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: khisamiddin@vail.ru
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.
