Rectangular Distribution Research Articles

Deconvolution of fractionation data to deduce consistent washability and partition curves for a mineral separator

A density partition curve for a hypothetical steady-state separator is applied to a known feed density distribution to give the density distributions of the product and reject. Density fractionation of each of these streams is then simulated, with the fractionator Ecarte Probable, EpX, set at a specific level, to produce a set of seven or more fractions of varying mass and increasing average density. This study then describes a new algorithm that attempts to recover the partition curve of the original steady-state separator, using only the three sets of limited fractionation data and the assumption that the form of the partition curve equation is known. The algorithm first uses a simple interpolation rule to convert each set of fractionation data into a cumulative density distribution. Then the feed density distribution and the partition curve parameters are simultaneously adjusted until a consistent set of feed, product and reject density distributions is found with minimum variation from the raw fractionation data. The algorithm was applied to a simple rectangular feed distribution, and then a more realistic distribution. In both cases the algorithm accurately determined the density cut point (D50) of the separator, even for poor quality fractionations. The accuracy of the determined separator Ep value depended on the fractionator EpX and the amount of near-density material. For the simple rectangular distribution, the algorithm under predicted the separator Ep, with the error being about 34% of the fractionator EpX. For the more realistic feed distribution, there was more scatter in the Ep values, but still the same general trend. The error increased when there was little near-density material. Increasing the number of flow fractions from 7 to 11 brought some improvement in accuracy. However, above 11 fractions there was no further significant improvement. Expressing the partition function in terms of D75 and D25 (instead of D50 and Ep) reduced the sensitivity of the algorithm to the initial guess values.

Effect of particle shape on the flotation kinetics of fine coking coal

Abstract Particle shape has a significant effect on the flotation process of fine mineral and fine coal particles. In this paper, the effects of particle shape on the flotation kinetics and behavior of fine coking coal of different size fractions were investigated. The coal particles with different shape properties were gained from the grinding and crushing products of the rod mill and jaw crusher, respectively. The shape parameter of the coal particle was measured by Image J software by analyzing the photos of coal particles taken by the microscope. The flotation tests of fine coking coal were done in a 0.5 L XFD flotation cell without any usage of flotation collector in order to study the natural floatability of coal particle. Six flotation models were used to assess how the particle shape affects the flotation behavior of fine coking coal of different size fractions. The results showed that the coal particle with a larger elongation ratio had a higher flotation recovery. The first-order model with rectangular distribution of floatability was found to be well consistent to the experimental data, and the modified flotation rate constant of the flotation process increased with the increase of elongation ratio.

Impact Pressure Distribution on a Monopile Structure Excited by Irregular Breaking Wave

Abstract The problem of impact pressure distribution on a monopole structure excited by irregular breaking waves is investigated. The analysis is performed by applying a numerical model that combines potential flow model with a Navier-Stokes/VOF solution. The temporal pressure distribution is analysed for two breaking wave cases characterized by the significant difference in the steepness of the wave front. The peak impact pressures are observed in the region below the overturning wave jet where the pressure increases rapidly resulting in a peak value of the slamming coefficient equal to Cs=2π. The vertical load distribution provided by the derived model is more realistic than a rectangular shape distribution applied in engineering practice. This is because the vertical load distribution strongly depends on breaking wave shape and it is difficult to uniquely approximate such a load distribution by a rectangle.

Freeform lens generation for quasi-far-field successive illumination targets

A predefined mapping to tailor one or more freeform surfaces is employed to build a freeform illumination system. The emergent rays from the light source corresponding to the prescribed target mesh for a pre-determined lighting distance are mapped by a point-to-point algorithm with respect to the freeform optics, which involves limiting design flexibility. To tackle the problem of design limitation and find the optimum design results, a freeform lens is exploited to produce the desired rectangular illumination distribution at successive target planes at quasi-far-field lighting distances. It is generated using numerical solutions to find out an initial starting point, and an appropriate approach to obtain variables for parameterization of the freeform surface is introduced. The relative standard deviation, which is a useful figure of merit for the analysis, is set up as merit function with respect to illumination non-uniformity at the successive sampled target planes. Therefore, the irradiance distribution in terms of the specific lighting distance range can be ensured by the proposed scheme. A design example of a freeform illumination system, composed of a spherical surface and a freeform surface, is given to produce desired irradiance distribution within the lighting distance range. An optical performance with low non-uniformity and high efficiency is achieved. Compared with the conventional approach, the uniformity of the sampled targets is dramatically enhanced; meanwhile, a design result with a large tolerance of LED size is offered.

Determining the Best Particle Size-Class for Flotation of a High Ash Coal

ABSTRACT The purpose of this study was to investigate the effect of particle size on coal flotation in terms of kinetic parameters such as flotation rate constant and maximum recovery. Six different flotation kinetic models were applied to data from tests using the MATLAB statistical analysis tool package to estimate the relationship between the flotation rate constant, the maximum combustible value recovery, and the particle size. Within the size range studied, the best flotation performance was obtained at −75 + 25 µ size fractions. The result shows that all kinetic models except the second-order model with rectangular distribution of flotabilities gave an excellent fit to the experimental data at various size fractions. From the statistical analysis, we conclude that coal flotation with combustible matter recovery as the response can be described with the first-order model with rectangular distribution of flotabilities, .

Orthotropic hydraulic permeability of arrays of parallel cylinders.

Approximate analytical methods are presented to calculate the overall orthotropic hydraulic permeability of a flow with low Reynolds number, passing through a bundle of parallel circular cylinders. Two particular distributions are considered: (i) arrays with ordered rectangular lattices and (ii) irregular nonrandom distributions for which the unit cell cross sections are elliptical. The standard unit cell models, originally developed by Happel and Kuwabara for a random distribution of cylinders, are adapted to the case of nonrandom distributions. The drag force on a representative cylinder in a direction perpendicular to its axis is obtained based on the standard unit cell model: the actual unit cell of rectangular or elliptical cross section is replaced with an "equivalent" cylindrical unit cell of diameter equal to the maximum width of the actual unit cell. Using the obtained drag forces and referring back to the original geometry of the unit cell, closed-form approximate expressions for the overall permeabilities in the perpendicular directions are obtained. Numerical comparisons with more sophisticated approaches confirm the good efficiency of the presented approach, especially in the range of low solid volume fraction, i.e., of high porosity. Previous studies have revealed that, for the parallel fluid flow, the variation of permeability with aspect ratio (or in general the lateral arrangement) of parallel cylinders is generally weak. These observations suggest that Happel's model for parallel permeability in a random distribution of cylinders could be a good approximation for parallel permeabilities in nonrandom distributions with the same volume fraction.

Effect of unitary sample shifted Laplacian and rectangular distributions in bearing fault identifications of induction motor

Fault identifications of bearings are complex process and require non-stationary signal processing methods along with the envelope and cepstral analysis techniques to demodulate the amplitude modulated signals. In this study, the fault signals from the drive end induction motor bearings are detected using proposed unitary sample shifted rectangular and Laplacian probability density functions (pdf). These functions are used to generate recursive frame-based crest factor (FCF), frame-based kurtosis (FKS) and frame-based energy (FE) time series. These time series substantially detect the impulses that depict the fault and shaft rotating frequencies effectively. Apart from detection of fault, the proposed method significantly suppresses the higher order modulated resonating frequencies without the use of non-coherent amplitude demodulation technique. Similarly, the spectral analyses are found to be observable in fault and shaft rotating frequencies identification using rectangular pdf. However, it is observed that the Laplacian pdf is more sensitive in detecting faults and the spectral leakages are lower than rectangular pdf for variable defect sizes. It is also found that the FCF and FKS time series are more probable in detecting fault, whereas FE is significant in depicting shaft rotation.

The equilibrium liquidus temperatures of rhenium–carbon, platinum–carbon and cobalt–carbon eutectic alloys

The eutectic alloys rhenium–carbon, platinum–carbon and cobalt–carbon have been proposed as reference standards for thermometry, with temperature and uncertainty values specified within the mise en pratique of the definition of the kelvin. These alloys have been investigated in a collaboration of eleven national measurement institutes and laboratories. Published results reported the point-of-inflection in the melting curve with extremely low uncertainties. However, to be considered as standards it is necessary to stipulate what phenomenon a temperature value has been ascribed to; specifically, this should be a thermodynamic state. Therefore, the data have been further evaluated and the equilibrium liquidus temperatures determined based on a consideration of limits and assuming a rectangular probability distribution. The values are: for rhenium–carbon 2747.91 ± 0.44 K, for platinum–carbon 2011.50 ± 0.22 K and for cobalt–carbon 1597.48 ± 0.14 K, with uncertainties at approximately a 95% coverage probability. It is proposed that these values could be used as the basis of thermodynamic temperature measurement at high temperatures (above 1300 K).

Optimized Base Station Location Planning for Indoor Positioning in Visible Light Communication System

AbstractVisible light communication (VLC) is an emerging optical communication technology, and indoor positioning of moving target devices is one of most important issues in the VLC system. In this study, we focused on LED base station location planning to improve the accuracy of the positioning method based on the received signal strength intensity (RSS) and the Kalman filter (KF). We developed a new base station location-planning scheme called equilateral triangle distribution. We proved theoretically that the equilateral triangle distribution scheme reduced the positioning error area when compared with the rectangular distribution scheme. The simulation results showed that the measuring error of the equilateral triangle distribution method using KF was less than 5 cm, even when the receiver was located at the boundary of the room. Our method verified that the VLC indoor positioning accuracy can be improved when LED base stations are distributed according to our equilateral triangle distribution scheme.

On Some Finite Range Life Distributions

Various reliability properties, including some characterization results, have been investigated for two finite range life distributions. One of these, namely, the exponentiated rectangular distribution, exhibits a bath-tub failure rate when the shape parameter has a value less than unity and, as expected, does not conform to certain results relating to ageing properties, which hold in the case of distributions with monotone failure rates. Moments as well as best percentile estimates of the shape parameter for this distribution have been worked out. Stress–strength reliability when stress and strength follow the same distribution of this type over the same range, but with different shape parameter values, has been derived and shown to be related to the Bhattacharyya distance between the two probability distributions. Distributions of parallel and standby system lives for two independent components, with this distribution having varying shape parameters, have been derived.

Investigation on the plasmon Talbot effect of finite-sized periodic arrays of metallic nanoapertures

We present an in-depth and systematical investigation on the plasmon Talbot effect of finite-sized two-dimensional (2D) periodic metallic nanoaperture arrays. The nanoaperture shapes, fill factor, lattice distribution, array size, film thickness, material property and polarization state of the incident light are considered, and the inherent influencing rules are summarized via the three-dimensional (3D) finite-difference time-domain (FDTD) numerical simulations. The nanoaperture shapes, fill factor or array size seems to express a tiny influence on Talbot effect, which shows a good agreement with our previously reported experimental results. Besides, square lattice brings out a much more uniform Talbot pattern than the triangular distribution, and the smaller array period should be taken to estimate the Talbot distance when it comes to a rectangular distribution. Furthermore, the thickness of Au film is suggested to within the range of 50~100 nm, which gives a broadest Talbot contour. It is also found out that the elliptical shape of hotspots is closely related to the linearly polarization state of the light source, showing an asymmetric electromagnetic field. The research contributes to a better understanding of the optical transmission features through periodic metallic nanoaperture arrays, which provides opportunities for the potential applications such as nanofabrication, optoelectronics, and imaging.

The new INRIM rotating encoder angle comparator (REAC)

A novel angle comparator has been built and tested at INRIM. The device is based on a double air bearing structure embedding a continuously rotating encoder, which is read by two heads: one fixed to the base of the comparator and a second fixed to the upper moving part of the comparator. The phase measurement between the two heads’ signals is proportional to the relative angle suspended between them (and, therefore, the angle between the base and the upper, movable part of the comparator). The advantage of this solution is to reduce the encoder graduation errors and to cancel the cyclic errors due to the interpolation of the encoder lines. By using only two pairs of reading heads, we have achieved an intrinsic accuracy of ±0.04″ (rectangular distribution) that can be reduced through self-calibration. The residual cyclic errors have shown to be less than 0.01″ peak-to-peak. The random fluctuations are less than 0.01″ rms on a 100 s time interval. A further advantage of the rotating encoder is the intrinsic knowledge of the absolute position without the need of a zeroing procedure. Construction details of the rotating encoder angle comparator (REAC), characterization tests, and examples of practical use are given.

Statistical Anisotropy in Imperfect Electromagnetic Reverberation

A probabilistic characterization is performed for a class of field anisotropy coefficients for local random electromagnetic fields. Isotropic stirring (ideal reverberation), anisotropic stirring (polarization bias), and incomplete stirring (line-of-sight coupling or direct illumination) are investigated individually. For small degrees of planar anisotropy or coupling, trapezoidal distributions are obtained, representing first-order departures from ideal rectangular distributions. Distributions for two definitions of total field anisotropy are found to be less sensitive than those for the individual planar anisotropies. The method and results are useful for the calibration and validation of reverberation chambers when field anisotropy requires a separate assessment. Theoretical results are compared to measured data in two chambers.

Target reconstruction algorithm for four-beam sheared coherent imaging

Sheared-beam imaging, which is a nonconventional coherent laser imaging technique, can be used to better solve the problem of taking pictures with high resolution for remote targets through turbulent medium than conventional optical methods. In the previous research on this technique, a target was illuminated by three coherent laser beams that were laterally arranged at the transmitter plane into an L pattern. In order to obtain a high quality image, a series of time-varying scattered signals is collected to reconstruct speckled images of the same object. To overcome atmospheric turbulence, multiple sets of three-beam laser should be emitted, which increases data acquisition time. In this paper, aiming at the quasi real-time problem of conventional sheared beam imaging technique, we use four-beam laser with rectangular distribution instead of the traditional L type sheared three-beam laser to illuminate the target. According to this, we propose a target reconstruction algorithm for four-beam sheared coherent imaging to reconstruct four target images simultaneously in one measurement, which can acquire high quality images by reducing the amount of measurement and the speckle noise. Meanwhile, it can greatly reduce the amount of beam switching in multi-group emission and improve the imaging efficiency. Firstly, the principle of the four-beam sheared coherent imaging technique is deduced. Secondly, in the algorithm, the speckle amplitude and phase difference frames can be extracted accurately by searching for the accurate positions of the beat frequency components. Based on the speckle phase difference frames, four sets of wavefront phases can be demodulated by the least squares method, and wavefront amplitude can be obtained by algebraic operation of speckle amplitude. The reconstructed wavefront is used for inverse Fourier transform to yield a two-dimensional image. A series of speckled images is averaged to form an incoherent image. Finally, the validity of the proposed technique is verified by simulations. From the simulation results, the image quality of the proposed method is better than that of the traditional method in the same amount of measurement. Furthermore, on the premise of the same image quality, the data acquisition amount of the proposed method is 2-3 times as large as that of the traditional method. In other words, compared with that of the traditional method, the data acquisition time of the proposed method is reduced at least by half and the algorithm processing time is less. It can be concluded that the proposed imaging technique can not only improve the efficiency of target reconstruction, but also present a better way of imaging the remote moving targets.

基于单目视觉和固定靶标的位姿测量系统

The position and orientation measurement system based on computer vision is extensively applied on robotics, motion control and precision detection systems. Using the minimum hardware resource, a localization system based on mono-vision and manual planar target was designed and the method of image matching and position resolving was also studied. Firstly, the object detection based on image matching was used to get the coordinate of the planar target in image. The matching was based on SIFT features and projection estimation to detect the target in image, and then the accurate coordinate of the target's center was calculated by some inherent shape information. A number of image samples were used to validate the accuracy and robustness of the image matching algorithm. Secondly, a new method to solve the PnP problem based on the rectangular distribution was proposed. The method uses the locations of target control points in the image coordinate system and the object coordinate system was used to get the relative position and orientation between the moving object and the camera. The experiment was conducted on the 5D precision displacement stage and results show that the system can achieve the accuracy level of mm in the range of 800 mm, which meets the project requirement.

Effects of Distribution Approximations to Expanded Uncertainty of Breakdown Strength of Mineral Insulating Oil in IEC 60156

Breakdown Strength test identifies the health condition of oil-immersed power transformers. The erratic and non-homogeneity behavior affects much the results of breakdown strength test. The inclusion of expanded uncertainty in test results of breakdown strength bounds the compensated test results in a defined confidence level, which is not defined in IEC 60156. This paper presents the analysis of the effects of distribution approximations used in the estimation of expanded uncertainty. The study involved Normal distribution (Gaussian), t-distribution (student distribution), Rectangular distribution and Triangular distribution in estimating an expanded uncertainty which incorporated Type A and Type B errors. The Expanded Uncertainty was based on a standard uncertainty multiplied by a coverage factor of k = 2, which provided a minimum confidence level of 95% in all approximated distributions. Normal distribution (Gaussian) showed the validity in the estimation of expanded uncertainty for a sufficiently large number of test results (of up to 25 tests per sample) while Student-t distribution showed invalidity for all test results. Since it is not a good practice to test insulating oil up to 25 tests, expanded uncertainty based on Normal distribution seemed to be valid for any test results less than 25. The inclusion of expanded uncertainty may predict the validity and avoid re-running of the test.

A New Class of Rectangular Distribution Properties and Application

In this article, we generalize the Rectangular distribution using the quadratic rank transmutation map studied by Shaw and Buckley (2007) to develop a transmuted Rectangular distribution. The expectation, variance, moments, reliability function, hazard rate function, Cumulative Hazard function, the moment generating function, and the characteristic function of the new distribution are provided. In addition, the model parameters are estimated by the maximum likelihood method. Finally, an application to generated data sets is illustrated.

Analysis of reverse heat transfer for conventional and optimized lubri-cooling methods during tangential surface grinding of ABNT 1020 steel

A numerical thermal model was developed to evaluate the heat flux which is conducted to a rectangular workpiece of steel plate ABNT 1020, thus making it possible to compute the maximum temperature in the grinding surface, taking into account the rectangular distribution of heat flux, the thermal properties of the grinding wheel conventional Al2O3, the piece to be machined and the lubri-refrigerating fluid. The finite volume method was employed for the discretization of the direct thermal problem from the heat diffusion equation associated with the two-dimensional problem of heat conduction in transient regime. The inverse thermal problem was solved by the Golden Section technique. The thermal flux, when compared to the conventional technique of method of application fluid, was reduced by 84.0% in the practices performed with cutting depth of 30µm, at 74.0% in practices with cutting depth of 45µm and 61.2% in the aggressive practices of 60µm, thus demonstrating the applicability of the optimized method for fluid application.

Flotation characteristics and flotation kinetics of fine wolframite

The flotation characteristics and kinetics of fine wolframite from Shizhuyuan Mine, Hunan, China were studied. Coarse and middle particles entered the concentrate easily. However, because of the reduction of middle particles as carriers and the poor collision and attachment between fine particles and air bubbles, it was difficult to float fine particles. A new kinetic model was proposed and compared with four common kinetic models, including a classical first-order model, a modified first-order model, a first-order kinetic model with a rectangular distribution of floatabilities and a second-order kinetic model with a rectangular distribution of floatabilities. The kinetic model evaluation showed that the author's model fitted the experimental data best. The new model was applied to industrial data in a dressing plant to verify its reliability, and a better imitative effect was obtained compared with other models investigated. The modified first-order model agreed well with experimental and industrial data. The optimum flotation kinetic equations of fine wolframite in the laboratory and dressing plant were ε=87.26−31.74e−3.446t−55.51e−0.879t and ε=85.76−44.67e−2.052t−41.09e−0.434t, respectively.

A comparison of the predictability of batch flotation kinetic models

Batch flotation test data of a mixture of pyrite and calcite were used to compare regression parameters of four kinetic model structures. The work included the use of unoxidized or a mixture of partially oxidized pyrite (by microwave irradiation). The objective of floating oxidized pyrite was to have mineral particles with different floatability, closer to a real situation. The models considered include: single rate constant, distributed rate constants (i.e. rectangular and gamma distributions), and a recently introduced approach based on fractional calculus. Such models were selected due to their good tradeoff between simplicity and accuracy. The regressions were performed (1) taking all the data points and comparing the mean square error (MSE) and adjusted correlation factor RAdj2 as indicators of the goodness of fit; and (2) taking the first data points while neglecting the last ones (from 1 to 3) and observing the variability of the model parameters and the prediction of maximum recovery (R∞). For the latter regression scheme, besides MSE and RAdj2, a predictive factor, E, was defined by subtracting the final measured recovery from the calculation obtained by the model. This allowed to measure the ability of each model to extrapolate the omitted points on the recovery vs. time curve.Results from this study showed that the single constant model had a satisfactory performance with the advantage of having the least parameters compared to the other structures. The gamma model was effective and robust. The rectangular model gave an acceptable goodness of fit but overestimated the maximum and final recovery. Finally, the fractional calculus approach gave the best goodness of fit, overall, but failed in predicting the maximum recovery, which occurred when the derivative order was greater than 1.