Elsevier

Computers & Chemistry

Volume 23, Issue 2, 30 March 1999, Pages 143-151
Computers & Chemistry

The kinetic evolutionary modeling of complex systems of chemical reactions

https://doi.org/10.1016/S0097-8485(99)00005-4Get rights and content

Abstract

To overcome the drawbacks of most available methods for kinetic analysis, this paper proposes a hybrid evolutionary modeling algorithm called HEMA to build kinetic models of systems of ordinary differential equations (ODEs) automatically for complex systems of chemical reactions. The main idea of the algorithm is to embed a genetic algorithm (GA) into genetic programming (GP) where GP is employed to optimize the structure of a model, while a GA is employed to optimize its parameters. The experimental results of two chemical reaction systems show that by running the HEMA, the computer can discover the kinetic models automatically which are appropriate for describing the kinetic characteristics of the reacting systems. Those models can not only fit the kinetic data very well, but also give good predictions.

Introduction

During a chemical engineering production process, in order to optimize the chemical reaction process, to restrain secondary reactions, to maximize the yield, and to improve the safety of operation, chemical engineers often need to carry out kinetic analysis for complex systems of chemical reactions. This is usually considered an arduous task. On one hand, chemical reactions involve complex stoichiometry and thermodynamics. On the other hand, heterogeneous reactions are concerned with mass diffusion, and the reaction rate is affected by the speed of agitation, by interfacial area, by diffusion coefficients and by many other factors. The traditional method for kinetic analysis is to build exact kinetic models only for simple chemical reactions, based on molecular kinetics. However due to the complexity of chemical reaction processes and the short market window of chemical products, it is usually not worthwhile for producers and developers to spend a lot of time studying the detailed mechanism of complex reactions. Hence, it is necessary to find approximate kinetic models for these complex reaction systems by using fast and effective methods, so as to provide a basis for the optimization of chemical reaction processes in further steps. At present, the most utilized approaches for kinetic analysis are the tendency modeling and approximation methods. The tendency modeling, i.e. the so-called gray model method (Filippi et al., 1986, Deng, 1985), which is based on a phenomenological approach, reduces the complex chemical reactions to simple kinetic equations by considering general laws in chemical reactions such as mass and energy balances, and then estimates a small number of kinetic parameters by using regression, integral methods (Himmelblau et al., 1967) and differential methods (Kennard and Melsen, 1985). Approximation methods, i.e. the so-called “black box” methods (Galvan et al., 1996, Cao et al., 1998), do not take into account the physical characteristics of practical reacting systems in principle but employ approximating functions including polynomials, trigonometric series, splines and so on, to fit experimental data by adjusting the parameters. Nevertheless, people often feel troubled when applying these two approaches to practical problems. Firstly, it is difficult to choose an appropriate kinetic model to describe the reacting system due to the large amount of complex kinetic data. Secondly, the estimation of parameters requires the modeler to have a rich mathematical knowledge and chemical professional skill. The computation process is also usually rather complicated.

Recently, due to the merits of self-adaptation, self-organization, self-learning, intrinsic parallelism and generality, evolutionary algorithms (EAs) have been successfully applied in a wide range of economic, engineering and scientific computations (Goldberg, 1989, Mitchell, 1996). The applications of EAs in chemistry are also very wide (Hibbert, 1993, Lucasicus and Kateman, 1994). EAs are adaptive methods for solving computational problems in many fields, which mimic the process of biological evolution and the mechanisms of natural selection and genetic variation. They use suitable codings to represent possible solutions to a problem, and guide the search by using some genetic operators and the principle of “survival of the fittest”. EAs originally consist of three branches, namely genetic algorithms (GAs), evolutionary programming (EP) and evolution strategies (ES). In the 1990s, a new branch called genetic programming (GP), was added to the group which was introduced by John Koza (Koza, 1992, Koza, 1994). GP is an extension of John Holland’s GA (Holland, 1975) in which the genetic population consists of computer programs of varying sizes and shapes. In standard GP, computer programs can be represented as parse trees, where a branch node represents an element from a function set, which usually contains some arithmetic operations and elementary functions of at least one argument, and a leaf node represents an element from a terminal set, which usually contains variables, constants and functions of no arguments. These symbolic programs are subsequently evaluated by running them on a set of “fitness cases”. Fitter programs are selected for recombination to create the next generation by using crossover and mutation. This step is iterated for some number of generations until the termination criterion of the run has been satisfied.

As the kinetic behaviors of most chemical reactions need to be described by a system of ordinary differential equations (ODEs), our research aims to build the kinetic ODEs model automatically based on the kinetic data by using GP. However, when we apply the standard GP to the modeling problem of ODEs, one major problem arises. Since the fitness value of a model depends largely upon the values of its parameters, a model with a favorable structure will have a great probability of being eliminated from the population during the evolution, if the randomly generated parameters are inappropriate. Consequently it is unlikely that we will obtain a highly accurate model for the system. Moreover the evolutionary process can be slow due to the large number of generations needed, as well as suffering from the “premature convergence” phenomenon. Besides, as an ODEs model is composed of multiple differential equations, the representation of single tree in standard GP is no longer appropriate to our problem. In order to overcome these drawbacks, we propose a hybrid evolutionary modeling algorithm called HEMA to build kinetic ODEs models automatically for complex systems of chemical reactions. The main idea of the algorithm is to embed a GA into GP, where GP is employed to optimize the structure of a model, while a GA is employed to optimize its parameters.

This paper is organized as follows. In Section 2, we present the structure of the HEMA and give its detailed descriptions. In Section 3, two examples of chemical reacting systems are used to test the effectiveness of the HEMA, and their experimental results and some discussions are also given here. Finally in Section 4, we give some conclusions.

Section snippets

HEMA

The HEMA mainly consists of two processes: one is the evolutionary modeling process used to optimize the structure of models based on GP, and the other is the parameter optimization process used to optimize the parameters of a model based on a GA. These two processes, accompanied by the simplification and normalization of the models, and the system prediction, constitute the framework of the HEMA. The structure of the HEMA can be described in pseudo code as follows:

  • Procedure HEMA;

  • begin

  • input the

Parameter settings and measures

To examine the effectiveness of the HEMA, we apply it to two chemical reaction systems with different numbers of variables and build the kinetic models of systems of ODEs for them. Twenty runs are conducted independently for each example and the best model is shown and discussed. All the experiments are performed on a Pentium II (266 MHz) using Visual C++ compilers. The parameter settings are as follows:

  • For the evolutionary modeling process we use the function set F={+, −, *, /, ^, sin, cos,

Conclusions

To overcome the drawbacks of most available methods for kinetic analysis, this paper proposes a hybrid evolutionary modeling algorithm called HEMA to build kinetic models of systems of ordinary differential equations (ODEs) automatically for complex systems of chemical reactions. The main idea of the algorithm is to embed a genetic algorithm (GA) into genetic programming (GP) where GP is employed to optimize the structure of a model, while a GA is employed to optimize its parameters.

Compared

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (no. 69635030) and National 863 High Technology Project of China. The authors would like to thank the anonymous referees for their helpful comments on the paper and to thank Dr Hugo de Garis for his efforts in improving the paper’s English and style.

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