Cumulative Frequency Distribution Function Research Articles

Hydrothermal carbonization of waste biomass to fuel: A novel technique for analyzing experimental data

This paper deals with a new stochastic approach to handling data from waste biomass hydrothermal carbonization. The dynamics of hydrochar properties are described using the concept of reaction time distributions. A set of cumulative frequency distribution functions is provided that could very well correlate the most disparate experimental data. The procedure for analyzing the results is detailed. The method is illustrated with experiments on batch runs with five different wastes from the agro-food industry. Isothermal reactions (200 °C) were performed up to 120 min at a constant 7/1 water/biomass ratio. The regression analyses fully confirm the correctness of the method. The use of dynamical van Krevelen plots is proposed. The approach allows also obtaining from lab-scale runs fundamental information for the correct waste-to-fuel process development at the industrial scale. The mathematics is not demanding and, unlike other methods, the difficulties substantially do not increase with the complexity of the modeled kinetic scheme.

Quantifying aperture, spacing and fracture intensity in a carbonate reservoir analogue: Crato Formation, NE Brazil

Abstract The characterization of fracture networks in carbonate reservoirs represents a primary approach to understand the processes of fluid flow. The study of outcrop analogues has been used to enhance the understanding of the relative contribution of fracture networks to reservoir permeability. The outcrop data can overcome scale limitations and/or lack of consistency of subsurface datasets (e.g. seismic, well logs, and cores). This paper presents the fracture characterization of the Aptian-Albian lacustrine laminites of Crato Formation (Araripe Basin, NE Brazil), which have been investigated as an analogue to some of the observed carbonate facies present in the pre-salt reservoir sequence of the Brazilian marginal basins. In order to provide fracture attributes for Discrete Fracture Network (DFN) modelling we investigate spacing and aperture distribution, length and intensity of fractures based on the use of traditional scanline (macro and microscale) method. The main fractures identified in the Crato Formation are shear and extensional fractures and with stylolites also present. In this study, we focused on vertical veins, which strike in two principal directions, NNW-SSE (Set 1), NE-SW (Set 2), and are filled or partially filled by calcite or gypsum. The relationship between length and aperture of vertical veins is linear. Apertures and spacings of fractures in both sets 1 and 2 follow power law and lognormal cumulative frequency distribution functions, respectively. Fractures of Set 2 are more likely to be clustered than are fractures of Set 1, which have a wider kinematic aperture. This structural database can be used to populate computational models that consider the widespread fracture system in the fluid flow simulation of carbonate reservoirs.

Evaluating the Agronomic Effectiveness of Fertiliser Products

One of the problems which arises when analysing and interpreting results from field trials designed to test the efficacy of fertilisers and fertiliser-type products on pastures and crops is the conundrum of Type 1 and Type 2 statistical errors: is the product having literally no effect, or is the trial not "powerful" enough to detect small differences. This problem can be objectively and pragmatically solved, when sufficient trial data are available, by using cumulative frequency distribution functions. In this paper we explain what cumulative frequency distribution functions are and their usefulness for determining the agronomic effectiveness of products. This technique is then applied to field trial data testing the effectiveness of a number of fertiliser and fertiliser-type products used in agriculture today. It is concluded that this approach provides a more objective basis for determining the efficacy or otherwise of fertilisers. Keywords: fertilisers, agronomic effectiveness, testing, field trials, cumulative frequency distribution.

Using radius frequency distribution functions as a metric for quantifying root systems

Root radius frequency distributions have been measured to quantify the effect of plant type, environment and methodology on root systems, however, to date the results of such studies have not been synthesised. We propose that cumulative frequency distribution functions can be used as a metric to describe root systems because (1) statistical properties of the frequency distribution can be determined, (2) the parameters for these can be used as a means of comparison, and (3) the analytical expressions can be easily incorporated into models that are dependent upon root geometry. We collated a database of 96 root radii frequency distributions and botanical and methodology traits for each distribution. To determine if there was a frequency distribution function that was best suited to root radii measurements we fitted the exponential, Rayleigh, normal, log-normal, logistic and Weibull cumulative distribution functions to each distribution in our database. We found that the log-normal function provided the best fit to these distributions and that none of the distribution functions was better or worse suited to particular shapes. We derived analytical expressions for root surface and volume and found that they are a valid, and simpler method for incorporating root architectural traits into analytical models. We also found that growth habit and growth media had a significant effect on mean root radius.

Modeling potential vadose-zone transport of nitrogen from onsite wastewater systems at the development scale

Water in the urban front-range corridor of Colorado has become an increasingly critical resource as the state faces both supply issues as well as anthropogenic degradation of water quality in several aquifers used for drinking water. A proposed development (up to 1100 homes over two quarter-quarter sections) at Todd Creek, Colorado, a suburb of Westminster located about 20 miles northeast of Denver, is considering use of onsite wastewater systems (OWS) to treat and remove domestic wastewater. Local health and environmental agencies have concerns for potential impacts to local water quality. Nitrogen treatment in the vadose zone and subsequent transport to ground water at a development scale is the focus of this investigation. The numerical model HYDRUS 1D was used, with input based on site-specific data and several transport parameters estimated from statistical distribution, to simulate nitrate concentrations reaching ground water. The model predictions were highly sensitive to mass-loading of nitrogen from OWS and the denitrification rate coefficient. The mass loading is relatively certain for the large number of proposed OWS. However, reasonable values for the denitrification rate coefficients vary over three orders of magnitude. Using the median value from a cumulative frequency distribution function, based on rates obtained from the literature, resulted in simulated output nitrate concentrations that were less than 1% of regulatory maximum concentrations. Reasonable rates at the lower end of the reported range, corresponding to lower 95% confidence interval estimates, result in simulated nitrate concentrations reaching groundwater above regulatory limits.

Climatological Thresholds of Daily Rainfall

This technical note summarizes the results from daily rainfall data analysis with relation to annual rainfall amounts. By investigating the behavior of correlation coefficients between the annual rainfall amounts and the within-the-year cumulative frequency distribution functions of wet days with various threshold rainfall depths, it was found that (1) the annual rainfall amount is weakly correlated with the number of either dry or wet days—thus, the zero rainfall depth is not a proper threshold for the explanation of dry or wet years; and (2) there exist certain thresholds of daily rainfall depth (the climatological thresholds) with which the cumulative frequency of wet days can be well related with the annual rainfall amount. Also with this threshold, the wet and dry years can be seen divided clearly into separate groups of the cumulative frequency distribution functions, without any overlaps, which were common when using the zero rainfall depth as a threshold. Because it is found that there exist certain thresholds to be used for explaining the annual rainfall amount changes, these might also be used effectively for the analysis of possible changes in flood or drought patterns caught by global warming.

Alternative statistical method for detecting changes in spontaneous activity of single olfactory receptor cells

An statistical method for analysis of event data analysis, common in electrophysiological investigations of activities of olfactory receptor neurons, is proposed. The method is based on the analysis of the slopes of cumulative frequency distribution function. During the spontaneous activity the distribution function is linear which indicate that probability density distribution is uniform. During the chemical stimulation, the density of action potentials is either increased (excitation--E) or decreased (suppression--S), causing the observable change of the slope. The slope at each event is evaluated by piecewise linear regression. The type (E or S) and time of change of the slope can be determined, providing information for further statistical analyses. Graphically supported computer program (in S-PLUS) is available for interactive or routine use.

Alternative statistical method for detecting changes in spontaneous activity of single olfactory receptor cells.

An statistical method for analysis of event data analysis, common in electrophysiological investigations of activities of olfactory receptor neurons, is proposed. The method is based on the analysis of the slopes of cumulative frequency distribution function. During the spontaneous activity the distribution function is linear which indicate that probability density distribution is uniform. During the chemical stimulation, the density of action potentials is either increased (excitation - E) or decreased (suppression -S), causing the observable change of the slope. The slope at each event is evaluated by piecewise linear regression. The type (E or S) and time of change of the slope can be determined, providing information for further statistical analyses. Graphically supported computer program (in S-PLUS) is available for interactive or routine use.

Free energy and surface tension of arbitrarily large Mackay icosahedral clusters

We present a model for predicting the free energy of arbitrarily large Mackay icosahedral clusters. van der Waals clusters are experimentally observed to be particularly stable at magic numbers corresponding to these structures. Explicit calculations on the vibrational states were used to determine the spectrum of fundamental frequencies for smaller (≤561) icosahedral clusters. The scaled cumulative frequency distribution function rapidly approaches a limiting function for large clusters. This function was used to predict zero-point energies and vibrational free energies for larger clusters (≳561 atoms). Combining these predictions with correlations for the moment of inertia and for the minimum potential energy of large clusters leads to free energies of arbitrary large clusters. The free energies are used to predict the chemical potential and surface tension as a function of size and temperature. This connects macroscopic properties to the microscopic atomic parameters.

The influence of porous medium characteristics and measurement scale on pore-scale distributions of residual nonaqueous-phase liquids

A series of experiments was performed to characterize the morphologic distribution of nonaqueous-phase liquids (NAPL's) at residual saturation, as a function of porous medium size. Morphologic characterization of NAPL distributions was accomplished using a novel in situ polymerization technique. The porous medium consisted of glass beads. Blob length, volume and shape characteristics were determined for each experiment, and pore size distributions were determined through capillary pressure-saturation experiments. Both the blob lenght and pore size distributions were fitted to a van Genuchten function. Both blob lenght and pressure-saturation data could be scaled with the same averaged porous medium characteristics. The blob length distributions were found to be wider than the pore size distributions. Estimates of representative elementary volumes (REV's) were generated from statistical analysis using a van Genuchten cumulative frequency distribution function for blob lenght and an empirical function for blob volume as a function of blob length. Simulations were also performed using a Monte Carlo method. The size of the REV needed for a given level of prediction of the residual saturation level was found to increase as a function of mean particle volume for the similar used in this study. Extrapolation of the REV analysis suggests that the size of an REV will increase rapidly as uniformity of the medium decreases. If this extrapolation holds true, significant uncertainty would exist in most determination of residual saturation for poorly sorted media that have been reported to date.

Linearized Water Application Function for Drip Irrigation Schedules

ABSTRACT Alinear cumulative frequency distribution function can be used as an approximation of the accumulated frequency S-curve determined from a normal distribution. The straight line function can be used to describe the irrigation scheduling parameters; percent of deficit, application efficiency and coefficient of variation by simple mathematical equations. Furthermore, a mathematical optimization can be developed to determine optimal water application based on irrigation system's uniformity, a reduction coefficient of a known crop under deficit irrigation situation, cost of water and the return of the crop yields.

Irregular Wave Runup and Overtopping

A cumulative frequency distribution function for the runup elevation of irregular waves is useful in the design of many coastal structures. Such a relationship would help a designer to quantify trade offs between structure geometry and allowable overtopping. The work of Saville and Battjes suggests that a Rayleigh distribution is a reasonable but somewhat conservative dierstribution for irregular wave runup elevations. This note gives a simple technique for establishing a Rayleigh runup distribution for given wave and structure characteristics and then extends the technique to estimate irregular wave overtopping rates.

Two-sample Kolmogorov-Smirnov test for truncated data

A nonparametric statistical test to compare two cumulative frequency distribution functions is presented that can be used even when both samples include censored data, as is often the case when comparing the survival of two groups of laboratory animals under conditions in which the experiment is terminated before all the animals die. (Such a design can produce considerable savings and is to be recommended.) The program calculates exact probabilities for both the one-sided and two-sided alternatives to the null hypothesis, applicable to the case of equal group size, as well as the corresponding general asymptotic values; a continuity correction is employed that markedly improves the asymptotic approximation. Expressions are stated in terms of two different but related statistics, and the one that utilizes more information in any particular set of data is selected for the probability calculations. All basic equations and definitions are provided.