Numerical methods for colorimetric calculations: A comparison of integration methods

Numerical methods for estimating colorimetric integrals using various data intervals are reviewed. Accuracy of Newton-Cotes and Gaussian numerical integration algorithms are compared. Tables of calorimetric data for performing Gauss-Legendre integration are presented. Appropriate numerical examples are presented and applicability to general and special cases is discussed. the B-spline method of integration is presented. Errors of integration are quantified and discussed. © 1992 John Wiley & Sons, Inc.     

On the optimal numerical time integration for DEM using Hertzian force models

The numerical accuracy of a selection of different time integration techniques used to solve particle motion is investigated using a normal collision employing the non-linear Hertzian contact force. The findings are compared against the linear force model where it has been found that the expected order of accuracy of higher-order integration schemes is not realised (Tuley et al., 2010). The proposed mechanism for this limitation has been cited as the errors in integration which occur across the force profile discontinuity. By investigating the characteristics of both the non-linear elastic and the non-linear damped Hertzian contact models, it has been found that higher orders of accuracy are recoverable and depends on the degree of the governing non-linear equation. The numerical errors of the linear and non-linear force models are however markedly different in character.     

Experimental and numerical simulation investigations on particle sampling for high-pressure natural gas

A novel Null-type sampling nozzle with six sampling holes was developed to measure the particle concentrations and size distributions in high-pressure natural gas flows under isokinetic sampling conditions. The isokinetic sampling performances of sampling nozzles were investigated by experiments and numerical simulations with standard κ–ε turbulent model. The maximum deviation between the experimental and numerical data was less than 4%. In order to ensure reliability and stability, sampling nozzles were calibrated in standard wind tunnel. Furthermore, this novel Null-type sampling nozzle was used to measure the separation efficiency of filter with an operation pressure of 6.0 MPa in a natural gas compressor station. The results of experiment and numerical simulation show that the new Null-type sampling nozzles have relatively smaller isokinetic sampling errors and reliable sampling efficiencies, which enable it competent to sample high-pressure natural gas.     

Model‐based corrections of geometric errors in multiangle measurements of gonio‐apparent coatings

Numerical corrections of colorimetric errors caused by positioning error and irregular geometry in multiangle measurements of gonio‐apparent coatings are presented. These corrections use a model function approximating the gonio‐reflection characteristics of the coating to convert the angular displacement due to those geometric errors to reflectance factor errors to be corrected as well as to estimate the angular displacement conversely from the reflectance factors. The performance of these corrections was evaluated for 11 different gonio‐apparent coatings using a multiangle geometry conforming to ASTM E2539 consisting of eight subgeometries to which artificial displacements were applied so as to present typical positioning errors or irregular geometries. The results show that both corrections significantly reduce the colorimetric errors for all specimens in all subgeometries sensitive to geometric errors. © 2015 Wiley Periodicals, Inc. Col Res Appl, 2015     

Mechanical properties of urea–formaldehyde foam

The mechanical behavior of urea-formaldehyde foam was studied to evaluate its potential for energy absorption applications. The apparent elastic modulus (E_f) as a function of foam density was obtained from force-deformation tests. The values of energy absorption capacity were derived from a numerical integration technique. Poisson's ratio (v) was determined by a method of uniaxial compression of cylindrical samples. An increase in foam density results in an increase in the apparent elastic modulus of the material and therefore in its energy absorption capacity. Poisson's ratio is independent of the foam density. The mechanical properties' values obtained can be incorporated in various analyses for predicting desired characteristics for energy absorption applications.     

A Method for Determining Bulk Density,Material Density,and Porosity of Melter Feed During Nuclear Waste Vitrification

Glassmelting efficiency largely depends on heat transfer to reacting glass batch (melter feed), which in turn is influenced by the bulk density (ρ_b) and porosity (?) of the reacting feed as functions of temperature (T). Neither ρ_b(T) nor ?(T) functions are readily accessible from direct measurements. For the determination of ρ_b, we monitored the profile area of heated feed pellets and calculated the pellet volume using numerical integration. For the determination of ?, we measured the material density of feeds quenched at various stages of conversion via pycnometry and then computed the feed density at heat‐treatment temperature using thermal expansion values of basic feed constituents.     

Estimation of nitric oxide emissions from an internal combustion engine

The analysis of nitric oxide, the principal nitrogen oxide in raw exhaust gas, is described by two different techniques. Concordance of analytical determinations using the Saltzman colorimetric solution method with the gas phase chemiluminescence method is demonstrated. By dilution of exhaust gas from an internal combustion engine with nitrogen it is shown that gas samples can be stored for periods of at least 30 min without significant change in sample composition with respect to nitric oxide. This procedure enables reliable estimations of nitric oxide in exhaust gas contained in a sealed transportable vessel to be performed at sites remote from the sample point, without introducing errors due to loss of nitric oxide by chemical reaction in the time between sampling and analysis.     

Spectral interpolation errors

Several CIE Technical Committees discussed the advantages and disadvantages of different spectral interpolation methods during the past years. Linear, Lagrange, cubic spline, and Sprague methods were under discussion. We have investigated the colorimetric differences caused by using the different interpolation methods, and showed that Lagrange, cubic spline, and Sprague methods provide very similar results for most well‐behaved spectra. As a further result, we found that the 5‐nm sampling and using the 5 nm tables of the CIE Technical Report “Colorimetry” is almost as good as an interpolation to 1‐nm sampling and colorimetric calculation using the 1‐nm tables of the CIE colorimetric standards. © 2005 Wiley Periodicals, Inc. Col Res Appl, 30, 348–353, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/col.     

Propagation of random errors in spectrophotometric colorimetry

Every physical quantity with an a priori range of numerical values constituting a continuum is subject to error in its measurement. It is important to report the highest amount by which any measured quantity might be in error. Optical radiation measurements are typiclly based on determination of the ratio of the instrumental reading of a calibrated standard to the instrumental reading of the test sample. There are random and systematic errors in both the instrumental readings and the calibration values. The uncertainty in a calculated value due to random error in its constituents can be determined using well known techniques of error propagation. Examples of error propagation through spectral reflectance factor measurements and colorimetric calculations are presented. The standard deviations of CIELAB coordinates for typical measurements can be as high as 0.258 due only to random errors in the calibration chain.     

The analysis of organic matter in coke oven emissions

A detailed analysis of coke oven emissions is reported. Measurements were made on samples collected at various workplace locations on the oven top. A new type of sampling system was used and is described. Results range from volatile monoaromatic compounds including benzene to polycyclic aromatic hydrocarbons (PAH) of six fused rings. Estimates of errors which may arise when sampling for PAH by filtration are given.     

原油水分和密度测定误差分析

对原油取样、水分和密度分析过程可能产生的误差进行了分析,并提出了减少原油水分和密度分析结果误差的方法。     

A COMPUTATIONAL METHOD FOR ESTIMATING DENSITIES OF NON-GAUSSIAN NONSTATIONARY UNIVARIATE TIME SERIES

Abstract. Kitagawa's (1987a) numerical integration formulae used to approximate the filtering, prediction and smoothing densities of nonlinear non-Gaussian state-space models are modified. The method involves integration by parts which permits the integration of the conditional system density and, possibly, the observational density prior to and independently of the computation of the filtering, prediction and smoothing densities. In addition to a substantial reduction in computing time and an increase in accuracy, this approach eliminates the necessity of incorporating dynamic adjustments to the filtering, prediction and smoothing processes to accommodate difficult noise densities.
Three numerical examples are presented. One example replicates Kitagawa's non-Gaussian state-space model; the second is a linear Gaussian model; and the third is a nonlinear non-Gaussian model. Comparisons of speed and accuracy between alternative methods and between three computers (personal computer, minicomputer and supercomputer) are made.     

Non-Gaussian Filter and Smoother Based on the Pearson Distribution System

The Pearson distribution system can represent wide class of distributions with various skewness and kurtosis. We develop a practical approach of using all types of its distribution system including the type-IV distribution which was difficult to implement.
We propose an easily implemented algorithm which uses less-memory and performs at a higher speed than other typical methods: using analytic approximation of successive conditional probability density functions for prediction and filtering by the Pearson distribution system in the case of both the system and observation noise being one-dimensional. By using the approximated probability density function and the numerical integration, we obtain mean, variance, skewness and kurtosis of the next distribution. We decide the next approximated distribution from the Pearson distribution system. We adopt these steps for the prediction, filtering and smoothing recursively. Our framework makes it possible to construct time series models with various noise distributions.
We apply our non-Gaussian filter to the estimation of non-Gaussian stochastic volatility models of the stock returns. We compare our method with the typical method.     

THE RELATIVE IMPORTANCE OF ERRORS OF SAMPLING AND MEASUREMENT IN CERAMIC RESEARCH1

The common use of average values based on measurements of a small sample of specimens in ceramic research is pointed out. The principal errors to which such averages are subject are described and the need for a method for proportioning sampling errors and errors of measurement shown. An equation giving the relation between sampling errors and errors of measurement is derived by expression of factors involved in statistical terms and combination of well-known statistical equations. This equation is used for constructing a chart by means of which questions involving sampling errors and errors of measurement can be readily solved. This chart also enables one to visualize the rôles played by sampling errors and errors of measurement in determining the accuracy of an average. The general applicability of the method is pointed out and its theoretical limitations are discussed.     

Exponential collocation of stiff reactor models

A collocation method is given for streamwise integration of stiff, nonlinear reactor models. Perturbation solutions of the model are used as basis functions. Low-order approximations give results analogous to the Method of Steady States, whereas higher approximations give more detailed solutions. The method uses a small set of mesh points, and selects them to minimize global truncation errors; this feature saves computation when the reactor model is complicated. The efficiency of the method is demonstrated by numerical examples for isothermal and adiabatic reactors.     

Algorithm for the accurate numerical solution of PBE for drop breakup and coalescence under high shear rates

Multiphase flows play an important role in the chemical and process industries and significant strides have recently been achieved in the design of such systems using the population balance models. However, some uncertainties still remain concerning the stability and accuracy of the numerical solution of such integro-differential equations.This paper proposes a new methodology for solving the discretized population balance equation by minimizing the finite domain errors that often arise while discretizing the drop size domain. It relies on the use of the size distribution sampling approach combined with a moving grid technique. In addition, an enhanced solution stability algorithm was proposed and which relies on monitoring the onset of errors in the various birth and death terms encountered in PBE. It consequently allows for corrective action to be undertaken before the errors propagate in an uncontrollable fashion, and was found to improve the stability and robustness of the solution method even under very high shear rate conditions.The proposed algorithm was tested using the model of Coulaloglou and Tavlarides (1977) under breakage and coalescence dominated conditions in low, moderate and high energy dissipation regions, and was found to provide a stable solution that accurately predicts the quasi-equilibrium Sauter mean diameter.     

Analytical Approximation Schemes for Mean Nucleation Rate in Turbulent Flows

Nucleation rate is a very sensitive function of the temperature and vapor mole fraction. Analytical approximation schemes for the mean nucleation rate in turbulent flows are derived using Laplace’s approximation method. The schemes only require the derivative of the nucleation rate function and the probability density function (pdf) of the vapor mole fraction and/or temperature at the point of maximum nucleation rate. Based on the relation between the mole fraction and temperature, i.e., linearly correlated or not, different approximation schemes are developed. Numerical examples are constructed to investigate the accuracy of these approximation. In the examples, the pdfs of mole fraction and/or temperature in various turbulent flows are assumed to come from the beta distribution with five distinct forms. The mean nucleation rate of dibutyl phthalate (DBP) aerosol in these turbulent flows are calculated from the approximation schemes, and compared with exact numerical integration. The relative errors are less than 1% for cases when nucleation rate diminishes at the bounds of temperature fluctuations, and no more than 50% for all studied examples. Furthermore, the approximation schemes are not sensitive to the precise form of the pdfs. Hence, these developed approximation schemes can be used to estimate the mean nucleation rate in a broad range of turbulent flows conveniently.

Copyright 2014 American Association for Aerosol Research     

Line Shapes in Gas Chromatography. An Improved Numerical Integration Method

Abstract

Line shapes in gas chromatography are modeled by numerical integration of the differential equations describing mass transfer. The numerical dispersion which usually introduces substantial error into such calculations is greatly reduced by the use of one of three so-called asymmetrical upwind algorithms for handling the advection term. The effects of the various parameters in the model, which assumes a Langmuir adsorption isotherm, are illustrated.     

Accurate tristimulus values from spectral data

Inter-instrument color measurement errors of up to 2 CIELAB units are typically introduced in the process of numerically integrating spectrophotometric data to obtain tristimulus values. These errors arise partly from inadequacies in the definition of the tristimulus values themselves, but mainly from failing to take into account the effect the instrument function has on the spectrophotometric data. Through using precise definition of the tristimulus values and through using source-observer weighting tables tailored to each specific instrument, it is practical to reduce this component of colorimetric error to less than 0.1 CIELAB unit for spectrophotometers with bandwidths up to 20 nm.     

A systematic approach to assessing measurement uncertainty for CO2 emissions from coal-fired power plants - Missing contributions from the Theory of Sampling (TOS)

An augmented measurement uncertainty approach for CO₂ emissions from coal-fired power plants with a focus on the often forgotten contributions from sampling errors occurring over the entire fuel-to-emission pathway is presented. Current methods for CO₂ emission determination are evaluated in detail, from which a general matrix scheme is developed that includes all factors and stages needed for total CO₂ determination, which is applied to the monitoring plan of a representative medium-sized coal-fired power plant. In particular sampling involved significant potential errors, as identified and assessed by the Theory of Sampling (TOS), which also shows how these can be eliminated and/or minimised. Since coal-related CO₂ emission calculations not only require analytical results of the carbon content of coal itself but also of the by-products fly ash and bottom ash, sampling procedures of these three materials were also given full attention. A systematic error (bias) is present in the current sampling approach, which increases the present uncertainty estimate unnecessarily. For both primary sampling and analytical sample extraction steps, random variations, which hitherto only have been considered to a minor extent, have now also been fully quantified and included in the overall uncertainty. Elimination of all identified sampling errors lead to modified CO₂ determination procedures, which indicate that the actual CO₂ emission is approximately 20,000 t higher than the present estimate. Based on extensive empirical sampling experiments, a fully comprehensive uncertainty estimate procedure has been devised. Even though uncertainties increased (indeed one particular factor is substantially higher, the so-called “emission factor”), the revised CO₂ emission budget for the case plant complies with the official pre-determined uncertainty levels maxima in the EU guidelines.