Different mathematical methods, including linearization, differential, integration and nonlinear least squares approximation (Newton-Marquardt method), were used to fit different kinetic equations, such as zero-order,...Different mathematical methods, including linearization, differential, integration and nonlinear least squares approximation (Newton-Marquardt method), were used to fit different kinetic equations, such as zero-order, first-order (i. e, membrane diffusion), second-order, parabolic-diffusion, Elovich, two-constant equations, to the experimental data of Pb2+ and Cu2+ adsorption on variable charge soils and kaolinite. Assuming each M2+ occupied two adsorption sites, two more equations, the so-called surface second-order equation and third-order equation were derived and compared with the above equations according to the fitting results, which showed that the second-order equation and surface second-order equation, being one equation in different expressions under some conditions, were better than the other equations in describing the Pb2+ and Cu2+ adsorption kinetics, and the latter was the best.展开更多
Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been anal...Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.展开更多
The present work aims to achieve a fast and accurate analytical solution of the point kinetics equations applied to subcritical reactors such as ADS (Accelerator-Driven System), assuming a linear reactivity and extern...The present work aims to achieve a fast and accurate analytical solution of the point kinetics equations applied to subcritical reactors such as ADS (Accelerator-Driven System), assuming a linear reactivity and external source variation. It was used a new set of point kinetics equations for subcritical systems based on the model proposed by Gandini & Salvatores. In this work it was employed the integrating factor method. The analytical solution for the case of interest was obtained by using only an approximation which consists of disregarding the term of the second derivative for neutron density in relation to time when compared with the other terms of the equation. And also, it is proposed an approximation for the upper incomplete gamma function found in the solution in order to make the computational processing faster. In addition, for purposes of validation and comparison a numerical solution was obtained by the finite differences method. Finally, it can be concluded that the obtained solution is accurate and has fast numerical processing time, especially when compared with the results of numerical solution by finite difference. One can also observe that the gamma approximation used achieve a high accuracy for the usual parameters. Thus we got satisfactory results when the solution is applied to practical situations, such as a reactor startup.展开更多
This study, based on the physico\|chemistry principle, establishes the mathematics model of the genesis of plagioclase oscillatory zoning. Based on the crystal growth principle, the plagioclase crystallization velocit...This study, based on the physico\|chemistry principle, establishes the mathematics model of the genesis of plagioclase oscillatory zoning. Based on the crystal growth principle, the plagioclase crystallization velocity equation has been put forward. This equation has such a feedback character as \!supersaturation—nucleation—depletion cycle". Small\|scale oscillatory zoning is attributable to self\|organization, and has no bearing on environmental factors. Its genesis is corresponding to a relatively static environment.展开更多
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 ...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.展开更多
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 st...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.展开更多
Isotope eff ects are pivotal in understanding silicate melt evaporation and planetary accretion processes.Based on the Hertz-Knudsen equation,the current theory often fails to predict observed isotope fractionations o...Isotope eff ects are pivotal in understanding silicate melt evaporation and planetary accretion processes.Based on the Hertz-Knudsen equation,the current theory often fails to predict observed isotope fractionations of laboratory experiments due to its oversimplified assumptions.Here,we point out that the Hertz-Knudsen-equation-based theory is incomplete for silicate melt evaporation cases and can only be used for situations where the vaporized species is identical to the one in the melt.We propose a new model designed for silicate melt evaporation under vacuum.Our model considers multiple steps including mass transfer,chemical reaction,and nucleation.Our derivations reveal a kinetic isotopic fractionation factor(KIFF orα)αour model=[m(^(1)species)/m(^(2)species)]^(0.5),where m(species)is the mass of the reactant of reaction/nucleation-limiting step or species of diffusion-limiting step and superscript 1 and 2 represent light and heavy isotopes,respectively.This model can eff ectively reproduce most reported KIFFs of laboratory experiments for various elements,i.e.,Mg,Si,K,Rb,Fe,Ca,and Ti.And,the KIFF-mixing model referring that an overall rate of evaporation can be determined by two steps jointly can account for the eff ects of low P_(H2)pressure,composition,and temperature.In addition,we find that chemical reactions,diffusion,and nucleation can control the overall rate of evaporation of silicate melts by using the fitting slope in ln(−ln f)versus ln(t).Notably,our model allows for the theoretical calculations of parameters like activation energy(E_(a)),providing a novel approach to studying compositional and environmental eff ects on evaporation processes,and shedding light on the formation and evolution of the proto-solar and Earth-Moon systems.展开更多
The distribution of continuous service time in call centers is investigated.A non-Maxwellian collision kernel combining two different value functions in the interaction rule are used to describe the evolution of conti...The distribution of continuous service time in call centers is investigated.A non-Maxwellian collision kernel combining two different value functions in the interaction rule are used to describe the evolution of continuous service time,respectively.Using the statistical mechanical and asymptotic limit methods,Fokker–Planck equations are derived from the corresponding Boltzmann-type equations with non-Maxwellian collision kernels.The steady-state solutions of the Fokker–Planck equation are obtained in exact form.Numerical experiments are provided to support our results under different parameters.展开更多
基金Project supported by the N ational Natural Science Foundation of China.
文摘Different mathematical methods, including linearization, differential, integration and nonlinear least squares approximation (Newton-Marquardt method), were used to fit different kinetic equations, such as zero-order, first-order (i. e, membrane diffusion), second-order, parabolic-diffusion, Elovich, two-constant equations, to the experimental data of Pb2+ and Cu2+ adsorption on variable charge soils and kaolinite. Assuming each M2+ occupied two adsorption sites, two more equations, the so-called surface second-order equation and third-order equation were derived and compared with the above equations according to the fitting results, which showed that the second-order equation and surface second-order equation, being one equation in different expressions under some conditions, were better than the other equations in describing the Pb2+ and Cu2+ adsorption kinetics, and the latter was the best.
文摘Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.
文摘The present work aims to achieve a fast and accurate analytical solution of the point kinetics equations applied to subcritical reactors such as ADS (Accelerator-Driven System), assuming a linear reactivity and external source variation. It was used a new set of point kinetics equations for subcritical systems based on the model proposed by Gandini & Salvatores. In this work it was employed the integrating factor method. The analytical solution for the case of interest was obtained by using only an approximation which consists of disregarding the term of the second derivative for neutron density in relation to time when compared with the other terms of the equation. And also, it is proposed an approximation for the upper incomplete gamma function found in the solution in order to make the computational processing faster. In addition, for purposes of validation and comparison a numerical solution was obtained by the finite differences method. Finally, it can be concluded that the obtained solution is accurate and has fast numerical processing time, especially when compared with the results of numerical solution by finite difference. One can also observe that the gamma approximation used achieve a high accuracy for the usual parameters. Thus we got satisfactory results when the solution is applied to practical situations, such as a reactor startup.
文摘This study, based on the physico\|chemistry principle, establishes the mathematics model of the genesis of plagioclase oscillatory zoning. Based on the crystal growth principle, the plagioclase crystallization velocity equation has been put forward. This equation has such a feedback character as \!supersaturation—nucleation—depletion cycle". Small\|scale oscillatory zoning is attributable to self\|organization, and has no bearing on environmental factors. Its genesis is corresponding to a relatively static environment.
文摘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.
基金Project(2014ZE56015)supported by Aeronautical Science Foundation of ChinaProject(51261020)supported by the National Natural Science Foundation of ChinaProject(Zk201001004)supported by the Open Fund of the Aeronautical Science and Technology Key Laboratory of Aeronautical Material Hot Processing Technology,China
文摘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.
基金supported by Chinese NSF project(42,130,114)the strategic priority research program(B)of CAS(XDB41000000)the pre-research Project on Civil Aerospace Technologies No.D020202 funded by Chinese National Space Administration(CNSA)and Guizhou Provincial 2021 Science and Technology Subsidies(No.GZ2021SIG).
文摘Isotope eff ects are pivotal in understanding silicate melt evaporation and planetary accretion processes.Based on the Hertz-Knudsen equation,the current theory often fails to predict observed isotope fractionations of laboratory experiments due to its oversimplified assumptions.Here,we point out that the Hertz-Knudsen-equation-based theory is incomplete for silicate melt evaporation cases and can only be used for situations where the vaporized species is identical to the one in the melt.We propose a new model designed for silicate melt evaporation under vacuum.Our model considers multiple steps including mass transfer,chemical reaction,and nucleation.Our derivations reveal a kinetic isotopic fractionation factor(KIFF orα)αour model=[m(^(1)species)/m(^(2)species)]^(0.5),where m(species)is the mass of the reactant of reaction/nucleation-limiting step or species of diffusion-limiting step and superscript 1 and 2 represent light and heavy isotopes,respectively.This model can eff ectively reproduce most reported KIFFs of laboratory experiments for various elements,i.e.,Mg,Si,K,Rb,Fe,Ca,and Ti.And,the KIFF-mixing model referring that an overall rate of evaporation can be determined by two steps jointly can account for the eff ects of low P_(H2)pressure,composition,and temperature.In addition,we find that chemical reactions,diffusion,and nucleation can control the overall rate of evaporation of silicate melts by using the fitting slope in ln(−ln f)versus ln(t).Notably,our model allows for the theoretical calculations of parameters like activation energy(E_(a)),providing a novel approach to studying compositional and environmental eff ects on evaporation processes,and shedding light on the formation and evolution of the proto-solar and Earth-Moon systems.
基金the Special Project of Yili Normal University(to improve comprehensive strength of disciplines)(Grant No.22XKZZ18)Yili Normal University Scientific Research Innovation Team Plan Project(Grant No.CXZK2021015)Yili Science and Technology Planning Project(Grant No.YZ2022B036).
文摘The distribution of continuous service time in call centers is investigated.A non-Maxwellian collision kernel combining two different value functions in the interaction rule are used to describe the evolution of continuous service time,respectively.Using the statistical mechanical and asymptotic limit methods,Fokker–Planck equations are derived from the corresponding Boltzmann-type equations with non-Maxwellian collision kernels.The steady-state solutions of the Fokker–Planck equation are obtained in exact form.Numerical experiments are provided to support our results under different parameters.
