Variants of application of the least squares method in Szyszkowski and Rosin–Rammler approximations

Abstract

Relevance. Caused by the need to develop and optimize the mathematical apparatus for processing the results of laboratory experiments and increasing the adequacy of the results obtained. Aim. To create alternative methods for finding the parameters of the Szyszkowski and Rosin–Rammler dependencies, which are subject to surfactant adsorption from an aqueous solution on solid adsorbents and deposition of suspended particles in sedimentation analysis. Methods. The main method for determining the parameters of two-parameter dependencies is the least squares method. The standard approach is based on finding the minimum of a function of two variables by computational methods of nonlinear programming. The equations, obtained by equating the derivatives of the objective function for each of the parameters to zero, are used as necessary conditions for the minimum of the objective function. The paper considers alternative approaches to obtaining explicit formulas and reduction to the solution of the transcendental equation. Results. For the two-parameter dependencies of Szyszkowski and Rosin–Rammler, the alternative approaches for determining unknown parameters are proposed. In the standard approach, solving the problem is based on numerical minimization of a function of two variables by nonlinear programming methods. The authors propose the approach, in which the Szyszkowski and Rosin–Rammler equations are subjected to some equivalent transformations so that the use of the necessary minimum conditions makes it possible to obtain a linear equation with respect to at least one of the required parameters. This leads to simplification of calculations, it is required to solve one transcendental equation numerically, the second parameter is then determined by an explicit formula. And for the Rosin–Rammler dependence, in one of the proposed variants, it was possible to obtain explicit formulas for finding both parameters.