# mathematical model

## Kinetic Regularities and Mathematical Modelling of Potassium Chloride Dissolution

The dissolution process of potassium chloride particles in the apparatus with two-blade mechanical stirrer was investigated and the mass transfer coefficient was determined. The experimental results were generalized by criterion dependence. The independence of the mass transfer coefficient from the solid particles diameter was confirmed. A countercurrent process of potassium salt dissolution in two apparatuses with a mechanical stirring was considered. A mathematical model for countercurrent dissolution was developed and the efficiency of this process was determined.

## Phase Equilibrium of Petroleum Dispersion Systems in Terms of Thermodynamics and Kinetics

The process of paraffin formation has been considered, including the peculiarities of the paraffin structure as a result of phase transitions with a decreasing temperature. Mathematical models for thermodynamic and kinetic calculations of the "solid-liquid" system phase equilibrium have been developed. To shift the "fuel oil-paraffin" balance towards the liquid, it is necessary to reduce the activity ratio of solid and liquid phases by introducing into the system a substance with a lower solubility parameter.

## Analysis of the State of Territorial Communities to Model Their Socio-Economic Development

The problems of development of united territorial communities, in particular unemployment and economic problems, are considered. Communities, in most cases, lack the resources to address economic and other issues. Therefore, it is necessary to create self-sufficient communities in which there are enough financial instruments for their own development. The mathematical model of the decision support system for the development of territorial communities using the agro-industrial sector was considered.

## Mathematical Modeling of Area Boundary of Biaxial Absolutely Elastic States of Wood

In the paper, the mathematical model for determination of the region boundary of biaxial elastic states of orthotropic materials is synthesized and the system of nonlinear algebraic equations for identification of its parameters is obtained. Using the continuous method of solution continuation concerning the best parameter and the Runge-Kutta method, the demarcation curves of absolutely elastic and non-elastic deformation regions for pine trees are depicted.

## THE ROLE OF NITROGEN OXIDES IN THE PRESENCE OF SULFUR DIOXIDE IN THE DYNAMICS OF THE CARBON DIOXIDE ABSORPTION OFCHLOROPHYLL - PRODUCING MICROALGAE

The effect of nitrogen oxides in the presence of sulfur oxide on the absorption of carbon dioxide by chlorophyll – producing microalgae  Chlorella was investigated. Experimental dependences of the dynamics of CO2 uptake by microalgae in the presence of NxOy alone and at the critical concentration of the SO2 photosynthesis inhibitor in the presence of NxOy are presented.

## Analysis and improvement of design diagrams and mathematical models of vibratory lapping machines

Problem statement. The development of energy-efficient and high-performance vibratory lapping machines demands the improvement of their design diagrams and calculation techniques. Purpose. The main objectives of this research consist in detailed analysis of existent design diagrams and mathematical models of vibratory lapping machines; designing the three-mass hanger-type structures of such machines providing circular oscillations of laps; derivation of differential equations describing the motion of their oscillatory systems.

## Advanced asymptotic approaches and perturbation theory methods in the study of the mathematical model of single-frequency oscillations of a nonlinear elastic body

A combination of asymptotic methods in nonlinear mechanics with basic techniques of perturbation theory to study a mathematical model of the nonlinear oscillation system is proposed in the paper.  The system under consideration describes the torsional vibrations of an elastic body, where its elastic properties are under the nonlinear law.  The relationships presented as the ordinary differential equations are obtained due to the proposed procedure.  Therefore, the main parameters of the single-frequency oscillations and the resonance conditions can be determined.  There are proposed applica

## Wave concept of motion in mathematical models of the dynamics of two-dimensional media studying

The methodology of the studying of dynamic processes in two-dimensional systems by mathematical models containing nonlinear equation of Klein-Gordon was developed. The methodology contains such underlying: the concept of the motion wave theory; the single - frequency fluctuations principle in nonlinear systems; the asymptotic methods of nonlinear mechanics. The aggregate content allowed describing the dynamic process for the undisturbed (linear) analogue of the mathematical model of movement.

## Mathematical Modelling of Heat Transfer System of Convective Heating Surfaces of TPP-210А Steam Boiler

A mathematical model and the respective structural scheme of the convective heating surfaces of the TPP-210А steam boiler were developed as a system of interconnected heat exchangers. The interconnected convective heating surfaces are regarded as the convective heat transfer system of the boiler.

## Integral conditions in the inverse problems of heat conduction

Thermal processes of new technological methods of heat treatment (thermocyclic, electropulse) of metals and alloys are considered in the paper.  Mathematical models of the temperature field in a moving tape and a wire with cyclically acting pulsed heat sources are considered.  Based on these models, the formulation of inverse problems for homogeneous and inhomogeneous thermal conductivity equations is proposed.  For each case (internal, external heat source or a combination), the appropriate method for solving the inverse problem is proposed.  The integral condition of heat balance is used