Search NASA⌕ Search

SEARCH · Search NASA

Results for “Kinetic Modelling”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 505 records · Page 28
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Approaching PetaVolts per Meter Plasmonics Using Structured Semiconductors

A newly uncovered class of plasmons in the strongly excited limit opens access to unprecedented Petavolts per meter electromagnetic fields with wide-ranging, transformative impact. Unlike conventional plasmons, such plasmons are constituted by non-perturbative, large-amplitude oscillations of the ultradense, delocalized free electron Fermi gas inherent in conductive media. Here structured semiconductors doped to have an appropriate conduction electron density are introduced to tune the properties of the Fermi gas for matched excitation of large-amplitude plasmons using readily available electron beams which enables immediate experimental validation. Specifically, an electrostatic, surface “crunch-in” plasmon is collisionlessly excited by the beam launched inside a tube. Strong excitation due to matching results in relativistic oscillations of the electron gas and unravels unique phenomena. Relativistically induced ballistic electron transport comes about due to relativistic multifold increase in the mean free path and also leads to unconventional heat deposition beyond Ohm’s law. This explains the absence of observed damage or solid-plasma formation in past experiments on conductive samples interacting with electron bunches shorter than 10-13 seconds. Furthermore, relativistic momentum leads to copious tunneling of electron gas across the surface, which then crunches inside the tube. Relativistic effects along with large, localized electron density variations underlying these modes necessitate kinetic approach to theoretical and computational modeling. Kinetic model presented here demonstrates experimental viability of observing tens of gigavolts per meter plasmonic fields excited by matching readily available electron beams to plasmons in semiconductors with 1018cm-3 free electron density, and paves the way for Petavolts per meter plasmonics.

42 ENGINEERING↗
DOE OSTI2021

Atomic-Level Features for Kinetic Monte Carlo Models of Complex Chemistry from Molecular Dynamics Simulations

The high computational cost of evaluating atomic interactions recently motivated the development of computationally inexpensive kinetic models, which can be parametrized from MD simulations of complex chemistry of thousands of species or other processes and accelerate the prediction of the chemical evolution by up to four order of magnitude. Such models go beyond the commonly employed potential energy surface fitting methods in that they are aimed purely at describing kinetic effects. So far, such kinetic models utilize molecular descriptions of reactions and have been constrained to only reproduce molecules previously observed in MD simulations. Therefore, these descriptions fail to predict the reactivity of unobserved molecules, for example in the case of large molecules or solids. In this work, we propose a new approach for the extraction of reaction mechanisms and reaction rates from MD simulations, namely the use of atomic-level features. Using the complex chemical network of hydrocarbon pyrolysis as example, it is demonstrated that kinetic models built using atomic features are able to explore chemical reaction pathways never observed in the MD simulations used to parametrize them, a critical feature to describe rare events. Atomic-level features are shown to construct reaction mechanisms and estimate reaction rates of unknown molecular species from elementary atomic events. Through comparisons of the model ability to extrapolate to longer simulation timescales and different chemical compositions than the ones used for parameterization, it is demonstrated that kinetic models employing atomic features retain the same level of accuracy and transferability as the use of features based on molecular species, while being more compact and parametrized with less data. We also find that atomic features can better describe the formation of large molecules enabling the simultaneous description of small molecules and condensed phases.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Bimolecular Peroxy Radical (RO 2 ) Reactions and Their Relevance in Radical Initiated Oxidation of Hydrocarbons

The kinetics of peroxy radical (RO 2 ) reactions have been of long-standing interest in atmospheric and combustion chemistry. Nevertheless, the lack of kinetic studies at higher temperatures for their reactions with other radicals such as OH has precluded the inclusion of this class of reactions in detailed kinetics models developed for combustion applications. In this work, guided by the limited room temperature experimental studies on selected alkyl-peroxy radicals and literature theoretical kinetics on the prototypical CH 3 O 2 + OH system, we have performed parametric studies on the effect of uncertainties in the rate coefficients and branching ratios to potential product channels for RO 2 + OH reactions at higher temperatures. Literature kinetics models were used to simulate autoignition delays, laminar flame speeds, and speciation profiles in flow and stirred reactors for a variety of common combustion-relevant fuels. Inclusion of RO 2 + OH reactions was found to retard autoignition in fuel-lean (φ = 0.5) mixtures of ethane and dimethyl ether in air. The observed effects were noticeably more pronounced in ozone-enriched combustion of ethane and dimethyl ether. The simulations also examined the influence of ozone doping levels, pressures, and equivalence ratios for both ethane and dimethyl ether oxidation. Sensitivity and flux analyses revealed that the RO 2 + OH reaction is a significant sink of RO 2 radicals at the early stage of autoignition, affecting fuel oxidation through RO 2 ↔ QOOH, RO 2 ↔ alkene + HO 2 , or RO 2 + HO 2 ↔ ROOH + O 2 . Additionally, the kinetic stability of the trioxide formed from RO 2 + OH reactions was investigated using master equation analyses. Last, we discuss other bimolecular reactions that are missing in literature kinetics models but are relevant to hydrocarbon oxidation initiated by external radical sources (plasma-enhanced, ozone-enriched combustion, etc.). In conclusion, the present simulations provide a strong motivation for better characterizing the bimolecular kinetics of peroxy radicals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Dynamic Metabolic Modeling of Denitrifying Bacterial Growth: The Cybernetic Approach

Denitrification is a multistage reduction process converting nitrate ultimately to nitrogen gas, carried out mostly by facultative bacteria. Modeling of the denitrification process is challenging due to the complex metabolic regulation that modulates sequential formation and consumption of a series of nitrogen oxide intermediates, which serve as the final electron acceptors for denitrifying bacteria. In this work, we examined the effectiveness and accuracy of the cybernetic modeling framework in simulating the growth dynamics of denitrifying bacteria in comparison with kinetic models. In four different case studies using the literature data, we successfully simulated diauxic and triauxic growth patterns observed in anoxic and aerobic conditions, only by tuning two or three parameters. In order to understand the regulatory structure of the cybernetic model, we systematically analyzed the effect of cybernetic control variables on simulation accuracy. The results showed that the consideration of both enzyme synthesis and activity control through u- and v-variables is necessary and relevant and that uvariables are of greater importance in comparison to v-variables. In contrast, simple kinetic models were unable to accurately capture dynamic metabolic shifts across alternative electron acceptors, unless an inhibition term was additionally incorporated. Therefore, the denitrification process represents a reasonable example highlighting the criticality of considering dynamic regulation for successful metabolic modeling.

cybernetic model↗
DOE OSTI2015

THREE-DIMENSIONAL KINETIC-MHD MODEL OF THE GLOBAL HELIOSPHERE WITH THE HELIOPAUSE-SURFACE FITTING

This paper provides a detailed description of the latest version of our model of the solar wind (SW) interaction with the local interstellar medium (LISM). This model has already been applied to the analysis of Lyα absorption spectra toward nearby stars and for analyses of Solar and Heliospheric Observatory/SWAN data. Katushkina et al. (this issue) used the model results to analyze IBEX-Lo data. At the same time, the details of this model have not yet been published. This is a three-dimensional (3D) kinetic-magnetohydrodynamical (MHD) model that takes into account SW and interstellar plasmas (including α particles in SW and helium ions in LISM), the solar and interstellar magnetic fields, and interstellar hydrogen atoms. The latitudinal dependence of SW and the actual flow direction of the interstellar gas with respect to the Sun are also taken into account in the model. It was very essential that our numerical code was developed in such a way that any numerical diffusion or reconnection across the heliopause were not allowed in the model. The heliospheric current sheet is a rotational discontinuity in the ideal MHD and can be treated kinematically. In the paper, we focus in particular on the effects of the heliospheric magnetic field and on the heliolatitudinal dependence of SW.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Exploring growth kinetics of carbon nanotube arrays by in situ optical diagnostics and modeling

Simple kinetic models of carbon nanotube growth have been able to successfully link together many experimental parameters involved in the growth of carbon nanotubes for practical applications including the prediction of growth rates, terminal lengths, number of walls, activation energies, and their dependences on the growth environment. The implications of recent experiments utilizing in situ monitoring of carbon nanotube growth on our past kinetic model are first reviewed. Then, sub-second pulsed feedstock gas introduction is discussed to explore the nucleation and initial growth of carbon nanotubes in the context of the kinetic model. Moreover, kinetic effects in "pulsed CVD" - using repeated pulsed gas introduction to stop and restart nanotube growth - are explored to understand renucleation, the origin of alignment in nanotube arrays, and incremental growth. Time-resolved reflectivity of the surface is used to remotely understand the kinetics of nucleation and the coordinated growth of arrays. This approach demonstrates that continuous vertically aligned single wall carbon nanotubes can be grown incrementally by pulsed CVD, and that the first exposure of fresh catalyst to feedstock gas is critical to nanotubes site density required for coordinated growth. Aligned nanotube arrays (as short as 60 nm) are shown to nucleate and grow within single, sub-second gas pulses. The multiple-pulse growth experiments (> 100 pulses) show that a high fraction of nanotubes renucleate on subsequent gas pulses.

Puretzky, Alexander A↗
DOE OSTI2025

Constraining the Low-Temperature Oxidation Mechanism of n -Hexanol through the Detection and Identification of C 6 Elusive Intermediates

Alcohol-based fuels are currently considered to be viable energy carriers for the transportation sector. Consequently, a comprehensive mechanistic understanding of the low-temperature oxidation of alcohols is essential for application in advanced low-temperature compression engines. Here, in this work, a multidimensional approach involving experimental investigations, kinetic modeling, and theoretical calculations was used to provide new insights into the low-temperature oxidation mechanism of a C 6 alcohol, n -hexanol (CH 3 (CH 2 ) 5 OH), through the detection and identification of elusive C 6 intermediates. The oxidation of n-hexanol was investigated in a jet-stirred reactor under stoichiometric conditions (ϕ = 1.0), an initial fuel concentration of 2%, a residence time of 2 s, a temperature range between 500 and 660 K, and a pressure of 700 Torr. The reactants, intermediates, and final products were detected and identified by means of molecular-beam mass spectrometry coupled with single-photon ionization employing tunable synchrotron-generated vacuum ultraviolet radiation. Chemical kinetic simulations were performed using a previously published kinetic model (Togbé et al., Energy Fuels 2010, 11, 5859−5875) to predict the reactivity of n-hexanol and elucidate the predominant formation pathways of the observed low-temperature species. Experimental photoionization efficiency curves in conjunction with ab initio calculations, enabled the identification of important low-temperature species, such as C 6 unsaturated alcohols, C 6 olefinic hydroperoxides, C 6 cyclic ethers, C 6 diones, and C 6 ketohydroperoxides. The results of this study provide valuable insight into the mechanism of the low-temperature oxidation chemistry of n -hexanol, contributing to the development of kinetic models for the low-temperature oxidation of n-hexanol and other long-chain linear alcohols.

alcohols↗
DOE OSTI2018

Nuclear yield reduction in inertial confinement fusion exploding-pusher targets explained by fuel-pusher mixing through hybrid kinetic-fluid modeling

In inertial confinement fusion implosion experiments involving fast laser heating of glass capsules containing gaseous nuclear fuel, some nuclear measurements cannot be rendered by standard hydrodynamics numerical simulations. Here, the calculated values of the nuclear yield are found to be too high by more than a decade in some cases. The first kinetic simulations of the enclosed reacting gas have bridged only half that gap. In this Rapid Communication, using a hybrid ion kinetic-fluid numerical model which takes into account the kinetic interaction between the fuel and the pusher, we reach full agreement with the experimental yield data. We thus unambiguously demonstrate that the kinetic behavior of the system, resulting in both nonthermodynamic equilibrium of the ions comprising the fuel and strong pusher-fuel mixing, is the correct interpretation of the data.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Observation of ionization trends in a laboratory photoionized plasma experiment at Z

We report experimental and modeling results for the charge state distribution of laboratory photoionized neon plasmas in the first systematic study over nearly an order of magnitude range of ionization parameter ξ ∝ F/N e . The range of ξ is achieved by flexibility in the experimental platform to adjust either the x-ray drive flux F at the sample or the electron number density N e or both. Experimental measurements of photoionized plasma conditions over such a range of parameters enable a stringent test of atomic kinetics models used within codes that are applied to photoionized plasmas in the laboratory and astrophysics. From experimental transmission data, ion areal densities are extracted by spectroscopic analysis that is independent of atomic kinetics modeling. The measurements reveal the net result of the competition between photon-driven ionization and electron-driven recombination atomic processes as a function of ξ as it affects the charge state distribution. Results from radiation-hydrodynamics modeling calculations with detailed inline atomic kinetics modeling are compared with the experimental results. There is good agreement in the mean charge and overall qualitative similarities in the trends observed with ξ but significant quantitative differences in the fractional populations of individual ions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2019

Exploring gasoline oxidation chemistry in jet stirred reactors

Recent decades have seen increasingly restrictive regulations applied to gasoline engines. Gasoline combustion chemistry must be investigated to achieve a better understanding and control of internal combustion engine efficiency and emissions. In this work, several gasoline fuels, namely the FACE (Fuel for Advanced Combustion Engines) gasolines, were selected as targets for oxidation study in jet-stirred reactors (JSR). The study is facilitated by formulating various gasoline surrogate mixtures with known hydrocarbon compositions to represent the real gasolines. Surrogates included binary mixtures of n-heptane and iso-octane, as well as more complex multi-component mixtures. Here, the oxidation characteristics of FACE gasolines and their surrogates were experimentally examined in JSR-1 and numerically simulated under the following conditions: pressure 1 bar, temperature 500–1050 K, residence time 1.0 and 2.0 s, and two equivalence ratios (Φ= 0.5 and 1.0). In the high temperature region, all real fuels and surrogates showed similar oxidation behavior, but in the low temperature region, a fuel’s octane number and composition had a significant effect on its JSR oxidation characteristics. Low octane number fuels displayed more low temperature reactivity, while fuels with similar octane number but a larger number of n-alkane components were more reactive. A gasoline surrogate kinetic model was examined with FACE gasoline experiments either measured in JSR-2, or taken from previous work under the following conditions: pressure 10 bar, temperature 530–1200 K, residence time 0.7 s, and three equivalence ratios (Φ = 0.5, 1.0 and 2.0). Comparison between FACE gasoline experimental results with surrogate model predictions showed good agreement, demonstrating considerable potential for surrogate fuel kinetic modeling in engine simulations.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

DME-Propane Blends Ignition Experiments and Modeling for Heavy-Duty Mixing Controlled Compression Engines

A blend of dimethyl ether (DME) and propane (C3H8) is being studied in a shock tube at heavy-duty engine conditions at 110 bar. Due to its intrinsic combustion properties, DME/propane blend can potentially replace diesel in mixing controlled compression ignition engines. A blend of DME/propane can reduce emissions in mixing controlled compression ignition in heavy-duty engines through modifications, which require simulations using a high-fidelity chemical kinetics model that can accurately predict the chemistry of the blend. An essential aspect of testing the chemical kinetics model is doing baseline fundamental chemistry studies on neat DME and propane, which include ignition delay time measurements. In this work, using a high-pressure shock tube, ignition delay times were gathered for DME/Propane blends at 110 bar diluted with AR to test chemical kinetic models published in the literature. These models include Aramco 3.0, NUIG V1.1, C3mech V3.3, and Dames et al. Comparisons with the experimental IDTs and models were conducted, and general agreement was observed. A sensitivity analysis was conducted, and important reactions were outlined.

Mohammed, Zuhayr Pasha [University of Central Flor↗
DOE OSTI2013

Electron attachment to C{sub 2} fluorocarbon radicals at high temperature

Thermal electron attachment to the radical species C{sub 2}F{sub 3} and C{sub 2}F{sub 5} has been studied over the temperature range 300–890 K using the Variable Electron and Neutral Density Attachment Mass Spectrometry technique. Both radicals exclusively undergo dissociative attachment to yield F{sup −}. The rate constant for C{sub 2}F{sub 5} shows little dependence over the temperature range, remaining ∼4 × 10{sup −9} cm{sup 3} s{sup −1}. The rate constant for C{sub 2}F{sub 3} attachment rises steeply with temperature from 3 × 10{sup −11} cm{sup 3} s{sup −1} at 300 K to 1 × 10{sup −9} cm{sup 3} s{sup −1} at 890 K. The behaviors of both species at high temperature are in agreement with extrapolations previously made from data below 600 K using a recently developed kinetic modeling approach. Measurements were also made on C{sub 2}F{sub 3}Br and C{sub 2}F{sub 5}Br (used in this work as precursors to the radicals) over the same temperature range, and, for C{sub 2}F{sub 5}Br as a function of electron temperature. The attachment rate constants to both species rise with temperature following Arrhenius behavior. The attachment rate constant to C{sub 2}F{sub 5}Br falls with increasing electron temperature, in agreement with the kinetic modeling. The current data fall in line with past predictions of the kinetic modeling approach, again showing the utility of this simplified approach.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Probing the antagonistic effect of toluene as a component in surrogate fuel models at low temperatures and high pressures. A case study of toluene/dimethyl ether mixtures

There is a dearth of experimental data which examine the fundamental low-temperature ignition (T < 900 K) behavior of toluene resulting in a lack of data for the construction, validation, and interpretation of chemical kinetic models for commercial fuels. In order to gain a better understanding of its combustion chemistry, dimethyl ether (DME) has been used as a radical initiator to induce ignition in this highly knock resistant aromatic, and its influence on the combustion of toluene ignition was studied in both shock tube and rapid compression machines as a function of temperature (624–1459 K), pressure (20–40 atm), equivalence ratio (0.5–2.0), and blending ratio (100% toluene, 76% toluene (76T/24D), 58% toluene (58T/42D), 26% toluene (26T/74D) and 100% DME). We use several literature chemical kinetic models to interpret our experimental results. For mixtures containing high concentrations of toluene at low-temperatures none of these are capable of reproducing experiment. This then implies an incomplete understanding of the low-temperature oxidation pathways which control its ignition in our experimental reactors, and by extension, in spark- (SI) and compression-ignition (CI) engines, and an updated detailed chemical kinetic model is presented for engineering applications. Model analyses indicate that although the initial fate of the fuel is dominated by single-step H-atom abstraction reactions from both the benzylic and phenylic sites, the subsequent fate of the allylic and vinylic radicals formed is much more complex. Further experimental and theoretical endeavors are required to gain a holistic qualitative and quantitative chemical kinetics based understanding of the combustion of pure toluene, toluene blends, and commercial fuels containing other aromatic components, at temperatures of relevance to SI and CI engines.

30 DIRECT ENERGY CONVERSION↗
DOE OSTI2021

Plasma-catalyst modeling for materials selection: challenges and opportunities in nitrogen oxidation

Reliable kinetic models are essential to rationalize observations and to guide the selection of appropriate plasma-catalyst combinations. Kinetic models coupling plasma- and heterogeneous-catalytic-chemistries, however, are primitive. Here, we combine reduced plasma chemistries and density functional theory parameterized surface reactions to model NO formation from N2 oxidation, an alternative nitrogen fixation process. We first show plasma excited species can enhance turnover frequencies of both Pt and Au. The enhancing potential is a function of plasma species and their densities. We then compare NO production between plasma/Pt and plasma/Au with an integral reactor model and explore the dependence of NO production as a function of plasma conditions, reactor configurations and chemical compositions. Plasma/Pt generates more NO than plasma/Au under most thermal and plasma parametric conditions. Plasma/Au outperforms plasma/Pt under high plasma excitations at low temperatures. Furthermore, these results highlight that selection of plasma-catalyst combinations should consider the properties of plasmas, materials and the coupling of the two.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2014

An Experimental and Chemical Kinetics Study of the Combustion of Syngas and High Hydrogen Content Fuels

An integrated and collaborative effort involving experiments and complementary chemical kinetic modeling investigated the effects of significant concentrations of water and CO 2 and minor contaminant species (methane [CH 4 ], ethane [C 2 H 6 ], NOX, etc.) on the ignition and combustion of HHC fuels. The research effort specifically addressed broadening the experimental data base for ignition delay, burning rate, and oxidation kinetics at high pressures, and further refinement of chemical kinetic models so as to develop compositional specifications related to the above major and minor species. The foundation for the chemical kinetic modeling was the well validated mechanism for hydrogen and carbon monoxide developed over the last 25 years by Professor Frederick Dryer and his co-workers at Princeton University. This research furthered advance the understanding needed to develop practical guidelines for realistic composition limits and operating characteristics for HHC fuels. A suite of experiments was utilized that that involved a high-pressure laminar flow reactor, a pressure-release type high-pressure combustion chamber and a high-pressure turbulent flow reactor.

10 SYNTHETIC FUELS↗
DOE OSTI2023

A High-Pressure Shock Tube Study of Hydrogen and Ammonia Addition to Natural Gas for Reduced Carbon Emissions in Power Generation Gas Turbines

Ignition delay times from undiluted mixtures of natural gas (NG)/H2/Air and NG/NH3/Air were measured using a high-pressure shock tube at the University of Central Florida. The combustion temperatures were experimentally tested between 1000-1500 K near a constant pressure of 25 bar. Mixtures were kept undiluted to replicate the same chemistry pathways seen in gas turbine combustion chambers. Recorded combustion pressures exceeded 200 bar due to the large energy release, hence why these were performed at the high-pressure shock tube facility. The data is compared to the predictions of the NUIGMech 1.1 mechanism for chemical kinetic model validation and refinement. An exceptional agreement was shown for stoichiometric conditions in all cases but strayed at lean and rich equivalence ratios, especially in the lower temperature regime of H2 addition and all temperature ranges of the baseline NG mixture. Hydrogen addition also decreased ignition delay times by nearly 90%, while NH3 fuel addition made no noticeable difference in ignition delay time. NG/NH3 exhibited similar chemistry to pure NG under the same conditions, which is shown in a sensitivity analysis, demonstrating hydrogen chemistry to be dominant in NG/H2 mixtures and hydrocarbon chemistry to be dominant in NG/NH3 mixtures. The reaction CH3 + O2 = CH3O + O is identified and suggested as a possible modification target to improve model performance. Increasing the robustness of chemical kinetic models via experimental validation will directly aid in designing next-generation combustion chambers for use in gas turbines, which in turn will greatly lower global emissions and reduce greenhouse effects.

Pierro, Michael↗
DOE OSTI2013

Thermal characterization and model free kinetics of aged epoxies and foams using TGA and DSC methods.

Two classes of materials, poly(methylene diphenyl diisocyanate) or PMDI foam, and cross-linked epoxy resins, were characterized using thermal gravimetric analysis (TGA) and differential scanning calorimetry (DSC), to help understand the effects of aging and %E2%80%9Cbake-out%E2%80%9D. The materials were evaluated for mass loss and the onset of decomposition. In some experiments, volatile materials released during heating were analyzed via mass spectroscopy. In all, over twenty materials were evaluated to compare the mass loss and onset temperature for decomposition. Model free kinetic (MFK) measurements, acquired using variable heating rate TGA experiments, were used to calculate the apparent activation energy of thermal decomposition. From these compiled data the effects of aging, bake-out, and sample history on the thermal stability of materials were compared. No significant differences between aged and unaged materials were detected. Bake-out did slightly affect the onset temperature of decomposition but only at the highest bake-out temperatures. Finally, some recommendations for future handling are made.

Cordaro, Joseph Gabriel↗
DOE OSTI2013

Wave speeds in the macroscopic extended model for ultrarelativistic gases

Equations determining wave speeds for a model of ultrarelativistic gases are investigated. This model is already present in literature; it deals with an arbitrary number of moments and it was proposed in the context of exact macroscopic approaches in Extended Thermodynamics. We find these results: the whole system for the determination of the wave speeds can be divided into independent subsystems which are expressed by linear combinations, through scalar coefficients, of tensors all of the same order; some wave speeds, but not all of them, are expressed by square roots of rational numbers; finally, we prove that these wave speeds for the macroscopic model are the same of those furnished by the kinetic model.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗