Mathematical methods for application of experimental adiabatic data – An update and extension

https://doi.org/10.1016/j.jlp.2014.11.014Get rights and content

Highlights

  • The Fisher's method for phi-factor correction is improved to enable accurate prediction of the adiabatic time scale.

  • New method for phi-factor correction of pressure data is proposed.

  • Limitations of the simplified method for phi-factor correction are revealed.

  • Limitations of the simplified method for predicting adiabatic time to maximum rate are stated.

  • Advantages of the kinetics-based simulation for adiabatic data analysis are demonstrated.

Abstract

The paper represents some results of comparative analysis of the methods used for processing and interpreting data of adiabatic calorimetry as well as applying it to practical situations. Specifically two approaches are compared – approximate method based on evaluation of simplified kinetics and a more comprehensive, simulation-based method that utilizes the evaluation of more detailed kinetic models.

The analysis is focused on two important types of data processing – correction of experimental results on thermal inertia (phi-factor correction) and estimation of adiabatic time to maximum rate (TMR).

The most widely cited method for phi-factor correction is considered and its improvement is proposed to enable more precise prediction of the adiabatic time scale. A procedure for phi-factor correction of pressure response is also proposed. The limitations of this enhanced Fisher's method are discussed by comparison with simulation-based method. All the illustrative materials are based on real examples.

As an example of application, the simplified method will be used to predict TMR and its limitations will be discussed.

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.csφTt=W(t,T,f(C¯)),t=0,T=Ton,C¯=C¯0where φ = 1 + (cbmb)/(csms), W(t,T,f(C¯)) is specific heat release rate generated by a reaction, f(C¯) 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:

  • -

    short introduction;

  • -

    kinetics evaluated using Arrhenius method and Expert method;

  • -

    results of φ-factor-correction by applying Standard and Enhanced Fisher methods and the Expert simulation-based method and comparison of the results;

  • -

    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.

References (14)

There are more references available in the full text version of this article.

Cited by (0)

View full text