Mathematical methods for application of experimental adiabatic data – An update and extension
Introduction
The main goal of this paper is to compare two approaches for analysis and application of experimental data derived from adiabatic calorimeters. The first approach is based on a popular simplified method (referred to here as Standard method) that is widely used. The second, involving a more comprehensive approach based on the sequential use of the kinetic models coupled with mathematical simulation, will be referred to as the Expert method.
The focus will be on three particular but very important types of data treatment, namely kinetics evaluation, correction of experimental data regarding thermal inertia, and estimation of adiabatic time to maximum rate, TMR. Several examples based on real experimental data will be used to reveal the potential of the Standard approach and to highlight possible errors that may arise when it is used beyond the limits of its applicability.
It should be noted that kinetics evaluation has been mentioned at the outset very deliberately. The fact is that all the data analysis methods, including the simplified one, require knowledge of reaction kinetics, at least the apparent activation energy and rate constant.
In the Standard methods the discussion will start with Fisher's method for thermal inertia (or φ-factor) correction (HarsNet, 2003, Fisher et al., 1992) and the method for calculation of induction period of thermal explosion (under adiabatic conditions) which from a practical viewpoint is analogous to adiabatic TMR prediction (Grewer, 1994). The treatment will include simple N-order kinetics (HarsNet, 2003, Fisher et al., 1992, McIntosh and Waldram, 2003).
Treatment of the Expert methodology will include an overview of a more comprehensive kinetic analysis and also briefly consider a simulation-based approach to φ-factor correction and TMR determination.
Section snippets
Kinetics evaluation
The kinetic approach that is most often used in adiabatic calorimetry is the Arrhenius linearization method. According to this, under certain simplifying assumptions, the thermal state of the sample container (or bomb) and the sample can be described by the heat balance equation.where φ = 1 + (cbmb)/(csms), is specific heat release rate generated by a reaction, stands for the function that defines dependency of reaction rate on
Backgrounds of the Expert approach
As mentioned earlier, evaluation of complex kinetic model, either self-accelerating or multi-stage, requires the use of non-linear optimization methods. Moreover, in the case of adiabatic data the assumption that temperature can be considered as independent variable, i.e. that it can be taken from the experiment, results in biased estimates of kinetic parameters (see Kossoy and Koludarova, 1995, Kossoy and Akhmetshin, 2007 for more details). Therefore the complete model comprising heat balance
Comparison of Standard and Expert approaches
Several real examples will be used to compare results of applying Standard and Expert approaches. In each case, the analysis will be presented as follows:
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short introduction;
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kinetics evaluated using Arrhenius method and Expert method;
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results of φ-factor-correction by applying Standard and Enhanced Fisher methods and the Expert simulation-based method and comparison of the results;
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Results of TMR determination using Frank-Kamenetskii method and Expert method and comparison of the results.
Correction of heat and gas production responses
The procedure for correction of self-heating and SHR due to changes in thermal inertia have been considered in detail but there are still more important variables that also require correction, for instance, heat release Q(t), gas generation G(t) and pressure P(T) and their rates.
The relations for correcting all the responses except pressure can be easily derived.
As far as the integral responses Q and G are concerned, both these quantities are independent of thermal inertia therefore it is
Discussion of the results
Though the spectrum of possible complex reactions is much wider than considered above, even the limited number of examples presented here allows several general conclusions to be drawn.
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