Some aspects of mathematical statistics as applied to nonisothermal kinetics

Restrictions of the simplest use of correlation and regression analysis to obtain a single-valued solution to the inverse kinetic problem are considered. The Coats-Redfern method is suggested as a version of nonlinear regression analysis to increase the unambiguity of the solution.

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Zusammenfassung Die Beschränkungen der einfachsten Anwendung der Korrelations- und Regressionsanalyse zur eindeutigen Lösung des inversen kinetischen Problems werden erörtert. Die Coats-Redfern-Methode wird als eine Version der nichtlinearen Regressionsanalyse zur Erhöhung der Eindeutigkeit der Lösung vorgeschlagen.

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Some aspects of mathematical statistics as applied to nonisothermal kinetics

The suggested unified approach is based on the consideration of the entire set of kinetic functions as some space. The analysis of this space provides four classes of kinetic functions, characterized by similar properties and close values of kinetic parameters. The solution of the inverse kinetic problem within the unified approach reduces to finding unambiguously the class of kinetic functions.

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Zusammenfassung Die vorgeschlagene vereinheitlichte NÄherung geht davon aus, da\ das gesamte System der kinetischen Gleichungen als Raum angesehen wird. Die Analyse dieses Raumes ergibt vier Klassen von kinetischen Funktionen, die durch Ähnliche Eigenschaften und geschlossene Werte der kinetischen Parameter charakterisiert sind. Die Lösung des inversen kinetischen Problems reduziert sich in der vereinheitlichten NÄherung auf das eindeutige Auffinden der Klasse der kinetischen Funktionen.

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Some aspects of mathematical statistics as applied to nonisothermal kinetics V

The paper gives a quantitative comparison of two methodological approaches to the solution of the inverse kinetic problem: the traditional approach and the nontraditional approach suggested by the authors. It is shown that the amount of information (in the sense of Shannon) obtained within the scope of the nontraditional approach is always greater than that obtained with the use of the traditional approach.

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Zusammenfassung Zwei methodologische Näherungen der Lösung des inversen kinetischen Problems werden quantitativ verglichen, nämlich die traditionelle und die von den Autoren vorgeschlagene nicht-traditionelle Näherung. Es wird gezeigt, daß mit der nicht-traditionellen Näherung erhaltene (im Sinne von Shannon verstandene) Informationsmenge immer größer als die durch Anwending der traditionellen Näherung erhaltene ist.

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Some aspects of mathematical statistics as applied to non-isothermal kinetics

As a numerical characteristic of the sensitivity of the kinetic parameter calculation method to the form of the kinetic function in non-isothermal kinetics, the value of the curvature of the line plotting residual dispersionvs. formal reaction order at the minimum point is suggested. The efficiency of this characteristic is exemplified.     

Some mathematical aspects of nonisothermal kinetics

The results of an attempt to derive correct nonisothermal kinetic equations from isothermal ones through the classical nonisothermal change (CNC) of the postulated primary kinetic equations are presented. An alternative possibility through use of the model of infinitesimal isothermal portions (MIIP) is discussed.

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Zusammenfassung Die Autoren stellen die Ergebnisse eines Versuchs vor, korrekte Gleichungen für die nicht-isotherme Kinetik durch den klassischen nichtisothermen Übergang aus den als primär postulierten isothermen kinetischen Gleichungen abzuleiten. Als alternative Möglichkeit wird das Modell der infinitesimalen isothermen Abschnitte diskutiert.

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Some fundamental aspects of the kinetics of flow-induced crystallization of polymers

Some indispensable fundamental aspects of the kinetics of polymer crystallization are summed up. Those aspects are of particular interest in the development of crystalline structures during polymer processing. In this connection also a monograph (Janeschitz-Kriegl 2009) must be mentioned, which has been written last year by the first author of this article. The present paper contains a selection of particularly interesting subjects. It turns out that many of these subjects have eluded a mathematical treatment. The pertinent list is given in the introduction. And those people, who feel pressed to produce computer programs, are exhorted to mind the content of the present article and also of the mentioned monograph in avoiding arbitrary assumptions.     

Isothermal and nonisothermal crystallization kinetics of irradiated nylon 1212

The crystallization behavior of nylon 1212, irradiated at ⁶⁰Co γ‐rays (50 kGy), was studied by a rheometer, polarized optical microscopy (POM), and differential scanning calorimeter (DSC). The results showed that irradiated nylon 1212 samples exhibited abnormal crystallization behavior during the crystallization process: The Avrami exponent n was calculated and was found to be in the range from 2.06–2.41 for isothermal crystallization, and from 2.67–4.91 for nonisothermal crystallization; the spherulite morphology also changed largely by polarized optical microscopy (POM); the crystallization activation energy ΔE for isothermal and nonisothermal crystallization process of irradiated nylon 1212 are determined to be 57.4 kJ/mol and 78.65 kJ/mol, respectively, which are lower than that of nonirradiated nylon 1212. At the same time, a new method by a combination of the Avrami and Ozawa equations was successfully applied to analyze the noncrystallization process of irradiated nylon 1212. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2326–2333, 2005     

Statistical decision theory applied to analytical chemistry : Part 1. The statistical decision model and its relation to branches of mathematical statistics

Statistical decision theory can be considered as an extension of point estimation, interval estimation and hypotheses testing. Basic elements of statistical decision models are discussed and the relation between decision theory and other branches of mathematical statistics is indicated. Some illustrative examples of application to analytical chemistry are given.     

New approximations of the temperature integral for nonisothermal kinetics

The accuracy and scope of application of previously reported approximations of the temperature integral were evaluated. The exact solution was obtained independently by solving the temperature integral numerically be Simpson's rule, the trapezoidal rule and the Gaussian rule. Two new approximations have been proposed: $$\begin{gathered} P(X) = e^{ - x} (1/X^2 )(1 - 2/X)/(1 - 5.2/X^2 ) \hfill \\ P(X) = e^{ - x} (1/X^2 )(1 - 2/X)/(1 - 4.6/X^2 ) \hfill \\ \end{gathered} $$ whereX=E/RT. The first equation gives higher accuracy, with a deviation of less than 1% and 0.1% from the exact solution forX≥7 andX≥10, respectively. The second equation has a wider scope of application, with a deviation of less than 1% forX≥4 and of less than 0.1% forX≥35.     

Model-free kinetics applied to sugarcane bagasse combustion

Vyazovkin's model-free kinetic algorithms were applied to determine conversion, isoconversion and apparent activation energy to both dehydration and combustion of sugarcane bagasse. Three different steps were detected with apparent activation energies of 76.1 ± 1.7, 333.3 ± 15.0 and 220.1 ± 4.0 kJ/mol in the conversion range of 2-5%, 15-60% and 70-90%, respectively. The first step is associated with the endothermic process of drying and release of water. The others correspond to the combustion (and carbonization) of organic matter (mainly cellulose, hemicellulose and lignin) and the combustion of the products of pyrolysis.     

The heating rate as a variable in non-isothermal kinetics. A new method to evaluate the nonisothermal kinetic parameters

The heat exchange calorimetry, hitherto developed for ordinary size samples, was modified for diminishing the sample size to one tenth or less of that commonly used in previous reports. Improvements were made with respect to the vessels for sample and reference, the stirrer of the sample solution, the thermistor and the calibration heater. The value of α, a constant relating to the heat transfer and critically affecting the sensitivity to smaller heat effects, was given an appropriately small value. The improved version of the calorimeter employed a sample vessel of a capacity of 6 cm³ and was suited to accomodate about 3 cm³ of a solution. The calorimeter proved to give sufficiently precise results when total heats ranging from 0.05 to 0.4 J were evolved.     

Some aspects on the kinetics of etherification in the preparation of CMC

Carboxymethylcellulose (CMC) was prepared from nine commercial dissolving pulps with different origins, viscosities and S₁₈-values. The rate of the etherification reaction and the final DS were studied with respect to the influence of pulp composition, pulp viscosity and S₁₈-value, and with respect to the sodium hydroxide and sodium monochloro-acetate charges. The possible influence on the final DS-value of the CMC of the washing procedure after the etherification was also investigated.     

Model-free kinetics applied to regeneration of coked alumina

Thermogravimetry is proposed to study the alumina catalyst regeneration deactivated by coke, after being used in the transformation of styrene in a fixed bed continuous flow reactor. The model-free kinetic approach has been applied to data for the thermal oxidation of carbonaceous deposits on the catalyst. The activation energy (E) was calculated as a function of a (conversion) and T (temperature), by using Vyazovkin model-free kinetic method, allowing to estimate time required to remove coke at a given temperature. This revised version was published online in August 2006 with corrections to the Cover Date.     

Potential of thermal analysis as applied to studying the kinetics of thermal degradation of polymers

The kinetics of thermal degradation of low-density polyethylene was studied by TGA and DSC at heating rates from 0.5 to 40 deg min^–1. Causes of significant discrepancies in the published effective kinetic constants of the overall reaction of thermal degradation of the polymers, determined using different experimental methods and different data treatment procedures, were analyzed. The possibility of using random break model as an alternative approach to describing polymer thermal degradation curves obtained by thermal analysis methods was demonstrated by the example of polyethylene.     

Isothermal and nonisothermal crystallization kinetics of nylon‐46

Isothermal and nonisothermal crystallization kinetics of nylon‐46 were investigated with differential scanning calorimetry. The equilibrium melting enthalpy and the equilibrium melting temperature of nylon‐46 were determined to be 155.58 J/g and 307.10 °C, respectively. The isothermal crystallization process was described by the Avrami equation. The lateral surface free energy and the end surface free energy of nylon‐46 were calculated to be 8.28 and 138.54 erg/cm², respectively. The work of chain folding was determined to be 7.12 kcal/mol. The activation energies were determined to be 568.25 and 337.80 kJ/mol for isothermal and nonisothermal crystallization, respectively. A convenient method was applied to describe the nonisothermal crystallization kinetics of nylon‐46 by a combination of the Avrami and Ozawa equations. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1784–1793, 2002     

Thermal analysis kinetics applied to flame retardant polycarbonate

The degradation kinetics of polycarbonate with flame retardant additive was investigated by means of thermogravimetric analysis. The samples were heated from 30 to 900°C in nitrogen atmosphere, with three different heating rates: 5, 10 and 20°C min^–1. The Vyazovkin model-free kinetics method was applied to calculate the activation energy (E _a) of the degradation process as a function of conversion and temperature. The results indicated that the polycarbonate without flame retardant additive starts to loose mass slightly over 380°C and the polycarbonate with flame retardant additive, slightly over 390°C (with heating rate of 5°C min^–1). The activation energy for flame retardant polycarbonate and normal polycarbonate were 190 and 165 kJ mol^–1, respectively.     

Concept of variable activation energy and its validity in nonisothermal kinetics

The concept of variable activation energy in solid-state kinetics under nonisothermal conditions has been suffering from doubt and controversy. Rate equations of nonisothermal kinetics of solid decomposition, which involve the factors of thermodynamics conditions, pressure of gaseous product, structure parameters of solid, and/or extent of conversion, are derived from the models of the interface reaction, the diffusion of gaseous product, and the nuclei growth of the solid product, respectively. The definition of the validity function in the rate equations represents the influence of the factors on the reaction rate. A function of variable activation energy depending on the validity function is also developed. The changing trend and degree of activation energy are extrapolated from the function of variable activation energy and based on the data of nonisothermal thermal decomposition of calcium carbonate. It is shown that the concept of variable activation energy is meaningfully applicable to solid-state reactions under nonisothermal conditions.     

Model-free kinetics applied to the vaporization of caprylic acid

The evaporation of octanoic (caprylic) acid was investigated by means of thermogravimetric analysis (temperature range: 300–600 K) under a nitrogen dynamic atmosphere (heating rates: 0.16, 0.31, 0.63, 1.25, 2.5, 5 and 10 K min^?1). Kinetic plots for a zero-order process were constructed based on the Arrhenius equation. The activation energy for the evaporation process was calculated via both the Arrhenius plot and Vyazovkin’s isoconversional model-free method.     

Isothermal and nonisothermal crystallization kinetics of novel odd-odd polyamide 9 11

A study on isothermal and nonisothermal crystallization kinetics of odd-odd polyamide 9 11 was carried out by differential scanning calorimetry (DSC). The equilibrium melting temperature of polyamide 9 11 was determined to be 199.1 °C. The Avrami equation was adopted to describe isothermal crystallization of polyamide 9 11. Nonisothermal crystallization was analyzed using both the Avrami relation modified by Jeziorny and the equation suggested by Mo. The isothermal and nonisothermal crystallization activation energies of polyamide 9 11 were determined to be −310.9 and −269.0 kJ/mol using the Arrhenius equation and the Kissinger method, respectively.