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.

Adsorption Kinetics of Pb^(2+) and Cu^(2+) on Variable Charge Soils and Minerals: Ⅱ. Equations for Describing Experimental Data

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.

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.

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）.

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.

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.

Comparative Study on the Kinetic Equations of Potassium Release from Soils

Elovicb, two-constant, parabolic diffusion, exponential, second-order, first-order and zero-order equations were used to describe the kinetic characteristics of potassium desorption from six paddy soils df Zhejiang Province in a constant electric field (44.4 V/ cm) of EUF. Results showed that the second-order and Elovich equations could describe the potassium desorption kinetics best, as evidenced by the highest correlation coefficients (r) and the lowest standard errors (SE). The first-order, two-constant and parabolic diffusion equations also described the K desorption kinetics well, as showed by the relatively high correlation coefficients and relatively low standard errors. The zero-order equation did not describe the K desorption satisfactorily with a relatively low correlation coefficient and relatively high standard error. However, the exponential equation could not be used to describe the K desorption kinetics, due to the lowest correlation coefficient and the highest standard error.

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.

Limitation of the Johnson-Mehl-Avrami equation for the kinetic analysis of crystallization in a Ti-based amorphous alloy

The primary crystallization of the Ti40Zr25NisCu9Bc18 amorphous alloy was studied by isochronal differential scanning calorimetry （DSC）. The activation energy was determined by the Kissinger-Akahim-Sunose method. Trying to analyze the crystallization kinetics of the Ti40Zr25NigCu9Be18 amorphous alloy by two different methods, it was found that the crystallization kinetics did not obey the Johnson-Mehl-Avrami equation. A modified method in consideration of the impingement effect was proposed to perform kinetic analysis of the isochronal crystallization of this alloy. The kinetic parameters were then obtained by the linear fitting method based on the modified kinetic equation. The results show that the isochronal crystallization kinetics of the amorphous Ti40Zr25Ni8CugBe18 alloy is heating rate dependent, and the discrepancy between the Johnson-Mehl-Avrami method and the modified method increases with the increase of heating rate.

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.

KINETIC MODELING OF ESTERIFICATION OF EPOXY RESIN IN THE PRESENCE OF TRIPHENYLPHOSPHINE FOR PRODUCING VINYL ESTER RESIN: MECHANISTIC RATE EQUATION

Due to its mechanical properties and ease of use, vinyl ester resin is enjoying increasing consideration. This resin normally is produced by reaction between epoxy resin and unsaturated carboxylic acid. In the present study, bis-phenol A based epoxy resin and methacrylic acid was used to produce vinyl ester resin. The reaction was conducted under both stoichiometric and non-stoichiometric conditions in the presence of triphenylphosphine as catalyst. The stoichiometric and non-stoichiometric experiments were conducted at 95, 100, 105 and 110℃ and at 90 and 95℃, respectively. The first order rate equation and mechanism based rate equation were examined. Parameters are evaluated by least square method. A comparison of mechanism based rate equation and experimental data show an excellent agreement. Finally, Arrhenius equation and activation energy were presented.

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.

Higher-order approximate solutions of fractional stochastic point kinetics equations in nuclear reactor dynamics

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.

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.

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.

Enzyme Kinetic Equations of Irreversible and Reversible Reactions in Metabolism

This paper compares the irreversible and reversible rate equations from several uni-uni kinetic mechanisms (Michaelis-Menten, Hill and Adair equations) and bi-bi mechanisms (single- and double- displacement equations). In reversible reactions, Haldane relationship is considered to be identical for all mechanisms considered and reversible equations can be also obtained from this rela- tionship. Some reversible reactions of the metabolism are also presented, with their equilibrium constant.

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.

Analytical solution of modified point kinetics equations for linear reactivity variation in subcritical nuclear reactors adopting an incomplete gamma function approximation

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.

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.