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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 325 records · Page 18
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Prediction of Self-Diffusion in Binary Fluid Mixtures Using Artificial Neural Networks

Artificial neural networks (ANNs) were developed to accurately predict the self-diffusion constants for individual components in binary fluid mixtures. The ANNs were tested on an experimental database of 4328 self-diffusion constants from 131 mixtures containing 75 unique compounds. The presence of strong hydrogen bonding molecules may lead to clustering or dimerization resulting in non-linear diffusive behavior. To address this, self- and binary association energies were calculated for each molecule and mixture to provide information on intermolecular interaction strength and were used as input features to the ANN. An accurate, generalized ANN model was developed with an overall average absolute deviation of 4.1%. Forward input feature selection reveals the importance of critical properties and self-association energies along with other fluid properties. Additional ANNs were developed with subsets of the full input feature set to further investigate the impact of various properties on model performance. The results from two specific mixtures are discussed in additional detail: one providing an example of strong hydrogen bonding and the other an example of extreme pressure changes, with the ANN models predicting self-diffusion well in both cases.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

AUTOIGNITION DELAY TIMES FOR REFORMATE GAS MIXTURES FROM METHANE GAS ENGINES

Methane slip is a prominent issue in natural gas reciprocating engines that are used in transportation and marine applications. The incomplete combustion that results in methane slip can be resolved with the introduction of hydrogen within the combustion mixture to improve methane oxidation and further enable combustion within the engine crevices where methane has previously remained unreacted. Steam methane reforming (SMR) is a common method used to produce hydrogen and can be used to design an onboard device to reduce methane slip from reciprocating engines. The development of this reformer device requires the validation of high-fidelity chemical kinetic models at the low temperatures of the crevice volumes of these engines. In this work, auto-ignition data is obtained using a shock tube at lean (φ—0.714 or λ—1.4) and stoichiometric (φ, λ = 1) equivalence ratios spanning a temperature range of 1042–1234 K at the 80-bar operating pressure of the test engine. Blends of methane, hydrogen, and reformate products from the SMR reaction are shock-heated in synthetic air, with the ignition delay time measured using an OH* chemiluminescence detector at 310 nm and a CH* detector at 430 nm. The experimental results are compared to several state-of-the-art chemical kinetic mechanisms from the literature. In general, most of the mechanisms show very good agreement with experiments at higher temperatures, with simulation results showing little deviation from experiments at lower temperatures. A sensitivity analysis was conducted, and the results reveal that the reaction H2 + CH3O2 = H + CH3O2H has a very significant role in determining low-temperature ignition delay times (IDTs) of SMR mixtures. These findings provide valuable insights into the chemical kinetics governing methane reformate combustion and contribute to the optimization of onboard reformer designs aimed at mitigating methane slip in natural gas-fueled engines.

Fraze, Matthew↗
DOE OSTI2021

Surfactants and cloud droplet activation: A systematic extension of Köhler theory based on analysis of droplet stability

The activation of aerosol particles to form cloud droplets, a necessary first step in cloud formation, controls much of the impact that aerosols have on clouds and climate. Recently, there has been a surge of interest in extending the Köhler theory of cloud droplet activation to include surface active (typically organic) as well as water-soluble (typically inorganic) aerosol components, but a systematic framework for doing this has yet to be developed. Here, we apply a droplet stability analysis to this end. Ideal and Szyszkowski–Langmuir surfactant models are analyzed to demonstrate the new approach, but the underlying theoretical framework is fundamental and model free. A key finding is that superficial densities at the cloud activation threshold (Köhler maximum) are significantly sub-monolayer, with fractional coverage ranging from 69% to 85% for the organic compounds and mixtures studied. The result, significant for model inventories of cloud condensation nuclei, is a weakening of the surfactant effect relative to expectations based on bulk sample measurements. Analytical results are obtained for the loci of Köhler maxima and applied to aerosol mixtures containing an arbitrary number of water-soluble and surfactant components.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Measurement of diamond nucleation rates from hydrocarbons at conditions comparable to the interiors of icy giant planets

We present measurements of the nucleation rate into a diamond lattice in dynamically compressed polystyrene obtained in a pump-probe experiment using a high-energy laser system and in situ femtosecond x-ray diffraction. Different temperature-pressure conditions that occur in planetary interiors were probed. For a single shock reaching 70 GPa and 3000 K no diamond formation was observed, while with a double shock driving polystyrene to pressures around 150 GPa and temperatures around 5000 K nucleation rates between 10 29 and 10 34 m –3 s –1 were recorded. These nucleation rates do not agree with predictions of the state-of-the-art theoretical models for carbon-hydrogen mixtures by many orders of magnitude. Our data suggest that there is significant diamond formation to be expected inside icy giant planets like Neptune and Uranus.

43 PARTICLE ACCELERATORS↗
DOE OSTI2022

Thermal properties of HfN-MoW surrogate cermet fuel for nuclear thermal propulsion

Here, the thermal properties of MoW-HfN, a surrogate cermet for MoW-UN nuclear thermal propulsion fuel, were characterized over a wide range of elevated temperatures. Thermal diffusivity, coefficient of thermal expansion (CTE), and heat capacity were measured. Optical and scanning electron microscopy were performed to characterize the microstructure and draw structure–property correlations. The thermal diffusivity was obtained using the laser flash method. Diffusivity values ranged from about 0.18 cm 2 /s at 200°C to 0.15 cm 2 /s at 1800°C. The CTE was measured using push-rod dilatometry up to 1600°C, giving values between 6.0 and 9.0 μm/m. A scientific rationalization of the effective material properties is made using the rule-of-mixtures and other effective properties models.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2021

Hydrogenous Content in Residues from Aqueous Chloride Pu and Am Recovery Operations

Byproduct residues generated from Aqueous Chloride Pu and Am recovery operations at Los Alamos National Laboratory have been investigated and determined to contain high weight % chloride salts and minimal hydrogen content. Hydrogen content is the result of limited surface area adsorption of ambient moisture. The assumption used in technical analysis NCS-TECH-19-029 to model potentially hydrogenous items as mixtures of polyethylene is shown to be overly conservative. Our results and literature precedent indicate that water adsorption of 1-3 % by weight is expected for hydroxide cakes and Pu oxides, and a conservative bounding assumption for these materials is 5% water by weight. LOH results for a dissolution heel stored long term showed 6.25% weight loss, which was conservatively assumed to be water. A conservative bounding assumption for dissolution heels is 8 wt.% water content. This equates to 0.55 % hydrogen by weight for hydroxide cakes, and 0.88% hydrogen by weight for dissolution heels. The bulk composition of dissolution heels reflects the composition of the original byproduct feed material, chloride salts of potassium, sodium, magnesium and calcium. The bulk composition of calcined hydroxide precipitation residues is composed of potassium chloride as a result of potassium hydroxide neutralization of hydrochloric acid solutions. Metal hydroxides generated during the hydroxide precipitation process are decomposed under the conditions of calcination to metal oxide forms.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2024

Reduced-order condensed-phase kinetic models for polyethylene, polypropylene and polystyrene thermochemical recycling

Thermochemical recycling of plastic waste (PW) into chemicals and energy vectors requires coupling particle and reactor-scale simulations to accurate condensed phase pyrolysis mechanisms for each constituent. This work proposes a methodology to derive reduced-order condensed-phase kinetic models from validated semi-detailed kinetic mechanisms. Two types of kinetic models are obtained for polyethylene (PE), polypropylene (PP) and polystyrene (PS): reduced semi-detailed models and multi-step fully lumped ones. These families offer different compromises between accuracy and computational cost. The former employ 50–100 gas + liquid species and describe both the radical degradation and the detailed carbon distribution of the products. Conversely, the latter involves 5–10 species per polymer tracking only the main petroleum cuts. The kinetic mechanisms are complemented by the definition of thermochemical properties of gas, liquid, and solid-phase species, accounting for phase-transitions through pseudo-chemical reactions. Model validations are performed by comparison with experimental data and the original semi-detailed mechanisms in terms of mass loss, heat fluxes and product distribution profiles. The resulting CHEMKIN-like condensed-phase models are attached as Supplementary Material and as a GitHub repository. Extending the proposed approach to other polymers and coupling it with existing subsets in the CRECK kinetic framework (e.g., biomass, PVC, PET) offers a powerful tool to model thermochemical recycling of PW and biomass/PW mixtures.

kinetics↗
DOE OSTI2026

Application of an Empirical Density Law via Python for Aqueous Plutonium Chloride Systems for MCNP6

Criticality safety models for aqueous plutonium chloride systems often contain a significant bias due to assumptions in material compositions. These systems are currently modeled as a fictitious metal-water mixture because little is known about the true solution density. Furthermore, no predictive density tools or capabilities for modeling aqueous plutonium chloride systems are approved for use at Los Alamos National Laboratory. Recent density measurements of this ternary system (PuCl 3 -HCl-H 2 O) have allowed for the development of a more realistic density law, which is applied in this work via an empirical method based in Python. This tool, entitled PuCS (Plutonium Chloride Solution tool) may be used to determine solution density and composition based on the plutonium content, acid content, and temperature for MCNP6 inputs. PuCS has been found to predict density within 2% of experimental data. In conclusion, MCNP6 calculations have found that crediting minimal amounts of free acid (0.5 M) may correspond to a ~12% decrease in peak reactivity in comparison to current modeling methods.

42 ENGINEERING↗
DOE OSTI2021

Investigation of post-injection strategies for diesel engine Catalyst Heating Operation using a vapor-liquid-equilibrium-based spray model

Most multidimensional engine simulations spend much time solving for non-equilibrium spray dynamics (atomization, collision, vaporization). However, their accuracy is limited by significant grid dependency, and the need for extensive calibration. This is critical for modeling cold-start diesel fuel post injections, which occur at low temperatures and pressures, far from typical model validation ranges. At the same time, resolving micron-scale spray phenomena would render full Eulerian multiphase calculations prohibitive. In this study, an improved phase equilibrium based approach was implemented and assessed for simulating diesel catalyst heating operation strategies. A phase equilibrium solver based on the model by Yue and Reitz [1] was implemented: a fully multiphase CFD solver is employed with an engineering-size engine grid, and fuel injection is modeled using the standard Lagrangian parcels approach. Mass and energy from the liquid parcels are released to the Eulerian multiphase mixture according to an equilibrium-based liquid jet model. An improved phase equilibrium solver was developed to handle large real-gas mixtures such as those from accurate chemical kinetics mechanisms. The liquid-jet model was improved such that momentum transfer to the Eulerian solver better reproduces the physical spray jet structure. Validation of liquid/vapor penetration predictions showed that the model yields accurate results with very limited tuning and low sensitivity to the few calibration constants. In-cylinder simulations of diesel catalyst heating operation strategies showed that capturing spray structure is paramount when short, transient injection pulses and low temperatures are present. Furthermore, the EP model provides improved predictions of post-injection spray structure and ignitability, while conventional spray modeling does not capture the increase of liquid penetration during the expansion stroke. Finally, the only important EP model calibration constant, Cliq, does not affect momentum transfer, but it changes the local charge cooling distribution through the local energy transfer, which makes it candidate to additional research. The results confirm that non-equilibrium spray processes do not need to be resolved in engineering simulations of high-pressure diesel sprays.

42 ENGINEERING↗
DOE OSTI2023

Reinforcement learning based hybrid bond-order coarse-grained interatomic potentials for exploring mesoscale aggregation in liquid–liquid mixtures

Exploring mesoscopic physical phenomena has always been a challenge for brute-force all-atom molecular dynamics simulations. Although recent advances in computing hardware have improved the accessible length scales, reaching mesoscopic timescales is still a significant bottleneck. Coarse-graining of all-atom models allows robust investigation of mesoscale physics with a reduced spatial and temporal resolution but preserves desired structural features of molecules, unlike continuum-based methods. Here, we present a hybrid bond-order coarse-grained forcefield (HyCG) for modeling mesoscale aggregation phenomena in liquid–liquid mixtures. The intuitive hybrid functional form of the potential offers interpretability to our model, unlike many machine learning based interatomic potentials. We parameterize the potential with the continuous action Monte Carlo Tree Search (cMCTS) algorithm, a reinforcement learning (RL) based global optimizing scheme, using training data from all-atom simulations. The resulting RL-HyCG correctly describes mesoscale critical fluctuations in binary liquid–liquid extraction systems. cMCTS, the RL algorithm, accurately captures the mean behavior of various geometrical properties of the molecule of interest, which were excluded from the training set. The developed potential model along with the RL-based training workflow could be applied to explore a variety of other mesoscale physical phenomena that are typically inaccessible to all-atom molecular dynamics simulations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

An experimental and kinetic modeling study of cyclopentane and dimethyl ether blends

Cyclopentane is a suitable naphthene, or cycloalkane, in a palette for multi-component gasoline surrogate fuels due to its presence in market fuels and its relevance to alkyl substituted cyclopentanes also present. However, the previous oxidation studies of cyclopentane have primarily focused on neat mixtures. Blending cyclopentane with dimethyl ether in this work therefore serves to inform our understanding of, and improve predictive models for, multi-component mixtures. Here, the auto-ignition of cyclopentane/dimethyl ether blends was studied in a high-pressure shock tube and in a rapid compression machine. A wide range of temperatures (650 – 1350 K) and elevated pressures of 20 and 40 bar were studied at equivalence ratios of 0.5, 1.0 and 2.0 in air for two blending ratios (30/70 and 70/30 mole% cyclopentane/di-methyl ether mixtures). A detailed kinetic model for cyclopentane was revised to capture the measured ignition delay times and apparent heat release rates in this study. Literature ignition delay time, jet-stirred reactor, and laminar burning velocity measurements of neat cyclopentane were used as additional validation. Improvements to the kinetic model were based on recent literature studies related to sub-models including cyclopentene and cyclopentadiene which allowed the removal of previous local rate-constant optimizations. Low temperature reactivity of cyclopentane was found to be controlled by the branching ratio between concerted elimination of HO͘ 2 and the strained formation of Q&#775OOH radicals in agreement with previous studies. In this study, the low branching ratio of Q&#775OOH formation increases the influence of a competing consumption pathway for cyclopentyl-peroxy (CPTO͘ 2 J) radicals. The sensitivity of the simulated ignition delay times to the formation of cyclopentyl hydroperoxide (CPTO 2 H), from CPTO͘ 2 J and HO͘ 2 , is discussed. The current model is used to analyze the influence of dimethyl ether on the reactivity of cyclopentane in the context of previous literature studies of dimethyl ether binary blends with ethanol and toluene.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Modeling hydrodynamic and biomass pyrolysis effects of recycled product gases in a bubbling fluidized bed reactor

Fast pyrolysis of biomass in a fluidized bed reactor is typically conducted in a nitrogen gas environment. Recycling product gas can improve the economics of operating such a system by reducing reliance on pure process streams, but much less is known about how recycling pyrolysis product gas may affect fluidization behavior and pyrolysis kinetics. Therefore, gas effects in a fluidized bed biomass pyrolysis reactor were investigated using engineering correlations, low-order models, and CFD simulations for N 2 , H 2 , CO, CO 2 , and CH 4 carrier gas mixtures. Here, our findings reveal viscosity of a gas mixture can be significantly underestimated depending on the model and correlation. Furthermore, fluidization characteristics such as U mf and gas-solid convective heat transfer can be greatly affected by the gas properties. By utilizing H 2 as the fluidizing gas (instead of N 2 ), while maintaining a constant fluidization ratio (U s /U mf ), the bio-oil yields can be increased ~5%. This is due to the lower density H 2 producing similar hydrodynamics as N 2 at higher gas flow rates. These higher flow rates result in shorter gas residence times, and as a result, less secondary reactions that convert bio-oil to light gases and char. Model results also suggest that bio-oil yield is not significantly affected by the type of carrier gas used, with bio-oil yield varying by ~2% across different gas mixtures while maintaining constant flow rate. In conclusion, this indicates that recycled pyrolytic gas can be used as the carrier gas for biomass pyrolysis.

09 BIOMASS FUELS↗
DOE OSTI2022

Application of Linear Additive Conditions for Near-Infrared Diffuse Reflectance Absorption Spectroscopy

Determining the homogeneity of material mixing in real time during product processing is critical for quality control. According to the Kubelka–Munk (K-M) function of diffuse reflectance absorption spectrum, absorbance (A) is approximately linear with the content of the components when the sample scattering coefficient (S) is in a certain range. The S is determined by the particle size of powder samples. Therefore, this study determined particle size ranges that satisfy linear additivity in near-infrared diffuse reflectance spectroscopy (NIRDRS). Thus, the proposed NIRDRS analysis technique can be used to determine the homogeneity of material mixes or analyze the percentages of the components in the mixture. In this study, vitamin B3 and vitamin C were used for preparing mixed samples with varying percentages. The experimental results revealed that linear additivity is satisfied when the powder particle size is in the range of less than 280, 280–450, and 450–900 μm. When the confidence level is 0.01, the actual mixed spectra are not significantly different from the “simulated mixed spectra” constructed by linear addition, with their relative deviations less than 1.08%. The absolute errors of the actual and analytic percentages were within 2.98% for each component in the mixtures. The above conclusions also hold for sorghum, which has a complex material composition. Statistical models cannot analyze the percentages of components in the mixture. In contrast, linear addition and direct calibration approach avoids the use of a large number of samples for statistical modeling and analyze the percentages of mixed samples. Meanwhile, it can be used to discriminate and analyze the material mixing uniformity by building a mechanistic model.

Feng, Zhiyue↗
DOE OSTI2026

Enhancing photoionization rate calculations in low-temperature plasmas using spectral methods

Photoionization plays a central role in the development of streamer discharges and other non-equilibrium plasma phenomena. It creates seed electrons, which are essential for positive streamer propagation, allowing the ionization front to move forward. Because of this, accurate modeling of photoionization is very important for predicting streamer behavior and plasma evolution. The photoionization process in air (N 2 – O 2 mixture) is often described by the Zheleznyak model (1982). This model is usually solved through Helmholtz-type equations that approximate the Zheleznyak photoionization model (Zheleznyak et al. 1982) as Partial Differential Equations (PDEs). Conventional numerical methods, such as the Finite Difference Method (FDM) or Finite Volume Method (FVM), are widely used to solve these equations. Although they are prevalent, the computational cost of these methods due to their need for matrix operations and iterative solver is demanding. To address this challenge, this work develops a spectral solver based on the Fast Fourier Transform (FFT) combined with Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) to calculate the photoionization rate efficiently in an axisymmetric cylindrical domain. This method naturally satisfies the boundary conditions used in the model and converts the PDE into algebraic ones in spectral space. Thus, avoids the need for iterative matrix solvers. When compared with FDM results, it is demonstrated that the new solver not only maintains accuracy, but also reduces the computational cost, showing a performance increase of approximately 100 compared to FDM over a wide range of problem sizes. The method is parallelized using Message Passing Interface (MPI) and has been integrated into a fluid plasma model for streamer simulation. Here, this FFT-based approach provides a fast and reliable alternative for calculating photoionization in fluid models, helping large-scale plasma simulations run faster and efficiently, and allows higher-resolution simulation without extra computational cost.

Axisymmetric system↗
DOE OSTI2025

A fluctuating hydrodynamics model for nanoscale surfactant-laden interfaces

A multispecies diffuse interface model is formulated in a fluctuating hydrodynamics framework for the purpose of simulating surfactant interfaces at the nanoscale. The model generalizes previous work to ternary mixtures, employing a Cahn-Hilliard free energy density combined with incompressible, isothermal fluctuating hydrodynamics where dissipative fluxes include both deterministic and stochastic terms. The intermolecular parameters in the free energy are chosen such that one species acts as a partially miscible surfactant. From Laplace pressure measurements, we show that in this model the surface tension decreases linearly with surfactant concentration, leading to Marangoni convection for interfaces with concentration gradients. In the capillary wave spectrum for interfaces with and without surfactant, we find that for the former, the spectrum deviates significantly from classical capillary wave theory, presumably due to Gibbs elasticity. In non-equilibrium simulations of the Rayleigh-Plateau instability, deterministic simulations showed that the surfactant delays pinching of a fluid cylinder into droplets. However, stochastic simulations indicate that thermal fluctuations disrupt the surfactant's stabilizing effect. Similarly, the spreading of a patch of surfactant, driven by Marangoni convection, was found to be partially suppressed by thermal fluctuations.

Capillary waves↗
DOE OSTI2020

Combined Experimental/Numerical Study of the Soot Formation Process in a Gasoline Direct-Injection Spray in the Presence of Laser-Induced Plasma Ignition

Here, combustion issued from an eight-hole, direct-injection spray was experimentally studied in a constant-volume pre-burn combustion vessel using simultaneous high-speed diffused back-illumination extinction imaging (DBIEI) and OH* chemiluminescence. DBIEI has been employed to observe the liquid-phase of the spray and to quantitatively investigate the soot formation and oxidation taking place during combustion. The fuel-air mixture was ignited with a plasma induced by a single-shot Nd:YAG laser, permitting precise control of the ignition location in space and time. OH* chemiluminescence was used to track the high-temperature ignition and flame. The study showed that increasing the delay between the end of injection and ignition drastically reduces soot formation without necessarily compromising combustion efficiency. For long delays between the end of injection and ignition (1.9 ms) soot formation was eliminated in the main downstream charge of the fuel spray. However, poorly atomized and large droplets formed at the end of injection (dribble) eventually do form soot near the injector even when none is formed in the main charge. The quantitative soot measurements for these spray and ignition scenarios, resolved in time and space, represents a significant new achievement. Reynolds-averaged Navier-Stokes (RANS) simulations were performed to assess spray mixing and combustion. An analysis of the predicted fuel-air mixture in key regions, defined based upon experimental observations, was used to explain different flame propagation speeds and soot production tendencies when varying ignition timing. The mixture analysis indicates that soot production can be avoided if the flame propagates into regions where the equivalence ratio (Φ) is already below 2. Reactive RANS simulations have also been performed, but with a poor match against the experiment, as the flame speed and heat-release rate are largely over estimated. This modeling weakness appears related to a very high level of turbulent viscosity predicted for the high-momentum spray in the RANS simulations, which is an important consideration for modeling ignition and flame propagation in mixtures immediately created by the spray.

42 ENGINEERING↗
DOE OSTI2022

Fractional vortices, Z 2 gauge theory, and the confinement-deconfinement transition

Here, in this Letter we discuss the classical three-dimensional XY model whose nearest-neighbor interaction is a mixture of cos (θ i -θ j ) (ferromagnetic) and cos 2 (θ i -θ j ) (nematic). This model is dual to a theory with integer and half-integer vortices. While both types of vortices interact with a noncompact U(1) gauge field, the half-integer vortices interact with an extra interaction mediated by a Z 2 gauge field. We shall discuss the confinement-deconfinement transition of the half-integer vortices, the Wilson and the 't Hooft loops, and their mutual statistics in path integral language. In addition, we shall present a quantum version of the classical model which exhibits these physics.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Inverse-Velocity Transformation Wall Model for Reacting Turbulent Hypersonic Boundary Layers

The present study builds on prior work by taking advantage of the novel framework proposed by Griffin et al. (hereafter referred to as the GFM) as a baseline. The model is progressively extended to multicomponent reacting mixtures, accounting for differential diffusion and finite-rate chemistry in a similar fashion to Di Renzo & Urzay and Di Renzo et al. The accuracy of the present approach, as well the prior model of Di Renzo & Urzay, is assessed in an a priori sense for the first time in a turbulent reacting boundary layer, using the boundary-layer data of Williams et al. Five species are included in the present analysis, i.e., N s = 5, namely N 2 , O 2 , NO, N and O, a neutral mixture most representative of dissociation/recombination phenomena for temperatures below 6000 K. A schematic of a flow over a wedge representative of the described configuration is presented in Figure 1. The brief is organized as follows: In Section 2, the wall-model equations and the computational framework are presented. In Section 3, the a priori results of the proposed model are described and compared to the extended EWM. Finally, in Section 4, some conclusions are offered.

97 MATHEMATICS AND COMPUTING↗