DETERMINATION OF A MATHEMATICAL MODEL OF THE EXTRACTION PROCESS USING FILTRATION AND MASS EXCHANGE EQUATIONS

61 Volume 5, Issue 10: Special Issue (EJAR) ISSN: 2181-2020 MPHAPP THE 6TH INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE “MODERN PHARMACEUTICS: ACTUAL PROBLEMS AND PROSPECTS” TASHKENT, OCTOBER 17, 2025 in-academy.uz DETERMINATION OF A MATHEMATICAL MODEL OF THE EXTRACTION PROCESS USING FILTRATION AND MASS EXCHANGE EQUATIONS Ilhamov Kh.Sh. Tashkent Pharmaceutical Institute, Tashkent city, Republic of Uzbekistan e-mail: [email protected] https://doi.org/10.5281/zenodo.17318408 Relevance: As in scientific and technical, industrial, social and other spheres, the study of processes through mathematical modeling based on digital technologies, which allows predicting their results, is becoming increasingly important in pharmaceutical production. The use of mathematical modeling, which saves both time and costs in the production of drugs or the creation of a new drug with an optimal composition, remains one of the pressing issues of pharmaceutical production. The purpose of the study: the main purpose of the article is to mathematically model the process of extracting useful substances from medicinal plant raw materials and to determine the important factors affecting the extraction process, the technological parameters of extraction and the characteristics of medicinal plant extracts, as well as to study the role and relevance of the use of mathematical modeling methods. Methods and techniques: The production of medicines based on medicinal plant collections requires the extraction of medicinal plant collections. Although there are several extraction methods, they can be divided into static and dynamic methods that can be carried out continuously or periodically. As is known, the crushed raw materials for extraction are loaded into a percolator (reactor) and lightly compressed. Then, an extractant is passed through the raw material layer at a certain speed, and this extractant leaves the extractor reactor enriched with biologically active and other extractive substances of the raw material. The efficiency of this process depends on the nature of the extractor, the hydrodynamic conditions of the process, the physical and mechanical properties of the raw material, the porosity of the layer, and many other factors. Therefore, determining the optimal values is very important for the technological parameters of the process. Based on these, we build a mathematical model of the process of washing the medicinal plant collection placed in a cylindrical reactor with an extractant. In this case, it is necessary to determine the optimal speed of the movement of the extractant through the layer of the medicinal plant collection, taking into account the particle size of the placed medicinal plant collection in relation to its placement density. We assume that the porous medium of the medicinal plant raw material in percolators and the extractor are mutually penetrating systems, and the extractant forms a filtration process with the plant raw material. In this case, we assume that the movement occurs only along the vertical z axis, and we assume that there is no movement along the horizontal x and y axes. For the mathematical model of the process, the model of interpenetrating systems of X.A. Rakhmatulin, for the filtration process, the equation of motion of the extractant in a porous medium according to Darcy's law and differential equations of mass exchange depending on temperature were obtained. After some mathematical transformations, a formula was obtained, which depends on the extractant pressure p0 for the separated substance, on Q - on the flow rate of the extractor, on the S-equivalent diameter of the porous mass, on the viscosity of the extractor, on the permeability of the porous mass consisting of medicinal plants and on other hydrodynamic parameters of the process. By adding to this series of differential equations, an idealized diffusion model of the extractor motion and differential equations of mass exchange depending on temperature, with equations taking into account boundary conditions, 62 Volume 5, Issue 10: Special Issue (EJAR) ISSN: 2181-2020 MPHAPP THE 6TH INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE “MODERN PHARMACEUTICS: ACTUAL PROBLEMS AND PROSPECTS” TASHKENT, OCTOBER 17, 2025 in-academy.uz we form a system of closed differential equations for the mathematical model of the extraction process. Results: The mathematical model of the extraction process, consisting of a system of differential equations with particular derivatives, was numerically solved in the MathCAD program, and the numerical results obtained were quite close to the experimental data, and it was calculated that the concentration of bioactive substances released during the extraction process reaches its maximum value at a temperature of 80°-90°C and a duration of the extraction process of 70-80 minutes. Conclusions: Solving the resulting equation allows us to determine the level of permeability of the extractant in the raw material layer. By increasing the pressure, we can change the flow rate of the extractant and achieve its maximum penetration into the medicinal plant collection. As a result, the hydraulic resistance of the medicinal plant raw material layer increases, and the release of bioactive substances increases due to the mass transfer process. The values of the hydrodynamic parameters of the process and their effect on the extraction allow us to determine the optimal parameters of the extraction process.