DETERMINATION OF THE KINETIC PARAMETERS OF THE PROCESS OF THERMAL DECOMPOSITION OF FUEL FROM DERIVATOGRAPHIC DATA

S. I. SMOLYANINOV, V. I. LYUZBIN, V. M. INGRIN, Ya. A. BELIKHMAER

(Presented by the organic section of the scientific-methodological seminar of KhTF)

The study of the kinetics and mechanism of thermal destruction of solid combustible minerals is most effective when a set of experimental data obtained by physical, physicochemical, and chemical methods is used.

At present, installations are becoming widespread that combine the determination of thermographic and thermogravimetric parameters with analysis of the volatile products of decomposition of the sample (above all, individual gas components) in a single experiment. In this case two different methods of sampling the vapor-gas products of the reaction are possible:

1) from a separate retort located next to the working crucible and equivalent to it in thermophysical terms;

2) by suction directly from the working crucible in an inert-gas medium.

It is evident that the latter method, for solving kinetic problems, is more accurate and flexible, provided that the analysis of the diluted reaction products is sufficiently efficient. In this case, if a constant rate of suction of the vapor-gas mixture is ensured, the concentration of each component in the gas is proportional to the rate of its formation.

To determine the kinetic parameters of the process of thermal decomposition of fuel from derivatographic data, curves of weight loss (TG) and relative rate of weight loss (DTG) are used. The scale of the ordinate of the DTG curve, characterizing the total rate of evolution of volatiles, can be readily found by graphical differentiation of the TG curve at the corresponding points.

The thermal decomposition of solid combustible minerals proceeds, to a certain extent, stepwise, which is expressed in the character of the TG and DTG curves, especially the latter, on which several maxima of the rates of weight loss are observed quite distinctly. On the basis of the available data, these distinguishable stages of the process of thermal decomposition of fuel may be interpreted as zones in which the decomposition of homogeneous groups of substances predominantly occurs or reactions of the same type proceed.

Thus, the initial parameters characterizing the kinetics of the processes under consideration may be:

1) the temperatures of the maxima of the rates of weight loss on the DTG curve;

2) the absolute rates of weight loss at the points of the maxima, referred to unit weight of the organic mass of the fuel;

c3) the release of volatiles in characteristic temperature intervals.

The temperatures of the extrema on the DTA curve are also sufficiently characteristic; however, for well-known reasons, the use of the latter for calculating kinetic parameters is extremely limited. The possibilities for a kinetic assessment of the process of thermal decomposition of a fuel increase greatly if the traditional information of derivatography is supplemented with data on the dynamics of the evolution of individual components of the volatile products.

The principal kinetic parameters—the activation energy \((E)\), the pre-exponential factor \((K_0)\), and the order of the process (reaction) \((n)\)—are calculated from the equations of nonisothermal kinetics:

\[ \omega_\tau=\frac{dx}{d\tau}=K_0 e^{-E/RT}(1-x)^n \tag{1} \]

and

\[ \omega_T=\frac{dx}{dT}=\frac{K^0}{b}\cdot e^{-E/RT}(1-x)^n, \tag{2} \]

where

\[ x \text{ is the degree of conversion in fractions of unity,} \]

\[ b \text{ is the heating rate.} \]

The initial experimental data for the calculation are the degree of conversion, or quantities proportional to it, determined at different temperatures and different heating rates, and the rates of the process \(\omega_T\) and \(\omega_\tau\).

The main difficulties in the successful solution of the problem posed lie precisely in the correct interpretation of the initial experimental quantities. The complexity of the composition of the initial substances and of the process of thermal decomposition itself has led to numerous attempts to transform the initial equations into forms convenient for using the characteristic parameters of derivatograms, to find conditions and points that increase the accuracy of measurements, and also to search for experimental and computational procedures that ensure a clearer differentiation of the individual stages of thermal decomposition [1].

Following the recommendation of Van Krevelen [2], an interpretation of the degree of conversion in terms of the weight loss at a given moment relative to the “limiting” yield of volatile substances is very popular. Apart from a certain conventionality of the latter concept, such a representation is too general. A more rigorous approach can be developed when it is possible to distinguish clearly individual independent segments on the DTG curves or on curves of the rates of formation of gas components. In this case the degree of conversion is determined from the ratio of the area under the curve at a given time to the total area. The desired quantity can be found when the yield of some individual product is recorded, for example carbon dioxide by the increase in the weight of the absorbent. Here too, however, caution should be exercised in attributing the observed weight loss to the given product, since when successive processes overlap one another the error may be considerable. Finally, if a calculation formula contains the ratio of degrees of conversion, then any quantities proportional to it may be used.

In addition to the methods set forth in the literature cited above, we propose several transformed equations which, in our opinion, are simpler and more convenient for calculating kinetic parameters from derivatographic data.

If experiments have been carried out at different heating rates, then the activation energy can be determined from the ordinary Arrhenius equation

\[ \mathrm{E}=4.575\frac{T_2\cdot T_1}{T_2-T_1}\lg\frac{K_2}{K_1}, \tag{3} \]

transforming it as follows. Let us replace \(K\), proceeding from the general kinetic equation in integral form,

\[ K\tau=\psi(x), \]

then

\[ \mathrm{E}=4.575\frac{T_2\cdot T_1}{T_2-T_1}\lg\left[\frac{\tau_1}{\tau_2}\times\frac{\varphi(x_2)}{\varphi(x_1)}\right] \tag{4} \]

or, for an equal degree of conversion at two different temperatures,

\[ \mathrm{E}=4.575\frac{T_2\cdot T_1}{T_2-T_1}\lg\frac{\tau_1}{\tau_2}. \tag{5} \]

But under heating at a constant rate

\[ T=T_0+b\tau, \tag{6} \]

where

\(b\) is the heating rate, and, consequently,

\[ \mathrm{E}=4.575\frac{T_2\cdot T_1}{T_2-T_1}\lg\frac{b_2}{b_1}\times\frac{T_1-T_0}{T_2-T_0}. \tag{7} \]

The present equation is convenient in that, when the order is unknown, it makes it possible to determine the activation energy of the reaction. In addition, it can be used over a wide range of temperatures and degrees of conversion, and also in the case where the activation energy proves to be a variable quantity. However, here attention should be paid to the need to establish or choose \(T_0\). A certain conventionality (logical premises or extrapolation of the curves) apparently cannot lead to a fundamental error. In those cases where the individual stages of decomposition of an organic substance do not overlap one another, \(T_0\) can be determined with sufficient accuracy.

If the reaction order is known, or if it can be assumed a priori, then on the basis of equation (3)

\[ \mathrm{E}=4.575\frac{T_2\cdot T_1}{T_2-T_1}\left(\lg\frac{w_2}{w_1}+n\lg\frac{1-x_1}{1-x_2}\right), \tag{8} \]

in which \(w\) is found from the general equation for the reaction rate

\[ w=K(1-x)^n. \tag{9} \]

It can be shown (see below) that the ratio of the degrees of conversion \((1-x)\) at definite points of the DTG and TG curves does not depend on the heating rate for reactions of a single order. Then the equation for determining the activation energy is substantially simplified.

For calculation it is convenient to use two points on the DTG curves: the temperature of the maximum rate of weight loss and the temperature at which one half of the maximum-rate value is reached. For these conditions the ratio of degrees of conversion is 2.22 for a first-order reaction and 1.74 for a second-order reaction. Then

\[ \mathrm{E}=3.0\frac{T_m\cdot T_{0.5m}}{T_m-T_{0.5m}},\quad (n=1) \tag{10} \]

and

\[ \mathrm{E}=3.6\frac{T_m\cdot T_{0.5m}}{T_m-T_{0.5m}},\quad (n=2), \tag{11} \]

or, in general form,

\[ E=M\frac{T_m\cdot T_{0.5m}}{T_m-T_{0.5m}}, \tag{12} \]

where, in accordance with equation (8),

\[ M=4.575\left(\lg 2+n\lg\frac{1-x_{0.5m}}{1-x_m}\right). \tag{13} \]

Figure 1 presents, calculated by us from the data of Van Hike and Yungten [3], the ratios of the degrees of conversion and the coefficient \(M\) entering formula (12).

Formulas (12) and (13), together with the graph in Fig. 1, can be used to determine the order of the reaction if \(E\) is found independently of \(n\).

Fig. 1. Degree of conversion at the maximum of the curve of the dependence of reaction rate on temperature (a) [3], the ratio of these degrees at the points of half-maximum and maximum (b, 1), and the coefficient \(M\) according to formula (13) (b, 2) at various reaction orders.

This method supplements the well-known proposals of Kissinger, Van Hike, and Yungten [3], according to which the shape of the curve of the dependence of reaction rate on temperature, and consequently also the degree of conversion at the maximum, are determined by the reaction order.

Determination of kinetic parameters at extremal points gives a large effect not only because these points are experimentally recorded most clearly, but also because many kinetic equations acquire in this case the simplest form.

Thus, we have shown that the left-hand side of the equation, when integrated over the limits from \(O\) to \(x_i\),

\[ \int_{0}^{x_1} \frac{dx}{(1-x)^n} = \int \frac{K}{b}\, dT \tag{14} \]

at the point of the maximum of the reaction rate is equal to 1 regardless of \(n\). This is easily verified by making the corresponding substitutions \(1-x_m\) and \(n\) (from Fig. 1).

Then, using the Coats and Redfern equation [1], obtained by approximate integration of the basic equation (2), we find the following relatively simple expression for \(K_0\):

\[ K_0 = \frac{bE}{RT_m^2}\cdot e^{\frac{E}{RT_m}}, \tag{15} \]

where the subscript \(m\) denotes the value of the given parameter at the maximum point.

An advantage of the present equation is also that, for calculating \(K_0\), it is not necessary to determine the degree of conversion.

REFERENCES

1. Reich L., Levi G. Dynamic thermogravimetric analysis in the destruction of polymers. In: New Methods in Polymer Research, Moscow, “Mir,” 1968, pp. 148–199.

2. Van Krevelen D. W., Van Heerden C., Huntjens F. I. Physicochemical aspects of the pyrolysis of coal and coal-like organic compounds. Fuel, vol. 30, No. 1, 1951, pp. 853–859.

3. Van Heek K. H., Juntgen H. Determination of kinetic parameters of a reaction in nonisothermal measurements. Berichte der Buns. Gesell. für Phys. Ch., vol. 72, No. 9/10, 1968, pp. 1223–1233.