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.

Aging kinetics of 14H-LPSO precipitates in Mg-Zn-Y alloy

Alloys with long-period stacking ordered structures(LPSO)have good properties and are highly regarded.Mg-Zn-Y alloy containing LPSO phase was prepared by the traditional casting method,and the aging heat treatment was performed at different temperatures and times.The microstructure and phase constitutions of the alloy were observed by means of optical microscopy and scanning electron microscopy methods.Results show the microstructure of as-cast Mg95.5Zn1.5Y3 mainly consists ofα-Mg,W phases and LPSO phases.During the aging treatment,fine lamellar-shaped 14H-LPSO phase is formed at the grain boundaries and precipitates from the supersaturated magnesium matrix,and the volume fraction increases as the aging time increases.By controlling the aging time,Mg-Zn-Y alloys with different volume fractions of 14H-LPSO phase were prepared.The aging kinetics equation of the 14H-LPSO phase is summarized,that is f=1-exp(-0.2705 t 0.6368).The phase transformation mechanism of 14H-LPSO in Mg95.5Zn1.5Y3 alloy can be described as the change of dislocation energy.

Two stages kinetics of municipal solid waste inoculation composting processes

In order to understand the key mechanisms of the composting processes, the municipal solid waste(MSW) composting processes were divided into two stages, and the characteristics of typical experimental scenarios from the viewpoint of microbial kinetics was analyzed. Through experimentation with advanced composting reactor under controlled composting conditions, several equations were worked out to simulate the degradation rate of the substrate. The equations showed that the degradation rate was controlled by concentration of microbes in the first stage. The degradation rates of substrates of inoculation Run A, B, C and Control composting systems were 13 61 g/(kg·h), 13 08 g/(kg·h), 15 671 g/(kg·h), and 10 5 g/(kg·h), respectively. The value of Run C is around 1 5 times higher than that of Control system. The decomposition rate of the second stage is controlled by concentration of substrate. Although the organic matter degradation rates were similar to all Runs, inoculation could reduce the values of the half velocity coefficient K \-m and could be more efficient to make the composting stable. Particularly, for Run C, the degradation rate is high in the first stage, and K \-m is low in the second stage. The results indicated that the inoculation was efficient for the composting processes.

Developed mathematical technique for fractional stochastic point kinetics model in nuclear reactor dynamics

Fractional stochastic kinetics equations have proven to be valuable tools for the point reactor kinetics model, where the nuclear reactions are not fully described by deterministic relations. A fractional stochastic model for the point kinetics system with multi-group of precursors,including the effect of temperature feedback, has been developed and analyzed. A major mathematical and inflexible scheme to the point kinetics model is obtained by merging the fractional and stochastic technique. A novel split-step method including mathematical tools of the Laplace transforms, Mittage–Leffler function, eigenvalues of the coefficient matrix, and its corresponding eigenvectors have been used for the fractional stochastic matrix differential equation. The validity of the proposed technique has been demonstrated via calculations of the mean and standard deviation of neutrons and precursor populations for various reactivities: step, ramp, sinusoidal, and temperature reactivity feedback. The results of the proposed method agree well with the conventional one of the deterministic point kinetics equations.

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.

Exchange Reaction Between Selenite and Hydroxyl Ion of Variable Charge Soil Surfaces: Ⅱ. Kinetics of Hydroxyl Release

A self-made constant pH automated titration instrument was used to study thekinetics of hydroxyl release during selenite reacting with variable charge soils. The rate ofhydroxyl release was very rapid at the first several minutes, then gradually slowed down, and atlast did not change any more. The experimental data was well fitted by the Langmuir kineticequation, arid with increasing selenite concentration or decreasing solution pH, the reaction lastedlonger, the maximum of hydroxyl release (x_m) increased, and the binding constant (k) decreased.The time of hydroxyl release with Xuwen latosol was much longer than that with Jinxian red soil.

Kinetics of Nitrogen and Potassium Uptake by Various Rice Cultivars at Different Nitrogen Levels from Purple Soil

A pot experiment was conducted to study the effect of nitrogen fertilizer on nitrogen and potassium uptake by four rice cultivars. Results showed that the quadatic parabola relationship between biomass of rice and nitrogen levels was observed, with the maximum biomass at the nitrogen level of 150 mg kg-1.The rates of nitrogen and potassium uptake by the four rice cultivars depended on growth stage and rice cultivar with the maximum rate of N in Shanyou-63 and maximum rate of K in Kaiyou-5 (hybrid rice),respedively. The kinetics of nitrogen and potasssium uptake by rice plant could be quantitstively described by the following equations: y = a + blogt, y = ab + t1/2 and y = ae-bt. The b value in the equations was correlated significantly to the rates of nitrogen and potassium uptake (NR and KR, r=0.901**～0.990**),suggesting that the b value could be used to distinguish the index of nitrogen and potassium uptake capacity of rice. The maximum values of nitrogen uptake by plant (b value) and apparent recovery of fertilizer nitrogen were observed in Shanyou-63, and the minimum value in Eryou-6078. However, the capacity of potassium uptake (b value) by Kaiyou-5 ranked first and that by Shanyou-63 second. There was a significant linear relationship between nitrogen level and nitrogen uptake by rice, but a quadratic parabola relationship was found between nitrogen level and patassium uptake by rice. The application of nitrogen fertilizer decreased the ratios of potassium to nitrogen uptake by rice plant. The greatest reduction in the ratio was observed at high nitrogen level, and the least reduction was found in Kaiyou-5 and Shanyou-63 due to their greater ability to absorb potassium.

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.

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.

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.

Study on Kinetics for Desulfurization of Model Diesel

In this study,by means of the experiments for desulfurization of model diesel through oxidative extraction,the changes associated with the rate of desulfurization of diesel and the mechanism for oxidation of sulfides in diesel were explored. Through studying the mechanism for oxidation of sulfides and the principle of solvent extraction,the kinetic equation of desulfurization via oxidative extraction were determined. By means of the evaluation of model parameters and curve fitting,the reaction order between organic sulfide and sulfone,the intrinsic oxidation rate constant of organic sulfide and sulfone,and the equilibrium constant between sulfone in model diesel and extractive solvent were determined. The experimental values of the desulfurization rate and the theoretical values of the corresponding model equation had closely demonstrated that the desulfurization reaction rate had high accuracy. And the reaction kinetics could provide an important basis for diesel desulfurization process in the future.

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.

LESSER KNOWN MIRACLES OF BURGERS EQUATION

This article is a short introduction to the surprising appearance of Burgers equation in some basic probabilistic models.

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.

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）.

Studies on Kinetics of Chlorination Reaction of Polyethylene

Through kinetic method, the reaction mechanism and the rate equations of chlorination of polyethylene are suggested in this paper. The rate of chlorination is second order with respect to the concentration of methylene remained and chlorine, respectively. Apparent changes in crystallinity during chlorination processes were determinated by differential scanning calorimetry(DSC). The mathematical model relating crystallinety with kinetic parameters and function of sequence length distribution are derived.

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.