Bulletin of the Tomsk Polytechnic Institute of the Order of the October Revolution and the Order of the Red Banner of Labor named after S. M. Kirov

Vol. 198　　　　　　　　　　　　　　　　　　　　　　　　　　　　1974

On Determining the Degree of Influence of Macrokinetic Factors on the Rate of a Heterogeneous Reaction

S. I. Smol’yaninov, V. M. Mironov

(Submitted by the scientific-methodological seminar of the Chemical-Technological Faculty)

The degree of influence of macrokinetic factors (pressure, temperature, porosity, etc.) on the rate of a heterogeneous reaction can be determined from the magnitude of the process efficiency factor, expressed by the equation:

\[ \eta = \frac{w'}{w_0}, \quad \text{where} \tag{1} \]

\(\eta\)—efficiency factor;
\(w'\)—reaction rate determined by macrofactors;
\(w_0\)—reaction rate not controlled under the same conditions by macrofactors.

In the present case the concept of the efficiency factor is somewhat broader than in the well-known work of E. Thiele [1], where it is based only on the influence of the rate of diffusion of reactants to the active surface of the catalyst on the observed reaction rate and is defined by the expression:

\[ \eta = \varphi \left( L \cdot \sqrt{\frac{\kappa C^{\,n-1}}{D}} \right), \quad \text{where} \tag{2} \]

\(\varphi\)—a certain function, the form of which is determined by the reaction order and the shape of the catalyst granules,
\(L\)—size of the catalyst grain,
\(\kappa\)—rate constant of the surface reaction,
\(n\)—order of this reaction,
\(C\)—concentration,
\(D\)—diffusion coefficient.

In expression (2) the efficiency factor can vary only from 0 to 1, whereas in expression (1) it can vary from 0 to any positive number, since the reaction rate under the influence of macrokinetic factors may not only decrease (as, for example, due to diffusion of reactants), but also increase under some combination of external conditions.

We propose an analytical method for calculating the efficiency factors of a heterogeneous process from the values of reaction rates at three temperatures, two of which must belong to the kinetic region of the process and are necessary for calculating the activation energy. Geometrically, this calculation is represented by the following scheme, as shown in Fig. 1.

In the coordinates \(\ln w\)—\(1/T\), the values of the logarithms of three known reaction rates are plotted, respectively, for three temperatures. Through two points lying in the kinetic region, the straight line \(AB\) is drawn; the tangent of its angle of inclination determines the activation energy of the reaction proceeding in the kinetic region. From point \(C\), parallel to the ordinate axis, we draw a straight line to its intersection with the continuation of the straight line \(AB\). The segment \(CM\) will be equal to the logarithm of the effectiveness factor, since, on the basis of (1),

\[ \ln \eta = \ln w' - \ln w_0 . \tag{3} \]

Fig. 1. Graphical determination of the effectiveness factor

The derivation of the analytical expression is somewhat more complicated. Suppose we are given the reaction rates \(w_1\), \(w_2\), and \(w_3\), occurring at temperatures \(T_1\), \(T_2\), and \(T_3\), respectively. Then

\[ w_1 = \omega_1 e^{-\frac{E}{RT_1}}, \tag{4} \]

\[ w_3 = \omega_3 e^{-\frac{E'}{RT_3}} \quad \text{and} \tag{5} \]

\[ \eta = \frac{\omega_3}{\omega_1}\cdot e^{\frac{E-E'}{RT_3}}, \tag{6} \]

where

\(\omega_1\), \(\omega_3\) are constants;

\(E\) is the activation energy of the reaction proceeding in the kinetic region;

\(E'\) is the activation energy of the reaction proceeding in the region of influence of macrofactors;

\(R\) is the gas constant;

\(T\) is the temperature.

For convenience, we take the logarithm of (4)—(6) and, expressing the values \(\ln \omega_1\) and \(\ln \omega_3\) from (4) and (5), substitute them into (6). After some simplifications we obtain

\[ \ln \eta = \ln \frac{w_3}{w_1} + \frac{E}{R}\cdot \frac{(T_1 - T_3)}{T_1 \cdot T_3}. \tag{7} \]

We determine the activation energy \(E\) from the value of the reaction rates in the kinetic region:

\[ E = \frac{R \cdot T_1 \cdot T_2}{T_1 - T_2}\cdot \ln \frac{w_1}{w_2}. \tag{8} \]

Substituting (8) into (7), we obtain

\[ \ln \eta = \ln \frac{w_3}{w_1} + \frac{T_2 (T_1 - T_3)}{T_3 (T_1 - T_2)}\cdot \ln \frac{w_1}{w_2} \tag{9} \]

or, in a form more convenient for calculation:

\[ \lg \eta = \lg \frac{w_3}{w_1} + \frac{(T_1 - T_3)}{T_3}\cdot \frac{T_2}{(T_1 - T_2)}\lg \frac{w_1}{w_2}. \tag{10} \]

Using the derived formula, we calculated the efficiency factor of the synthesis process from carbon monoxide and water vapor at atmospheric and moderate pressure for various temperatures. The results are presented in Fig. 2 as plots of the dependence of the efficiency factor on

Fig. 2. Dependence of the efficiency factor of the synthesis process from CO and water vapor on temperature.

— atmospheric pressure,
— — — pressure 10 atm

temperature. The conclusions about the stronger influence of diffusion processes on the rate of hydrocarbon formation with increasing pressure, which follow from the character of the dependence presented, confirm the assumptions made by us earlier [2].

In conclusion, it is necessary to point out that, by the proposed formula, one can qualitatively judge the complexity of the mechanism of the reaction under study. Thus, at $\eta = 1$ the reaction in the temperature interval under study obeys the Arrhenius law; at $\eta < 1$, either the processes of mass and heat transfer exert a strong influence on the rate of chemical interaction, or the process proceeds by a consecutive mechanism; at $\eta > 1$, the process represents a complex set of parallel reactions.

Conclusions

An equation has been proposed for calculating the efficiency factor of a heterogeneous process. The initial data for the calculation are the values of the reaction rate obtained at three different temperatures.

References

1. E. W. Thiele. Ind. Eng. Chem., 34, No. 7, 916–920, 1939.
2. S. I. Smolyaninov, V. M. Mironov, and V. A. Kornyakov. Izvestiya TPI (in press).