The role of diffusion in propylene polymerization

The role of monomer diffusion in the polymerization of propylene by organometallic catalysis was examined by use of mathematical models which couple the rate of diffusion through the polymer film surrounding the catalyst with the rate of surface reaction. An approximate form of a second-order, integrated rate equation was used to describe the disappearance of active sites on the surface. For the most conservative model conceivable, it was estimated that the particle size would have to be 10–100 times the size for the catalysts presently in use before diffusion time would be significant. The size of the catalysts was determined by photomicrographs and nitrogen adsorption surface areas. The surface areas for three different catalysts were 7, 20–21 and 35 m.²/g., respectively. The kinetic model without the diffusion term was used satisfactorily to correlate productivity data. The characteristic decline in reaction rate was examined in terms of the decay of active sites on the surface of the catalyst. The rate of decay was determined to be second order with respect to the site concentration. The kinetic model indicates that the total polymerization time for a specified productivity is the sum of the monomer diffusion time and the surface reaction time. The model derived by use of an approximate second-order decay function is unique because of the additivity of diffusion and reaction times, which is not the case when the second-order function is used rigorously.     

A Monte-Carlo simulation of diffusion and reaction in zeolites

Investigations of two-dimensional models to simulate diffusion and reaction in zeolites with a Monte-Carlo method are presented herein. New results of the simulation of single component diffusion and binary diffusion and a possible method to rescale the obtained diffusion coefficients in real units is presented. The estimation of an accurate activation energy was found to be the most important factor, the other parameters were estimated by simple assumptions.Complex reactions such as a consecutive reaction and xylene isomerisation were simulated with a new model. With this model the selective behaviour of the two reactions was investigated. A Type-III selectivity according to Wheeler was found for the consecutive reaction. For xylene isomerisation, reaction paths were simulated and compared with experimental reaction paths. The composition of the thermodynamic equilibrium was obtained for high transition probabilities and the relative rate constants from the literature could be confirmed. According to the results of the simulations one can conclude that a ratio of the diffusion coefficients of o-, m-, and p-xylene=1:1:1000 is too large to explain the experimentally observed distribution of the xylene isomers and a difference of only one order of magnitude was found.     

焦炭溶损反应动力学及其模型研究

利用未反应核收缩模型对高炉焦炭与CO2的反应动力学进行了研究，建立了以可测参数（R）表达的焦炭与CO2的反应动力学关系式。并对反应速率常数和有效扩散系数、表观反应活化能和有效扩散活化能及反应过程中各步骤阻力进行了分析。结果表明，（1）焦炭与CO2的反应符合未反应核收缩模型。（2）反应的表观活化能Ea=124.5kJ/mol，有效扩散活化能ED=642.4 kJ/mol；界面化学反应的阻力随反应温度升高而增加；残余灰层内的内扩散传质阻力相对比例随温度升高而下降。（3）焦炭溶损反应在低温区主要受内扩散控制，随着温度升高，反应由外扩散、化学反应和内扩散三步控制；当进入高温区，反应进行一段时间后主要受内扩散控制。     

A kinetic study of the oxidation of uranium dioxide

Summary A kinetic model for the oxidation of a UO₂ pellet is suggested by considering the mass transfer and the diffusion of oxygen molecules. The kinetic parameters were estimated by a fitting of the experimental data. The activation energies for the chemical reaction and the product layer diffusion were calculated from the kinetic model. The suggested model explains well the oxidation behavior of UO₂.     

Diffusion and Reaction Rates of the Yttrium Aluminium Garnet Synthesis using Different Techniques

The mathematical model of the yttrium aluminium garnet synthesis presented in this article. The model based on a system of non-stationary diffusion equations containing a non-linear term related to kinetics of reaction. Using computer-simulation tools and known experimental results we estimated the diffusion and reaction rates of the synthesis. Also it was shown that diffusion rate is a limited stage of the synthesis.     

Modeling of nonlinear boundary value problems in enzyme-catalyzed reaction diffusion processes

A mathematical model of steady state mono-layer potentiometric biosensor is developed. The model is based on non stationary diffusion equations containing a non linear term related to Michaelis-Menten kinetics of the enzymatic reaction. This paper presents a complex numerical method (He’s variational iteration method) to solve the non-linear differential equations that describe the diffusion coupled with a Michaelis-Menten kinetics law. Approximate analytical expressions for substrate concentration and corresponding current response have been derived for all values of saturation parameter α and reaction diffusion parameter K using variational iteration method. These results are compared with available limiting case results and are found to be in good agreement. The obtained results are valid for the whole solution domain.     

氧化锌颗粒脱硫中固体扩散的动力学分析

用电子探针显微分析 (EPMA)对ZnO基脱硫剂的脱硫颗粒径向硫分布行为进行了考查 ,对颗粒最外层粒子的实验结果分别用未反应收缩核模型及改良收缩核模型进行了动力学处理 ,给出了固体扩散活化能的参数估值。通过与热重研究的比较 ,进一步证实了固体粒子扩散的作用。     

Effect of micropores on the effective diffusion coefficient

The effective diffusion coefficient for catalysts differing in their porous structure has been derived from experimental data on H₂S conversion in the Claus reaction. The effective diffusion coefficient increases under conditions of catalyst deactivation due to sulfur condensation in micropores. A mathematical model is suggested to describe the micropore effect on the effective diffusion coefficient.     

O2和CO表面催化反应的活性位分布模型

对于O_2和CO表面催化反应，建立了一个新的不可逆Monte Carlo模拟模型。在 二维格子中，引进了表面活性位和非活性位的要领。模型假设，一定浓度的活性位 随机分布在非活性位上，形成了活性位分布的二维格子模型反应器，并在ZGB模型 的基础上，考虑了氧原子和CO分子的表面扩散，特别是引进了吸附粒子的定向表面 扩散。其中，活性位和活性位最近邻是表面吸附物质反应的活性中心，而非活性位 的作用是通过表面扩散传质。当活性位浓度C_a = 1且考虑扩散时，模型还原为增 加了扩散的ZGB模型。当活性位浓度C_a = 1且只考虑氧的扩散时，模拟结果表明， 扩散几率达到某一数值（0.3）时，二级相变点完全消失。当活性位浓度C_a逐渐减 小时，单位活性位产生的CO_2的速率不断增大，这表明活性位的利用率提高了。     

Influence of Ag addition on the thermal behavior of the Cu-11 mass% Al alloy

In this work the influence of Ag additions on the thermal behavior of the Cu-11 mass% Al alloy was studied using differential scanning calorimetry, in situ X-ray diffractometry and scanning electron microscopy. The results indicated that changes in the heating rate shift the peak attributed to α phase formation to higher temperatures, evidencing the diffusive character of this reaction. The activation energy value for the α phase formation reaction, obtained from a non-isotherm kinetic model, is close to that corresponding to Cu atoms self diffusion, thus confirming that this reaction is dominated by Cu atoms diffusion through the martensite matrix.     

A model for the propagation of a redox reaction through microcrystals

Chronoamperometry of reversible redox reactions with the insertion of cations into solid particles immobilised at an electrode surface is analysed theoretically using a semiinfinite planar diffusion model. A coupled diffusion of electrons and ions within the crystal lattice is separated in two differential equations. The redox reaction is initiated by the polarisation of the three-phase boundary, where the crystal is in contact with both the electrode and the solution. From this contact line the redox reaction advances on the surface and into the crystal body by the diffusion of ions and conductance of electrons. The effects of the geometry and conductivity of the particles on the current are discussed. Received: 28 December 1996 / Accepted: 17 February 1997     

Spillover in monomer–monomer reactions on supported catalysts: dynamic mean-field study

The kinetics of a $A_1+A_2\rightarrow A_1A_2$ reaction on supported catalysts is investigated numerically using a phenomenological model which includes: the bulk diffusion of reactants from a bounded vessel towards the adsorbent and the product bulk one into the same vessel, adsorption and desorption of reactants molecules, and surface diffusion of adsorbed particles. The model is based on the Langmuir–Hinshelwood surface reaction mechanism coupled with the Eley–Rideal step. The model based only on the Langmuir–Hinshelwood mechanism is also studied. Simulations were performed using the finite difference technique. Three cases of reactants adsorption are considered: each reactant can adsorb on the active in reaction catalyst surface and inactive support, one of reactants adsorbs on the catalyst surface while the other one adsorbs on the support, both reactants adsorb only on the support. The surface diffusion and catalytic surface size influence on the catalytic reactivity of a supported catalyst is studied.     

Permeation of CO2 and N2 through glassy poly(dimethyl phenylene) oxide under steady- and presteady-state conditions

Glassy polymers are often used for gas separations because of their high selectivity. Although the dual-mode permeation model correctly fits their sorption and permeation isotherms, its physical interpretation is disputed, and it does not describe permeation far from steady state, a condition expected when separations involve intermittent renewable energy sources. To develop a more comprehensive permeation model, we combine experiment, molecular dynamics, and multiscale reaction–diffusion modeling to characterize the time-dependent permeation of N₂ and CO₂ through a glassy poly(dimethyl phenylene oxide) membrane, a model system. Simulations of experimental time-dependent permeation data for both gases in the presteady-state and steady-state regimes show that both single- and dual-mode reaction–diffusion models reproduce the experimental observations, and that sorbed gas concentrations lag the external pressure rise. The results point to environment-sensitive diffusion coefficients as a vital characteristic of transport in glassy polymers.     

Decoupling Study of Melt Polycondensation of Polycarbonate: Experimental and Modeling of Reaction Kinetics and Mass Transfer in Thin Films Polycondensation Process

During the melt polycondensation process of polycarbonate, reaction and mass transfer are deeply coupled owing to relatively high melt viscosity. In this work, the polycondensation reaction kinetics and mass transfer behavior of volatile phenol are decoupling studied in detail by using thin‐film experiments with 250–280 °C, 10–1000 Pa and 0.085–0.68 mm film thickness. A realistic apparent rate model coupled the reaction kinetics with thermodynamic equilibrium and diffusion behavior is developed to describe the polycondensation process, while the diffusion characteristic of small molecule (phenol) is further obtained based on penetration theory. The obtained polycondensation equilibrium constant ranges from 0.3 to 0.55, while the activation energy and pre‐exponential factor of temperature‐dependent diffusion coefficients of phenol are 87.9 kJ mol⁻¹ and 5.08 × 10² m² s⁻¹, respectively. It is also observed that the overall apparent rate of polycarbonate (PC) polycondensation process increases with higher temperature, lower pressure, and thinner film thickness. Coupling the reaction kinetics with mass transfer, the predictions of the realistic apparent rate model are in quite satisfactory agreement with experimental data.     

Fragmentation of diffusion zone in high-temperature oxidation of copper

Using thermogravimetry, microscopy and X-ray diffraction, high-temperature (600-900 °C) oxidation of copper wires and plates has been studied. An abrupt decrease in reaction rate after complete consumption of metal phase but long before reaching equilibrium has been observed. This phenomenon is connected to an irregular character of the development of the reaction diffusion zone. In contrast to the usually applied layer model, initially formed oxide layers separate into numerous aggregates of Cu₂O crystals chaotically scattered throughout the zone between thinner layers of CuO grains. Such fragmentation of the diffusion zone is induced by macro- and microcracks formed in copper scale under influence of mechanical stresses at metal-oxide phase boundary due to the difference in molar volume between copper and its oxides. The pattern of cracks provides channels of fast diffusion and maintains the reaction rate at high level but only until the source of crack formation remains in action.     

Modeling of electrocatalytic processes at conducting polymer modified electrodes

A mathematical model of electrocatalytic processes taking place at conducting polymer modified electrodes has been developed. The model takes into account the diffusion of solution species into a polymer film, diffusion of charge carriers within the film, and a chemical redox reaction within the film. The space- and time-resolved profiles for reactant and charge carrier concentration within the film, as well as dependencies of electric current on the concentration of solute species, reaction rate constant and thickness of a polymer layer have been obtained and discussed. It has been shown that, even at a relatively fast diffusion of charge carriers within the conducting polymer film, exceeding the diffusion rate of reactant by two orders of magnitude, electrocatalysis of solute species at conducting polymer modified electrodes proceeds within the polymer film rather than at the outer polymer/solution interface, i.e., electrocatalytic conversion follows a redox-mechanism rather than metal-like one. Based on the results obtained, optimization of reaction system parameters could be made for any particular case to get an optimum efficiency or reactant to product conversion.      

Modeling Investigation of Volume Variation Kinetics of Fast Response Hydrogels

Abstract

Fast response hydrogels have attracted great attention recently. Three kinds of models with different complexity are available in the literature to investigate their volume variation kinetics. A phenomenal model based on second‐order reaction kinetics provides a simple method to correlate the experimental kinetic data. A power law model can be employed to evaluate different volume variation mechanisms according to the diffusion exponential coefficient. The multi‐field model incorporates mechanical deformation with the diffusion of ion species. The volume variation rate is dependent on both hydrogel molecular design and operation conditions.     

扩散控制的固相反应动力学模型

在许多由固相反应生成合金和无机材料的生产过程中,扩散传质控制着反应的过程。但是,对扩散控制固相反应系统的机理和动力学研究还很不成熟。Jander首先把平面扩散抛物线方程用于球体内部的扩散过程,导出了Jander方程,由于平面扩散截面积不变和球体扩     

Sorption kinetics for the removal of aldehydes from aqueous streams with extractant impregnated resins

The sorption kinetics for the removal aldehydes from aqueous solutions with Amberlite XAD-16 and MPP particles impregnated with Primene JM-T was investigated. A model, accounting for the simultaneous mass transfer and chemical reaction, is developed to describe the process. It is based on the analogy to the diffusion and reaction in a stagnant liquid sphere, but corrected for the porosity and particle properties influencing the diffusion. The developed model describes the kinetic behavior of the process in the low concentration region rather well. However, in the high concentration region, larger discrepancies are observed. Initially, the influence of the flow rate was investigated to eliminate the effect of the external mass transfer. The influence of the particle morphology was investigated for both physical and reactive sorption. Physical sorption experiments were used to determine the factor τ that takes the particle properties influencing the diffusion into account. It was shown that the diffusion is faster in XAD-16 than in MPP impregnated systems. Reaction rate constant k _x was determined by fitting the model to the experimental data. Sorption of benzaldehyde appears to be significantly slower (k _x ∼10⁻⁴ l/mol s) than the sorption of pentanal (k _x ∼10⁻³ l/mol s) due to the slower chemical reaction. The influence of the particle size was investigated for the sorption of pentanal with XAD-16. It was observed that the particle size does influence the diffusion term, but does not have an effect on the reaction rate. On the other hand, the extractant loading influences the reaction rate slightly in the low concentration region, whereas the initial concentration of the solute has more pronounced effect.     

The Modulation of Desulphurization Properties of Calcium Oxide by Alkali Carbonates

Kinetics of CaO desulphurization reaction and the effects of alkali carbonates on it have been investigated by thermogravimetric analysis. A grain model was applied successfully to describe the kinetic behavior of the reactions. The activation energy of surface reaction and that of the product layer diffusion were determined by using the model. It was found that the overall desulphurization rate was controlled initially by surface chemical reaction and, in a later stage, by product layer diffusion. Addition of alkali carbonates can decrease the activation energy of the surface chemical reaction, with increasing effectiveness in the order of potassium, sodium and lithium. Such a property of alkali carbonates has also been demonstrated on a raw coal. The process is discussed in terms of a working mechanism of solid-state ionic diffusion. This revised version was published online in August 2006 with corrections to the Cover Date.