ON THE TEMPERATURE ERROR IN DETERMINING THE KINETIC CONSTANTS OF SOLID FUEL IN A DUST–AIR FLOW

E. A. KON’KOV

(Submitted by the scientific seminar of the Department of Boiler Construction and Boiler Installations)

A pulverized-coal flame, as an object of study in determining the kinetic properties of solid fuel, has a substantial drawback—nonisothermality. In the corresponding investigations, isothermal conditions are often ensured only approximately, although it can be shown that an inaccurate estimate of the reaction temperature leads not only to large quantitative errors in determining the indices of reactivity, but also qualitatively distorts the process under study.

As is known [1], the quantities \(k_0\) and \(E\)*), which characterize the kinetic properties of the fuel, are determined by graphical solution of the equation

\[ \ln k = \ln k_0 - \frac{E}{RT}. \tag{1} \]

In accordance with this method, the quantity \(k\) may be regarded as an empirical constant, and \(T\) as an operating parameter.

Owing to the impossibility of directly determining the reaction temperature of the particles, it is often identified with the temperature of the gas stream or calculated on the basis of the heat balance. We shall call this indirectly found temperature \(T_1\) the apparent temperature. Putting

\[ T_1 = T \pm \Delta T, \tag{2} \]

we have

\[ \ln k = \ln k_0 - \frac{E}{R(T \pm \Delta T)}. \tag{3} \]

According to some data [2], \(\Delta T = \text{const}\) may be assumed over a sufficiently wide temperature interval of flame burnout.

Differentiating equations (1) and (3) with respect to temperature and writing the resulting expressions in difference form, we obtain

\[ \frac{\Delta(\ln k)}{\Delta T} = -\frac{E}{R}\cdot \frac{1}{T}, \tag{4} \]

*) or, respectively, \(a_0\) and \(E_{\mathrm{eff}}\), if the reactivity is evaluated by the value of the reaction gas-exchange coefficient.

\[ \frac{\Delta(\ln k)}{\Delta T}=-\frac{E_1}{R}\cdot\frac{1}{(T\pm\Delta T)}. \tag{5} \]

The quantity \(\ln k\) is determined from experiment, i.e., it is an empirical constant and, obviously, does not depend on the apparent temperature. Consequently, the left-hand sides of equations (5) and (4) are equal. Solving equations (5) and (4) simultaneously under this condition and, in the course of the transformations, discarding powers of \(\Delta T\) greater than unity, it is not difficult to obtain

\[ E_1=E\left(1\pm2\frac{\Delta T}{T}\right). \tag{6} \]

If \(E\) is the activation energy calculated from the true temperature, and \(E_1\) from the apparent reaction temperature, then, obviously,

\[ E_1=E\pm\Delta E, \tag{7} \]

where

\[ \Delta E=2\cdot\Theta\cdot E,\quad \Theta=\frac{\Delta T}{T}. \]

We shall regard the temperature interval for investigating kinetic constants as very small. Then it is not difficult to see (cf. (1) and (3))

\[ \Delta(\ln k_0)=\frac{E}{RT}-\frac{E_1}{R(T\pm\Delta T)}. \tag{8} \]

Using (7), we obtain

\[ \Delta(\ln k_0)=\frac{E}{RT}\cdot\Theta, \tag{9} \]

and, expanding \(\frac{1}{1\pm\Theta}\) in a series and limiting ourselves to the first two terms of the expansion, we finally have

\[ \Delta(\ln k_0)=\frac{E}{RT}\cdot\Theta. \tag{10} \]

It follows from (10) that the relative error in estimating the value of the preexponential factor when the reaction temperature is determined inaccurately is

\[ \frac{\Delta k_0}{k_0}=\exp\Delta(\ln k_0)=\exp\frac{E}{RT}\cdot\Theta. \tag{11} \]

Considering the present possibilities for experimental determination of kinetic constants, the upper limit of the attainable temperature (determined by the resistance of reaction tubes made of stainless steel 1Kh18N9T) may be taken as equal to \(\sim 1500^\circ\text{K}\). If the temperature of the burning particles is set equal to the temperature of the gas stream, and the diameter of the reaction vessel is taken greater than the second critical diameter [3], then the error in determining the reaction temperature may be \(200\text{--}300^\circ\) [4].

If one even takes \(\Delta T=100^\circ\), then according to (7), for

\[ E\approx125\cdot10^3\ \text{kJ/mole} \]

\[ \Delta E=\pm E\frac{2\cdot100}{1500}\approx16600\ \text{kJ/mole}. \]

* In the case of a significant magnitude of this interval, the question may be reduced to considering the average temperature over it.

Such an error almost covers the range of variation of the activation energies of various fuels [3], which is \(\sim 28000\) kJ/mol. For the pre-exponential factors, according to (11), we have

\[ \frac{\Delta k_0}{k_0} = \exp \frac{125 \cdot 10^3}{8.36 \cdot 1500} \cdot \frac{100}{1500} \cdot 100\% \approx 200\%. \]

The numerical errors cited, despite their considerable magnitude, do not exhaust the matter. Indeed, as can be seen from (7), in the presence of a temperature error the quantity \(E_1\) is a linear function of temperature, but by no means a constant quantity, as it should be in the absence of an error in estimating the reaction temperature.

This, in turn, leads to the fact that the dependence \(\ln k = f\left(\frac{1}{T}\right)\) acquires a nonlinear character (Fig. 1). In particular, for \(\Delta T > 0\) this dependence is similar to the analogous dependence for a combustion process with diffusion inhibition. Such an influence of the temperature error is of great importance in establishing the boundaries of the regime regions of reaction (kinetic, diffusion, or intermediate) [5].

Fig. 1. Character of the dependence of the combustion-rate constant on temperature in the presence of a temperature error.

For \(\Delta T < 0\), dependence (3) becomes similar to the same dependence for a heterogeneous process complicated by the parallel occurrence of homogeneous reactions [6], which, obviously, will be the cause of incorrect conclusions concerning the nature of the process under study and practical measures for its intensification.

The kinetic aspect considered for the burnout process does not limit the desirability of ensuring isothermal conditions in the experimental combustion of a pulverized-coal flame. As is known, modern mathematical means for studying the latter require reducing it to a hydraulically one-dimensional scheme, which can likewise be accomplished only in a thermally homogeneous medium.

Conclusions

1. The influence of temperature error on the results of kinetic studies by studying a pulverized-coal flame has been analyzed.

2. It has been shown that deviation from isothermal conditions leads not only to gross errors in estimating the values of the kinetic constants obtained, but also substantially distorts the qualitative content of the processes being studied.

LITERATURE

1. B. V. Kantorovich. Fundamentals of the Theory of Combustion and Gasification of Solid Fuel. Publishing House of the Academy of Sciences of the USSR, Moscow, 1958.

1. A. B. Reznyakov. On the question of the combustion region of pulverized-coal particles in a collector. Izv. AN Kaz. SSR, Energy Series, no. 12, 1957.

2. T. V. Vilensky. Study of ignition in a dust–air flow. Author’s abstract of Candidate’s dissertation. Moscow, 1964.

3. A. B. Reznyakov. Combustion of a pulverized-coal flame as a polydisperse system. Alma-Ata, 1958.

4. Walker, Rusinko, Austin. Reactions of carbon with gases. In: Reactions of Carbon with Gases, ed. E. S. Golovina. Foreign Literature Publishing House, Moscow, 1963.

5. G. M. Panchenkov and V. P. Lebedev. Chemical Kinetics and Catalysis. Moscow State University Press, 1961.