Problem Solution Domain Research Articles

Solution of a two-dimensional parabolic model problem in a degenerate angular domain

In this paper, the boundary value problem of heat conduction in a domain was considered, boundary of which changes with time, as well as there is no the problem solution domain at the initial time, that is, it degenerates into a point. To solve the problem, the method of heat potentials was used, which makes it possible to reduce it to a singular Volterra type integral equations of the second kind. The peculiarity of the obtained integral equation is that it fundamentally differs from the classical Volterra integral equations, since the Picard method is not applicable to it and the corresponding homogeneous integral equation has a nonzero solution.

Discretization of an unsteady spatial model describing the solidification process of an ingot in a mold of a continuous casting machine

Continuous casting machines (CCM) provide high quality metal. Optimization of technological processes, at present, cannot be based only on an empirical approach based on a generalization of production experience. This is due to the fact that each caster is a complex technical system with its own specifics and features. The improvement of continuous casting technology is based, first of all, on the creation of mathematical models describing technological processes taking into account many technological and structural factors. When modeling crystallization processes, most researchers, in order to simplify the solution of the problem, consider stationary plane-parallel models. Such models cannot describe the studied phenomenon well in principle and therefore have a limited scope. Therefore, the development of methods for solving the unsteady spatial model of continuous casting is of great scientific interest and expands the possibilities of solving very complex boundary value problems with phase transitions. The finite element method (FEM) is one of the most universal numerical methods for solving differential and integral equations and their systems. FEM leads to a system of algebraic equations. The article describes the use of FEM for numerical modeling of the solidification process of an ingot in a continuous casting mold with the aim of conducting computational experiments. The discretization process of the system of differential equations and the problem solution domain is presented. In addition, an algorithm of finite-difference approximation of non-stationary members of the equations is described. To sample the region, three-dimensional finite elements in the form of a parallelepiped with nodes located at the vertices of the elements are used. The solution is in the space of piecewise linear functions.

Mathematical modeling of heat transfer in a closed two- phase thermosyphon

We suggested a new approach for describing heat transfer in thermosyphons and determining the characteristic temperatures. The processes of thermogravitation convection in the coolant layer at the lower cap, phase transitions in the evaporation zone, heat transfer as a result of conduction in the lower cap are described at the problem statement. The main assumption, which was used during the problem formulation, is that the characteristic times of steam motion through the thermosyphon channel are much less than the characteristic times of thermal conductivity and free convection in the coolant layer at the lower cap of the thermosyphon. For this reason, the processes of steam motion in the thermosyphon channel, the condensate film on the upper cap and the vertical walls were not considered. The problem solution domain is a thermosyphon through which heat is removed from the energy-saturated equipment. The ranges of heat flow changes were chosen based on experimental data. The geometric parameters of thermosyphon and the fill factors were chosen the same as in the experiments (height is 161 mm, diameter is 42 mm, wall thickness is 1.5 mm, ε=4-16%) for subsequent comparison of numerical simulation results and experimental data. In the numerical analysis it was assumed that the thermophysical properties of thermosyphon and coolant caps do not depend on temperature; laminar flow regime was considered. The dimensionless equations of vortex, Poisson and energy transfer for the liquid coolant under natural convection and the equations of thermal conductivity for the lower cap wall are solved by the method of finite differences. Numerical simulation results showed the relationship between the characteristic temperatures and the heat flow supplied to the bottom cap of thermosyphon. The results of the theoretical analysis are in satisfactory agreement with the known experimental data.

Simulation of Industrial Wastewater Treatment from the Suspended Impurities into the Flooded Waste Mining Workings

The paper is dedicated to the mathematical model of slurry wastewater treatment and disposal in a flooded mine working. The goal of the research is to develop and analyze the mathematical model of suspended impurities flow and distribution. Impurity sedimentation model is under consideration. Due to the sediment compaction problem solution domain can be modified. The model allows making a forecast whether volley emission is possible. Numerical simulation results for “Kolchuginskaya” coal mine presented. Impurity concentration diagrams in outflow corresponding to the real full-scale data obtained. Safely operation time mine workings like a wastewater treatment facility are estimated. The carried out calculations demonstrate that the method of industrial wastewater treatment in flooded waste mine workings can be put into practice but it is very important to observe all the processes going on to avoid volley emission of accumulated impurities.

Simulation of Domestic and Industrial Wastewater Disposal in Flooded Mine Workings

The paper is dedicated to the mathematical model of domestic and industrial wastewater treatment and disposal in a flooded mine working. The goal of the research is to develop and analyze the mathematical model of suspended impurities flow and distribution. Impurity sedimentation model is under consideration. Due to the sediment compaction problem solution domain can be modified. Impurities flow and distribution patterns are presented.