Integral Operator Solving Process of the Boundary Value Problem of Abstract Kinetic Equation with the First Kind of Critical Parameter and Generalized Periodic Boundary Conditions

In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.

Similarities and Differences Between Freundlich Kinetic Equation and Two-Constant Equation

A mathematical expression of Freundlich kinetic equation, 1nS=A'+B'1nt, is presented, and the physical meanings of its parameters are indicated. Although the Freundlich kinetic equation and the two-constant equation are the same in the form, the derivation of the Freundlich kinetic equation is precise, while the deriVation of the two-constant equation has some contradictions and is unreasonable. And it is suggested that the Freundlich kinetic equation should have priority over the two-constant equation to be used.

Comparison of Seven Kinetic Equations for K Release and Application of Kinetic Parameters

Corn field experiments with two treatments, NP and NPK, where N in the form of urea, P in the form of calcium phosphate, and K in the form of KCl were applied at rates of 187.5, 33.3, and 125 kg ha^-1, respectively, on soils derived from Quaternary red clay were conducted in the hilly red soil region of Zhejiang Province, China. Plant grains and stalks were collected for determination of K content. Seven equations were used to describe the kinetics of K release from surface soil samples taken before the corn experiments under electric field strengths of 44.4 and 88.8 V cm^-1 by means of electro-ultrafiltration （EUF） and to determine if their parameters had a practical application. The second-order and Elovich equations excellently described K release; the first-order, power function, and parabolic diffusion equations also described K release well; but the zero-order and exponential equations were not so good at reflecting K release. Five reference standards from the field experiments, including relative grain yield （yield of the NP treatment/yield of the NPK treatment）, relative dry matter yield （dry matter of the NP treatment/dry matter of the NPK treatment）, quantity of K uptake in the NP treatment （no K application）, soil exchangeable K, and soil HNO3-soluble K, were used to test the effectiveness of equation parameters obtained from the slope or intercept of these equations. Correlations of the ymax （the maximum desorbable quantity of K） in the second-order equation and the constant b in the first-order and Elovich equations to all five reference standards were highly significant （P ≤ 0.01）. The constant a in the power function equation was highly significant （P 〈 0.01） for four of the five reference standards with the fifth being significant （P ≤ 0.05）. The constant b in the parabolic equation was also significantly correlated （P ≤ 0.05） to the relative grain yield and soil HNO3-soluble K. These suggested that all of these parameters could be used to estimate the soil K supplying capacity and the crop response to K fertilizer.

Reaction Kinetic Equation for Char Combustion of Underground Coal Gasification

Based on the quasi-steady-state approximation, the dynamic equation of char combustion in the oxidation zone of underground coal gasification (UCG) was derived. The parameters of the dynamic equation were determined at 900℃ using a thermo-gravimetric (TG) analyzer connected to a flue gas analyzer and this equation. The equation was simplified for specific coals, including high ash content, low ash content, and low ash fusibility ones. The results show that 1) the apparent reaction rate constant increases with an increase in volatile matter value as dry ash-free basis,2) the effective coefficient of diffusion decreases with an increase in ash as dry basis, and 3) the mass transfer coefficient is independent of coal quality on the whole. The apparent reaction rate constant, mass-transfer coefficient and effective coefficient of diffusion of six char samples range from 7.51×104 m/s to 8.98×104 m/s, 3.05×106 m/s to 3.23×106 m/s and 5.36×106 m2/s to 8.23×106 m2/s at 900℃, respectively.

A GPU-based general numerical framework for plasma simulations in terms of microscopic kinetic equations with full collision terms

We have proposed a general numerical framework for plasma simulations on graphics processing unit clusters based on microscopic kinetic equations with full collision terms.Our numerical algorithm consistently deals with both long-range(classical forces in the Vlasov term)and short-range(quantum processes in the collision term)interactions.Providing the relevant particle masses,charges and types(classical,fermionic or bosonic),as well as the external forces and the matrix elements(in the collisional integral),the algorithm consistently solves the coupled multi-particle kinetic equations.Currently,the framework is being tested and applied in the field of relativistic heavy-ion collisions;extensions to other plasma systems are straightforward.Our framework is a potential and competitive numerical platform for consistent plasma simulations.

Kinetic equations characterizing double reversible transformations in heating CuZnAlMnNi shape memory alloy

The apparent activation energies and frequency factors of thedouble reversible transformations occurring in heating CuZnAlMnNIshape memory alloy (SMA) were deduced as ΔE_x→M = 62. 597 8 KJ/mol, ΔE_M → A = 153. 92 KJ/Mol, A_x→M = 5.2232 × 10~9S^-1, andA_ M → A = 2.3251 × 10~23 S^-1, respectively. The kinetic equationsof the two transformations due- Ing heating were establishedsimultaneously.

Kinetic Equation for Internal Oxidation of Cu-Al Alloy Spheres

The kinetics of internal oxidation of Cu-Al alloy spheres, containing up to 2.214% mole fraction Al was investigated in the temperature range 1 023 K to 1 273 K, and the depth of internal oxidation was measured in the microscopy. A kinetic equation was derived to describe the internal oxidation of Cu-A1 alloy spheres, which was checked experimentally by means of oxidation depth measurements. The results show that the derived equation is exact enough to describe the kinetics of internal oxidation of Cu-Al alloy spheres. Based on this equation and the oxidation depth measurements, the permeability of oxygen in solid copper has been obtained. Investigation also shows that in the process of internal oxidation, there is no evidence for preferential diffusion along grain boundaries.

TBE EFFIGIENGIES OF THREE INITIATORS OF Fe^(2+)—H_2O_2, K_2S_2O_8, KMnO_4 AND TBE KINETIC EQUATION IN THE GRAFTING OF AGRYLAMIDE ONTO STARCH

A comparative study is made on the efficiencies of three initiators of Fe^(2+)-H_2O_2, K_2S_2O_8 and KMnO_4 in the grafting of acrylamide onto starch. Of the three initiators the best one is KMnO_4. Moreover, the kinetic equation is also studied.

KINETIC EQUATIONS WITH VARIATION OF LIQUID REACTANT CONCENTRATION

For two common types of liquid-solid heterogeneous reactions,the kinetic equations have been established which involved both the variation of liquid reactant concentration and the va- riation of solid reactant geornetry with the reaction time.The experimental results show that the kinetic equations are more accurate and reasonable than those appeared in previous litera- tures.Moreover,they are also suitable for gas-solid heterogeneous reactions in principle.

Statistical deduction and experimental verification on kinetic equations for the curing reactions of epoxy resins/amines

Based on three typical mechanisms (second-order, third-order and competitive mechanisms) for the curing reactions of the epoxy resins with amines, a pair of the kinetic equations (for primary and secondary aminations) was presented to explain the uniformity and relationship among the three different kinetic mechanisms of the reactions. The presented macro-equations were deduced from the kinetic micro-equations by the statistics method. And the constitutive equations were verified by experimental data at different reaction times and temperatures (95°C, 60°C and 39°C), taking diglycidyl ether of bisphenol A (DGEBA) /ethyleneamine (EA) as a model.

Kinetic equation for internal oxidation of Cu-Al alloy cylinders

The kinetics of internal oxidation of Cu-Al alloy cylinders, containing up to 2.214mol% Al, were investigated in the temperature range of 1023 K to 1273 K, and the depth of internal oxidation was measured in the microscopy. A kinetic equation was derived to describe the internal oxidation of Cu-Al alloy cylinders. For the internal oxidation of Cu-Al alloys employed in the synthesis of alumina dispersion strengthened copper, the kinetic equation can be simplified. The derived equation was checked experimentally by means of oxidation depth measurements and the results show that the derived equation is exact enough to describe the kinetics of internal oxidation of Cu-Al alloy cylinders. Based on this equation and the oxidation depth measurements, the permeability of oxygen in solid copper was obtained. Investigation also shows that there is no evidence for preferential diffusion along grain boundaries in the process of internal oxidation.

Kinetic Equations of Potassium Desorption and the Ap－plication of Equation Constants

Elovich, parabolic diffusion, power function and exponential equations were used to describe K desorptionkinetics of 12 soils in a constant electric field of electro-ultrafiltration (EUF). Results showed that the Elovich,parabolic diffusion and power function equations could describe K desorption kinetics well owing to their highcorrelatfon coefficients and low standard errors; but the exponential equation was not suitable to be usedin this study due to its relatively low correlation coefficients and relatively high standard errors. This workestablished successfully the relationships between the constants (slope or intercept) of kinetic equations andthe barley responses to K fertilizer in the multiple-site field experiments and K-supplying status of soils, theconstants of Elovich, parabolic diffusion and power function equations were very significantly or significantlycorrelated to the soil available K, relative yield of barley and K uptake of barley in NP plot. It was suggestedthat the kinetic equation constants could be used to estimate K-supplying power of soils.

Accelerated procedure to solve kinetic equation for neutral atoms in a hot plasma

By reaching the first wall of a fusion reactor, charged plasma particles, electrons and ionsare recombined into neutral molecules and atoms of hydrogen isotopes. These speciesrecycle back into the plasma volume and participate, in particular, in charge–exchange(cx) collisions with ions. As a result, hot atoms with chaotically directed velocities aregenerated and some of them hit the wall. Statistical Monte Carlo methods often usedto model the behavior of cx atoms are too time-consuming for comprehensive parameter studies. Recently1 an alternative iteration approach to solve one-dimensional kineticequation2 has been significantly accelerated, by a factor of 30–50, by applying a passmethod to evaluate the arising integrals from functions, involving the ion velocity distribution. Here, this approach is used by solving a two-dimensional kinetic equation,describing the transport of cx atoms in the vicinity of an opening in the wall, e.g.,the entrance of a duct guiding to a diagnostic installation. To assess the erosion rateand lifetime of the installation, one need to know the energy spectrum of hot cx atomsescaping from the plasma into the duct. Calculations are done for a first mirror of molybdenum under plasma conditions expected in a fusion reactor like DEMO.3,4 The resultsof kinetic modeling are compared with those found by using a diffusion approximation5relevant for cx atoms if the time between cx collisions with ions is much smaller thanthe time till the ionization of atoms by electrons. The present more exact kinetic consideration predicts a mirror erosion rate by a factor of 2 larger than the approximatediffusion approach.

Gas-kinetic numerical method for solving mesoscopic velocity distribution function equation

A gas-kinetic numerical method for directly solving the mesoscopic velocity distribution function equation is presented and applied to the study of three-dimensional complex flows and micro-channel flows covering various flow regimes. The unified velocity distribution function equation describing gas transport phenomena from rarefied transition to continuum flow regimes can be presented on the basis of the kinetic Boltzmann-Shakhov model equation. The gas-kinetic finite-difference schemes for the velocity distribution function are constructed by developing a discrete velocity ordinate method of gas kinetic theory and an unsteady time-splitting technique from computational fluid dynamics. Gas-kinetic boundary conditions and numerical modeling can be established by directly manipulating on the mesoscopic velocity distribution function. A new Gauss-type discrete velocity numerical integra- tion method can be developed and adopted to attack complex flows with different Mach numbers. HPF paral- lel strategy suitable for the gas-kinetic numerical method is investigated and adopted to solve three-dimensional complex problems. High Mach number flows around three-dimensional bodies are computed preliminarilywith massive scale parallel. It is noteworthy and of practical importance that the HPF parallel algorithm for solving three-dimensional complex problems can be effectively developed to cover various flow regimes. On the other hand, the gas-kinetic numerical method is extended and used to study micro-channel gas flows including the classical Couette flow, the Poiseuillechannel flow and pressure-driven gas flows in twodimensional short micro-channels. The numerical experience shows that the gas-kinetic algorithm may be a powerful tool in the numerical simulation of microscale gas flows occuring in the Micro-Electro-Mechanical System （MEMS）.

Numerical simulation of metal evaporation based on the kinetic model equation and the direct simulation Monte Carlo method

Metal evaporation on the basis of the kinetic model equations(BGK and S-model) and the direct simulation Monte Carlo(DSMC) method was investigated computationally under the circumstances of collimators existing or not. Numerical data of distributions of number density, bulk velocity and temperature were reported over a wide range of evaporation rate.It was shown that these results reached a good agreement for the case of small evaporation rate, while the deviations became increasingly obvious with the increase of evaporation rate, especially when the collimators existed. Moreover, the deposition thickness over substrate obtained from the kinetic model equations were inaccurate even though the evaporation rate was small. All of the comparisons showed the reliability of the kinetic model equations, which require less computational cost at small evaporation rate and simple structure.

Asymptotic Solutions of the Kinetic Boltzmann Equation and Multicomponent Non-Equilibrium Gas Dynamics

In the article correct method for the kinetic Boltzmann equation asymptotic solution is formulated, the Hilbert’s and Enskog’s methods are discussed. The equations system of multicomponent non- equilibrium gas dynamics is derived, that corresponds to the first order in the approximate (asym- ptotic) method for solution of the system of kinetic Boltzmann equations.

An Algorithm to Optimize the Calculation of the Fourth Order Runge-Kutta Method Applied to the Numerical Integration of Kinetics Coupled Differential Equations

The kinetic electron trapping process in a shallow defect state and its subsequent thermal- or photo-stimulated promotion to a conduction band, followed by recombination in another defect, was described by Adirovitch using coupled rate differential equations. The solution for these equations has been frequently computed using the Runge-Kutta method. In this research, we empirically demonstrated that using the Runge-Kutta Fourth Order method may lead to incorrect and ramified results if the numbers of steps to achieve the solutions is not “large enough”. Taking into account these results, we conducted numerical analysis and experiments to develop an algorithm that determines the smallest non-critical number of steps in an automatic way to optimize the application of the Runge-Kutta Fourth Order method. This algorithm was implemented and tested in a variety of situations and the results have shown that our solution is robust in dealing with different equations and parameters.

Dynamic spheroidization kinetics behavior of Ti-6.5Al-2Zr-1Mo-1V alloy with lamellar microstructure

Abstract： The dynamic spheroidization kinetics behavior of Ti-6.5Al-2Zr-1Mo-1V alloy with a lamellar initial microstructure was studied by isothermal hot compression tests in the temperature range of 750-950℃ and strain rates of 0.001-10 s^-1. The results show that the spheroidized fraction of alpha lamellae increases with the increase of temperature and the decrease of strain rate. The spheroidization kinetics curves predicted by JMAK equation agree well with experimental ones. The corresponding SEM and TEM observations indicate that the dynamic spheroidization process can be divided into two stages. The primary stage is boundary splitting formed by two competing mechanisms which are dynamic recrystallization and mechanical twin. In the second stage, the penetration of beta phase into the alpha/alpha grain boundaries is dominant, which is controlled in nature by diffusion of the chemical elements such as Al, Mo and V.

Preparation of nano-TiO_2/diatomite-based porous ceramics and their photocatalytic kinetics for formaldehyde degradation

Diatomite-based porous ceramics were adopted as carriers to immobilize nano-TiO2 via a hydrolysis-deposition technique. The thermal degradation of as-prepared composites was investigated using thermogravimetric-differential thermal analysis, and the phase and microstructure were characterized by X-ray diffraction, Fourier transform infrared spectroscopy, and transmission electron microscopy. The results indicated that the carriers were encapsulated by nano-TiO2 with a thickness of 300-450 nm. The main crystalline phase of TiO2 calcined at 650~C was anatase, and the average grain size was 8.3 nm. The FT-IR absorption bands at 955.38 cm1 suggested that new chemical bonds among Ti, O, and Si had formed in the composites. The photocatalytic （PC） activity of the composites was investigated un- der UV irradiation. Furthermore, the photodegradation kinetics of formaldehyde was investigated using the composites as the cores of an air cleaner. A kinetics study showed that the reaction rate constants of the gas-phase PC reaction of formaldehyde were k = 0.576 mg＇m3·min^-1 and K = 0.048 m3/mg.