Quantifying uncertainty in chemical systems modeling

This study compares two techniques for uncertainty quantification in chemistry computations, one based on sensitivity analysis and error propagation, and the other on stochastic analysis using polynomial chaos techniques. The two constructions are studied in the context of H₂? O₂ ignition under supercritical‐water conditions. They are compared in terms of their prediction of uncertainty in species concentrations and the sensitivity of selected species concentrations to given parameters. The formulation is extended to one‐dimensional reacting‐flow simulations. The computations are used to study sensitivities to both reaction rate pre‐exponentials and enthalpies, and to examine how this information must be evaluated in light of known, inherent parametric uncertainties in simulation parameters. The results indicate that polynomial chaos methods provide similar first‐order information to conventional sensitivity analysis, while preserving higher‐order information that is needed for accurate uncertainty quantification and for assigning confidence intervals on sensitivity coefficients. These higher‐order effects can be significant, as the analysis reveals substantial uncertainties in the sensitivity coefficients themselves. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 368–382, 2005     

Uncertainty analysis applied to thermodynamic models and process design - 1. Pure components

A simple model is proposed to estimate the critical temperature and critical pressure of hydrocarbons in the range of C₅-C₃₆ with parameters determined using weighted linear least squares and weighted non-linear least squares taking into consideration the experimental uncertainty in the data as well as in the correlating parameters. The correlation model was parameterized using the normal boiling point and specific gravity at 60 °F. The uncertainties of parameters and associated covariance matrix necessary for error propagation calculations are reported and a comprehensive evaluation of acentric factors uncertainties based on the experimental vapor pressures was conducted. In addition, a simple sensitivity analysis designed to determine how the uncertainty of properties used for calculations based on the equations of state propagate thorough the model and affect the final results. The normal boiling points of two pure components, n-hexane and n-dodecane were calculated using an equation of state and the estimated error in the calculations is presented together with estimated uncertainties for the prototype pressure-temperature envelopes for two binary mixtures of methane-n-hexane and methane-n-dodecane.     

Status of uncertainty assessment in k 0-NAA measurement: anything still missing?

Several approaches to quantifying measurement uncertainty in k ₀-based neutron activation analysis (k ₀-NAA) are reviewed, comprising the original approach, the spreadsheet approach, the dedicated computer program involving analytical calculations and the two k ₀-NAA programs available on the market. Two imperfectness in the dedicated programs are identified, their impact assessed and possible improvements presented for a concrete experimental situation. The status of uncertainty assessment in k ₀-NAA is discussed and steps for improvement are recommended. It is concluded that the present magnitude of measurement uncertainty should further be improved by making additional efforts in reducing uncertainties of the relevant nuclear constants used.     

Uncertainty estimation in measurement of pKa values in nonaqueous media: A case study on basicity scale in acetonitrile medium

The difficulties in estimating uncertainty of pK_a values determined in nonaqueous media are reviewed and two different uncertainty estimation approaches are presented and applied to the pK_a values of the compounds on a previously established self-consistent spectrophotometric basicity scale in acetonitrile. One approach is based on the ISO GUM methodology (the “ISO GUM” approach) and involves careful analysis of the uncertainty sources and quantifying the respective uncertainty components. The second approach is based on the standard-deviation-like statistical parameter that has been used for characterization of the consistency of the scale (the “statistical” approach). It is demonstrated that the ISO GUM approach somewhat overestimates the uncertainty. The statistical approach is based on long-term within-laboratory statistical data and it is demonstrated that it underestimates the uncertainty. In particular it neglects the laboratory bias effects that are taken into account at least to some extent by the ISO GUM approach. Thus, together these two approaches allow to “bracket” the uncertainties of the pK_a values on the scale. The uncertainties of the pK_a values are defined in two different ways. Definition (a) includes the uncertainty of the pK_a of the reference base (anchor base of the scale) pyridine. Definition (b) excludes it. It is demonstrated that both definitions have their virtues. Definition (a) leads to the uncertainty ranges of 0.12-0.22 and 0.12-0.14 pK_a units at standard uncertainty level for different bases using the ISO GUM and statistical approach, respectively. Definition (b) leads to the uncertainty ranges of 0.04-0.19 and 0.02-0.08 pK_a units, respectively. The uncertainty of the pK_a of a given base is dependent on the quality of the measurements involved and on the distance from the reference base on the scale. The importance of the correlation between the pK_a values of bases belonging to the same scale is stressed.     

Estimation of the efficiencies and uncertainties of the extraction and cleanup steps of pesticide residue determination in cucumber using <Superscript>14</Superscript>C-carbaryl

The estimation of the uncertainty of measurement has, nowadays, become an integral part of analytical results. The uncertainty and efficiency of extraction and cleanup are very important components of a pesticide residue analytical method. In this work, for a quick review of extraction and cleanup efficiency and to evaluate the individual analysis steps during method adaptation, ¹⁴C-carbaryl was applied at all fortification levels. Then, further analyses, such as ethylacetate extraction, filtration, evaporation and cleanup, were performed. The calibration of gel chromatograph column, performed with both ¹⁴C-carbaryl and fortification mixture (dichlorvos, malathion and chlorpyrifos), showed that pesticide fractions came through the column between the 8- and 23-mL fractions. The overall recovery of ¹⁴C-carbaryl after the extraction and cleanup step was 0.91, with the relative uncertainty of 0.084. Using the “bottom-up” approach, the uncertainty of extraction u _cEX and cleanup u _cGPC were 0.033 and 0.107, respectively. The combined standard uncertainty u _c associated with the described analytical method was 0.112. Similar values were obtained using the alternative “top-down” approach: uncertainty of extraction u _ct1 was 0.039, uncertainty of cleanup u _ct2 was 0.108 and the combined standard uncertainty u _cAV was 0.081. Both approaches showed that the uncertainty of cleanup was the main source of combined standard uncertainty.     

Determination of uncertainty in analytical measurements from collaborative study results on the analysis of a phenoxymethylpenicillin sample

The correct interpretation of a measurement result requires knowledge about its uncertainty. Depending on the conditions under which the analyst is operating, different operational definitions of uncertainty have been proposed. They include: within-laboratory uncertainty, reproducibility uncertainty, bias-included uncertainty and absolute uncertainty. Here we consider the evaluation of the reproducibility uncertainty derived from the results obtained in an inter-laboratory experiment. Nine laboratories participated in an inter-laboratory study for the analysis of phenoxymethylpenicillin. The analyses consisted of a Karl-Fischer water determination, an acid-base titration to assay phenoxymethylpenicillin and a liquid chromatography (LC) method to determine 4-hydroxyphenoxymethylpenicillin and other impurities. The experimental set-up allowed to obtain for each determination s_r² and s_L² as estimates of the repeatability variance (σ_r²) and the between-laboratory variance (σ_L²), respectively. The reproducibility uncertainties for the different assays were then derived from these estimates.     

Flow reactor studies of methyl radical oxidation reactions in methane‐perturbed moist carbon monoxide oxidation at high pressure with model sensitivity analysis

New rate constant determinations for the reactions CH₃ + HO₂ → CH₃O + OH (1) CH₃ + HO₂ → CH₄ + O₂ (2) CH₃ + O₂ → CH₂O + OH (3) were made at 1000 K by fitting species profiles from high‐pressure flow reactor experiments on moist CO oxidation perturbed with methane. These reactions are important steps in the intermediate‐temperature burnout of hydrocarbon pollutants, especially at super‐atmospheric pressure. The experiments used in the fit were selected to minimize the uncertainty in the determinations. These uncertainties were estimated using model sensitivity coefficients, derived for time‐shifted flow reactor experiments, along with literature uncertainties for the unfitted rate constants. The experimental optimization procedure significantly reduced the uncertainties in each of these rate constants over the current literature values. The new rate constants and their uncertainties were determined to be, at 1000 K: k₁ = 1.48(10)¹³ cm³ mol⁻¹ s⁻¹ (UF = 2.24) k₂ = 3.16(10)¹² cm³ mol⁻¹ s⁻¹ (UF = 2.89) k₃ = 2.36(10)⁸ cm³ mol⁻¹ s⁻¹ (UF = 4.23) There are no direct and few indirect measurements of reactions ( 1 ) and ( 2 ) in the literature. There are few measurements of reaction ( 3 ) near 1000 K. These results therefore represent an important refinement to radical oxidation chemistry of significance to methane and higher alkane oxidation. The model sensitivity analysis used in the experimental design was also used to characterize the mechanistic dependence of the new rate constant values. Linear sensitivities of the fitted rate constants to the unfitted rate constants were given. The sensitivity analysis was used to show that the determinations above are primarily dependent on the rate constants chosen for the reactions CH₃ + CH₃ + M → C₂H₆ + M and CH₂O + HO₂ → HCO + H₂O₂. Uncertainties in the rate constants of these two reactions are the primary contributors to the uncertainty factors given above. Further reductions in the uncertainties of these kinetics would lead to significant reductions in the uncertainties in our determinations of k₁, k₂, and k₃. © 2001 John Wiley & Sons, Inc. Int J Chem Kinet 33: 75–100, 2001     

Cavity-Enhanced Saturation Spectroscopy of Molecules with sub-kHz Accuracy

Saturation spectroscopy is frequently used to obtain sub-Doppler measurement of atomic and molecular transitions. Optical resonant cavities can be used to enhance the effective absorption path length, and the laser power inside the cavity as well to saturate very weak ro-vibrational transitions of molecules. Three different cavity-enhanced methods, cavity enhanced absorption spectroscopy, cavity ring-down spectroscopy, and noise-immune cavity enhanced optical heterodyne molecular spectroscopy (NICE-OHMS), were compared by measuring the Lamb dip of a C₂H₂ line at 1.4 μm using a cavity with a finesse of 120000. The center of the line was determined by different cavity-enhanced methods, each giving a sub-kHz (δv/v≈10^-12) statistical uncertainty. The sensitivity and precision of different methods were analyzed and compared. As demonstrated in this study, the NICE-OHMS method is the most sensitive one, but more investigation on the systematic uncertainty is necessary before its application in metrology studies toward a sub-kHz accuracy.     

Fluorescence and intersystem crossing as a function of rotational energy on the S1 state of benzene

The effect of rotational excitation on the electronic relaxation of the S₁ state of “isolated” benzene has been studied by examination of resonance fluorescence generated when narrow-band exciting light is swept through the rotational envelopes of four S₁ ← S₀ absorption bands. To within the ten percent uncertainty in measurement, the fluorescence yields are independent of the position of the exciting light within an absorption band. Thus neither radiative decay nor intersystem crossing to the triplet state displays large sensitivity to molecular rotation, at least when excited state relaxation proceeds from rotational distributions of moderate diversity.     

电感耦合等离子体质谱法测定石墨样品中钒含量的不确定度评价

依据测量不确定度的评价与表示方法,建立了电感耦合等离子体质谱法测定石墨样品中钒含量的不确定度评价的数学模型,并对各不确定度分量进行分析,当样品中V₂O₅含量为0.215%时,其扩展不确定度为0.035%.通过分析发现,曲线拟合、回收率以及重复性试验对测定结果有较大影响.评价方法可为石墨样品中钒的测定结果准确性和置信度提供理论依据.     

Thermal conductivity of Al2O3/water nanofluids

A considerable number of studies can be found on the thermal conductivity of nanofluids in which Al₂O₃ nanoparticles are used as additives. In the present study, the aim is to measure the thermal conductivity of very narrow Al₂O₃ nanoparticles with the size of 5 nm suspended in water. The thermal conductivity of nanofluids with concentrations up to 5 % is measured in a temperature range between 26 and 55 °C. Using the experimental data, a correlation is presented as a function of the temperature and volume fraction of nanoparticles. Finally, a sensitivity analysis is performed to assess the sensitivity of thermal conductivity of nanofluids to increase the particle loading at different temperatures. The sensitivity analysis reveals that at a given concentration, the sensitivity of thermal conductivity to particle loading increases when the temperature increases.     

[Ga3+8Sm3+2, Ga3+8Tb3+2] Metallacrowns are Highly Promising Ratiometric Luminescent Molecular Nanothermometers Operating at Physiologically Relevant Temperatures

Nanothermometry is the study of temperature at the submicron scale with a broad range of potential applications, such as cellular studies or electronics. Molecular luminescent-based nanothermometers offer a non-contact means to record these temperatures with high spatial resolution and thermal sensitivity. A luminescent-based molecular thermometer comprised of visible-emitting Ga³⁺/Tb³⁺ and Ga³⁺/Sm³⁺ metallacrowns (MCs) achieved remarkable relative thermal sensitivity associated with very low temperature uncertainty of S_r=1.9 % K⁻¹ and δT<0.045 K, respectively, at 328 K, as an aqueous suspension of polystyrene nanobeads loaded with the corresponding MCs. To date, they are the ratiometric molecular nanothermometers offering the highest level of sensitivity in the physiologically relevant temperature range.     

Validation of a simplified field-adapted procedure for routine determinations of methyl mercury at trace levels in natural water samples using species-specific isotope dilution mass spectrometry

A field-adapted procedure based on species-specific isotope dilution (SSID) methodology for trace-level determinations of methyl mercury (CH₃Hg⁺) in mire, fresh and sea water samples was developed, validated and applied in a field study. In the field study, mire water samples were filtered, standardised volumetrically with isotopically enriched CH₃²⁰⁰Hg⁺, and frozen on dry ice. The samples were derivatised in the laboratory without further pre-treatment using sodium tetraethyl borate (NaB(C₂H₅)₄) and the ethylated methyl mercury was purge-trapped on Tenax columns. The analyte was thermo-desorbed onto a GC-ICP-MS system for analysis. Investigations preceding field application of the method showed that when using SSID, for all tested matrices, identical results were obtained between samples that were freeze-preserved or analysed unpreserved. For DOC-rich samples (mire water) additional experiments showed no difference in CH₃Hg⁺ concentration between samples that were derivatised without pre-treatment or after liquid extraction. Extractions of samples for matrix–analyte separation prior to derivatisation are therefore not necessary. No formation of CH₃Hg⁺ was observed during sample storage and treatment when spiking samples with ¹⁹⁸Hg²⁺. Total uncertainty budgets for the field application of the method showed that for analyte concentrations higher than 1.5 pg g^–1 (as Hg) the relative expanded uncertainty (REU) was approximately 5% and dominated by the uncertainty in the isotope standard concentration. Below 0.5 pg g^–1 (as Hg), the REU was >10% and dominated by variations in the field blank. The uncertainty of the method is sufficiently low to accurately determine CH₃Hg⁺ concentrations at trace levels. The detection limit was determined to be 4 fg g^–1 (as Hg) based on replicate analyses of laboratory blanks. The described procedure is reliable, considerably faster and simplified compared to non-SSID methods and thereby very suitable for routine applications of CH₃Hg⁺ speciation analysis in a wide range of water samples.     

Uncertainty Quantification and Sensitivity Analysis for the Electrical Impedance Spectroscopy of Changes to Intercellular Junctions Induced by Cold Atmospheric Plasma

The influence of pertinent parameters of a Cole-Cole model in the impedimetric assessment of cell-monolayers was investigated with respect to the significance of their individual contribution. The analysis enables conclusions on characteristics, such as intercellular junctions. Especially cold atmospheric plasma (CAP) has been proven to influence intercellular junctions which may become a key factor in CAP-related biological effects. Therefore, the response of rat liver epithelial cells (WB-F344) and their malignant counterpart (WB-ras) was studied by electrical impedance spectroscopy (EIS). Cell monolayers before and after CAP treatment were analyzed. An uncertainty quantification (UQ) of Cole parameters revealed the frequency cut-off point between low and high frequency resistances. A sensitivity analysis (SA) showed that the Cole parameters, R₀ and α were the most sensitive, while R_inf and τ were the least sensitive. The temporal development of major Cole parameters indicates that CAP induced reversible changes in intercellular junctions, but not significant changes in membrane permeability. Sustained changes of τ suggested that long-lived ROS, such as H₂O₂, might play an important role. The proposed analysis confirms that an inherent advantage of EIS is the real time observation for CAP-induced changes on intercellular junctions, with a label-free and in situ method manner.     

Uncertainty of measurement for the determination of fluoride in water and wastewater by direct selective electrode potentiometry

A procedure to estimate the uncertainty of measurement applied to the fluoride determination of waters and wastewaters matrices by selective electrode potentiometry was implemented based on Eurachem Guide. The major sources of uncertainty were identified as the calibration standard solutions, fluoride concentration obtained by potential interpolation of the regression line and the precision. However the relative uncertainties depend on the anion concentration levels. The methodology proposed was presented to two fluoride concentration levels that are in the range of surface water samples (C _sample=1.12 mgF l⁻¹) and of wastewater matrices (C _sample=101.4 mgF l⁻¹). The expanded uncertainties calculated were 0.40 and 9.1 mg l⁻¹ for low and high concentration levels, respectively, using the reproducibility uncertainty as precision evaluation. The relative expanded uncertainty was around ±10% for the highest concentration, which can be considered acceptable for the ion selective electrode potenciometric methods and ±36% for the lowest concentrations. In this case the sample fluoride content is very close to the limit of quantification which has a relative uncertainty of about ±30%. If the repeatability was used in spite of duplicate analysis the same conclusions were obtained (C _sample=1.12 ± 0.39 mgF l⁻¹ and C _sample=101.4 ± 7.0 mgF l⁻¹). Although the calculated expanded uncertainties and consequently the combined uncertainty, do not vary significantly in the cases where it was used the repeatability or reproducibility for evaluating the precision, each relative variances uncertainty contributions do. When the repeatability is used to determine the combined uncertainty, the CSS and uncertainties contributions are the most dominant ones. However, if reproducibility is used, relative uncertainty variance contributions are distributed among CSS, C _F, and precision. In both cases, the contribution increases and r _CSS contribution decreases with the increasing of the concentration level. The precision variance contribution is only significant in the case where the reproducibility is used, and increases with the increasing of the concentration level. The uncertainty in the result calculated using the proposed methodology (C _sample ± U _sample = 2.17 ± 0.42 mgF l⁻¹) is in satisfactory agreement with the estimated expanded uncertainty obtained using the relative reproducibility standard deviation obtained in interlaboratory studies ().     

Uncertainty evaluation of the thermal expansion of simulated fuel

Thermal expansions of simulated fuel (SS1) are measured by using a dilatometer (DIL402C) from room temperature to 1900 K. The main procedure of an uncertainty evaluation was followed by the strategy of the UO₂ fuel. There exist uncertainties in the measurement, which should be quantified based on statistics. Referring to the ISO (International Organization for Standardization) guide, the uncertainties of the thermal expansion are quantified in three parts—the initial length, the length variation, and the system calibration factor. Each part is divided into two types. The A type uncertainty is derived from the statistical iterative measurement of an uncertainty and the B type uncertainty comes from a non-statistical uncertainty including a calibration and test reports. For the uncertainty evaluation, the digital calipers had been calibrated by the KOLAS (Korea Laboratory Accreditation Scheme) to obtain not only the calibration values but also the type B uncertainty. The whole system, the dilatometer (DIL402C), is composed of many complex sub-systems and in fact it is difficult to consider all the uncertainties of sub-systems. Thus, a calibration of the system was performed with a standard material (Al₂O₃), which is provided by NETZSCH. From the above standard uncertainties, the combined standard uncertainties were calculated by using the law of a propagation of an uncertainty. Finally, the expanded uncertainty was calculated by using the effective degree of freedom and the t-distribution for a given confidence level. The uncertainty of the thermal expansion for a simulated fuel was also compared with those of UO₂ fuel.     

Changes in the Transfer Function of a flow microcalorimeter

Flow microcalorimeters are used to determine thermodynamic properties of liquid mixtures, the accuracy of these measures depends on the right calibration of the instrument. In this work the system is identified by means of the transfer functions of the two poles, it is proven that the first time constant and the sensitivity change with the value of rc _p f of the injected liquids (r - density, c _p - heat capacity, f - injection flow), and that the sensitivities obtained in the electrical and chemical calibrations are different for the same value of rc _p f because the dissipation in each case does not occur in the same place. As a summary of the calibration carried out, it is proposed a sensitivity value of 313±4 mV W^-1 for rc _p f<15 mW K^-1 that permits to make thermal measures with an uncertainty of 3%. This revised version was published online in July 2006 with corrections to the Cover Date.     

Estimation of uncertainty in the determination of nitrogen oxides emissions

This paper presents a methodology for estimation of uncertainty on a reference test method for the determination of nitrogen oxides concentration in gaseous emissions from stationary sources. As a first stage for identification of uncertainty sources, the test method is carefully reviewed in detail. Afterwards, these sources are quantified, bearing in mind its partial uncertainty, allowing the determination of the combined uncertainty and, finally, the expanded uncertainty. The calculation procedure was implemented into an excel calculation file. Using this file and considering several numerical applications from real situations, uncertainities around 15 mg/Nm³ over determined concentrations of 350 mg/Nm³ of NO _x (expressed as NO₂) were obtained.     

Applications of k 0 NAA at FRM II with high f values

The research reactor FRM II offers different irradiation facilities with highly thermalized neutron flux. 3 facilities for the k ₀ neutron activation analysis (k ₀ NAA) will be introduced shortly. The influence of flux parameter α on the concentration calculation of samples irradiated in a neutron field with very high ratio of thermal to epi-thermal neutron flux f > 1,000 are here investigated. Even for the most k ₀ isotopes with big Q ₀ values, the uncertainty of a concentration calculation without α correction is <3 %, when the f value larger than 3,000. The uncertainty is about 5 % for the isotope ⁹⁶Zr in this case. The k ₀ library of the computer program MULTINAA is updated. A standard reference material IAEA/soil-7 was analyzed to verify the k ₀ NAA at FRM II.     

Classification of biodegradable materials using QSAR modelling with uncertainty estimation§

Abstract

The ability to determine the biodegradability of chemicals without resorting to expensive tests is ecologically and economically desirable. Models based on quantitative structure–activity relations (QSAR) provide some promise in this direction. However, QSAR models in the literature rarely provide uncertainty estimates in more detail than aggregated statistics such as the sensitivity and specificity of the model’s predictions. Almost never is there a means of assessing the uncertainty in an individual prediction. Without an uncertainty estimate, it is impossible to assess the trustworthiness of any particular prediction, which leaves the model with a low utility for regulatory purposes. In the present work, a QSAR model with uncertainty estimates is used to predict biodegradability for a set of substances from a publicly available data set. Separation was performed using a partial least squares discriminant analysis model, and the uncertainty was estimated using bootstrapping. The uncertainty prediction allows for confidence intervals to be assigned to any of the model’s predictions, allowing for a more complete assessment of the model than would be possible through a traditional statistical analysis. The results presented here are broadly applicable to other areas of modelling as well, because the calculation of the uncertainty will clearly demonstrate where additional tests are needed.