Determination of the Parameters of Kinetic Equations for the Reaction of Alkaline Oxymethylation of Phenol from Experimental Data
Abstract Generated abstract
This paper estimates parameters for kinetic equations describing the alkaline oxymethylation of phenol, using a reaction scheme based on the Manasse mechanism and simplifying assumptions about irreversibility, reaction order, formaldehyde consumption, and catalyst dependence. The authors formulate a system of differential equations for phenol, formaldehyde, and intermediate and final methylolphenol products, with temperature dependence represented by Arrhenius parameters. Sixteen parameters were fitted to experimental data from 15 experiments at three temperatures by minimizing squared concentration deviations, using a modified gradient method followed by a modified pattern search program on an M-20 computer. The study obtains approximate parameter values, finds stronger sensitivity to preexponential factors than to activation energies, and concludes that the model and parameter estimates require further refinement because of experimental errors, possible multi-extremality, and likely oversimplified catalyst and formaldehyde dependences.
Full Text
Determination of the Parameters of the Kinetic Equations for the Reaction of Alkaline Oxymethylation of Phenol from Experimental Data
V. G. Martynenko, V. P. Lopatinskii
(Presented by the scientific-methodological seminar of the Faculty of Chemical Technology)
As the kinetic model in compiling the mathematical description of the process of obtaining phenol alcohols, the reaction-mechanism scheme proposed by Manasse [1] was adopted.
[Reaction-mechanism scheme shown: phenol and its oxymethylated derivatives, with reaction arrows labeled \(k_1, k_2, k_3, k_4, k_5, k_6, k_7, k_8\).]
For the derivation of the kinetic equations, the following assumptions were adopted:
- All reactions are irreversible.
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The order of the elementary stages corresponds to the stoichiometry, with the exception of the formation of dioxydiphenylmethane, which proceeds according to a first-order reaction [2].
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In the range of initial-component concentrations and temperatures studied, no consumption of formaldehyde as a result of the Cannizzaro reaction occurs.
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The rates of the individual stages are proportional to the concentration of the catalyst. Taking the adopted assumptions into account, the equations for the consumption of the initial reactants and the accumulation of the reaction products are written as:
\[ \frac{d x_1}{d t}=-x_1 \cdot x_8 \cdot C \cdot (k_1+k_2); \]
\[ \frac{d x_2}{d t}=x_8 \cdot C \cdot (k_1 \cdot x_1-k_3 \cdot x_2); \]
\[ \frac{d x_3}{d t}=x_8 \cdot C \cdot (k_1 x_1-k_4 \cdot x_3-k_5 \cdot x_3); \]
\[ \frac{d x_4}{d t}=x_8 \cdot C \cdot (k_3 \cdot x_2+k_4 \cdot x_3-k_6 \cdot x_4); \tag{1} \]
\[ \frac{d x_5}{d t}=x_8 \cdot C \cdot (k_5 \cdot x_3-k_7 \cdot x_5); \]
\[ \frac{d x_6}{d t}=x_8 \cdot C \cdot (k_6 \cdot x_4+k_7 \cdot x_5)-2k_8 \cdot x_6 \cdot C; \]
\[ \frac{d x_7}{d t}=k_8 \cdot C \cdot x_6; \]
\[ \frac{d x_8}{d t}=-x_8 \cdot x_1 \cdot C \cdot (k_1+k_2)-x_8 \cdot C \cdot (k_3 x_2+k_4 x_3+k_5 \cdot x_3+k_6 \cdot x_4+ \]
\[ +k_7 \cdot x_5)+k_8 \cdot x_6 \cdot C; \]
with \(t=0\): \(x_1=x_{10}\); \(x_8=x_{80}\); \(x_2=x_3=x_4=x_5=x_6=x_7=0\). Here: \(x_1,\ldots,x_8\) are, respectively, the concentrations of phenol; paraoxybenzyl alcohol; saligenin; 2,4-dimethylolphenol; 2,6-dimethylolphenol; 2,4,6-trimethylolphenol; \(3,3',5,5'\)-tetramethylol-4,4′-dioxydiphenylmethane; and formaldehyde. \(C\) is the concentration of the catalyst, sodium hydroxide. \(k_1,\ldots,k_8\) are the rate constants of the elementary stages of the reaction, whose dependence on temperature is expressed by the Arrhenius equation:
\[ k_i=a_i \cdot \exp\left(-\frac{E_i}{R\cdot T}\right);\qquad i=1\ldots 8; \tag{2} \]
where \(a_i, E_i\) are the preexponential factor and the activation energy of the \(i\)-th reaction. The preexponential and exponential factors were selected simultaneously by means of a modified gradient method. This method satisfactorily overcomes the difficulties associated with the presence of “ravines” in the function being minimized, and is characterized by comparatively rapid convergence under the assumption that the function is unimodal. In contrast to the ordinary gradient method, the step size is changed depending on the ratio of the number of successful and unsuccessful steps at the last three points of the “descent” [3]. When the minimum value had been found, in the region of which the convergence of the method decreased significantly, the further search for the constants was carried out with the aid of
of the “pattern search” method [4], which underwent certain modifications. The modification includes scaling the increments of the various parameters in the process of gradient descent so that the influence of a unit increment of any of the variables would be the same. Thus an attempt was made to give the level surfaces of the function a spherical shape—symmetric in the direction of any independent variable. To determine the indicated 16 parameters, 15 experiments were selected, 5 for each of three temperatures: 30, 70, and 80°C [2], [5]. The experiments contain considerable errors, associated with the fact that the quantitative analysis of the phenol alcohols formed in the reaction has not yet been sufficiently developed. The indicated parameters were chosen from the condition of minimizing the sum of squared deviations:
\[ \sigma^2=\sum_{j=1}^{N}\sum_{i=1}^{n}\left(\bar{x}_i^j-x_i^j\right)^2;\qquad N=15;\quad n=8; \tag{3} \]
where \(x_i^j\) is the concentration of the \(i\)-th component in the \(j\)-th experiment. The overbar indicates membership in the experimental data. A program for minimizing the function \(\sigma^2\) by means of the “pattern search” method was developed for the “M-20” computer. The values of the parameters and of the minimized function, found by means of the modified gradient and “pattern search” methods, are presented in Table 1.
Table 1
| Parameters | Initial approximation | Gradient method Point close to the optimum |
“Pattern search” method Point close to the optimum |
|---|---|---|---|
| \(a_1\) | \(6.2\cdot10^{-6}\) | \(4.56\cdot10^{-2}\) | \(1.52\cdot10^{-4}\) |
| \(E_1\) | 20.5 | 21.35 | 17.3 |
| \(a_2\) | \(1.05\cdot10^{-5}\) | \(5.02\cdot10^{-2}\) | \(1.86\cdot10^{-4}\) |
| \(E_2\) | 21.0 | 20.9 | 14.3 |
| \(a_3\) | \(7.5\cdot10^{-6}\) | \(1.21\cdot10^{-2}\) | \(2.98\cdot10^{-4}\) |
| \(E_3\) | 20.7 | 21.6 | 7.2 |
| \(a_4\) | \(7.3\cdot10^{-6}\) | \(5.28\cdot10^{-3}\) | \(1.57\cdot10^{-4}\) |
| \(E_4\) | 21.0 | 21.3 | 6.02 |
| \(a_5\) | \(8.7\cdot10^{-6}\) | \(1.24\cdot10^{-3}\) | \(1.2\cdot10^{-4}\) |
| \(E_5\) | 20.6 | 21.35 | 20.9 |
| \(a_6\) | \(9.1\cdot10^{-6}\) | \(3.09\cdot10^{-3}\) | \(7.94\cdot10^{-5}\) |
| \(E_6\) | 20.5 | 21.0 | 19.4 |
| \(a_7\) | \(4.17\cdot10^{-5}\) | \(1.45\cdot10^{-9}\) | \(4.17\cdot10^{-6}\) |
| \(E_7\) | 21.0 | 20.6 | 23.3 |
| \(a_8\) | \(4.4\cdot10^{-7}\) | \(2.24\cdot10^{-5}\) | \(4.11\cdot10^{-8}\) |
| \(E_8\) | 20.5 | 21.05 | 21.0 |
| Values of \(\sigma^2\) | 2.7871 | 0.342 | 0.0238 |
As a result of the work, approximate values of the parameters of the kinetic equations were obtained. It was found that the magnitude of the change in the pre-exponential factors has a greater influence on the minimized function than does the exponent. The validity of the kinetic model can in practice be judged by checking the reliability and constancy of the obtained values of \(E_i\). It is rather difficult to use \(a_i\) for such a check, since there is no complete confidence in the perfection of the kinetic model or in the necessary accuracy of the experiments. There are grounds to believe that the values of the pre-exponential factors contain errors associated with the assumptions adopted, namely: the reaction rates have a more complex dependence on the catalyst concentration, and the reaction has a fractional order with respect to formaldehyde. The values of the activation energies are in partial agreement with the literature data [5]. It should be noted that, in view of the revealed multi-extremal character of the function \(\sigma^2\), the “ravine” method becomes more effective when the additions made are taken into account. Analysis of the data obtained leads to the conclusion that a more accurate estimation of the parameters and further refinement of the adopted model are necessary.
The authors express their gratitude to the staff of the Mathematical Modeling Laboratory of the L. Ya. Karpov Physicochemical Institute for kindly providing the description and instructions for the gradient method with modifications.
REFERENCES
- O. Manasse, Ber., 35, 3846, 1902.
- J. H. Freeman, C. W. Lewis. J. Amer. Chem. Soc. 76, 2080—2087, 1954.
- G. M. Ostrovskii, Yu. M. Volin. Methods for the Optimization of Chemical Reactors. Khimiya Publishing House, Moscow, 1967.
- I. M. Gel'fand, M. L. Tsetlin. Uspekhi matematicheskikh nauk, 17, issue 1, 1962.
- T. V. Vetoshkina. Dissertation, NIIKhP, K., 1966.