Proceedings of the Tomsk Polytechnic Institute Named after S. M. Kirov, Holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Vol. 257　　　　　　　　　　　　　　　　　　　　　　　　　1973

On the Calculation of Temperature Conditions in Chemical Reactors

A. I. Ruban, V. N. Klimova

The problem is solved of computing, from known kinetic equations, a temperature regime and initial concentrations that ensure an extremum of a certain quality function. For example, the quality criterion for an ideal-displacement reactor is often the maximum product yield.

In the present work it is shown that an effective way of solving the above-stated problem is the joint use of the Ritz method [1], adjoint equations [2], and Fibonacci numbers [3, 4].

At the first stage, the infinite-dimensional problem is reduced by the Ritz method to a finite-dimensional one, and from the solution of the adjoint equations the gradient direction with respect to the sought parameters is determined: the initial conditions of the kinetic equations and the coefficients in the expansion of the temperature dependence on contact time. Then, along the gradient direction, the extremum of the quality function is found with any prescribed accuracy by means of an iterative procedure based on the use of Fibonacci numbers. At the new point of the parameter space all calculations are repeated from the beginning, and we obtain an iterative algorithm for solving the problem.

Let us consider an example—the determination of the optimal temperature profile ensuring the maximum yield of product \(P\) in an ideal-displacement reactor for the reversible first-order reaction \(A \rightleftarrows P\).

The mathematical description of the reactor has the form

\[ \frac{dC(l)}{dl}=+K_1(T)[C_0-C(l)]-K_2(T)C(l), \qquad C(0)=0 \tag{1} \]

where \(C_0\) is the concentration of the feed at the reactor inlet, \(C(l)\) is the concentration of product \(P\) along the length of the reactor, \(l \in [0,1]\); a restriction is imposed on the reactor temperature, \(T(l)\le T_0\).

In this example, one can pass to a finite-dimensional problem by using—

using the maximum principle [1, 2]. Then, after the corresponding transformations, we obtain the equation for the temperature at \(T(\ell)\le T_0\)

\[ \frac{dT_{\mathrm{opt}}}{d\ell} = -RT_{\mathrm{opt}}^{2}\, \frac{K_2(T_{\mathrm{opt}})C_0}{E_1\bigl(C_0-C(\ell)\bigr)} . \tag{2} \]

Since a constraint is imposed on \(T(\ell)\), and the solution of (2) is a monotonically decreasing function, in the interval \([0,\ell_1]\) \(T_{\mathrm{opt}}(\ell)=T_0\), while in the interval \([\ell_1,1]\) it is determined by equation (2). Thus, the unknown parameter in this problem remains \(\ell_1\).

The search for \(\ell_1\) is carried out by means of a finite-step iterative procedure based on the use of Fibonacci numbers [3, 4]:

\[ u_1,u_2,\ldots,u_n, \quad \text{where} \quad u_1=u_2=1 \quad \text{and for } n>2 \quad u_n=u_{n-1}+u_{n-2}. \]

For a given \(n\), the error in calculating the value \(\ell_1\) does not exceed

\[ \frac{1}{u_{n+2}}. \]

For \(n=7\), a number of variants were calculated, two of which are given below.

I.

\[ K_1=0.5\cdot 10^8 \exp\left\{-\frac{10000}{RT}\right\}, \quad K_2=10^8 \exp\left\{-\frac{12000}{RT}\right\}, \quad C_0=1,\quad T=500^\circ\mathrm{C}. \]

As a result of the adjustment, \(\ell_1=0.46\pm 0.03\). Consequently, the optimal temperature on the segment \([0;0.46]\) is equal to the maximum, \(500^\circ\mathrm{C}\), and on the segment \([0.46;1]\), in accordance with equation (2), it decreases by \(40.4^\circ\mathrm{C}\).

II.

\[ K_1=0.5\cdot 10^8 \exp\left\{-\frac{9000}{RT}\right\}, \quad K_2=10^8 \exp\left\{-\frac{10000}{RT}\right\}, \quad C=1,\quad T_0=500^\circ\mathrm{C}. \]

For \(n=7\), \(\ell_1=0.577\), and in the interval \([0.577;1]\) the temperature profile falls to \(448.7^\circ\mathrm{C}\).

Thus, a successful combination of the maximum principle and a search procedure based on Fibonacci numbers makes it possible to solve, in a fairly simple way, certain problems of finding optimal temperature regimes for chemical reactors.

References

1. A. A. Fel’dbaum. Foundations of the Theory of Optimal Automatic Systems. “Nauka,” Moscow, 1966.

2. G. M. Ostrovskii, Yu. I. Volin. Methods of Optimization of Chemical Reactors. “Khimiya,” Moscow, 1967.

3. D. J. Wilde. Optimum Seeking Methods. “Nauka,” Moscow, 1967.

4. G. N. Vorob’ev. Fibonacci Numbers. “Nauka,” Moscow, 1969.