Simple Kinetic Equation Research Articles

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.

Galactic Bar Resonances with Diffusion: An Analytic Model with Implications for Bar–Dark Matter Halo Dynamical Friction

The secular evolution of disk galaxies is largely driven by resonances between the orbits of “particles” (stars or dark matter) and the rotation of non-axisymmetric features (spiral arms or a bar). Such resonances may also explain kinematic and photometric features observed in the Milky Way and external galaxies. In simplified cases, these resonant interactions are well understood: for instance, the dynamics of a test particle trapped near a resonance of a steadily rotating bar is easily analyzed using the angle-action tools pioneered by Binney, Monari, and others. However, such treatments do not address the stochasticity and messiness inherent to real galaxies—effects that have, with few exceptions, been previously explored only with complex N-body simulations. In this paper, we propose a simple kinetic equation describing the distribution function of particles near an orbital resonance with a rigidly rotating bar, allowing for diffusion of the particles’ slow actions. We solve this equation for various values of the dimensionless diffusion strength Δ, and then apply our theory to the calculation of bar–halo dynamical friction. For Δ = 0, we recover the classic result of Tremaine and Weinberg that friction ultimately vanishes, owing to the phase mixing of resonant orbits. However, for Δ > 0, we find that diffusion suppresses phase mixing, leading to a finite torque. Our results suggest that stochasticity—be it physical or numerical—tends to increase bar–halo friction, and that bars in cosmological simulations might experience significant artificial slowdown, even if the numerical two-body relaxation time is much longer than a Hubble time.

Prediction of end-dialysis serum sodium concentration in severely hyponatremic kidney failure patients undergoing conventional hemodialysis using sodium kinetic equation

Background and ObjectivesConventional hemodialysis (HD) for kidney failure patients with severe hyponatremia may be complicated by rapid correction of hyponatremia, which increases the risk of osmotic demyelination syndrome. A simple sodium kinetic equation was effective in prediction of end-dialysis serum Na+ in severely hyponatremic kidney failure patient treated with continuous venovenous hemofiltration, but was not tested in conventional HD. The aim of this study was to assess the validity of this equation when used in conventional HD.MethodsTwenty conventional HD sessions were delivered to 12 kidney failure patients with severe hyponatremia (serum Na+ < 120 mEq/L). The target change in serum Na+ was 4 mEq/L. The DNa.t/V that obtained this change was predetermined according to the sodium kinetic equation and monitored in real time by online clearance monitoring software embedded in dialysis machine. The dialysis session was terminated once the target DNa.t/V was achieved.ResultsThe mean observed and predicted serum Na+ were 119.80 ± 3.42 mEq/L and 119.45 ± 3.12 mEq/L, respectively. Bland–Altman plot analysis revealed a mean difference ± SD of 0.33 ± 1.26 mEq/L, and 95% limits of agreement of − 2.13 to 2.83. The imprecision in prediction of end-dialysis serum Na+ was 2.52 mEq/L. The small difference and clinically insignificant 95% limits of agreement indicate a good agreement between the observed and predicted serum Na+.ConclusionThe sodium kinetic equation was effective in prediction of end-dialysis serum Na+ in kidney failure patients with severe hyponatremia.

Swelling of couscous grains under saturated conditions

The article presents a grain-scale experimental approach for the study of a model that swells granular food material. During the swelling phase, both the evolution of particle size using image analysis and the spatially resolved field of diffusion coefficient of water inside the grains were studied using MRI measurements. A concentrically penetration of water in the cross section of the grains was shown on images. The experimental results were compared to classical models from the literature used to predict the swelling of polymeric materials. A simple first-order kinetic equation was shown to be adequate to describe the size evolution during swelling. Thus, the results suggested that the swelling dynamics was a water uptake-limiting mechanism because of the existence of a gelatinous layer at the grain surface formed during the manufacturing process, as previously reported in the literature. • MRI is an effective tool to investigate water distribution in granular food materials. • Water diffusion coefficient in grains is heterogeneously distributed in space. • Classical diffusive models from the literature cannot predict swelling dynamics. • Results suggest that the swelling dynamics is a water uptake-limiting mechanism.

Weak Langmuir turbulence in disordered multimode optical fibers

We consider the propagation of temporally incoherent waves in multimode optical fibers (MMFs) in the framework of the multimode nonlinear Schr\"odinger (NLS) equation accounting for the impact of the natural structural disorder that affects light propagation in standard MMFs (random mode coupling and polarization fluctuations). By averaging the dynamics over the fast disordered fluctuations, we derive a Manakov equation from the multimode NLS equation, which reveals that the Raman effect introduces a previously unrecognized nonlinear coupling among the modes. Applying the wave turbulence theory on the Manakov equation, we derive a very simple scalar kinetic equation describing the evolution of the multimode incoherent waves. The structure of the kinetic equation is analogous to that developed in plasma physics to describe weak Langmuir turbulence. The extreme simplicity of the derived kinetic equation provides physical insight into the multimode incoherent wave dynamics. It reveals the existence of different collective behaviors where all modes self-consistently form a multimode spectral incoherent soliton state. Such an incoherent soliton can exhibit a discrete behavior characterized by collective synchronized spectral oscillations in frequency space. The theory is validated by accurate numerical simulations: The simulations of the generalized multimode NLS equation are found in quantitative agreement with those of the derived scalar kinetic equation without using adjustable parameters.

Electrical conductivity under shear flow of molten polyethylene filled with carbon nanotubes: Experimental and modeling

AbstractThis work aims to describe the conductivity evolution of polymer composites (polyethylene filled with carbon nanotubes) during a shearing deformation. Rheo‐electric measurements were carried out to observe the shear‐induced fillers network modification. Extended steady shear forces the conductivity to evolve asymptotically to a steady level attesting to an equilibrium between structuring and break up mechanisms in the melted polymer. Numerous experiments were conducted to cover a wide range of shear rate from 0.05 to 10 s−1 and for carbon nanotubes concentrations between 1.3 and 2.9 vol%. A model is proposed to predict the conductivity evolution under shear deformation using a simple kinetic equation inserted in a percolation law. Structuring parameter was found to be solely dependent on the temperature whereas shear induced modification terms were found to be mostly driven by the shear rate and the fillers content.

Effective quantum kinetic theory for spin transport of fermions with collsional effects

We systematically derive the collision term for the axial kinetic theory, a quantum kinetic theory delineating the coupled dynamics of the vector/axial charges and spin transport carried by the massive spin-1/2 fermions traversing a medium. We employ the Wigner-function approach and propose a consistent power-counting scheme where the axial-charge distribution function, a non-conserved quantity for massive particles, is accounted as the first-order quantity in the ħ expansion, while the vector-charge distribution function the zeroth-order quantity. This specific power-counting scheme allows us to organize a reduced ħ expansion for the collision term and to formally identity the spin- diffusion effect and the spin-polarization effect at the same order. We confirm that the obtained collisional axial kinetic theory smoothly reduces to the chiral kinetic theory in the massless limit, serving as a consistency check. In the absence of electromagnetic fields, we further present the simplified axial kinetic equations suitable for tracking dynamical spin polarization of heavy and light fermions, respectively. As an application to the weakly coupled quark-gluon plasma at high temperature, we compute the spin-diffusion term for massive quarks within the leading-log approximation. The formal expression for the first- order terms provides a path toward evaluation of the spin polarization effect in quantum chromodynamics.

Polymorphic Transition of Pre-oriented Polybutene-1 under Tensile Deformation: In Situ FTIR Study

Deformation-induced phase transition of Form II to Form I in polybutene-1 (PB-1) has been investigated by time-resolved Fourier transform infrared (FTIR) spectroscopy over a wide temperature range from 25 °C to 105 °C. The initial film sample containing orientated lamellae is prepared by pre-stretching of PB-1 melt followed by solidification. This is to realize a homogeneity of subsequent deformation at the mesoscale of lamellar stacks by avoiding large-scale spherulites. The deformation induced phase transition is recognized to occur with two stages: first, Form II undergoes the lamellar fragmentation, slipping or local melting after yielding to activate its transition to Form I, which may be realized by releasing the restrictions on chains translational movements in crystalline phase; second, the phase transition proceeds with a continuous dissipation of external work and determines the tensile mechanical response of film. To quantify the relationship between crystalline transition of Form II to Form I and external tensile field, a simple kinetic equation is well established based on FTIR measurement. The equation can describe not only the dependence of crystal transitional degree on applied specific work, but also the retardation effect of elevating temperature on phase transition.

Optimized Reforming of Biomass Derived Gas Based on Thermodynamic and Kinetics Analysis with Activated Carbon Fibers Supported Ni-Al2O3

Thermodynamics and kinetics are the two key factors in optimizing the biomass derived gas reforming to obtain syngas. In this study, the effects of components of simulated biomass derived gas (CH4, CO2, H2, CO, N2, and steam), reforming conditions, and different reactions on products distribution were investigated under thermodynamic equilibrium using Gibbs free energy minimization method so as to optimize the reforming conditions and provide theoretical basis for further reforming of biomass derived gas. The results showed that temperature and steam play an important role in manipulating the ratio of H2/CO during reforming. Changing the temperature and the amount of steam can promote the reverse water-gas shift reaction in different directions; thereby the syngas yield can be increased, and the H2/CO ratio can be adjusted so as to meet different application requirements. Furthermore, the macroscopic kinetics analysis with a simple power-law type kinetic equation over our previous reported catalyst Ni-Al2O3/ACF (activated carbon fibers) used for the reforming of simulated biomass derived gas was obtained. The influence of carbon fibers on the catalytic performance of nickel-based alumina catalyst was explained in terms of activation energy.

Enzymolysis kinetics and structural-functional properties of high-intensity ultrasound-assisted alkali pretreatment ovalbumin

ABSTRACT This research investigated the enzymolysis kinetics of high-intensity ultrasound (HIU)-assisted alkali pretreatment Ovalbumin (OVA) through using Alcalase, and its structural and functional properties. The structural characteristics of OVA were performed with sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE), surface hydrophobicity, Fourier Transform Infrared, and differential scanning calorimetry. SDS-PAGE analysis indicated that the pretreatment sample showed proportion of lower molecular mass components (14.3–20.1 kDa). There was an increase in the intensity of fluorescence spectrum and changes in functional groups after the pretreatment on OVA compared with the control (without treatment), which might be the reason for the significantly enhancing efficiency of the enzymatic hydrolysis. The thermal denaturation temperature of enzymatic hydrolysis of OVA after pretreatment was 95.1°C, which was 17.5°C higher than that of control. As compared to OVA (control), the enzymatic hydrolysis with HIU-assisted alkali pretreatment was found to significantly improve protein solubility, emulsifying properties, foaming ability (FA) and foam stability, oil, and water absorption capacity (WAC). Especially emulsifying activity index, FA, WAC, and oil absorption capacity were increased by 228.7%, 88.5%, 102.6%, and 67.4%, respectively (P< 0.05). A simplified kinetic equation for the enzymolysis model was deduced to successfully describe the enzymatic hydrolysis of pretreated OVA. The results proved the effects of the pretreatment on the substrate enhanced the enzymolysis process and this influence had relation to the structure changes. Furthermore, the enzymatic hydrolysis process for other proteins demonstrated the applicability of the model.

Theoretical research on the transport and ionization rate coefficients in glow discharge dusty plasma

The electron kinetic model for investigating the transport and ionization rate coefficients of argon glow discharge dusty plasma is developed from the Boltzmann equation. Both of the electron–neutral and electron–dust collisions are considered as collision terms in the kinetic equation. The kinetic equation is simplified by employing the local approximation and nonlocal approach under the same discharge conditions, and the corresponding simplified kinetic equations are known as local and nonlocal kinetic equations respectively. Then the electron energy distribution function (EEDF) is obtained by numerically solving the local and nonlocal kinetic equations and the dust charging equations simultaneously. Based on the obtained EEDFs, the effective electron temperature, electron mobility, electron diffusion coefficient and ionization rate coefficient are calculated for different discharge conditions. It is shown that the EEDFs calculated from the local kinetic model clearly differ from the nonlocal EEDFs and both the local and nonlocal EEDFs are also clearly different with Maxwellian distributions. The appearance of dust particles results in an obvious decrease of high energy electrons and increase of low energy electrons when axial electric field is low. With the increase of axial electric field, the influence of dust particles on the EEDFs becomes smaller. The electron mobility and diffusion coefficients calculated on the basis of local and nonlocal EEDFs do not differ greatly to the dust-free ones. While, when dust density nd = 106 cm−3, the electron mobility increases obviously compared with the dust-free results at low axial electric field and decreases with the increasing axial electric field until they are close to the dust-free ones. Meanwhile, electron diffusion coefficients for dusty case become smaller and decrease with the increasing axial electric field. The ionization rate coefficients decrease when dust particles are introduced and they approach the dust-free results gradually with the increasing axial electric field.

Accurate and efficient computation of the Boltzmann equation for Couette flow: Influence of intermolecular potentials on Knudsen layer function and viscous slip coefficient

The Couette flow is one of the fundamental problems of rarefied gas dynamics, which has been investigated extensively based on the linearized Boltzmann equation (LBE) of hard-sphere molecules and simplified kinetic model equations. However, how the different intermolecular potentials affect the viscous slip coefficient and the structure of Knudsen layer remains unclear. Here, a novel synthetic iteration scheme (SIS) is developed for the LBE to find solutions of Couette flow accurately and efficiently: the velocity distribution function is first solved by the conventional iterative scheme, then it is modified such that in each iteration i) the flow velocity is guided by an ordinary differential equation that is asymptotic-preserving at the Navier–Stokes limit and ii) the shear stress is equal to the average shear stress. Based on the Bhatnagar–Gross–Krook model, the SIS is assessed to be efficient and accurate. Then we investigate the Knudsen layer function for gases interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones potentials, subject to diffuse-specular and Cercignani–Lampis gas-surface boundary conditions. When the tangential momentum accommodation coefficient (TMAC) is not larger than one, the Knudsen layer function is strongly affected by the potential, where its value and width increase with the effective viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits similarities among different values of TMAC when the intermolecular potential is fixed. For Cercignani–Lampis boundary condition with TMAC larger than one, both the viscous slip coefficient and Knudsen layer function are affected by the intermolecular potential, especially when the “backward” scattering limit is approached. With the asymptotic theory by Jiang and Luo (2016) [14] for the singular behavior of the velocity gradient in the vicinity of solid surfaces, we find that the whole Knudsen layer function can be well fitted by the power series ∑n=02∑m=02cn,mxn(xln⁡x)m, where x is the distance to the solid surface. Finally, the experimental data of the Knudsen layer profile are explained by the LBE solution with proper values of the viscosity index and TMAC.

Tuning the morphology of self-assisted GaP nanowires

Patterned arrays of self-assisted GaP nanowires (NWs) were grown on a Si substrate by gas source molecular beam epitaxy using various V/III flux ratios from 1–6, and various pitches from 360–1000 nm. As the V/III flux ratio was increased from 1–6, the NWs showed a change in morphology from outward tapering to straight, and eventually to inward tapering. The morphologies of the self-assisted GaP NWs are well described by a simple kinetic equation for the NW radius versus the position along the NW axis. The most important growth parameter that governs the NW morphology is the V/III flux ratio. Sharpened NWs with a stable radius equal to only 12 nm at a V/III flux of 6 were achieved, demonstrating their suitability for the insertion of quantum dots.

Fluid dynamics of out of equilibrium boost invariant plasmas

By solving a simple kinetic equation, in the relaxation time approximation, and for a particular set of moments of the distribution function, we establish a set of equations which, on the one hand, capture exactly the dynamics of the kinetic equation, and, on the other hand, coincide with the hierarchy of equations of viscous hydrodynamics, to arbitrary order in the viscous corrections. This correspondence sheds light on the underlying mechanism responsible for the apparent success of hydrodynamics in regimes that are far from local equilibrium.

Direct measurement of ˙OH and HO2˙ formation in ˙R + O2 reactions of cyclohexane and tetrahydropyran.

Formation of the key general radical chain carriers, ˙OH and HO2˙, during pulsed-photolytic ˙Cl-initiated oxidation of tetrahydropyran and cyclohexane are measured with time-resolved infrared absorption in a temperature-controlled Herriott multipass cell in the temperature range of 500-750 K at 20 Torr. The experiments show two distinct timescales for HO2˙ and ˙OH formation in the oxidation of both fuels. Analysis of the timescales reveals striking differences in behavior between the two fuels. In both cyclohexane and tetrahydropyran oxidation, a faster timescale is strongly related to the "well-skipping" (˙R + O2 → alkene + HO2˙ or cyclic ether + ˙OH) mechanism and is expected to have, at most, a weak temperature dependence. Indeed, the fast HO2˙ formation timescale is nearly temperature independent both for cyclohexyl + O2 and for tetrahydropyranyl + O2 below 700 K. A slower HO2˙ formation timescale in cyclohexane oxidation is shown to be linked to the sequential ˙R + O2 → ROO˙ → alkene + HO2˙ pathway, and displays a strong temperature dependence mainly from the final step (with energy barrier ∼32.5 kcal mol-1). In contrast, the slower HO2˙ formation timescale in tetrahydropyran oxidation is surprisingly temperature insensitive across all measured temperatures. Although the ˙OH formation timescales in tetrahydropyran oxidation show a temperature dependence similar to the cyclohexane oxidation, the temperature dependence of ˙OH yield is opposite in both cases. This significant difference of HO2˙ formation kinetics and ˙OH formation yield for the tetrahydropyran oxidation can arise from contributions related to ring-opening pathways in the tetrahydropyranyl + O2 system that compete with the typical ˙R + O2 reaction scheme. This comparison of two similar fuels demonstrates the consequences of differing chemical mechanisms on ˙OH and HO2˙ formation and shows that they can be highlighted by analysis of the eigenvalues of a system of simplified kinetic equations for the alkylperoxy-centered ˙R + O2 reaction pathways. We suggest that such analysis can be more generally applied to complex or poorly known oxidation systems.

Measurement of the Influence of the Microstructure of Alumina‐Supported Cobalt Catalysts on their Activity and Selectivity in Fischer–Tropsch Synthesis by using Steady‐State and Transient Kinetics

AbstractFive alumina‐supported Co catalysts with different particle sizes and structures were synthesized. The catalysts showed different activities during long‐term Fischer–Tropsch experiments. Steady‐state isotopic transient kinetic analysis (SSITKA) 12CO/H2→13CO/H2 experiments were performed during these long‐term runs. The number of active sites for CO adsorption and activation was estimated through the summation of all surface intermediates derived from the SSITKA results. Rather than comparing turnover frequencies, a kinetic analysis was performed. A simple kinetic rate equation based on the in situ number of active sites described the steady‐state CO conversion over the five catalysts adequately. Thus the difference in the catalytic performance could not be attributed to a difference in particle size, phase orientation (face‐centered cubic or hexagonal close packed), or Pt‐promotion effect but instead was only because of the number of reduced Co atoms exposed during the reaction. Similarly, the selectivity depended on the CO conversion level and temperature and not on the catalyst structure.

The thermalization of soft modes in non-expanding isotropic quark gluon plasmas

We discuss the role of elastic and inelastic collisions and their interplay in the thermalization of the quark–gluon plasma. We consider a simplified situation of a static plasma, spatially uniform and isotropic in momentum space. We focus on the small momentum region, which equilibrates first, and on a short time scale. We obtain a simple kinetic equation that allows for an analytic description of the most important regimes. The present analysis suggests that the formation of a Bose condensate, expected when only elastic collisions are present, is strongly hindered by the inelastic, radiative, processes.

A simple kinetic equation of swarm formation: Blow-up and global existence

In the present paper we identify both blow-up and global existence behaviors for a simple but very rich kinetic equation describing of a swarm formation.

Modeling and kinetics study of Bisphenol A (BPA) degradation over an FeOCl/SiO2 Fenton-like catalyst

A heterogeneous Fenton-like catalyst was used for the oxidation of Bisphenol A (BPA), which is a typical endocrine disruptor. A simple semi-empirical kinetic equation was developed to define the effects of experimental conditions (initial concentration of H2O2, initial concentration of BPA, temperature and the catalyst loading) on the reaction rate. The plausible pathway of the degradation process was proposed. A pseudo-first-order reaction rate expression with respect to BPA concentration was developed to fit all data for these experiments, and the equation of kobs (observed rate constant) was obtained with the apparent activation energy of 42.2kJ/mol. In order to reduce the operational cost, the values of parameters (temperature, initial concentration of H2O2 and pH) that optimize the catalytic performance were studied by means of a response surface methodology. Finally, a plausible mechanism was proposed on the basis of kinetics and optimization studies, which illustrates the reaction process and optimization in a microscopic view.

Enzymolysis kinetics and structural characteristics of rice protein with energy-gathered ultrasound and ultrasound assisted alkali pretreatments

This research investigated the structural characteristics and enzymolysis kinetics of rice protein which was pretreated by energy-gathered ultrasound and ultrasound assisted alkali. The structural characteristics of rice protein before and after the pretreatment were performed with surface hydrophobicity and Fourier transform infrared (FTIR). There was an increase in the intensity of fluorescence spectrum and changes in functional groups after the pretreatment on rice protein compared with the control (without ultrasound and ultrasound assisted alkali processed), thus significantly enhancing efficiency of the enzymatic hydrolysis. A simplified kinetic equation for the enzymolysis model with the impeded reaction of enzyme was deduced to successfully describe the enzymatic hydrolysis of rice protein by different pretreatments. The initial observed rate constants (Kin,0) as well as ineffective coefficients (kimp) were proposed and obtained based on the experimental observation. The results showed that the parameter of kin,0 increased after ultrasound and ultrasound assisted alkali pretreatments, which proved the effects of the pretreatments on the substrate enhancing the enzymolysis process and had relation to the structure changes of the pretreatments on the substrate. Furthermore, the applicability of the simplified model was demonstrated by the enzymatic hydrolysis process for other materials.