The mathematical model and numerical calculation method of efflorescence

The materials of the article consider the main methodological provisions of calculations and optimization of NPP power unit equipment parameters using mathematical modeling methods. For the effective implementation of tasks related to determining the optimal parameters and structures of NPP power unit equipment using mathematical modeling and well-developed multifactor optimization methods, it is necessary to fulfill a number of requirements at their productions. Practice shows that it is impractical to optimize with the help of a single mathematical (simulation) model the entire set of parameters characterizing a given power unit, because with such a formulation, the optimization tasks are often mutually incorrect due to a significant discrepancy in the accuracy of various source information, unequal influence of parameters on the target function, specific differences in the mathematical description of various units and elements of the power unit. In order to effectively optimize the parameters of NPP power units, it is necessary to create a system of interconnected mathematical models, which include: a group of detailed mathematical models of individual units and elements of power unit equipment; more generalized mathematical models for the main equipment of power units built on their basis; complete mathematical model of power units.

Наведено результати теоретичного та експериментального дослідження процесу розчинення калійної руди, основу якої складають сполуки калій хлориду та калій сульфату. Подано результати експериментального дослідження розчинення калійної солі у реакторі з механічним та пневматичним перемішуванням, який імітував одну з комірок промислового апарата. Фізично змодельовано цей процес у реакторі з механічним перемішуванням. Процес розчинення проведено за низьких чисел обертів, які становили 80 об/хв, що імітувало розчинення у промисловому шнековому апараті-розчиннику, у якому тверда фаза переміщається шнеком по дну апарата, а рідина рухається над поверхнею твердої фази, при цьому тверда фаза не переходить у зважений стан. Транспортування твердої солі шнеком перемішує її та оновлює поверхню контакту солі з рідиною. Визначення коефіцієнта масовіддачі проведено під час розчинення калійної руди у насиченому розчині натрій хлориду за відсутності у ньому іонів калію, так і за їх наявності, що відповідає промисловим умовам розчинення і є необхідною умовою розчинення для кристалізації калійної солі. Наведено основні параметри процесу розчинення калійної солі у насиченому розчині натрій хлориду. На основі рівняння масовіддачі визначено коефіцієнт масовіддачі під час механічного перемішування та пневматичного барботування розчину. Досліджено розчинення калійної солі за умов механічного перемішування та додаткової подачі нагрітого повітря. Наведено методики розрахунку системи апаратів-розчинників у протитечійно-прямотечійній схемі розчинення. Для промислового розчинника на основі рівнянь матеріального балансу та кінетики складено математичну модель, яка описує процес розчинення. Розглянуто систему трьох апаратів-розчинників, кожен з яких розділений вертикальною перегородкою на дві комірки, що імітують реактор ідеального перемішування. У кожному апараті фази рухаються прямотечійно. У системі трьох апаратів рух твердої та рідкої фаз відбувається протитечійно. Рішення математичної моделі представлено у графічному вигляді.

The article deals with peculiarities of application of mathematical methods and models in management of agrarian enterprises. Mathematical modeling is found to be a universal and effective tool for studying the internal laws of certain phenomena and processes. It has been stated that since the solution of economic and mathematical models it is necessary to take into account certain system characteristics and a close connection of qualitative system parameters and how they are resolved. It is noted that a specialist who develops a particular model should have a thorough knowledge of agrarian economics and modeling theory and perfect to have mathematical tools. It is proved that the application of mathematical modeling methods economic and manufacturing processes make efficient use existing resource base in the process of achieving the agrarian strategic goals businesses and take into account significant sources of uncertainty and minimize them negative economic consequences. The arsenal of economic and mathematical methods and models has been systematized used in marketing management decisions business processes. The features by which the economic and mathematical models are classified are presented used in agricultural production. The ways of constructing the optimization models used in agriculture. The process of making managerial decisions on business processes is considered agricultural production using the optimization model fertilizer application. It is noted that, from a mathematical point of view, some economic processes in agribusinesses are the same and can be described using typical models. It is established that by means of economic-mathematical modeling it is possible to solve specific economic problems and practical problems in the field management of agricultural production and economic activity enterprises. It is concluded that the application of mathematical tools methods and models make it possible to substantiate the adoption process better management decisions when planning and finding promotion reserves efficiency of agricultural enterprises in crisis conditions and dynamic changes in the environment.

Modelling of photo-electrocatalytic behaviour of TiO2 nanotubes under solar light irradiation

Floating solar photovoltaic (FPV) systems provide an effective approach to increase renewable energy production while avoiding competition for land resources. This paper introduces a mathematical modeling framework for the automated identification and preliminary assessment of water bodies suitable for FPV deployment. The methodology combines satellite remote sensing data (Sentinel-2, Landsat-8) with geospatial analysis and mathematical modeling techniques. Water bodies are delineated using the Normalized Difference Water Index (NDWI), after which filtering criteria—such as minimum surface area, seasonal stability, vegetation cover, and proximity to infrastructure—are applied. A multi-criteria mathematical scoring model is developed to evaluate and rank the suitability of candidate sites. Preliminary experiments confirm the ability of the proposed model to detect and prioritize high-potential areas, thereby reducing the need for extensive field surveys. The research advances the integration of mathematical modeling and geospatial methods for renewable energy planning, with ongoing work directed toward refining model accuracy and environmental adaptability.

It is known that the study and research of existing problems of medicine using modern information technologies is the most relevant today. Mathematical modeling of the activities of the regulatory mechanisms of living systems at the organismal, organ, cellular and molecular-genetic levels is one of the promising areas in the field of medical biology. The scientific research discusses the results of a quantitative study of the regulatory mechanisms of liver cells and hepatitis B viruses based on mathematical and computer models by using the Matlab application package. The study of the quasi-stationary state of the liver cell under the pressure of hepatitis B viruses using mathematical modeling methods is one of the urgent tasks. The use of mathematical modeling to analyze the interaction of the regulatory mechanisms of molecular genetic systems of liver cells and hepatitis B viruses makes it possible to analyze the main forms of infectious viral hepatitis B in a quasi-stationary state of liver cells. The results of a qualitative study of the equations of hepatitis B regulators in the quasi-stationary state of liver cells are presented.

Compositions of the equ‘diprated solid-liquid phases in the A1-CUA12, Al-Si and Pb-Sn Systems were exam ined by using Cameca microprobe microanalyser. Ali the solid and liquid phases were found to be uniform for the Systems. The m easured consentraticns of the solid and liquid phases in the A1-CU A12 system, the solid phases in the Al-Si system, the solid Pb and the liquid PbSn phases in the Pb-Sn system were in agreem ent w ith the phases diagram values. However the m easured compositions of the solid Sn phase and liquid Al-Si phases were found to be quite different from the phase diagram values. The difference for the solid Sn phase came from the semaring effect. and the liquid AlSi phase m ight came from the very different obsorbsi- tion coefficient for A lK a and S İK a lines. The corrected consantration of th e Iiquid AlSi phase was also in good agreem ent w ith the phase diagram values.

In recent decades in Romania, no flood management plan has been implemented in natural riverbeds, although there are known areas that face repeated floods such as the Siret River basin. Practically every year, floods produce uncontrolled erosions and landslides in certain areas, followed by the deposition of sediments, usually on agricultural land, compromising crops indefinitely. This paper analyzes the natural transport capacity of the Siret River based on direct measurements and data recorded during the floods of 2005 and 2020. The mathematical model of the sediment transport is presented, starting with the upstream zone, from the confluence with its main tributaries: Bistrita, Trotus, and Cracau. The recorded flood hydrographs are used in this analysis to model the sediment transport for variable flow rates. The upstream flood hydrograph, the steady downstream level, and the initial riverbed cross-sections represent the boundary conditions. The mathematical model is numerically tested for the risk zones by determining the modifications of the riverbed cross-sections. The variation in time of the liquid and solid phases allows the estimation of the longitudinal riverbed shape with the floodable surfaces. To mitigate the effects of floods—and to protect the population, agricultural lands, and environment—some solutions are finally proposed.

The issues related to the study of the possibilities of mathematical and software modeling of digital algorithms for discrete Fourier transform without algorithmic multiplication operations are considered. The relevance of the study is due to the need to reduce the computational complexity of digital signal processing algorithms. The purpose of the study is to formalize a numerical method for the multi–stage discrete Fourier transform of a digital complex signal. The methods of mathematical and software modeling of numerical methods of digital signal processing were used in the study. The results of the study showed and confirmed the possibility of modeling a numerical method of multi-stage discrete Fourier transform of a digital complex signal based on numerical methods of difference digital filtering with integer difference coefficients of various orders of difference and a numerical method of reducing direct calculations of complex functions to perform simple operations of addition and shift. The concept of a multi-stage discrete Fourier transform and the concept of this transformation are defined and described. Its mathematical model is proposed and the numerical method of such transformation is formalized. It is noted that it allows to reduce the computational complexity of the hardware and software implementation of digital signal processing through the use of numerical methods of differential digital filtering, directed enumeration and comparative analysis of acceptable sets of numerical values of integer coefficients, which provides the possibility of reducing the error of the numerical method of multi-stage discrete Fourier transform. To reduce the computational complexity of his algorithms for isolating the harmonic components of a digital complex signal, the principles of dividing its frequency spectrum into narrow-band spectral components, frequency shifting of these components and their low-frequency difference digital filtering without performing arithmetic multiplication operations are applied.

In the modern world, the introduction of mathematical methods in various subject areas. Today, many disciplines use methods of mathematical statistics, modeling, forecasting, and optimal planning. A student of a technical University majoring in «Mining», as a future engineer, often has to take into account many factors and rely on complex criteria for the effectiveness of ways to achieve final goals. Various modeling methods help you quickly find solutions to various problems. We offer students at the technical University to teach not only the fundamentals of higher mathematics, but also to teach them to build mathematical models and solve applied problems. Mathematical modeling helps students to see mathematics in life and future profession. It contributes to the development of mathematical competence of students, which is one of the components of their professional competence.

The paper consists of three sections and is of an informational and generalizing nature, indicating promising areas for further research. The first section "R-functions method in mathematical modeling of convective heat transfer in fuel cartridges with fuel elements" is devoted to the use of new constructive tools of the R-functions method for mathematical and computer modeling of fuel elements packings with different types of symmetry, as well as the study of convective heat transfer in fuel elements grids and the effect of the type of packing on the distribution of velocity and temperature. An octahedral cartridge with 37 fuel elements packed according to three patterns (cyclic, checkerboard and in-line) is considered. It is noted that when constructing the equations of a cartridge with bundles of fuel elements using the new method, the number of R-operations and, accordingly, the calculation time are significantly reduced. An analysis of the obtained results allows to conclude that the maximum temperature is obtained with cyclic packing. The scheme of the reactor, the cartridges of which are hexagonal casings, with 91 fuel elements placed in each of them both with checkerboard and cyclic packing, is also considered. In the second section "Thermal-hydraulic calculation of fuel elements cartridges in case of violation of the rods packing symmetry", a hexagonal fuel cartridge with 169 fuel elements and checkerboard packing is considered. An increase in temperature is analyzed in case of violation of the packing symmetry while maintaining the parallelism of the rods, as well as in case of a curvature of one of them. The third section "R-functions, fuel element with polyzonal finning of the shell and heat transfer during fluid motion" is focused on the construction of equations for various finning surfaces of fuel elements and the study of hydrodynamic and temperature fields in case of polyzonal finning of the shell. At the same time, using the apparatus of tensor analysis, a transition to a curvilinear non-orthogonal (helical) coordinate system was made. It is noted that mathematical modeling and the associated computer experiment are indispensable in cases where a full-scale experiment is impossible or difficult to conduct for one reason or another. In addition, working with mathematical model of the process and the computational experiment make it possible to investigate the properties and behavior of the process in various situations relatively quickly and without significant expenses. The reliability of the methods, results and conclusions is confirmed by comparison with the information given in the references, the results of the analysis of the numerical convergence of solutions and the calculation of the residual.

The most soluble component of Portland cement-based cement stone is calcium hydroxide. The intensity of liquid corrosion of concrete is determined by the rate of leaching of cement stone, that is, the rate of removal of calcium hydroxide from the concrete body. When the temperature decreases, the solubility of calcium hydroxide increases, and therefore, the rate of corrosion processes that occurring in concrete under the influence of liquid media increases. Reducing the water absorption of concrete allows to control the intensity of mass transfer processes occurring during corrosion in concrete and affects the durability of concrete and reinforced concrete products. The results of the study of the kinetics of cement concrete leaching showed that in the samples with hydrophobic additives state close to equilibrium in liquid corrosion is achieved earlier. This is due to the fact that the content of “free calcium hydroxide” in hydrophobized samples is less, while the possibility of developing corrosion of concrete is reduced. The data obtained as a result of the experiment make it possible to determine the concentration value for the portable component “free calcium hydroxide” over the entire thickness of the concrete structure at any time and with the help of a mathematical model of concrete corrosion of the first type to calculate the mass transfer parameters and establish the service life of the concrete product. The introduction of hydrophobic additives in the composition of cement concrete and temperature control during operation of the concrete product allows providing the required durability and reliability of structures made of concrete.

This paper presents modern methods of mathematical modeling, which are widely used in the development of new inhalation and intranasal drugs, including those necessary for the treatment of socially significant diseases, which include: tuberculosis, bronchial asthma, and mental and behavioral disorders. Based on the conducted studies, it was revealed that the methods of mathematical modeling used in the development of drugs are fragmented, and there is no single approach that would combine the existing methods. The results presented in the work should contribute to the development of a unified multiscale model as a new approach in mathematical modeling that contributes to the accelerated development and introduction to the market of new drugs with high bioavailability and the required therapeutic efficacy.

The variety of fabric structures is divided into four parts as wovens, knitts, braids and nonwovens. Comparing with other fabrics, woven fabrics display both good dimensional stability in the warp and weft directions and highest cover yarn packing density. One of the most important features for the characterization of woven fabric quality and fabric performance is tensile properties of fabric strength. Even though the end products of spinning and weaving are woven fabrics, they are raw materials for clothing and other industries such as composites and medical textiles. Every piece of woven fabric is an integration of warp and weft yarns through intersection. The extent of this intersection is largely dependent on the friction between fibres and yarns. The study of woven fabric mechanics is came across in a work reported by Haas in the German aerodynamic literature in 1912 during development of airships. In English literature, the paper by Peirce (1937) who is the pioneer in the investigation of tensile deformation of woven fabrics presented a geometrical and a mathematical force model of the plain-weave structure which is highly theoretical, both of which have been used extensively and modified by subsequent workers in the field. Many researchers modified his geometrical model to analyze tensile behaviour. Considerable progress has been made over the last century in the development of the theory of geometrical structure and mechanical properties of fabrics. The mathematical modelling of fabric stress– strain relationships is a very tough topic. During the last 60 years, many outstanding textile scientists have dedicated to their talents to this field. The development of mathematical models for woven fabrics is an extremely complicated and difficult task due to the large numbers of factors on which the behaviour of the fabric depends. Usually, a mathematical model requires a large number of assumptions, covering missing knowledge or inability to express some of the relevant factors. Mathematical models based on the fundamental mechanics of woven fabrics often fail to yield satisfactory results, as it is hardly possible to incorporate all the complexities in the model. Moreover, the application range of mathematical models is also very specific. Therefore, it is necessary to introduce a different approach for the mathematical modelling of fabric constitutive equations. With fabric, fundamental distinctions may be made between three kinds of modelling, namely: predictive, descriptive and numerical models. The predictive models which form most of the existing research into fabric mechanics are based

Novel mathematic models for quantitative transitivity of quality-markers in extraction process of the Buyanghuanwu decoction