Mathematical considerations for nonisothermal kinetics in thermal decomposition
Introduction
Kinetic analysis of thermal decomposition processes has focused the interest of a large number of researches all along the modern history of thermal decomposition. The interest is fully justified: On one side, kinetic data are essential for designing any kind of device in which the thermal decomposition takes place. On the other side, kinetics is intrinsically related with the decompositions mechanisms. The knowledge of the mechanisms allows to postulate kinetic equations or vice versa, kinetics is the starting point to postulate mechanisms for the thermal decomposition. A 20 years old, but excellent and nowadays plenty useful review by Antal [1] illustrates previous comments. For a more recent review on kinetic analysis see Conesa et al. [2].
Literature dedicated specifically to kinetics of thermal decomposition is really extensive: a simple search using SciFinder provides more than 500 references containing exactly the sentence “kinetic pyrolysis” more than 6000 containing both concepts “kinetics” and “pyrolysis” simultaneously (more than 600 in the last 2 years). These numbers increases with more general searches, for example there are more than 20,000 references including “Kinetics, Thermal Decomposition”. Even in papers not directly dedicated to kinetics it is not uncommon to find some kinetic analysis.
Although kinetics studies can be performed in very different devices, thermogravimetry (TG) is, by large, the most used technique. For the sake of simplicity but without loss of generality the following discussions are referred to TG analysis albeit they can be extended to almost any other kind of experimental devices.
Despite of the huge amount of literature and kinetic data it is not uncommon to find papers that:
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Present some deficiencies or at least very discussible approaches (correlation models with no physical meaning, or at least with very difficult interpretation) and therefore, in which it is not possible any kind of extrapolation.
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The model (kinetic equations) is not clearly specified.
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Data are not correctly treated, especially when derivative of TG (DTG) curves are used.
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Everything is correct but the mathematical procedure employed is far from the state of the art, meanly based on the calculation capacity of modern personal computers. Of course, this “brute force” approach is not important for the final reader if he/she is only interested in the final results, but in some situations it could be very time consuming.
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No data at all are provided about quality of the fitted parameters.
In this paper we review some of most relevant mathematical aspect behind the TG–DTG kinetic analysis, and provide some suggestions and recommendations for getting reliable results, as well as an overview of the modern optimization techniques that can be used for fitting the experimental and modeling values.
In this work we only deal with the mathematical aspects from the treatment of the “raw TG data” until the final set of kinetic parameters are obtained. We will not discuss experimental errors (incorrect temperature measure, heat transfer limitations and so on) but those due to the precision of the apparatus. In the next sections we will treat the following aspects:
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Data treatment.
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Mathematical models: specification of kinetic equations and kinetic models.
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Fitting data to kinetic models: differential and Integral methods.
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Quality of the fit and the fitted parameters.
Section snippets
Influence of errors in temperature
In a TG experiment, a modern equipment typically registers hundreds or thousands of experimental points; usually total weight (or a related magnitude), temperature and time. Temperature and time are related by a ‘program’ set by the user. Although the equipment control system tries to adjust the actual temperature to fit the programmed one, some times important differences appear (for example due to endothermic or exothermic reactions, some induction time in the furnace, etc.). However, if the
Mathematical models
It seems evident that the correct specification of the kinetic model is mandatory in a kinetic analysis. In this work we do not deal with the problem of whether the model is able or not to describe the experiments, or if it is based on mechanistic considerations and so on. Vyazovkin and Lesnikovich [15], [16], Conesa et al. [2], Varhegyi et al. [17] and Koga et al. [18] among others studied deeply these problems. What we want to remark here is that simple affirmations like “kinetic constants (k0
Fitting data to kinetic models
The use of thermal analysis to measure reaction rates dates from the beginning of 20th century. Some good reviews of literature previous to 1970 can be found in the works of Sestak [26] (methods of kinetic data evaluation for isothermal and non-isothermal TG curves) and Flynn and Wall [27] (general treatment of thermogravimetry polymers).
From the very beginning, the methods for kinetic analysis were classified in integral and differential methods.
Choosing the points to fit
As we commented above, when we performed a TG/DTG analysis, we select a subset of the thousands of points that we get from the thermobalance, usually around 100–200 points. Frequently those points are equally spaced in time or temperature. However, some care must be taken with the selection of points. It is common to see TG or DTG curves with a rather large number of points in the initial and final flat zones of the curves and a relatively small number of points in the zones of most interest.
Acknowledgements
Authors gratefully acknowledge financial support provided by projects PPQ2002-00567, PPQ2002-10548-E and PPQ2002-01734.
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