Chile during the period 1992 to 2009.

Jan 1, 2011

By J. M. Wempen, R. D. WEYHER, M. K. McCarter

Beam theory is a proxy for the behavior of intact, deforming roof strata above underground openings. Strong, stiff, intact strata have the ability to store significant strain energy when underground openings have large spans, as is typical of longwall operations. When strata fail, they may release some or all of this strain energy. Released energy is partitioned between fracture, thermal and seismic energy. The tensile failure surface in simply supported beams is predicted to be free of shear stress. This prediction and the kinematic inability of crack slip during fracture make significant dissipation of energy by friction unlikely.  In this study, the failure energy partitioning of specimens from the Nugget Sandstone formation is examined. Tensile fractures in specimens were induced using three-point loading tests. Elastic waves emanating from fracture events were recorded using an array of accelerometers. Accelerometers were placed within two crack lengths of the fracture surface, and the acceleration records are similar to strong motion seismic records. Spectral analysis was performed and used to develop synthetic, well-behaved, acceleration signals. Synthetic velocity signals were used to estimate radiated seismic energy generated by fracture. These estimates of radiated seismic energy demonstrate a potential means to constrain the minimum seismic partition. Fracture testing was performed on additional samples to estimate the fracture toughness of the sandstone. Synthesis of this data demonstrates a potential means to constrain a maximum seismic partition. Bounding the seismic partition for this type of fracture event could aid the detection and evaluation of seismicity emanating from longwall operations. Having better information about the location and magnitude of mining induced seismicity within longwall overburdens could lead to better understanding of extraction sequencing and rock behavior in general. These energy estimates also support the hypothesis that dissipation of thermal energy by friction is small in bending tensile failure.

By George J. Dvorznak, Robert W. Waytulonis

The Bureau of Mines is conducting a research project in cooperation with the Mine Safety and Health Administration (MSHA). Mining companies, and equipment manufacturers are also participating through the Colorado Mining Association (CMA). The objectives of this cooperative research project are to determine the practicality of coupling a dry exhaust conditioning system (dry system) and diesel particulate filter (DPF) to make an integrated control system, and to determine the effectiveness and durability of the integrated system in an underground coal mine. Diesel-powered mining machines must be equipped with control devices to reduce surface and exhaust temperatures to safe levels and prevent flames and sparks from being emitted to the mine atmosphere. Diesel exhaust-gas water scrubbers have been successfully used for many years but they have several disadvantages; they are large, consume large quantities of water, and require frequent maintenance to remain functional. Additionally, diesel exhaust is a source of noxious gases and particulate matter. Particulate emissions are composed primarily of small carbon particles with absorbed hydrocarbons and they are almost entirely respirable in size. Even an engine in good mechanical condition can emit up to 400 mg/cu m of respirable particulate matter. A dry system for control of diesel fire and explosion hazards, coupled to a DPF, promises to be a complete control system for diesel engines. The dry system cools the exhaust and suppresses sparks and flame with a heat exchanger and flame and spark traps. The conditioner is smaller, consumes no water, and requires less maintenance than a comparable water scrubber. The DPF uses a ceramic substrate as a wall-flow filter and is very efficient at collecting exhaust particles. Average particulate mass collection efficiency is 90%. This report describes a cooperative research project to combine the dry system with the DPF, conduct performance tests on the integrated system at Bureau and MSHA 1aboratory facilities, and conduct field tests of the system on a production machine in an underground coal mine.

Jan 1, 1987

By Andres Salazar, Carlos Delgado Robles, Richard K. Preece

The paper discusses the evolution of the geometallurgical model at the Escondida mine in northern Chile, the world’s largest copper producer. It highlights the transition from early metallurgical forecasting methods to a comprehensive model that includes spatial distribution of metallurgical forecasts. This model helps in planning and adapting to the variability in ore characteristics, such as hardness and mineralogy, which affect processing performance.

Dec 1, 2023

By [Miroslaw] M. Hajdasinski

The Payback Period (PBP) is one of the most widely applied project evaluation criteria 'in the mineral industry. As much as this criterion is liked by practitioners, it is also criticized by academicians who seriously question its validity. Inspired by this criticism, the author of this paper examines the classical definition of the PBP criterion, and formulates a new one designed for both investment and borrowing projects. It is demonstrated that the PBP, in its revised version, is free from all the shortcommings of the old one, and is fully compatible with the Net Present Value (NPV) criterion, as well as with all the other NPV compatible project evaluation criteria. Since the traditional approach to the comparison of mutually exclusive projects (MEP1s) by means of the PBP criterion has been incorrect, the paper also outlines the correct approach to this problem. Finally, owing to the fact that the PBP depends on the assumed interest rate, and using a specially developed computer program, the PBP is graphically presented, in six numerical examples, as a function of the interest rate.

Jan 1, 1989