Error in Estimating the Kinetics of Nonisothermal Pyrolysis of Solid Fuels
Abstract Generated abstract
This paper examines errors in determining kinetic parameters for the nonisothermal pyrolysis and thermal decomposition of solid fuels under linear temperature programming. The authors argue that conventional processing of differential and integral thermogravimetric or volumetric curves often treats projected reaction paths rather than the actual path in time, temperature, and conversion space, producing distortions that vary with heating rate. Using a geometric representation of the reaction path, they derive correction relationships for the effect of heating rate and note when an approximate form is acceptable. The study also shows that stopping experiments at a fixed final temperature changes the achieved conversion with heating rate, and recommends instead comparing experiments at a fixed degree of conversion.
Full Text
Errors in Estimating the Kinetics of Nonisothermal Pyrolysis of Solid Fuels
V. M. Mironov, Ya. A. Belikhmaer
(Presented by the Scientific-Methodological Seminar of the KhTF)
In the technology of thermal processing of solid combustible minerals, one very often has to deal with nonisothermal processes, the study of whose kinetics, even for reactions of simple types, is associated with great difficulties. However, the present problem is considerably simplified when the kinetics of nonisothermal reactions is considered under conditions of a linear rise in temperature.
Extensive investigations carried out on the kinetics of the thermal decomposition of solid substances, as well as of solid combustible minerals, under conditions of linear temperature programming have led to the appearance of several methods for processing differential and integral thermogravimetric and volumetric curves [1—9]. But in most cases the authors of the above-cited works did not take into account the projection distortions of these curves.
The purpose of the present work is to analyze the errors arising in the processing of experimental data on the thermal decomposition of solid fuels under conditions of a linear rise in temperature.
We shall use the concept of the reaction path, defining by this expression the dependence of the degree of conversion of one of the reaction components on time under conditions of varying temperature. In this case the reaction path will lie on a certain surface in the coordinates
\[ \alpha = f(\tau, T), \]
where
\[ \alpha \text{ is the degree of conversion,} \]
\[ \tau \text{ is time,} \]
\[ T \text{ is temperature.} \]
Thus, we shall consider the reaction path in a three-dimensional coordinate system. It is convenient to define the surface of a chemical reaction by level lines on the plane \(\tau OT\). With a linear rise in temperature, the reaction path will lie on the trace of a cutting plane perpendicular to the plane \(\tau OT\), and will be characterized by the points of intersection of the trace line with the level lines. Thus, in Fig. 1 the lines \(R_1, R_2, R_3\) correspond to the ideal course of a chemical reaction under linear temperature programming.
In kinetic studies, it is usually not the reaction path itself that is considered, but its projection onto the \(O\tau\) axis. With such an approach, possible projection distortions of the actual parameters of the reaction-path curve are not taken into account. The nature of such distortions is clearly seen from Fig. 1. This figure presents the change in the parameters of the diffe-
Fig. 1. Distortion of the projection of the reaction path when the heating rate changes
differential and integral curves of the reaction path as a function of the heating rate. It is seen that, as the heating rate increases, the integral curve \((\alpha-\tau)\) narrows and its curvature increases at all points.
For the differential curves \(\left(\dfrac{d\alpha}{d\tau}-\tau\right)\), an increase in the heating rate leads to an increase in the maximum value of the reaction rate with a simultaneous decrease in the limits of its occurrence. It is clear that the areas under the curves, because of the constancy of the limits of the degrees of conversion, must remain unchanged \((S_1=S_2=S_3)\). Since, in order to find the kinetic parameters of the reaction, it is necessary to exclude the influence of the heating rate on the parameters of the differential and integral curves of the reaction path, it is most correct to consider the reaction path itself, lying in the plane \(R\). For this we shall use
by the following elementary transformations. In triangle OAB (Fig. 1), \(\operatorname{tg}\beta\) represents the heating rate \(a\).
\[ \operatorname{tg}\beta = a. \]
From this triangle,
\[ \mathrm{OA} = \mathrm{OB}\cdot \frac{1}{\cos\beta} \]
or
\[ \mathrm{OA} = \mathrm{OB}\cdot \sqrt{1+\operatorname{tg}^{2}\beta}, \text{ i.e.} \]
\[ \mathrm{OA} = \mathrm{OB}\sqrt{1+a^{2}}. \tag{1} \]
For \(a \gg 1\),
\[ \mathrm{OA} = \mathrm{OB}\cdot a. \tag{2} \]
Analysis of equations (1) and (2) shows that at heating rates of more than \(5\ \text{deg/min}\), the error in replacing equation (1) by equation (2) is less than 2%. When working with heating rates below \(5\ \text{deg/min}\), only equation (1) should be used.
Sometimes [9] the investigation of the process of thermal decomposition of solid fuels is carried out in such a way that the process is interrupted upon reaching a certain temperature at various heating rates. The errors that arise in this case are also characterized quite fully (Fig. 1). It follows from Fig. 1 that, when the reaction is “frozen” upon reaching a certain temperature \(T_k\), the final degree of conversion is a function of the heating rate, and this degree of conversion is always lower in experiments with higher heating rates. Therefore, interrupting the process upon reaching a certain temperature actually reduces the reaction time in experiments with a high heating rate. In such a case it would be more correct to “freeze” the process upon reaching a certain degree of conversion, without imposing a restriction on the final temperature.
Conclusions
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An analysis has been made of the errors encountered in processing experimental data on the kinetics of thermal decomposition of solid substances under conditions of linear temperature programming.
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Methods for eliminating these errors have been proposed.
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