Kinetic Equation Research Articles

Asymptotic-Preserving Neural Networks for Multiscale Kinetic Equations

Asymptotic-Preserving Neural Networks for Multiscale Kinetic Equations

Ethyl acetate production by Fischer esterification: use of excess of acetic acid and complete separation sequence

Abstract In this work, production of ethyl acetate (EtAc) using Fischer esterification between acetic acid (AcOH) and ethyl alcohol (EtOH) is presented. Batch kinetics was developed with EtOH as the limiting reactant to avoid the problems associated with unreacted ethanol in the final product. Regression was performed to estimate the parameters of the Langmuir Hinshelwood type kinetic equation. Reaction mass obtained during kinetics was subjected to separation and purification. Molecular sieves (MS) were used to remove water. Distillation (simple and extractive distillation – with entrainer, dimethyl sulfoxide, DMSO) was used to obtain almost pure AcOH, EtOH, EtAc and DMSO. Future work in reactive separation environment is indicated. Use of EtOH as the limiting reactant, use of MS for water removal and complete separation sequence are important features of this work.

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.

#1605 Development of an app for a science-based prescription of haemodialysis

Abstract Background and Aims The delivery of an adequate haemodialysis (HD) dose (i.e., a target single-pool Kt/V (spKt/V) of 1.4 per session) is often challenging, especially in patients with huge urea distribution volumes (V). On the other hand, patients with an extremely small V, such as children, are at higher risk of disequilibrium syndrome, which is caused by an excessive or overly rapid correction of urea levels, particularly during the first HD session. The treatment prescription often neglects many elements contributing to the final delivered HD dose, whereas a systematic and comprehensive evaluation of all adjustable parameters could improve the management of patients, not only facilitating the achievement of treatment goals, but also avoiding short- and long-term complications. We aimed at developing a web app capable of predicting the delivered HD dose, given a wide range of parameters, and using renowned kinetic equations relevant to urea clearance in HD. Method The platform used to develop our app is based on Angular, an open-source framework to develop web applications. Specifically, it can be used to realise single-page apps and it is only for frontend. This fits our case, since a backend is not required due to the absence of huge databases or complex operations. The application is grounded on a stepper component, a paradigm to guide the user step-by-step during the data entry process. It consists of three phases, namely (1) patient, (2) haemodialyser, (3) target spKt/V. The last section is also for output. A modification of any input triggers a real-time output change. Results We developed a free access user-friendly web app, through which HD professionals can simulate a treatment prescription and obtain an estimate of the delivered dose in terms of spKt/V. The app is structured in three sections and all calculations are based on validated formulas and kinetic equations applicable to HD: The output “expected spKt/V” changes in real time, after any modification of the input parameters. The result can thus be compared to the treatment goal (i.e., 1.4 for a standard HD session; 0.55 for a first treatment, aiming at reducing the risk of disequilibrium syndrome), as suggested by international guidelines and expert opinions. Conclusion We developed a user-friendly web app, which will be freely accessible online by HD professionals, aimed at predicting the delivered HD dose (spKt/V) considering a wide range of patient- and prescription-related parameters. The application of such a tool in everyday clinical practice, particularly in cases requiring a more accurate definition of the actual delivered dose, could improve the management of patients through a personalisation of the treatment. This could enhance prescription adjustments aimed at achieving an adequate spKt/V in patients receiving an insufficient HD dose; on the other hand, it could help preventing disequilibrium syndrome in patients with very small urea distribution volumes.

A new generalized beta function associated with statistical distribution and fractional kinetic equation

Several authors have extensively investigated beta function, hypergeometric function and confluent hypergeometric function, their extensions and generalizations due to their several application in many areas of engineering, probability theory and science. The main purpose of this paper is to present a new generalization of extended beta function, hypergeometric function and confluent hypergeometric function with the help of m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formula and summation formulas. We also examine generalized Caputo fractional derivative operator with the help of m-parameter Mittag-Leffler function with associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function and characteristic function and cumulative distribution are derived. Further, we derive the solution of fractional Kinetic equation involving generalized hypergeometric function.

Once-in-a-lifetime encounter models for neutrino media: From coherent oscillations to flavor equilibration

Collective neutrino oscillations are typically studied using the lowest-order quantum kinetic equation, also known as the mean-field approximation. However, some recent quantum many-body simulations suggest that quantum entanglement among neutrinos may be important and may result in flavor equilibration of the neutrino gas. In this work, we develop new quantum models for neutrino gases in which any pair of neutrinos can interact at most once in their lifetimes. A key parameter of our models is γ=μΔz, where μ is the neutrino coupling strength, which is proportional to the neutrino density, and Δz is the duration over which a pair of neutrinos can interact each time. Our models reduce to the mean-field approach in the limit γ→0 and achieve flavor equilibration in time t≫(γμ)−1. These models demonstrate the emergence of coherent flavor oscillations from the particle perspective and may help elucidate the role of quantum entanglement in collective neutrino oscillations. Published by the American Physical Society 2024

Kinetic equation for stochastic vector bundles

The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker–Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker–Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.

Ultra-low frequency vibration energy harvesting of piezoelectric vibration systems with an adjustable device

Energy harvesting from external Ultra-low frequency vibrations is currently attracting a lot of attention. However, for the same system, it is difficult to satisfy the need for more efficient energy harvesting in a variety of external environments. Accordingly, an adjustable gearing device connected to the energy harvesting system is used herein for adapting energy harvesting to a wide range of external environments. The device uses four symmetrical inclined springs. One end of the springs is connected to a gear and the other end to a mass element. The geometrical parameters of the system can be varied by rotating the gears, thus varying the dynamic stiffness of the system. The kinetic equations of the system are derived and their nonlinear terms are Taylor-expanded. The amplitude-frequency response curve of the system is obtained by solving using the harmonic balance method. Multi-color plots of RMS values of induced voltage and electric power are obtained using the fourth-order Runge-Kutta method. The results show that the device has excellent adjustable performance. Adjusting the rotation angle of the gears can change the soft and hard characteristics of the system's dynamic stiffness and the intrinsic frequency, thus improving the system's energy harvesting performance in different environments.

Verification of wave turbulence theory in the kinetic limit

Using the 1D Majda-McLaughlin-Tabak model as an example, we develop numerical experiments to study the validity of the wave kinetic equation (WKE) at the kinetic limit (i.e., small nonlinearity and large domain). We show that the dynamics converge to the WKE prediction, in terms of the closure model and energy flux, when the kinetic limit is approached. When the kinetic limit is combined with a process of widening the inertial range, the theoretical Kolmogorov constant can be recovered numerically to a very high precision. Published by the American Physical Society 2024

Preparation of MOR/P(S‐co‐VPh) nanofiber composite membrane and its adsorption properties for rubidium ions

AbstractRubidium (Rb), a rare and precious metal, is widely used in high‐tech fields. In this work the structural advantages of mordenite (MOR) and group affinity of phenolic hydroxyl polymers (poly (styrene‐4‐hydroxystyrene) (P(S‐co‐VPh)) are combined to prepare MOR/P(S‐co‐VPh) nanofiber composite membranes with high selectivity and adsorption specificity for Rb+ by electrospinning. The test results show that MOR is successfully anchored on the nanofiber substrate and maintained its complete crystal structure. Adsorption experiments show that the adsorption capacity of Rb+ increases with the increase of MOR content, and pH = 9.0, T = 25°C are the best adsorption conditions. K+ shows the most obvious interference with the adsorption capacity decreased from 60.03 to 39.33 mg/g. The adsorption process follows with the Langmuir isothermal model and pseudo‐second‐order kinetic equation. The chemical adsorption based on the coordination reaction is the control step, and the intraparticle diffusion also has an important effect on the adsorption rate. The recycling test indicates that the prepared nanofiber composite membrane not only has efficient adsorption of Rb+, but also exhibits excellent regeneration and stability, which helps to achieve its widespread application and long‐term operation in harsh environments.

Numerical investigation of drop–film interactions with a thixotropic liquid

We investigate numerically the influence of thixotropic effects on the impact of a drop onto a thin film, a fundamental process in many technical systems. Direct numerical simulations are performed with a Volume-of-Fluid (VOF) method based multiphase flow solver whose capabilities are expanded in order to enable simulations of a thixotropic liquid. The thixotropic behavior is modeled by a rate kinetic equation for the structural integrity of the assumed microstructure of the liquid. The corresponding structural parameter is described by an additional VOF-variable. After a validation of the implementations, we vary systematically the two parameters of the thixotropic model for a selected impact scenario in order to identify thixotropic effects during the impact and on the overall impact morphology. The two parameters are the mutation number Mu=texp/tθ as the ratio of the experimental time scale to the time scale of the structural rebuilding and the parameter β, which describes the effectivity of the shear-induced structural disintegration. The parameter study leads to a regime map with three different regimes. For Mu>10, the liquid behaves purely shear-thinning. High shear rates during the early stages of the impact lead to a low apparent viscosity at the crown base and to an enhanced crown growth. For Mu<0.1, the liquid behaves irreversible thixotropic or rheodestructing, respectively. Structural rebuilding is negligible and every deformation leads to a further disintegration of the microstructure. In this regime, a thin region of disintegrated microstructure develops within the liquid, spanning from the location of high shear stresses at the bottom into the crown rim. In between these two regimes, purely thixotropic effects become significant. A complex microstructure develops during the impact, in which features of both regimes occur combined, leading to a pronounced viscosity gradient along the crown wall. A comparison of the resulting maximum crown heights reveals that various combinations of Mu and β values can lead to the same maximum crown height whereas the crown shapes prior to this point in time can be very different.

Parallel expansion of a fuel pellet plasmoid

The problem of the assimilation of a cryogenic fuel pellet injected into a hot plasma is considered. Due to the transparency to ambient particles of the plasmoid, the localised region of high-density plasma created by ionisation of the ablated pellet material, electrons reach a ‘quasiequilibrium’ (QE) state which is characterised by a steady-state on the fastest collisional time scale. The simplified electron kinetic equation of the QE state is solved. Taking a velocity moment of the higher-order electron kinetic equation, which is valid on the expansion time scale, permits a fluid closure, yielding an evolution equation for the macroscopic parameters describing the QE distribution function. In contrast to the Braginskii equations, the closure does not require that electrons have a short mean free path compared with the size of density perturbations, and permits an anisotropic and highly non-Maxwellian distribution function. As the QE distribution function accounts for both trapped and passing electrons, the self-consistent electric potential that causes the expansion can be properly described, in contrast to earlier models of pellet plasmoid expansion with an unbounded potential. The plasmoid expansion is simulated using both a Vlasov model and a cold-fluid model for the ions. During the expansion plasmoid ions and electrons obtain nearly equal amounts of energy; as hot ambient electrons provide this energy in the form of collisional heating of plasmoid electrons, the expansion of a pellet plasmoid is expected to be a potent mechanism for the transfer of energy from electrons to ions on a time scale shorter than that of ion–electron thermalisation.

Individual particle approach to diffusive shock acceleration. Effect of the non-uniform flow velocity downstream of the shock

The momentum distribution of particles accelerated at strong non-relativistic shocks may be influenced by the spatial distribution of the flow speed around the shock. This phenomenon becomes evident in the cosmic-ray-modified shock, where the particle spectrum itself determines the profile of the flow velocity upstream. However, the effects of a non-uniform flow speed downstream are unclear. Hydrodynamics indicates that the spatial variation of flow speed over the length scales involved in the acceleration of particles in supernova remnants (SNRs) could be noticeable. In the present paper, we address this question. We initially followed Bell's approach to particle acceleration and then also solved the kinetic equation. We obtained an analytical solution for the momentum distribution of particles accelerated at the cosmic-ray-modified shock with spatially variable flow speed downstream. We parameterised the downstream speed profile to illustrate its effect on two model cases: the test particle and non-linear acceleration. The resulting particle spectrum is generally softer in Sedov SNRs because the distribution of the flow speed reduces the overall shock compression accessible to particles with higher momenta. On the other hand, the flow structure in young SNRs could lead to harder spectra. The diffusive properties of particles play a crucial role as they determine the distance the particles can return from to the shock, and, as a consequence, the flow speed that they encounter downstream. We discuss a possibility that the plasma velocity gradient could be (at least partially) responsible for the evolution of the radio index and for the high-energy break visible in gamma rays from some SNRs. We expect the effects of the gradient of the flow velocity downstream to be prominent in regions of SNRs with higher diffusion coefficients and lower magnetic field, that is, where acceleration of particles is not very efficient.

Development of catalytic zero-valent iron incorporated PAN catalytic film for efficient degradation of organic matters

Catalytic films work well in degradation of organic matters. However, catalytic activity and stability of films are challenging factors. A nanoscale zero-valent iron (NZVI) incorporated porous PAN fiber (Fe-PAN) film was thus developed through a one-step cryogenic auxiliary electrospinning method. The Fe-PAN film overcame the problem in the traditional multistep preparation process. The excellent intrinsic properties of the polymer in the film were maintained. It exhibited high catalytic activity (> 95% conversion in just 4 min) and excellent stability and reusability, due to the synergistic interaction between PAN and NZVI. The degradation process was optimized by the Box-Behnken design, leading to the optimal condition: pH = 2.8, temperature = 56 °C, and oxidant concentration = 4.2 mmol/L. The degradation followed the 2nd order kinetic equation and was due to the reactions by ·OH and O2-· radicals. This study demonstrates the great potentials of the Fe-PAN film for industrial applications.

Расчеты сжатия сферической слоистой системы ударными волнами с учетом переноса тепла излучением в различных приближениях

Numerical modeling is one of the basic tools to investigate physical phenomena that occur in materials compressed by shock waves. A study of the behavior of shock waves using simplified models is helpful in the analysis of more complex systems, for instance, in the problems of inertial thermonuclear fusion on laser facilities. Mathematical simulation of the nonstationary radiative heat transfer in a spectral kinetic statement is rather complicated because the system of equations is nonlinear and large in size. Generally, the kinetic transport equation is solved in 7D phase space, which requires huge computational resources. The initial system is often approximated under certain assumptions but this immediately causes questions as to whether the simplified model is applicable to particular calculations. In this study, several economic radiative heat transfer models were implemented in a 2D hydrodynamic code. Based on these models, test calculations were performed to simulate the compression of a layered spherical system by shock waves, taking into account radiative heat transfer. When the shock wave reaches the center of the sphere, it is focused and reflected. In the neighborhood of the focal point of a converging wave, temperature gradients increase; therefore, thermal conduction and radiation become the main mechanisms of energy dissipation to be considered when taking into account heat transfer. Maximum densities and temperatures at the center of the sphere and their average values in its regions were calculated. In addition, the time when shock and heat waves cross the sphere center was determined, and their behavior before and after focusing at the sphere center was estimated. It is shown that solutions to such problems can be found much faster and with adequate accuracy when applying simplified models.

Relaxation and asymptotic expansion of controlled stiff differential equations

The control of relaxation-type systems of ordinary differential equations is investigated using the Hamilton–Jacobi–Bellman equation. Firstly, we recast the model as a singularly perturbed dynamics which we embed in a family of controlled systems. Then we study this dynamics together with the value function of the associated optimal control problem. We provide an asymptotic expansion in the relaxation parameter of the value function. We also show that its solution converges toward the solution of a Hamilton–Jacobi–Bellman equation for a reduced control problem. Such systems are motivated by semi-discretisation of kinetic and hyperbolic partial differential equations. Several examples are presented including Jin–Xin relaxation.

Microstructure Evolution and Homogenization Kinetics of 15Cr–30Ni–Fe Heat‐Resistant Alloy

The as‐cast microstructure, microsegregation of alloying elements, and microstructure evolution during homogenization of the newly designed 15Cr–30Ni–Fe heat‐resistant alloy are studied. Ti, Nb, Si, and Mo segregate at the interdendritic region. The brittle eutectic Laves phase contains Fe, Ti, Ni, Nb, Si, and Mo, which is identified as the Fe2Ti Laves phase. The activation energy for the Fe2Ti Laves phase dissolution is close to the activation energy for Ti diffusion in the γ matrix based on the Johnson–Mehl–Avrami–Kolmogorov analysis. The back‐diffusion of Ti to the γ matrix is the limiting step for the dissolution of Fe2Ti Laves phase. The kinetic equation of homogenization considering lattice parameters and pre‐exponential factor correction is established. After the double‐stage homogenization at 980 °C/10 h + 1140 °C/4 h, the volume fraction of the Laves phase decreases from 5.11% to 0.26%, and the large eutectic Laves phase is completely dissolved. The reduction in the standard deviation of microhardness at different soaking times effectively reflects the gradual decrease in the degree of microsegregation of alloying elements. The average grain size of the alloy after homogenization is 410 ± 63.1 μm, which is beneficial to control the further refinement of grain during the subsequent hot working.

Direct Synthesis of Calcium Lactate through the Reaction of Glycerol with Calcium Hydroxide Catalyzed by Bimetallic AuCu/SiO2 Nanocatalysts

Bimetallic AuCu/SiO2 nanosized catalysts were prepared using the wet chemical reduction technique. From among Au0.1–1.5Cu10/SiO2 catalysts, the Au0.5Cu10/SiO2 catalyst gave the highest yield of calcium lactate of 87% at a glycerol conversion of 96% when the reaction of glycerol with calcium hydroxide at a mole ratio of calcium hydroxide to glycerol of 0.8:1 was conducted under an anaerobic atmosphere at 200 °C for 2 h. The interactions between metallic Au0 and Cu0 nanoparticles facilitate calcium lactate formation. The simulation of glycerol consumption rate with an empirical power-function reaction kinetics equation yielded a reaction activation energy of 44.3 kJ∙mol−1, revealing that the catalytic reaction of glycerol with calcium hydroxide to calcium lactate can be conducted by overcoming a mild energy barrier. The synthesis of calcium lactate through the catalytic reaction of glycerol with calcium hydroxide on a bimetallic AuCu/SiO2 nanosized catalyst under a safe anaerobic atmosphere is an alternative to the conventional calcium lactate production technique through the reaction of expensive lactic acid with calcium hydroxide.

Kinetics Parameters Estimation for the Coupled Neutron-Photon Point Kinetics Model in Monte Carlo Eigenvalue Calculations

The delayed photoneutron effect should be considered in the point kinetics equation (PKE) applied to D2O and beryllium-moderated or -reflected systems because of its importance on reactor kinetics. This paper proposes neutron-photon-coupled PKE with a clear definition of adjoint-weighted kinetics parameters after providing the mathematical derivation from the neutron and photon transport and precursor equations. The Monte Carlo algorithms to estimate the kinetics parameters are induced from their definition and implemented in the eigenvalue calculation module of McCARD. They are verified through homogeneous infinite-medium problems with few-group neutron and photon cross sections by comparing with analytic solutions. The proposed PKE and kinetics parameters estimation methods are applied for the kinetics analyses of the artificial control rod drop transient problem in the RB reactor. Photonuclear physics is also implemented in a time-dependent Monte Carlo simulation to calculate the reference data of neutron amplitude in the core to be compared with the PKE results. As good agreement is shown in the neutron amplitude, it is validated that the developed PKE is capable of predicting the reactor kinetics considering photoneutrons.

Vacuum evaporation and condensation thermodynamics and evaporation kinetics of pure silver

A comprehensive analysis of the characteristics of vacuum evaporation and condensation of pure silver is crucial for advancing vacuum separation and purification technology for crude silver. This study analyzes the vacuum evaporation and condensation thermodynamics and evaporation kinetics of pure silver. The results reveal a substantial increase in the saturated vapor pressure and theoretical maximum evaporation rate of pure silver with increasing temperature. When the temperature is in the range of 1373 ∼ 1773 K and the pressure is in the range of 0.1 ∼ 10 Pa, the average molecular free path is between 0.514 ∼ 66.415 cm. Moreover, the mean molecular free path of silver increases with increasing temperature and decreasing pressure. Besides, the silver triple point occurs at a temperature of 1234.78 K and a pressure of 0.343 Pa. The average evaporation rate of pure silver demonstrates a linear relationship with temperature, and the relationship with pressure conforms to the Logistic model. Notably, the vacuum evaporation of pure silver is a first-order reaction. Furthermore, in this study, the kinetic equation of vacuum evaporation of pure silver is established. In the temperature range of 1473–1673 K, the gas phase mass transfer is the rate-controlling step rather than the mass transfer in the gas phase boundary layer. As the temperature increases, the rate-controlling step gradually changes into gas relative flow mass transfer and gas-liquid interface evaporation mass transfer.