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Using the Diffusion Equation to Predict CST Resin Breakthroughs - 19051
The crystalline silicotitanate, CST, resin manufactured by UOP Honeywell is well-suited for single-use decontamination of caustic liquid wastes at Hanford and Savannah River Sites. There are many measurements of column loading and breakthrough for CST resin decontamination of Cs-137 in various Hanford and SRS waste simulants. Such measurements provide a very good basis for estimating CST resin demand and these more refined predictions make the CST column loading as efficient as possible given the very expensive CST resin. Although there are a variety of tools for predicting chromatographic resin column performance, [1-4], there is a particularly useful method that is suitable as a simple spreadsheet model [4]. This spreadsheet model is an analytical error function for the mass transfer zone (mtz) diffusion width in resin columns and is well-suited for resin breakthrough and column loading. This paper describes just such an error function spreadsheet model and how well the error function agrees with more traditional models and fits the observed small scale breakthrough curves for CST that can then better predict large scale column loading efficiency. The very high sorption of Cs onto CST tends to be slower than that of other lower sorption resins and as a result of the slow CST kinetics, there is also a marked asymmetry in many CST breakthrough curves. The CST breakthrough asymmetry is largely due to the distribution of CST particle sizes because smaller particles sorb much faster than larger particles. It is then the smaller particles that determine the leading mtz at a decontamination factor of 1,000 breakthrough and therefore the decontamination efficiency for a given loading. The larger CST particles correspondingly determine the tailing mtz width and therefore the overall loading efficiency and it is important to model both leading and tailing mtz widths to best optimize CST for a particular Cs-137 decontamination application. (authors)
Nodal variational principle for excited states
It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. In conclusion, simple numerical illustrations of the error functions are presented.
Halogen Thermochemistry Assessed with Density Functional Theory: Systematic Errors, Swift Corrections and Effects on Electrochemistry
Abstract Despite its sizable errors, density functional theory (DFT) is extensively used to evaluate thermochemical properties of gases, liquids and their interfaces with solids. As numerous halogen‐containing compounds appear as reactants, products and/or electrolytes in electrochemical reactions, and ionic effects are currently an active area of research, it is important to evaluate the accuracy of DFT for halogen thermochemistry. Herein, we assess the formation energies of interhalogens, hydrogen halides, diatomic and atomic halogens and their ions using six widespread functionals at the GGA, meta‐GGA and hybrid levels. We observe that DFT errors with respect to experiments are correlated with the electronegativity of the species and there are systematic trends across functionals, such that swift corrections were devised. Specifically, the average of the mean absolute errors for the six functionals decreased from 0.19 eV before the corrections to 0.08 eV after them. Besides, the overall maximum absolute error (MAX) decreased from 0.76 to 0.44 eV and the average of the MAXs decreased from 0.51 to 0.24 eV. Finally, we illustrate the qualitative and quantitative impact of gas‐phase errors on the predictions of surface Pourbaix diagrams.
Effects of Dufour and fractional derivative on unsteady natural convection flow over an infinite vertical plate with constant heat and mass fluxes
In this paper, we analyze the effects of Dufour number and fractional-order derivative on unsteady natural convection flow of a viscous and incompressible fluid over an infinite vertical plate with constant heat and mass fluxes. The fractional constitutive model is obtained using fractional calculus approach. The Caputo fractional derivative operator is used in this problem. The dimensionless system of equations has been solved by employing Laplace transformation technique. Closed form solutions for concentration, temperature and velocity are presented in the form of Wright function and complementary error function. Effects of fractional and physical parameters on temperature and velocity profiles are illustrated graphically.
Using Density-Corrected DFT to Understand Density-Driven and Functional-Dependent Errors in Ab Initio Simulations of the Hydrated Electron
The hydrated electron, an excess electron in liquid water, plays a crucial role in a plethora of chemical processes, motivating extensive research efforts to characterize its structure, dynamics, and reactivity in solution. Recent theoretical approaches to understanding this intriguing object have involved ab initio simulations based on density functional theory (DFT). Although DFT allows for the study of hydrated electron reactivity and quantum mechanical behavior, it is well-known that anionic systems can suffer from significant density-driven errors (DDEs). Density-corrected DFT (DC-DFT) provides a framework to mitigate such errors; the method reduces DDEs by replacing the self-consistent (SC) density associated with a given density functional with the Hartree–Fock (HF) density. Since HF densities tend to be more localized than DFT SC densities, the DC-DFT scheme significantly improves errors in calculations where the SC density is spuriously delocalized. Here, we investigate how the use of density correction affects the calculated properties of the DFT-simulated (PBEh) hydrated electron, a particularly challenging diffuse anionic system to simulate. First, we analyze charge delocalization in a system consisting of a model octahedral hydrated electron water cluster (the so-called Kevan structure) along with a spatially separated sulfur atom. We show that the use of density correction indeed reduces DDEs in comparison to a standard DFT global hybrid functional. We then propagate molecular dynamics trajectories of the hydrated electron using DC-DFT, where we find that DC further localizes electron density in the cavity region, a signature of reduced charge delocalization. Unfortunately, the decreased radius of gyration of the spin density and corresponding tightening of the local solvation structure from density correction causes predicted observables to deviate further from experimental measurements than when density correction is not employed. Here, we argue that DC’s worse agreement with experiment results from the removal of a fortuitous cancellation of errors that is intrinsic to the PBEh functional. This indicates that the difficulties with DFT to simulate hydrated electrons are primarily due to the inherent approximations in DFT rather than to density-driven errors.
High Impedance Fault Detection Through Quasi-Static State Estimation: A Parameter Error Modeling Approach
This paper presents a model for detecting high impedance faults using parameter error modeling and a two step per-phase weighted-least squares state estimation process. The proposed scheme leverages the use of Phasor Measurement Units and synthetic measurements to identify per-phase power flow and injection measurements which indicate a parameter error through ?2 Hypothesis Testing applied to the composed measurement error. Although current and voltage waveforms are commonly analyzed for high-impedance fault detection, wide area power flow and injection measurements, which are already inherent to the state estimation process, also show promise for real-world high-impedance fault detection applications. The error distributions after detection share the measurement function error spread observed in proven parameter error diagnostics and can be applied to high-impedance fault identification. Further, this error spread across measurement functions related to the fault will be clearly discerned from measurement error. Case studies are performed on the IEEE 33-Bus Distribution System along with the proposed model in Simulink.
Fractional dynamics of an erbium-doped fiber laser model
In this paper we investigate the model of the time-fractional dynamics of an erbium-doped fiber laser model (TFDEFL) with Liouville–Caputo (LC), Caputo–Fabrizio–Caputo (CFC) and Atangana–Baleanu–Caputo (ABC) time-fractional derivatives. We employ the homotopy analysis transform method (HATM) to calculate approximate solutions for the TFDEFL model. This method gives the solution in the form of a rapidly convergent series that can ensure the convergence in solving the resultant series. We study the convergence analysis of HATM by computing the interval of convergence through the h-curves, the residual error function and the average residual error, respectively. We also show the effectiveness and accuracy of this method by comparing the approximate solutions based upon the LC, CFC and ABC time-fractional derivatives.
New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations
In this paper, we extend the model of the Korteweg–de Vries (KDV) and Korteweg–de Vries–Burger’s (KDVB) to new model time fractional Korteweg–de Vries (TFKDV) and time fractional Korteweg–de Vries-Burger’s (TFKDVB) with Liouville–Caputo (LC), Caputo–Fabrizio (CF), and Atangana-Baleanu (AB) fractional time derivative equations, respectively. We utilize the q-homotopy analysis transform method (q-HATM) to compute the approximate solutions of TFKDV and TFKDVB using LC, CF and AB in Liouville–Caputo sense. We study the convergence analysis of q-HATM by computing the Residual Error Function (REF) and finding the interval of the convergence through the h-curves. Also, we find the optimal values of h so that, we assure the convergence of the approximate solutions. The results are very effective and accurate in solving the TFKDV and TFKDVB.
Delocalization Errors in Density Functional Theory Are Essentially Quadratic in Fractional Occupation Number
Approximate functionals used in practical density functional theory (DFT) deviate from the piecewise linear behavior of the exact functional for fractional charges. This deviation causes excess charge delocalization, which leads to incorrect densities, molecular properties, barrier heights, band gaps, and excitation energies. Here, we present a simple delocalization function for characterizing this error and find it to be almost perfectly linear vs the fractional electron number for systems spanning in size from the H atom to the C 12 H 14 polyene. This causes the delocalization energy error to be a quadratic polynomial in the fractional electron number, which permits us to assess the comparative performance of 47 popular and recent functionals through the curvature. The quadratic form further suggests that information about a single fractional charge is sufficient to eliminate the principal source of delocalization error. Generalizing traditional two-point information like ionization potentials or electron affinities to account for a third, fractional charge-based data point could therefore permit fitting/tuning of functionals with lower delocalization error.
Optimization of spray breakup model parameters for predicting fuel spray and film characteristics in gasoline direct injection engines
This study investigated the behavior of gasoline direct injection (GDI) sprays using computational fluid dynamics (CFD). The authors developed an approach to identify optimal spray breakup model parameters by evaluating an error function across numerous simulations, with the goal of minimizing discrepancies from experimental data. Using the optimal setup, the simulated spray matched well with projected liquid volume distributions, liquid penetration, and spray width measured in a constant-pressure continuous-flow chamber. To further validate the approach, the same setup was tested across various fuels, injectors, and operating conditions. Subsequently, the optimal setup, along with a recently developed spray-wall interaction model, were applied to a direct-injection spark-ignited engine under late-injection conditions to predict and evaluate fuel film formation and evolution at varying engine coolant temperatures. Here, with the centrally mounted injector directing the spray toward the piston, simulations indicated that the spray tends to impinge on the piston surface. The proposed simulation framework also accurately captured the aggregate film area on the piston surface, aligning with previously published experimental results. Moreover, simulations showed that increasing the coolant temperature from cold start conditions (333 K) to warm conditions (363 K) reduced the fuel mass deposited on the piston by roughly 50%. Furthermore, for the spray-guided engine configuration studied in this work, the CFD model predicted minimal film deposition on the spark plug electrodes regardless of the coolant temperatures due to a relatively weak in-cylinder flow during the compression phase.
Multilabel proportion prediction and out-of-distribution detection on gamma spectra of short-lived fission products
In the machine learning problem of multilabel classification, the objective is to determine for each test instance which classes the instance belongs to. In this work, we consider an extension of multilabel classification, called multilabel proportion prediction, in the context of radioisotope identification (RIID) using gamma spectra data. We aim to not only predict radioisotope proportions, but also identify out-of-distribution (OOD) spectra. We achieve this goal by viewing gamma spectra as discrete probability distributions, and based on this perspective, we develop a custom semi-supervised loss function that combines a traditional supervised loss with an unsupervised reconstruction error function. Our approach was motivated by its application to the analysis of short-lived fission products from spent nuclear fuel. In particular, we demonstrate that a neural network model trained with our loss function can successfully predict the relative proportions of 37 radioisotopes simultaneously. The model trained with synthetic data was then applied to measurements taken by Pacific Northwest National Laboratory (PNNL) to conduct analysis typically done by subject-matter experts. Here, we also extend our approach to successfully identify when measurements are OOD, and thus should not be trusted, whether due to the presence of a novel source or novel proportions.
Calculation of Nuclear Reactor Cooling Tower Performance With Limited Data Streams
Monitoring of cooling tower performance in a nuclear reactor facility is necessary to ensure safe operation; however, instrumentation for measuring performance characteristics can be difficult to install and may malfunction or break down over long duration experiments. This paper describes employing a thermodynamic approach to quantify cooling tower performance, the Merkel model, which requires only five parameters, namely, inlet water temperature, outlet water temperature, liquid mass flowrate, gas mass flowrate, and wet bulb temperature. Using this model, a general method to determine cooling tower operation for a nuclear reactor was developed in situations when neither the outlet water temperature nor gas mass flowrate are available, the former being a critical piece of information to bound the Merkel integral. Furthermore, when multiple cooling tower cells are used in parallel (as would be in the case of large-scale cooling operations), only the average outlet temperature of the cooling system is used as feedback for fan speed control, increasing the difficulty of obtaining the outlet water temperature for each cell. To address these shortcomings, this paper describes a method to obtain individual cell outlet water temperatures for mechanical forced-air cooling towers via parametric analysis and optimization. In this method, the outlet water temperature for an individual cooling tower cell is acquired as a function of the liquid-to-gas ratio (L/G). Leveraging the tight tolerance on the average outlet water temperature, an error function is generated to describe the deviation of the parameterized L/G to the highly controlled average outlet temperature. The method was able to determine the gas flowrate at rated conditions to be within 3.9% from that obtained from the manufacturer’s specification, while the average error for the four individual cooling cell outlet water temperatures were 1.6 °C, -0.5 °C, -1.0 °C, and 0.3 °C.
Impact of point spread function higher moments error on weak gravitational lensing – II. A comprehensive study
ABSTRACT Weak lensing is one of the most powerful probes for dark matter and dark energy science, although it faces increasing challenges in controlling systematic uncertainties as the statistical errors become smaller. The point spread function (PSF) needs to be precisely modeled to avoid systematic error on the weak lensing measurements. The weak lensing biases induced by errors in the PSF model second moments, i.e. its size and shape, are well-studied. However, Zhang et al. showed that errors in the higher moments of the PSF may also be a significant source of systematics for upcoming weak lensing surveys. Therefore, this work comprehensively investigate the modelling quality of PSF moments from the 3rd to 6th order, and propagate the PSFEx higher moments modelling error in the HSC survey data set to the weak lensing shear–shear correlation functions and their cosmological analyses. The overall multiplicative shear bias associated with errors in PSF higher moments can cause an ∼0.1σ shift on the cosmological parameters for LSST Y10, while the associated additive biases can induce 1σ uncertainties in cosmology parameter inference for LSST Y10, if not accounted. We compare the PSFEx model with PSF in Full FOV, and find similar performance in modelling the PSF higher moments. We conclude that PSF higher moment errors of the future PSF models should be reduced from those in current methods, otherwise needed to be explicitly modeled in the weak lensing analysis.
Adaptive Time Step Control for Multirate Infinitesimal Methods
Multirate methods have been used for decades to temporally evolve initial-value problems in which different components evolve on distinct time scales, and thus use of different step sizes for these components can result in increased computational efficiency. Generally, such methods select these different step sizes based on experimentation or stability considerations. For problems that evolve on a single time scale, adaptivity approaches that strive to control local temporal error are widely used to achieve numerical results of a desired accuracy with minimal computational effort, while alleviating the need for manual experimentation with different time step sizes. However, there is a notable gap in the publication record on the development of adaptive time step controllers for multirate methods. In this paper, we extend the single-rate controller work of Gustafsson [ACM Trans. Math. Software, 20 (1994), pp. 496-517] to the multirate method setting. Specifically, we develop controllers based on polynomial approximations to the principal error functions for both the "fast" and "slow" time scales within multirate infinitesimal (MRI) methods. We additionally investigate a variety of approaches for estimating the errors arising from each time scale within MRI methods. We then numerically evaluate the proposed multirate controllers and error estimation strategies on a range of multirate test problems, comparing their performance against an estimated optimal performance. Through this work, we combine the most performant of these approaches to arrive at a set of multirate adaptive time step controllers that robustly achieve desired solution accuracy with minimal computational effort.
Explicit solution for a two-phase fractional Stefan problem with a heat flux condition at the fixed face
A generalized Neumann solution for the two-phase fractional Lamé–Clapeyron–Stefan problem for a semi-infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a restriction on data is satisfied. The fractional derivative in the Caputo sense of order α∈(0,1) respect on the temporal variable is considered in two governing heat equations and in one of the conditions for the free boundary. Furthermore, we find a relationship between this fractional free boundary problem and another one with a constant temperature condition at the fixed face and based on that fact, we obtain an inequality for the coefficient which characterizes the fractional phase–change interface obtained in Roscani and Tarzia (Adv Math Sci Appl 24(2):237–249, 2014). We also recover the restriction on data and the classical Neumann solution, through the error function, for the classical two-phase Lamé–Clapeyron–Stefan problem for the case α=1.
Exploring the dependence of gas cooling and heating functions on the incident radiation field with machine learning
ABSTRACT Gas cooling and heating functions play a crucial role in galaxy formation. But, it is computationally expensive to exactly compute these functions in the presence of an incident radiation field. These computations can be greatly sped up by using interpolation tables of pre-computed values, at the expense of making significant and sometimes even unjustified approximations. Here, we explore the capacity of machine learning to approximate cooling and heating functions with a generalized radiation field. Specifically, we use the machine learning algorithm XGBoost to predict cooling and heating functions calculated with the photoionization code cloudy at fixed metallicity, using different combinations of photoionization rates as features. We perform a constrained quadratic fit in metallicity to enable a fair comparison with traditional interpolation methods at arbitrary metallicity. We consider the relative importance of various photoionization rates through both a principal component analysis (PCA) and calculation of SHapley Additive exPlanation (shap) values for our XGBoost models. We use feature importance information to select different subsets of rates to use in model training. Our XGBoost models outperform a traditional interpolation approach at each fixed metallicity, regardless of feature selection. At arbitrary metallicity, we are able to reduce the frequency of the largest cooling and heating function errors compared to an interpolation table. We find that the primary bottleneck to increasing accuracy lies in accurately capturing the metallicity dependence. This study demonstrates the potential of machine learning methods such as XGBoost to capture the non-linear behaviour of cooling and heating functions.
A Differentiable Hydrology Approach for Modeling With Time-Varying Parameters
In the analysis of observational data with mathematical models, it is often desirable to let one or more model parameters vary with time to account for changing environmental conditions or to offer more flexibility. A standard approach for dynamic models is to allow for the number of parameters to grow as a function of time, leading to a high-dimensional inference problem for long data records. We propose using Hamiltonian Monte Carlo, a gradient-based Markov chain Monte Carlo (MCMC) method, to perform Bayesian inference for time-varying parameters (TVP) in hydrology models. As derivatives of model error functions with regard to parameters are not available in closed form, we implemented the GR4J rainfall-runoff models in Theano/PyMC3, and Jax/NumPyro, allowing for the application of automatic differentiation techniques. A simulation experiment assessing the viability of these methods for recovering underlying temporal variation in parameters indicates that a discrete Gaussian random walk-based prior appears to be best suited across a range of scenarios while the continuous-time Gaussian process compares relatively poorly. In an analysis of 20 years of daily real-world streamflow records from the Model Parameter Estimation Experiment (MOPEX) hydrology data set with increasing data sparsity and at multiple temporal resolutions, we find that TVP inference with gradient-based MCMC is a flexible and reliable approach for analyzing hydrology models with dynamic parameter sets.