Calculation of Experimental Values of Quality Indicators of Chemical-Technological Processes
Abstract Generated abstract
This paper addresses the determination of maximum or minimum quality indicators in chemical-technological processes after experimental response functions have been approximated by first- or second-degree polynomials. It proposes a simplified method for quadratic programming problems with bounded input variables, reducing a second-degree response surface to canonical form by Lagrange’s method and then examining extrema on the faces of a multidimensional rectangular admissible region. The authors emphasize that, for problems with up to three variables, the procedure requires only a small finite number of steps and is computationally accessible to process engineers. The method is reported to have been applied to calculating extreme quality indicators in hydrocracking processes for producing pure benzene, naphthalene, and mesitylene.
Full Text
CALCULATION OF EXTREMAL VALUES OF QUALITY INDICATORS FOR CHEMICAL-TECHNOLOGICAL PROCESSES
A. I. Ruban, L. A. Granskaya, G. D. Litvyakova
After conducting experiments in the region of interest and obtaining in it an adequate mathematical description of the response function in the form of a first- or second-degree power polynomial, it is necessary to find such a set of input variables that ensures the maximum (or minimum) of the response function. At the same time, restrictions are imposed on the input variables, singling out the region of interest to the experimenter. In this case we have a problem of either linear or quadratic programming. If the first of these is solved comparatively simply, the second requires complex solution algorithms, whose implementation even on a digital computer entails considerable difficulties.
In the present work, a simple and effective method is proposed for solving problems of quadratic programming. Its idea reduces to the following.
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The second-degree polynomial describing the response surface is reduced to canonical form by Lagrange’s method. The canonical form makes it possible to investigate the surface for an extremum and to determine the coordinates of the center of the surface. Most often, the extremum of interest to the researcher lies on the boundary of the region of variation of the input variables, and then the problem reduces to investigating the boundary. Let us consider the most common case, when the admissible region is a multidimensional rectangle.
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The response surface is investigated on each face of the rectangle. This problem is analogously reduced to the search for an extremum with respect to one independent variable. After a finite and small number of steps (when the number of variables is $\leq 3$), the global one is selected from the extrema obtained, and this is the solution of the problem.
The approach described above is simple from a computational standpoint and accessible to any process engineer. The algorithm was used to calculate the extreme values of the quality indicator for hydrocracking in the production of pure benzene, naphthalene, and mesitylene.