Identification of the effective distribution function for determination of the distributed activation energy models using the maximum likelihood method: Isothermal thermogravimetric data

The new procedure for identification of the effective distribution function for determination of the distributed activation energy models, which is based on use the maximum likelihood method (MLM), was established. The five different continuous probability functions (exponential, logistic, normal, gamma, and Weibull probability functions (the extended set of distributions)) were used for searching the best reactivity model for two heterogeneous processes: (a) the isothermal reduction process of nickel oxide under hydrogen atmosphere and (b) the isothermal degradation process of bisphenol‐A polycarbonate (Lexan) under nitrogen atmosphere. The MLM showed that for both processes, the most suitable reactivity model represents the Weibull distribution model. It was concluded that the values of Arrhenius parameters (ln A and E_a), evaluated from the Weibull distribution model, represent the effective kinetic values for both considered processes. This procedure enables identification the suitable distribution model for considered process only from the experimental data (based on the shapes of obtained integral kinetic curves), and this fact represents the advantage of established analysis. The established mathematical procedure, which is based on the MLM, can be applied as the preliminary analysis for evaluating the distribution of activation energies for complex heterogeneous processes. © 2008 Wiley Periodicals, Inc. Int J Chem Kinet 41: 27–44, 2009     

Kinetics of the thermal degradation of poly(lactic acid) obtained by reactive extrusion: Influence of the addition of montmorillonite nanoparticles

A reactive extrusion-calendering process was used in order to manufacture sheets with a nominal thickness of 1 mm of poly(lactic acid) and its nanocomposite with 2.5% of an organo-modified montmorillonite. During processing, the properties of the melt were stabilized and enhanced by the addition of 0.5% of a styrene-acrylic multi-functional-epoxide oligomeric reactive agent. The general analytical equation has been used in order to evaluate the kinetic parameters of the thermal degradation of poly(lactic acid) obtained by reactive extrusion and its nanocomposite. Various empirical and theoretical solid-state mechanisms have been tested to elucidate the best kinetic model. In order to reach this goal, master plots have been constructed by means of standardized conversion functions. Given that it is not always easy to visualize the best accordance between experimental and theoretical values of standardized conversion functions, a new index has been developed to quantitatively discern the best mechanism. By doing that, it has been possible to determine the right activation energy of the thermal degradation. It has been demonstrated that the best theoretical mechanism was the random scission of macromolecular chains within the polymer matrix. This was also in accordance with an empirical kinetic model based on an autocatalytic kinetic model. The presence of montmorillonite nanoparticles has been beneficial and has enhanced the thermal stability of poly(lactic acid).     

Identification of the effective distribution function for determination of the distributed activation energy models using Bayesian statistics: Application of isothermal thermogravimetric data

The new procedure for identification of the effective distribution function for determination of the distributed activation energy models, which is based on the Bayesian statistics, has been established. The five different continuous probability functions (exponential, logistic, normal, gamma and Weibull probability functions (the extended set of distributions)) were used for searching the most appropriate reactivity model for two heterogeneous processes: (a) the isothermal reduction process of nickel oxide under hydrogen atmosphere and (b) the isothermal degradation process of bisphenol‐A polycarbonate (Lexan) under nitrogen atmosphere. Using the Bayes weights, it was shown that for both processes, the most suitable distributed reactivity model is the Weibull distribution model. The kinetic parameters (ln A, E_a) attached with the Weibull distribution model were calculated for both investigated processes, using three different computational methods (the maximum likelihood method (MLM), the nonlinear regression analysis (NRA), and the posterior mean (the expected value of scale parameter η, E(η)). It was shown that there is an excellent agreement between the values of kinetic parameters calculated by the MLM, NRA, and E(η) approaches. Using Bayes weights, it is possible to discriminate between different probability models and to quantify how well a distribution fits the experimental data. For the formal reactivity model comparison, the use of the (nonnormalized) Jeffreys prior is recommended. © 2010 Wiley Periodicals, Inc. Int J Chem Kinet 42: 641–658, 2010     

Application of a single model to study the adsorption equilibrium of prednisolone on six carbonaceous materials

The knowledge of sorption processes of nonelectrolytes in solution by solid adsorbents implies the study of kinetics, equilibrium, and thermodynamic functions. However, quite frequently the equilibrium isotherms are studied by comparing them with those corresponding to the Giles et al. classification (1); these isotherms are also analyzed by fitting them to equations based on thermodynamic or kinetic criteria, and even to empirical equations. Nevertheless, information obtained is more coherent and satisfactory if the adsorption isotherms are fitted by using an equation describing the equilibrium isotherms according to the kinetic laws. These mentioned laws would determine each one of the unitary processes (one or more) which condition the global process. In this paper, an adsorption process of prednisolone in solution by six carbonaceous materials is explained according to a previously proposed single model, which allows to establish a kinetic law which fits satisfactorily most of C vs t isotherms (2). According to the above-mentioned kinetic law, equations describing sorption equilibrium processes have been deducted, and experimental data points have been fitted to these equations; such a fitting yields to different values of adsorption capacity and kinetic equilibrium constants for the different processes at several temperatures. However, in spite of their practical interest, these constants have no thermodynamic signification. Thus, the thermodynamic equilibrium constant (K) has been calculated by using a modified expression of the Gaines et al. equation (3). Global average values of the thermodynamic functions have also been calculated from the K values. Information related to variations of DeltaH and DeltaS with the surface coverage fraction was obtained by using the corresponding Clausius-Clapeyron equations.     

Kinetic analysis of the complex process of poly(vinyl alcohol) pyrolysis using a new coupled peak deconvolution method

A peak deconvolution procedure used for the analysis of data corresponding to simultaneous overlapping processes begins with separation of individual processes using functions such as Gaussian, Lorentzian, Weibull, and Fraser–Suzuki (FS) followed by application of kinetic analysis methods to the separated peaks. We propose a coupled peak deconvolution procedure to link the parameters of the FS functions of similar peaks in two DTG curves obtained at different linear heating rates, so that the coordinates of each peak can be obtained in a constrained manner. The proposed technique is a kinetic deconvolution method rather than a pure mathematical deconvolution technique. To analyze individual peaks in our study, the non-parametric kinetic and Freidman’s isoconversional methods have been applied to determine kinetic triplet of each process. This technique has been tested with both simulated and experimental data. Using this technique, the effects of molecular weight and degree of hydrolysis of polyvinyl alcohol (PVA) samples on reaction mechanism and activation energy of thermal degradation were studied. The presence of acetate group in the PVA samples causes thermal stability, decreases the rate of main reactions, and increases the activation energy. The results of this study may help tailor heat-resistant materials with proper choice of polymer characteristics.     

Kinetic modelling of the reduction of SiCl4 in an arc heater for the production of silicon

The conversion of SiCl₄ into Si has been achieved from reduction by a hydrogen plasma produced in an arc heater. As the results (conversion yield about 60%) are far from chemical equilibrium predictions, a kinetic model is proposed using the few kinetic data available in the literature and a temperature history of the reactants deduced from measurements of the temperatures and velocities of the flowing chemical mixture.     

Calculation of the effect of substitution on the yield and topological parameters of hyperbranched polymers synthesized by the cyclotrimerization of mono- and bifunctional monomers

A kinetic model allowing for the effect of substitution has been developed for the synthesis of hyperbranched polymers through the cocyclotrimerization of mono- and bifunctional monomers. The calculation has been performed by means of a mathematical apparatus for generating functions. New theoretical relationships have been obtained that enable one to predict critical conversion and variations in the structural and molecular-mass parameters of hyperbranched polymers during reaction in relation to the initial monomer ratio, relative reactivities of functional groups, and variations in the reactivities during the process.     

Application of a single model to study the adsorption kinetics of prednisolone on six carbonaceous materials

The knowledge of the adsorption processes of nonelectrolytes from liquid solution on solid materials involves the study of their kinetic and equilibrium aspects as well as the understanding of their thermodynamic functions. However, in most published papers adsorption isotherms are analyzed by using the Giles classification and other proposed equations which are either empirical or based on kinetic or thermodynamic criteria. Our opinion is that both the kinetic and the equilibrium studies must be complementary and that, in general, equations describing the adsorption isotherms come from the kinetic laws governing the different partial processes which determine the global process. These kinetic laws may be derived from single models. In this paper a single model is proposed, which makes it possible to establish a kinetic law satisfactorily fitting a great number of C (concentration) vs t (time) isotherms. This model has been applied to study the adsorption process of prednisolone by six carbonaceous materials from ethanol solution, the specific adsorption rate, and the activation thermodynamic functions being calculated. The results obtained have also been used to analyze the influence of the intraparticle diffusion on the kinetics of the process.     

Electrochemical Impedance Analysis of Thermogalvanic Cells

Thermogalvanic cells(also known as thermo-electrochemical cells) that convert waste heat energy to electricity are a new type of energy conversion device. However, the electron transfer kinetics and mass transfer of redox couples have not been thoroughly studied. Here, the ion reaction and charge transport in thermogalvanic cells are investigated by electrochemical impedance analysis. We first propose the detailed impedance model followed experimental verification on three types of electrode materials. Parameters including kinetic rate constants and ion diffusion coefficients for the electrodes are obtained by fitting the impedance data. Our study shows explicitly that impedance analysis can provide useful information on selecting suitable electrode materials for thermogalvanic cells.     

Cobalt Ferrite from Citrate Precursor by Self-Propagating Combustion Reaction

A Mangelsdorf's approach to modeling the epoxy-amine cure kinetics has been developed. Analysis of the data by means of Mangelsdorf's approach makes it possible not only to determine the reaction rate constant and the heat of epoxy ring opening, but also to elucidate the reaction mechanism. However, to model the kinetic curves obtained by the calorimetric method for the complicated reaction should be derived an equation expressing the rate of change of the heat with time, as a function of the reaction rate and the extent of conversion. In a detailed examination the thermokinetic data, we found that glassy state transition is kinetically feasible. Using data available in literature, the kinetic model for epoxy-amine cure reaction was developed. Our treatment of glass formation is based on the picture of the reaction system as a miscible mixture of two structurally different liquids. This approach is similar to that presented by Bendler and Shlesinger as a Two-Fluid model. In the application of this model to reaction kinetics, we believe the explanation of glass structure formation lies in the splitting of the homogeneous mixture into two liquid phases. This revised version was published online in August 2006 with corrections to the Cover Date.     

Kinetic method by using calorimetry to mechanism of epoxy-amine cure reaction

A Mangelsdorf's approach to modeling the epoxy-amine cure kinetics has been developed. Analysis of the data by means of Mangelsdorf's approach makes it possible not only to determine the reaction rate constant and the heat of epoxy ring opening, but also to elucidate the reaction mechanism. However, to model the kinetic curves obtained by the calorimetric method for the complicated reaction should be derived an equation expressing the rate of change of the heat with time, as a function of the reaction rate and the extent of conversion. In a detailed examination the thermokinetic data, we found that glassy state transition is kinetically feasible. Using data available in literature, the kinetic model for epoxy-amine cure reaction was developed. Our treatment of glass formation is based on the picture of the reaction system as a miscible mixture of two structurally different liquids. This approach is similar to that presented by Bendler and Shlesinger as a Two-Fluid model. In the application of this model to reaction kinetics, we believe the explanation of glass structure formation lies in the splitting of the homogeneous mixture into two liquid phases.     

Isothermal and non-isothermal pyrolysis kinetics of Kapton polyimide

The kinetics involved in the thermal decomposition of Kapton^® polyimide 100HN under nitrogen atmosphere were studied by applying various fitting techniques to the isothermal and non-isothermal gravimetric data. The correlation of the reaction mechanism fitting, the analytical model fitting and the isoconversional method to these data was examined in relation to the kinetic parameters and the kinetic predictions. The mechanisms for solid-state reactions fit the isothermal data very well but result in highly uncertain values for the kinetic parameters when applied to the non-isothermal data. Isoconversional methods show that the apparent activation energy depends on the extent of conversion but do not provide information for the reaction order and the pre-exponential factor. Three single heating-rate analytical models by Coats-Redfern, MacCallum-Tanner and van Krevelen were analysed using the non-isothermal data. A multi-heating rate model is proposed and its validity is compared to the single-heating rate models on the basis of kinetic predictions.     

Comparison of global models of sub-bituminous coal devolatilization by means of thermogravimetric analysis

Coal gasification and combustion are strongly dependent on devolatilization step. Aim of this work is to obtain the parameters of global kinetics of devolatilization of a sub-bituminous coal with high sulfur content. The kinetic parameters are obtained by means of TG experimental data, and applying different approaches to extrapolate the data to industrial relevant conditions. The simpler method is a model-free one which supposes a single step process whose Arrhenius kinetic parameters (A and E _a) have to be determined. Another common approach is the distributed activation energy model (DAEM) which assumes a series of first order parallel reactions occurring and sharing the same pre-exponential factor, A, with a continuous distribution of the activation energy. For the fitting of the experimental data, a numerical solution to DAEM and two approximate methods have been evaluated. Moreover, the results of these kinetic methods based on empirical approaches were compared with simulated data obtained using a more complex model based on percolation theory with cross-linking mechanism and vapor–liquid equilibrium (chemical percolation devolatilization, CPD model), which allows to simulate the coal pyrolysis from volatile yield data.     

Influence of composition of reaction mixture on selectivity in oxidation of aromatic compounds on oxide catalysts

A general outline is given of a kinetic model of oxidation of a hydrocarbon under the conditions of coexistence on the catalyst surface of sections of different oxidation levels. An analytical dependence has been obtained of the selectivity of the process and conversion on the composition of the reaction mixture. A qualitative agreement has been established between the theoretical and experimental dependences of selectivity and conversion on the ratio of the benzene and oxygen concentrations in the reaction mixture.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 1, pp. 105–108, January–February, 1985.     

A statistical approach to characterize the viscoelastic creep compliances of a vinyl ester polymer

The objective of this study was to develop a model to predict the viscoelastic material functions of a vinyl ester (VE) polymer with variations in its experimentally obtained material properties under combined isothermal and mechanical loading. Short-term tensile creep experiments were conducted at three temperatures below the glass transition temperature of the VE polymer, with 10 replicates for each test configuration. The measured creep strain versus time responses were used to determine the creep compliances using the generalized viscoelastic constitutive equation with a Prony series representation. The variation in the creep compliances of a VE polymer was described by formulating the probability density functions (PDFs) and the corresponding cumulative distribution functions (CDFs) of the creep compliances using a two-parameter Weibull distribution. Both Weibull scale and shape parameters of the creep compliance distributions were shown to be time and temperature dependent. Two-dimensional quadratic Lagrange interpolation functions were used to characterize the Weibull parameters to obtain the PDFs and, subsequently, the CDFs of the creep compliances for the complete design temperature range during steady state creep. At each test temperature, creep compliance curves were obtained for constant CDF values and compared with the experimental data. The predicted creep compliances of the selected VE polymer in the design space are in good agreement with the experimental data for all three test temperatures.     

Modeling the oxidative coupling of methane： Heterogeneous chemistry coupled with 3D flow field simulation

The oxidative coupling of methane (OCM) over titanate perovskite catalyst has been developed by three-dimensional numerical simulations of flow field coupled with heat transfer as well as heterogeneous kinetic model. The reaction was assumed to take place both in the gas phase and on the catalytic surface. Kinetic rate constants were experimentally obtained using a ten step kinetic model. The simulation results agree quite well with the data of OCM experiments, which were used to investigate the effect of temperature on the selectivity and conversion obtained in the methane oxidative coupling process. The conversion of methane linearly increased with temperature and the selectivity of C2 was practically constant in the temperature range of 973–1073 K. The study shows that CFD tools make it possible to implement the heterogeneous kinetic model even for high exothermic reaction such as OCM.     

Evaluating the logistic mixture model on real and simulated TG curves

The aim of this paper is to evaluate and explain the fitting of dynamic TG curves by a mixture of logistic functions. This model assumes that more than one physical process may be involved in each mass loss step and that each physical process may extend along all the experiment. One of the main sources of difficulties in TG is that, very often, different stages of decomposition substantially overlap each other. Several real and simulated TG curves were analysed in this paper. An optimal fitting of the TG curves was obtained by a mixture of logistics. In many cases the optimal fitting reproduces accurately the TG curve. Accordingly, the TG curve can be understood as a sum of parallel reactions, where each single reaction is represented by one or a small number of logistic components. Additionally, making use of the analytical derivative of the fitting, a mixture of Arrhenius reaction order equations was applied to the same curves. In all the cases, the fitting obtained with the mixture of Arrhenius was worse than the obtained with the mixture of logistics. A software was developed to automatically perform these tasks. The physical meaning of the fitting was explained.     

Curing degree prediction for S-TBBS-DPG natural rubber by means of a simple numerical model accounting for reversion and linear interaction

The paper presents a simple numerical model able to provide directly kinetic constants and reliable numerical rheometer curves for S-TBBS-DPG natural rubber. The approach is suitable to calculate the kinetic constants and maximum torque (MH) at any S-TBBS-DPG concentration, following a 3D mathematical interpolation/extrapolation procedure, when kinetic constants on few grid points of S-TBBS-DPG concentrations are available. In particular, the possibility to estimate with sufficient accuracy the behavior of natural rubber at any intermediate concentration of S-TBBS-DPG having engineering relevance has been proved, calibrating the model by means of simple closed form standard best fitting on few experimental data. The model used is a three kinetic parameters one, derived from the well known Han's and co-workers approach, where constants have been evaluated normalizing experimental rheometers curves following the commonly accepted Sun and Isayev method. The procedure has been validated against experimentally obtained rheometer curves by means of inverse analysis, exhibiting excellent prediction capabilities. The approach may be used for practical purposes in order to avoid expensive and cumbersome experimental investigations.     

Comparative study of kinetic modeling for the oxidative coupling of methane by genetic and marquardt algorithms

Overall kinetic studies on the oxidative coupling of methane, OCM, have been conducted in a tubular fixed bed reactor, using perovskite titanate as the reaction catalyst. The appropriate operating conditions were found to be: temperature 750-775 ℃, total feed flow rate of 160 ml/min, CH4/O2 ratio of 2 and GHSV of 100 min-1. Under these conditions, C2 yield of 28% was achieved. Correlations of the kinetic data have been performed with lumped rate equations for C2 and COx formation as functions of temperature, O2 and CH4 partial pressures. Six models have been selected among the common lumped kinetic models. The selected models have been regressed with the experimental data which were obtained from the Catatest system by genetic algorithm in order to obtain optimized parameters. The kinetic coefficients in the overall reactions were optimized by different numerical optimization methods such as: the Levenberg-Marquardt and genetic algorithms and the results were compared with one another. It has been found that the Santamaria model is in good agreement with the experimental data. The Arrhenius parameters of this model have been obtained by linear regression. It should be noted that the Marquardt algorithm is sensitive to the first guesses and there is possibility to trap in the relative minimum.     

Modeling the kinetics of step homopolymerization reactions--an assessment of different approaches

A critical analysis of two models used to describe the kinetics of step homopolymerization has been undertaken. The classical second-order kinetic model and the more widely adopted Mathew et al. oligomer precipitation model were tested against data published in the literature. The classical model, which predicts an identical molecular weight distribution to that obtained by Flory using a statistical approach, is based on a single rate constant and provided an excellent fit (R²>0.99) to the experimental data. Derived rate constants exhibited logical trends. The Mathew et al. oligomer precipitation model, on the other hand, requires three fitting parameters. This model also fitted the data well but in many cases yielded either negative rate constants for the purported termination step or illogical parametric trends. It was concluded that the classical model was superior to the precipitation model in describing the kinetics and reaction mechanisms of step homopolymerization.