5 research outputs found
CO2-free coal-fired power generation by partial oxy-fuel and post-combustion CO2capture: Techno-economic analysis
Among the carbon capture and storage (CCS) technologies suitable for power generation plants, partial oxy-combustion coupled with post combustion CO2capture is gaining interest, since such a hybrid configuration could allow to reduce the size and enhance the performance of post-combustion CO2capture by operating combustion with air enriched with oxygen and reducing the dilution of flue gas. Moreover, partial oxy-combustion is a potential candidate for the retrofit of existing steam plants because it could be based on an almost conventional boiler and requires a smaller CO2capture section. This work presents the results of a comparative techno-economic analysis of a 1000 MWthpartial oxy-combustion plant based on an ultra-supercritical pulverized coal combustion power plant integrated with a post-combustion CO2capture system and geological storage in saline aquifer. In particular, plant performance is assessed by using simulation models implemented through Aspen Plus 7.3 and Gate Cycle 5.40 commercial tools, whereas economic performance are evaluated on the basis of the expected annual cash flow. The analysis shows that, for new plants, this hybrid approach is not feasible from the economic point of view and full oxy-combustion potentially remains the most profitable technology even if, in the short-term period, the lack of commercial experience will continue to involve a high financial risk
Application of life cycle assessment in process design: case study on SOβ abatement technologies in the PGM sector
Platinum group elements (PGEs) are increasingly being used in a variety of environmentally-related technologies such as catalysts and catalytic converters which have strong expected growth to meet environmental and technological challenges this century. The platinum industry is actively seeking to progress its commitment to sustainability principles by reducing the negative impacts of their mining and mineral processing operations. Technical innovation to improve future plant designs, as well as the development of management policies, guidelines and protocols for efficient operation of process plants has therefore become a strategic priority for the South African platinum industry. The industry has also made an effort to understand the environmental impacts of its products from mine to metal, using life cycle methods. However, very limited research has been done to investigate what environmental value could be created if strategic and design decisions in minerals processing were life cycle based, particularly in the context of PGMs. Seminal work by Stewart (1999) investigating the environmental life cycle consideration for design-related decision making in the minerals industry has not led to significant adoption. Forbes et al. (2000) analysed metal processing using LCA and were able to identify opportunities for improved environmental performance. They however did not explore how it would be incorporated into the decision making cycle. Therefore, the main objective of this research is to determine whether life cycle assessment could help inform design decision making in the minerals industry. In the years 2002-2008 several PGM-producing companies commissioned new SOβ scrubbing technologies to meet the regulations that had been set to prevent the release of excessive amounts of sulphur dioxide from smelters in the Rustenburg area, a mining town located in the North West Province of South Africa. Using these cleanup process retrofits as case studies, this dissertation aims to determine whether the introduction of LCA as an environmental analysis tool would have provided additional value to the decision makers. The case study approach that was chosen compared and assessed the performance of SOβ abatement technologies and the effect of efficiencies chosen on environmental performance by using life cycle assessment modelling. By doing the life cycle assessment on the different options that the companies had, it was possible to evaluate the indirect environmental impacts that could have been overlooked during the design decision making process. In addition , experts who were involved in the design processes of the SOβ abatement retrofits were interviewed to establish: i) how the design decisions were made and ii) whether the life cycle based insights into technology performance would have been of use in the design work. The goal of the life cycle assessment was to identify whether there were design decisions that induced environmental burden shifting when platinum smelters in the Rustenburg area added SOβ abatement technologies to their processes, which could have been avoided had the LCA perspective been taken into account. The assessments considered two key variables, namely extent of recovery and technology choice
Modelling of flue gas desulfurization process by sorbent injection into the pulverized coal-fired utility boiler furnace
Environmental problems during energy conversion from coal into electric power are of great
importance and must be addressed as such. Before undertaking measures to improve existing
utility boilers, or during planning and building new plants, detailed analysis are required,
considering both techno-economic and the environmental issues. During the middle of the last
century a rapid development of computers started, and at the same time computers became
affordable and available to the end user. Thus, the 21st century becomes the era that will be
marked by significant changes in computer structure, possibilities and use. Advances in
computer development allowed for improvement of the computational methods in mechanical
engineering and in other fields as well. Process control and plant design with the aid of
computers are becoming everyday task and allow dealing with engineering problems that
have previously been unsolvable and required empirical approach.
One of the major contributors to environmental pollution is the emission of pollutants from
large stationary sources, that is, more precisely, from the pulverized coal powered utility
steam boilers. The subject of research in the dissertation is numerical modelling of complex
processes in utility boiler furnace during direct injection of pulverized calcium-based sorbent
(limestone, or lime) into the furnace for sulfur oxides reduction, with the model development,
as well as numerical analysis and optimization of the processes as the primary goals. Process
is well known in theory, however, as it can be found in the literature, the sorbent behavior
during the furnace sorbent injection is still not understood enough, and thus on the full-scale
plants the efficiency of the process significantly varies. Problems and the causes of significant
drops in efficiency can be attributed to the poor process control. Numerical modeling allows
for investigation of furnace behavior during various configurations of the sorbent injection
process, before any changes are made at the plant itself, which is of primary importance
during analysis and decision making about directions of the changes and upgrades of the
existing plants, and can give good ideas about the design of the new plants.
Developed software for three-dimensional furnace calculation includes differential model of
flow and heat transfer processes, combustion reactions model, nitrogen oxides formation and
destruction reactions model, and two selected and optimized models of sorbent particle
reactions with sulfur oxides from furnace gasses, applied within the comprehensive model of
furnace processes. A k-Ξ΅ model is used for turbulence modeling, while the radiative heat
exchange is modelled by using the six fluxes model. Two-phase gas-particle turbulent flow is
modeled with Euler-Lagrangian approach. Interaction between gas phase and particles is
treated by PSI-Cell method, with transport equations for gas phase having source terms that
takes into account the particles influence.
Significance of development and application of such a software for calculations is mostly
notable in possibility to perceive and analyze processes inside of the furnace which cannot be
analyzed and (the entire system cannot be) predicted by other means. Understanding the
behavior of the boiler furnace during certain operation regimes, with the use of various fuels,
as well as under modifications such as the furnace sorbent injection is of great importance,
and represents a prerequisite for achieving efficient, reliable and environmentally friendly
boiler operation with compromises between the three, important but to some extent opposed
conditions.
Particular attention is devoted to the modeling of pulverized sorbent furnace injection,
regarding that a primary goal is investigation of possibility to reduce sulfur oxides emission
by means of direct sorbent injection into the boiler furnace. Problem is approached through
several phases, starting with the analysis of selected models of calcination, sintering and
sulfation reactions, their stability and behavior in two-dimensional simulated reactors with
focus on comparison with available experimental results in order to validate the models
implementation. In further study, models are implemented in three-dimensional numerical
code for simulation of in-furnace processes, with particular interest to observe, beside the
sorbent influence on sulfur oxides content, the influence it has on the furnace exiting gas
temperature and other relevant process parameters in the furnace.
During the research, a complex numerical study of the furnace sorbent injection possibilities
and accompanying phenomena was performed. Sorbent injection was simulated through the
burner tiers, and through the special injection ports above the burner tiers, individually and in
combination. Process was analyzed for several fuels with different heating values and varied
sulfur content, and various the impacts of different operation regimes and combustion
configurations on the gaseous combustion products at the furnace exit were shown. Influence
of wide range of desulfurization process parameters was considered, such as: sorbent injection
position and particle distribution, particle temperature history and residence time, local gas
temperature within the furnace, calcium β sulfur molar ratio, local sulfur oxides concentration,
local oxygen concentration, etc. Conclusions were drawn considering possibilities for direct
sorbent injection into the pulverized coal fired boiler furnace, as well as suggestions were
given on optimal furnace sorbent injection configuration, depending on the boiler operation
parameters.
The developed software includes a user interface for easier data input for the case-study boiler
furnace, allowing for easier boiler analysis, and provides engineering staff with a tool for an
efficient software control, with the purpose of considering and analyzing better the furnace
sorbent injection technology and its potential applications in the utility boiler furnaces.ΠΠΊΠΎΠ»ΠΎΡΠΊΠΈ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠΈ ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΠΏΡΠ΅ΡΠ²Π°ΡΠ°ΡΠ° Π΅Π½Π΅ΡΠ³ΠΈΡΠ΅ ΡΠ°Π΄ΡΠΆΠ°Π½Π΅ Ρ ΡΠ³ΡΡ Ρ Π΅Π»Π΅ΠΊΡΡΠΈΡΠ½Ρ
Π΅Π½Π΅ΡΠ³ΠΈΡΡ ΡΡ ΠΎΠ΄ ΠΈΠ·ΡΠ·Π΅ΡΠ½ΠΎΠ³ Π·Π½Π°ΡΠ°ΡΠ° ΠΈ ΠΏΠΎΡΠ²Π΅ΡΡΡΠ΅ ΠΈΠΌ ΡΠ΅ ΠΏΠΎΡΠ΅Π±Π½Π° ΠΏΠ°ΠΆΡΠ°. ΠΡΠ΅ ΠΏΡΠ΅Π΄ΡΠ·ΠΈΠΌΠ°ΡΠ°
ΠΊΠΎΠ½ΠΊΡΠ΅ΡΠ½ΠΈΡ
ΠΌΠ΅ΡΠ° Π½Π° ΡΠ½Π°ΠΏΡΠ΅ΡΠ΅ΡΡ ΠΏΠΎΡΡΠΎΡΠ΅ΡΠΈΡ
ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠ°, ΠΈΠ»ΠΈ ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΠΏΠ»Π°Π½ΠΈΡΠ°ΡΠ° ΠΈ
ΠΈΠ·Π³ΡΠ°Π΄ΡΠ΅ Π½ΠΎΠ²ΠΈΡ
ΠΏΠΎΡΡΠ΅Π±Π½ΠΎ ΡΠ΅ ΠΈΠ·Π²Π΅ΡΡΠΈ Π΄Π΅ΡΠ°ΡΠ½Π΅ Π°Π½Π°Π»ΠΈΠ·Π΅, ΠΊΠ°ΠΊΠΎ ΡΠ΅Ρ
Π½ΠΎ-Π΅ΠΊΠΎΠ½ΠΎΠΌΡΠΊΠ΅, ΡΠ°ΠΊΠΎ ΠΈ
Π°Π½Π°Π»ΠΈΠ·Π΅ ΡΡΠΈΡΠ°ΡΠ° Π½Π° ΠΆΠΈΠ²ΠΎΡΠ½Ρ ΡΡΠ΅Π΄ΠΈΠ½Ρ. Π‘ΡΠ΅Π΄ΠΈΠ½ΠΎΠΌ ΠΏΡΠΎΡΠ»ΠΎΠ³ Π²Π΅ΠΊΠ° ΠΎΡΠΏΠΎΡΠ΅ΠΎ ΡΠ΅ ΡΠ±ΡΠ·Π°Π½ ΡΠ°Π·Π²ΠΎΡ
ΡΠ°ΡΡΠ½Π°ΡΠ°, ΡΠ· ΠΈΡΡΠΎΠ²ΡΠ΅ΠΌΠ΅Π½ΠΎ ΠΏΠΎΡΠ΅ΡΡΠΈΡΠ΅ΡΠ΅ ΠΈ Π΄ΠΎΡΡΡΠΏΠ½ΠΎΡΡ ΠΊΡΠ°ΡΡΠ΅ΠΌ ΠΊΠΎΡΠΈΡΠ½ΠΈΠΊΡ, a 21. Π²Π΅ΠΊ ΡΠ΅
ΡΡΠΎΠ»Π΅ΡΠ΅ ΠΊΠΎΡΠ΅ ΡΠ΅ ΠΎΠ±Π΅Π»Π΅ΠΆΠΈΡΠΈ ΠΈ Π²Π΅Ρ ΠΎΠ±Π΅Π»Π΅ΠΆΠ°Π²Π°ΡΡ Π·Π½Π°ΡΠ°ΡΠ½Π΅ ΠΏΡΠΎΠΌΠ΅Π½Π΅ Ρ ΡΡΡΡΠΊΡΡΡΠΈ ΡΠ°ΡΡΠ½Π°ΡΠ°,
ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈΠΌΠ° ΠΈ ΡΠΏΠΎΡΡΠ΅Π±ΠΈ. ΠΠ°ΠΏΡΠ΅Π΄Π°ΠΊ Ρ ΡΠ°Π·Π²ΠΎΡΡ ΡΠ°ΡΡΠ½Π°ΡΠ° ΠΎΠΌΠΎΠ³ΡΡΠΈΠΎ ΡΠ΅ ΡΠ°Π·Π²ΠΎΡ Π½ΠΎΠ²ΠΈΡ
ΠΏΡΠΎΡΠ°ΡΡΠ½ΡΠΊΠΈΡ
ΠΌΠ΅ΡΠΎΠ΄Π° Ρ ΠΌΠ°ΡΠΈΠ½ΡΡΠ²Ρ ΠΊΠ°ΠΎ ΠΈ Π΄ΡΡΠ³ΠΈΠΌ ΠΎΠ±Π»Π°ΡΡΠΈΠΌΠ°. ΠΠΎΡΠ΅ΡΠ΅ ΠΏΡΠΎΡΠ΅ΡΠ° ΠΈ ΠΏΡΠΎΡΠ΅ΠΊΡΠΎΠ²Π°ΡΠ΅ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠ° ΡΠ· ΠΏΡΠΈΠΌΠ΅Π½Ρ ΡΠ°ΡΡΠ½Π°ΡΠ° ΠΏΠΎΡΡΠ°ΡΡ Π½Π°ΡΠ° ΡΠ²Π°ΠΊΠΎΠ΄Π½Π΅Π²Π½ΠΈΡΠ° Ρ ΠΊΠΎΡΠΎΡ ΡΠ΅ ΠΌΠΎΠ³ΡΡΠ΅ ΡΠ΅ΡΠΈΡΠΈ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ΅ ΠΊΠΎΡΠΈ ΡΡ ΡΠ°Π½ΠΈΡΠ΅ Π±ΠΈΠ»ΠΈ Π½Π΅ΡΠ΅ΡΠΈΠ²ΠΈ ΠΈ ΠΏΡΠΈΡΡΡΠΏΠ°Π»ΠΎ ΠΈΠΌ ΡΠ΅ ΠΈΡΠΊΡΡΡΠΈΠ²ΠΎ
Π΅ΠΌΠΏΠΈΡΠΈΡΡΠΊΠΈ.
ΠΠ΅Π΄Π°Π½ ΠΎΠ΄ ΠΎΠ²ΠΈΡ
ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΡΠ΅ΡΡΠ΅ ΠΈ ΠΏΡΠΎΠ±Π»Π΅ΠΌ Π΅ΠΌΠΈΡΠΈΡΠ΅ ΡΡΠ΅ΡΠ½ΠΈΡ
ΡΠ΅Π΄ΠΈΡΠ΅ΡΠ° ΠΈΠ· ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΈΡ
ΠΈΠ·Π²ΠΎΡΠ° Π²Π΅Π»ΠΈΠΊΠΈΡ
ΠΊΠ°ΠΏΠ°ΡΠΈΡΠ΅ΡΠ°, ΠΎΠ΄Π½ΠΎΡΠ½ΠΎ, Ρ Π½Π°ΡΠ΅ΠΌ ΠΊΠΎΠ½ΠΊΡΠ΅ΡΠ½ΠΎΠΌ ΡΠ»ΡΡΠ°ΡΡ ΠΈΠ· Π΅Π½Π΅ΡΠ³Π΅ΡΡΠΊΠΈΡ
ΠΏΠ°ΡΠ½ΠΈΡ
ΠΊΠΎΡΠ»ΠΎΠ²Π° Π½Π° ΡΠ³ΡΠ΅Π½ΠΈ ΠΏΡΠ°Ρ
. ΠΡΠ΅Π΄ΠΌΠ΅Ρ ΠΏΡΠΎΡΡΠ°Π²Π°ΡΠ° Ρ ΠΎΠ²ΠΎΡ Π΄ΠΈΡΠ΅ΡΡΠ°ΡΠΈΡΠΈ ΡΠ΅ΡΡΠ΅
Π½ΡΠΌΠ΅ΡΠΈΡΠΊΠΎ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ°ΡΠ΅ ΡΠ»ΠΎΠΆΠ΅Π½ΠΈΡ
ΠΏΡΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ ΠΏΠ°ΡΠ½ΠΎΠ³ ΠΊΠΎΡΠ»Π° ΠΏΡΠΈ ΡΠ½ΠΎΡΠ΅ΡΡ
ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Π½Π° Π±Π°Π·ΠΈ ΠΊΠ°Π»ΡΠΈΡΡΠΌΠ° (ΠΊΡΠ΅ΡΡΠ°ΠΊΠ°, ΠΈΠ»ΠΈ ΠΊΡΠ΅ΡΠ°) Π΄ΠΈΡΠ΅ΠΊΡΠ½ΠΎ Ρ Π»ΠΎΠΆΠΈΡΡΠ΅ ΡΠ°Π΄ΠΈ
ΡΠΌΠ°ΡΠ΅ΡΠ° Π΅ΠΌΠΈΡΠΈΡΠ΅ ΠΎΠΊΡΠΈΠ΄Π° ΡΡΠΌΠΏΠΎΡΠ°, Π° ΡΠΈΡΠ΅Π²ΠΈ Π΄ΠΈΡΠ΅ΡΡΠ°ΡΠΈΡΠ΅ ΡΡ ΡΠ°Π·Π²ΠΎΡ ΠΌΠΎΠ΄Π΅Π»Π°, ΠΊΠ°ΠΎ ΠΈ
Π½ΡΠΌΠ΅ΡΠΈΡΠΊΠ° Π°Π½Π°Π»ΠΈΠ·Π° ΠΈ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΡΠ° ΠΎΠ²ΠΈΡ
ΠΏΡΠΎΡΠ΅ΡΠ°. ΠΡΠΎΡΠ΅Ρ ΡΠ΅ ΠΏΠΎΠ·Π½Π°Ρ, Π°Π»ΠΈ, ΠΊΠ°ΠΎ ΡΡΠΎ ΡΠ΅
ΠΌΠΎΠΆΠ΅ ΠΏΡΠΎΠ½Π°ΡΠΈ Ρ Π»ΠΈΡΠ΅ΡΠ°ΡΡΡΠΈ, ΠΏΠΎΠ½Π°ΡΠ°ΡΠ΅ ΡΠΎΡΠ±Π΅Π½ΡΠ° ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΡΠ½ΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΠ΅ ΡΠ΅ ΠΈ
Π΄Π°ΡΠ΅ Π½Π΅Π΄ΠΎΠ²ΠΎΡΠ½ΠΎ ΠΏΠΎΠ·Π½Π°Ρ ΠΏΡΠΎΡΠ΅Ρ, ΠΈ Π½Π° ΡΡΠ²Π°ΡΠ½ΠΈΠΌ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠΈΠΌΠ° Π΅ΡΠΈΠΊΠ°ΡΠ½ΠΎΡΡ ΠΏΡΠΎΡΠ΅ΡΠ°
Π·Π½Π°ΡΠ°ΡΠ½ΠΎ Π²Π°ΡΠΈΡΠ° ΠΈΠ·ΠΌΠ΅ΡΡ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠ° ΠΈΡΡΠ΅ ΠΈΠ»ΠΈ ΡΠ»ΠΈΡΠ½Π΅ ΡΠ½Π°Π³Π΅. ΠΡΠΎΠ±Π»Π΅ΠΌΠ΅ ΠΈ ΡΠ·ΡΠΎΠΊΠ΅ Π·Π½Π°ΡΠ°ΡΠ½ΠΈΡ
ΡΠ°Π·Π»ΠΈΠΊΠ° Ρ Π΅ΡΠΈΠΊΠ°ΡΠ½ΠΎΡΡΠΈ ΠΌΠΎΠ³ΡΡΠ΅ ΡΠ΅ ΡΡΠ°ΠΆΠΈΡΠΈ Ρ Π»ΠΎΡΠ΅ΠΌ Π²ΠΎΡΠ΅ΡΡ ΠΏΡΠΎΡΠ΅ΡΠ°. ΠΡΠΌΠ΅ΡΠΈΡΠΊΠΎ
ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ°ΡΠ΅ Π½Π°ΠΌ ΠΎΠΌΠΎΠ³ΡΡΠ°Π²Π° Π΄Π° ΠΈΡΠΏΠΈΡΠ°ΠΌΠΎ ΠΏΠΎΠ½Π°ΡΠ°ΡΠ΅ Π»ΠΎΠΆΠΈΡΡΠ° ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΡΠ°Π·Π»ΠΈΡΠΈΡΠΈΡ
ΠΎΡΠ³Π°Π½ΠΈΠ·Π°ΡΠΈΡΠ° ΠΏΡΠΎΡΠ΅ΡΠ° Π²Π΅Π·Π°Π½ΠΈΡ
Π·Π° ΡΠ½ΠΎΡΠ΅ΡΠ΅ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΠ΅, ΠΏΡΠ΅ Π±ΠΈΠ»ΠΎ ΠΊΠ°ΠΊΠ²ΠΈΡ
ΠΈΠ·ΠΌΠ΅Π½Π°
Π½Π° ΠΏΠΎΡΡΠΎΡΠ΅ΡΠ΅ΠΌ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΡ, ΡΡΠΎ ΡΠ΅ ΠΎΠ΄ ΠΈΠ·ΡΠ·Π΅ΡΠ½ΠΎΠ³ Π·Π½Π°ΡΠ°ΡΠ° ΠΏΡΠΈ Π°Π½Π°Π»ΠΈΠ·Π°ΠΌΠ° ΠΈ ΠΎΠ΄Π»ΡΡΠΈΠ²Π°ΡΡ ΠΎ
ΠΏΡΠ°Π²ΡΠΈΠΌΠ° Ρ ΠΊΠΎΡΠΈΠΌΠ° ΡΡΠ΅Π±Π° Π²ΡΡΠΈΡΠΈ ΠΈΠ·ΠΌΠ΅Π½Π΅ Π½Π° ΠΏΠΎΡΡΠΎΡΠ΅ΡΠΈΠΌ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠΈΠΌΠ°, ΠΎΠ΄Π½ΠΎΡΠ½ΠΎ Π΄Π°ΡΠΈ
ΡΠΌΠ΅ΡΠ½ΠΈΡΠ΅ ΠΏΡΠΈ Π΄ΠΈΠ·Π°ΡΠ½Ρ Π½ΠΎΠ²ΠΈΡ
.
Π Π°Π·Π²ΠΈΡΠ΅Π½ ΡΠ΅ ΡΠΎΡΡΠ²Π΅Ρ Π·Π° ΠΏΡΠΎΡΠ°ΡΡΠ½ ΠΏΡΠΎΡΠ΅ΡΠ° Ρ ΡΡΠΎΠ΄ΠΈΠΌΠ΅Π½Π·ΠΈΠΎΠ½Π°Π»Π½ΠΎΠΌ ΠΊΠΎΡΠ»ΠΎΠ²ΡΠΊΠΎΠΌ Π»ΠΎΠΆΠΈΡΡΡ
ΠΊΠΎΡΠΈ ΡΠΊΡΡΡΡΡΠ΅ Π΄ΠΈΡΠ΅ΡΠ΅Π½ΡΠΈΡΠ°Π»Π½ΠΈ ΠΌΠΎΠ΄Π΅Π» ΡΡΡΡΡΠ½ΠΎΡΠ΅ΡΠΌΠΈΡΠΊΠΈΡ
ΠΏΡΠΎΡΠ΅ΡΠ°, ΠΌΠΎΠ΄Π΅Π» ΡΠ΅Π°ΠΊΡΠΈΡΠ°
ΡΠ°Π³ΠΎΡΠ΅Π²Π°ΡΠ°, ΡΠ΅Π°ΠΊΡΠΈΡΠ° Π½Π°ΡΡΠ°ΡΠ°ΡΠ° ΠΈ Π΄Π΅ΡΡΡΡΠΊΡΠΈΡΠ΅ ΠΎΠΊΡΠΈΠ΄Π° Π°Π·ΠΎΡΠ°, ΠΈ Π΄Π²Π° ΠΎΠ΄Π°Π±ΡΠ°Π½Π° ΠΈ
ΠΎΠΏΡΠΈΠΌΠΈΠ·ΠΎΠ²Π°Π½Π° ΠΌΠΎΠ΄Π΅Π»Π° ΡΠ΅Π°ΠΊΡΠΈΡΠ° ΡΠ΅ΡΡΠΈΡΠ° ΡΠΎΡΠ±Π΅Π½ΡΠ° ΡΠ° ΠΎΠΊΡΠΈΠ΄ΠΈΠΌΠ° ΡΡΠΌΠΏΠΎΡΠ° ΠΈΠ· Π»ΠΎΠΆΠΈΡΠ½ΠΈΡ
Π³Π°ΡΠΎΠ²Π°, ΠΏΡΠΈΠΌΠ΅ΡΠ΅Π½Π° Ρ ΡΠ»ΠΎΠΆΠ΅Π½ΠΎΠΌ ΠΌΠΎΠ΄Π΅Π»Ρ ΠΏΡΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ. Π£ΠΏΠΎΡΡΠ΅Π±ΡΠ°Π²Π° ΡΠ΅ οΏ½ β οΏ½
ΠΌΠΎΠ΄Π΅Π» ΡΡΡΠ±ΡΠ»Π΅Π½ΡΠΈΡΠ΅, Π΄ΠΎΠΊ ΡΠ΅ Π·Π° ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ°ΡΠ΅ ΡΠ°Π΄ΠΈΡΠ°ΡΠΈΠΎΠ½Π΅ ΡΠ°Π·ΠΌΠ΅Π½Π΅ ΡΠΎΠΏΠ»ΠΎΡΠ΅ ΠΊΠΎΡΠΈΡΡΠΈ ΠΌΠΎΠ΄Π΅Π»
ΡΠ΅ΡΡ ΡΠ»ΡΠΊΡΠ΅Π²Π°. ΠΠ²ΠΎΡΠ°Π·Π½ΠΈ Π³Π°Ρ-ΡΠ΅ΡΡΠΈΡΠ΅ ΡΡΡΠ±ΡΠ»Π΅Π½ΡΠ½ΠΈ ΡΠΎΠΊ ΡΠ΅ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ° ΠΏΡΠΈΠΌΠ΅Π½ΠΎΠΌ EulerLagrange-ΠΎΠ²ΠΎΠ³ ΠΏΠΎΡΡΡΠΏΠΊΠ°. ΠΠ½ΡΠ΅ΡΠ°ΠΊΡΠΈΡΠ° ΠΈΠ·ΠΌΠ΅ΡΡ Π³Π°ΡΠΎΠ²ΠΈΡΠ΅ ΡΠ°Π·Π΅ ΠΈ ΡΠ΅ΡΡΠΈΡΠ° ΡΠ΅ ΡΡΠ΅ΡΠΈΡΠ°
ΠΏΠΎΠΌΠΎΡΡ PSI-Cell ΠΌΠ΅ΡΠΎΠ΄Π΅, ΠΎΠ΄Π½ΠΎΡΠ½ΠΎ Ρ ΡΡΠ°Π½ΡΠΏΠΎΡΡΠ½ΠΈΠΌ ΡΠ΅Π΄Π½Π°ΡΠΈΠ½Π°ΠΌΠ° Π·Π° Π³Π°ΡΠ½Ρ ΡΠ°Π·Ρ ΠΏΠΎΡΡΠΎΡΠ΅
ΠΈΠ·Π²ΠΎΡΠ½ΠΈ ΡΠ»Π°Π½ΠΎΠ²ΠΈ ΠΊΠΎΡΠΈΠΌΠ° ΡΠ΅ ΡΠ·ΠΈΠΌΠ° Ρ ΠΎΠ±Π·ΠΈΡ ΡΡΠΈΡΠ°Ρ ΡΠ΅ΡΡΠΈΡΠ°.
ΠΠ½Π°ΡΠ°Ρ ΡΠ°Π·Π²ΠΎΡΠ° ΠΈ ΠΏΡΠΈΠΌΠ΅Π½Π΅ ΠΎΠ²Π°ΠΊΠ²ΠΎΠ³ ΡΠΎΡΡΠ²Π΅ΡΠ° Π·Π° ΠΏΡΠΎΡΠ°ΡΡΠ½ ΡΠ΅ ΠΎΠ³Π»Π΅Π΄Π° Ρ ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈ
ΡΠ°Π³Π»Π΅Π΄Π°Π²Π°ΡΠ° ΠΈ Π°Π½Π°Π»ΠΈΠ·Π΅ ΠΏΡΠΎΡΠ΅ΡΠ° ΡΠ½ΡΡΠ°Ρ Π»ΠΎΠΆΠΈΡΡΠ° ΠΊΠΎΡΠ΅ Π½Π° Π΄ΡΡΠ³ΠΈ Π½Π°ΡΠΈΠ½ Π½ΠΈΡΠ΅ ΠΌΠΎΠ³ΡΡΠ΅
Π°Π½Π°Π»ΠΈΠ·ΠΈΡΠ°ΡΠΈ Π½ΠΈΡΠΈ ΠΏΡΠ΅Π΄Π²ΠΈΠ΄Π΅ΡΠΈ ΠΏΠΎΠ½Π°ΡΠ°ΡΠ΅ ΡΠΈΡΡΠ΅ΠΌΠ° Π΄ΡΡΠ³ΠΈΠΌ ΡΠ΅Π΄Π½ΠΎΡΡΠ°Π²Π½ΠΈΡΠΈΠΌ ΠΌΠ΅ΡΠΎΠ΄Π°ΠΌΠ°.
ΠΠΎΠ·Π½Π°Π²Π°ΡΠ΅ ΠΏΠΎΠ½Π°ΡΠ°ΡΠ° ΠΊΠΎΡΠ»ΠΎΠ²ΡΠΊΠΎΠ³ Π»ΠΎΠΆΠΈΡΡΠ° ΠΏΡΠΈ ΠΎΠ΄ΡΠ΅ΡΠ΅Π½ΠΈΠΌ ΡΠ°Π΄Π½ΠΈΠΌ ΡΠ΅ΠΆΠΈΠΌΠΈΠΌΠ°, ΡΠ·
ΡΠΏΠΎΡΡΠ΅Π±Ρ ΡΠ°Π·Π»ΠΈΡΠΈΡΠΈΡ
Π³ΠΎΡΠΈΠ²Π°, ΠΊΠ°ΠΎ ΠΈ ΠΏΡΠΈ ΠΌΠΎΠ΄ΠΈΡΠΈΠΊΠ°ΡΠΈΡΠ°ΠΌΠ° ΠΏΠΎΠΏΡΡ ΡΠ½ΠΎΡΠ΅ΡΠ° ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ
Π»ΠΎΠΆΠΈΡΡΠ΅ ΡΠ΅ ΠΎΠ΄ ΠΈΠ·ΡΠ·Π΅ΡΠ½ΠΎΠ³ Π·Π½Π°ΡΠ°ΡΠ°, ΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²ΡΠ° ΠΏΡΠ΅Π΄ΡΡΠ»ΠΎΠ² Π·Π° ΠΏΠΎΡΡΠΈΠ·Π°ΡΠ΅ Π΅ΡΠΈΠΊΠ°ΡΠ½ΠΎΠ³,
ΠΏΠΎΡΠ·Π΄Π°Π½ΠΎΠ³ ΠΈ Π΅ΠΊΠΎΠ»ΠΎΡΠΊΠΈ ΠΏΡΠΈΡ
Π²Π°ΡΡΠΈΠ²ΠΎΠ³ ΡΠ°Π΄Π° ΡΠ· ΠΊΠΎΠΌΠΏΡΠΎΠΌΠΈΡΠ΅ ΠΊΠΎΡΠΈ ΠΈΠ· ΡΠ° ΡΡΠΈ Π±ΠΈΡΠ½Π°, Π°Π»ΠΈ
Π΄ΠΎΠ½Π΅ΠΊΠ»Π΅ ΡΡΠΏΡΠΎΡΡΡΠ°Π²ΡΠ΅Π½Π° Π·Π°Ρ
ΡΠ΅Π²Π° ΠΏΡΠΎΠΈΠ·ΠΈΠ»Π°Π·Π΅.
ΠΠ²Π΄Π΅ ΡΠ΅ ΠΏΠΎΡΠ΅Π±Π½Π° ΠΏΠ°ΠΆΡΠ° ΠΏΠΎΡΠ²Π΅ΡΠ΅Π½Π° ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ°ΡΡ ΡΠ½ΠΎΡΠ΅ΡΠ° ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ
Π»ΠΎΠΆΠΈΡΡΠ΅ ΠΏΠ°ΡΠ½ΠΎΠ³ ΠΊΠΎΡΠ»Π°, Ρ ΠΎΠ±Π·ΠΈΡΠΎΠΌ Π΄Π° ΡΠ΅ Π³Π»Π°Π²Π½ΠΈ ΡΠΈΡ ΠΏΡΠΎΠ²Π΅ΡΠ° ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈ ΡΠΌΠ°ΡΠ΅ΡΠ°
Π΅ΠΌΠΈΡΠΈΡΠ΅ ΠΎΠΊΡΠΈΠ΄Π° ΡΡΠΌΠΏΠΎΡΠ° ΠΏΠΎΠΌΠΎΡΡ Π΄ΠΈΡΠ΅ΠΊΡΠ½ΠΎΠ³ ΡΠ½ΠΎΡΠ΅ΡΠ° ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΠ΅.
Π Π΅ΡΠ°Π²Π°ΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΠΏΡΠΈΡΡΡΠΏΡΠ΅Π½ΠΎ ΡΠ΅ ΠΊΡΠΎΠ· Π΅ΡΠ°ΠΏΠ΅, ΠΏΠΎΡΠ΅Π² ΠΎΠ΄ ΠΏΡΠΎΠ²Π΅ΡΠ΅ ΠΈΠΌΠΏΠ»Π΅ΠΌΠ΅Π½ΡΠ°ΡΠΈΡΠ΅
ΠΎΠ΄Π°Π±ΡΠ°Π½ΠΈΡ
ΠΌΠΎΠ΄Π΅Π»Π° ΡΠ΅Π°ΠΊΡΠΈΡΠ° ΠΊΠ°Π»ΡΠΈΠ½Π°ΡΠΈΡΠ΅, ΡΠΈΠ½ΡΠ΅ΡΠΎΠ²Π°ΡΠ° ΠΈ ΡΡΠ»ΡΠ°ΡΠΈΠ·Π°ΡΠΈΡΠ΅ ΡΠ΅ΡΡΠΈΡΠ΅
ΡΠΎΡΠ±Π΅Π½ΡΠ°, ΠΏΡΠΎΠ²Π΅ΡΠ΅ ΡΠΈΡ
ΠΎΠ²Π΅ ΡΡΠ°Π±ΠΈΠ»Π½ΠΎΡΡΠΈ ΠΈ ΠΏΠΎΠ½Π°ΡΠ°ΡΠ° Ρ Π΄Π²ΠΎΠ΄ΠΈΠΌΠ΅Π½Π·ΠΈΠΎΠ½Π°Π»Π½ΠΈΠΌ ΠΊΠ°Π½Π°Π»ΠΈΠΌΠ°
ΠΊΠΎΡΠΈΠΌΠ° ΡΠ΅ ΡΠΈΠΌΡΠ»ΠΈΡΠ°ΡΡ ΡΠ΅Π°ΠΊΡΠΎΡΠΈ ΠΈ Ρ ΠΊΠΎΡΠΈΠΌΠ° ΡΠ΅ ΠΏΠΎΡΠ΅Π±Π½ΠΎ ΠΏΠΎΡΠ²Π΅ΡΠ΅Π½Π° ΠΏΠ°ΠΆΡΠ° ΠΏΠΎΡΠ΅ΡΠ΅ΡΡ
ΡΠ΅Π·ΡΠ»ΡΠ°ΡΠ° ΡΠ° Π΄ΠΎΡΡΡΠΏΠ½ΠΈΠΌ Π΅ΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½ΡΠ°Π»Π½ΠΈΠΌ ΡΠ΅Π·ΡΠ»ΡΠ°ΡΠΈΠΌΠ° ΡΠ°Π΄ΠΈ Π²Π°Π»ΠΈΠ΄Π°ΡΠΈΡΠ΅ ΠΌΠΎΠ΄Π΅Π»Π°.
ΠΠ°Π΄Π°ΡΠ΅ ΡΡ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΏΡΠΈΠΌΠ΅ΡΠ΅Π½ΠΈ Ρ ΡΡΠΎΠ΄ΠΈΠΌΠ΅Π½Π·ΠΈΠΎΠ½Π°Π»Π½ΠΎΠΌ Π½ΡΠΌΠ΅ΡΠΈΡΠΊΠΎΠΌ ΠΊΠΎΠ΄Ρ Π·Π° ΡΠΈΠΌΡΠ»Π°ΡΠΈΡΡ
ΠΏΡΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ ΠΏΠ°ΡΠ½ΠΎΠ³ ΠΊΠΎΡΠ»Π° ΠΈ ΡΡ ΡΠ΅ ΠΏΠΎΡΠ΅Π±Π½ΠΎ ΠΈΠ½ΡΠ΅ΡΠ΅ΡΠ°Π½ΡΠ½ΠΎ Π±ΠΈΠ»ΠΎ ΠΏΠΎΡΠΌΠ°ΡΡΠ°ΡΠΈ, ΠΏΠΎΡΠ΅Π΄
ΡΡΠΈΡΠ°ΡΠ° ΡΠΎΡΠ±Π΅Π½ΡΠ° Π½Π° ΡΠ°Π΄ΡΠΆΠ°Ρ ΠΎΠΊΡΠΈΠ΄Π° ΡΡΠΌΠΏΠΎΡΠ°, ΠΈ ΡΡΠΈΡΠ°Ρ Π½Π° ΠΈΠ·Π»Π°Π·Π½Π΅ ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ΅ ΠΈ Π΄ΡΡΠ³Π΅
ΡΠ΅Π»Π΅Π²Π°Π½ΡΠ½Π΅ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠ΅ ΠΏΡΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ.
Π£ ΡΠΎΠΊΡ ΠΈΡΡΡΠ°ΠΆΠΈΠ²Π°ΡΠ° ΠΈΠ·Π²Π΅Π΄Π΅Π½Π° ΡΠ΅ ΠΎΠ±ΠΈΠΌΠ½Π° Π½ΡΠΌΠ΅ΡΠΈΡΠΊΠ° Π°Π½Π°Π»ΠΈΠ·Π° ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈ ΡΠ½ΠΎΡΠ΅ΡΠ°
ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΠ΅ ΠΈ ΠΏΡΠ°ΡΠ΅ΡΠΈΡ
ΠΏΠΎΡΠ°Π²Π°. Π‘ΠΈΠΌΡΠ»ΠΈΡΠ°Π½ΠΎ ΡΠ΅ ΡΠ½ΠΎΡΠ΅ΡΠ΅ ΠΊΡΠΎΠ· Π΅ΡΠ°ΠΆΠ΅
Π³ΠΎΡΠΈΠΎΠ½ΠΈΡΠΊΠΈΡ
ΠΏΠ°ΠΊΠ΅ΡΠ°, ΠΊΠ°ΠΎ ΠΈ ΠΊΡΠΎΠ· ΠΏΠΎΡΠ΅Π±Π½Π΅ ΠΎΡΠ²ΠΎΡΠ΅ ΠΈΠ·Π½Π°Π΄ Π³ΠΎΡΠΈΠΎΠ½ΠΈΡΠΊΠΈΡ
ΠΏΠ°ΠΊΠ΅ΡΠ°, ΠΏΠΎΡΠ΅Π΄ΠΈΠ½Π°ΡΠ½ΠΎ
ΠΈ Ρ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΈΡΠΈ. ΠΠ½Π°Π»ΠΈΠ·ΠΈΡΠ°Π½Π΅ ΡΡ ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈ ΠΏΡΠΎΡΠ΅ΡΠ° ΡΠ° Π²ΠΈΡΠ΅ Π³ΠΎΡΠΈΠ²Π°, ΡΠ° ΡΠ°Π·Π»ΠΈΡΠΈΡΠΈΠΌ
ΡΠΎΠΏΠ»ΠΎΡΠ½ΠΈΠΌ ΠΌΠΎΡΠΈΠΌΠ° ΠΈ ΡΠ°Π΄ΡΠΆΠ°ΡΠΈΠΌΠ° ΡΡΠΌΠΏΠΎΡΠ° ΠΈ ΠΏΡΠΈΠΊΠ°Π·Π°Π½ΠΈ ΡΡ ΡΡΠΈΡΠ°ΡΠΈ ΠΊΠΎΡΠ΅ ΡΠ°Π·Π»ΠΈΡΠΈΡΠΈ ΡΠ°Π΄Π½ΠΈ
ΡΠ΅ΠΆΠΈΠΌΠΈ ΠΈ ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΡΠ΅ ΡΠ°Π³ΠΎΡΠ΅Π²Π°ΡΠ° ΠΈΠΌΠ°ΡΡ Π½Π° ΡΠ°Π΄ΡΠΆΠ°ΡΠ΅ Π³Π°ΡΠΎΠ²ΠΈΡΠΈΡ
ΠΏΡΠΎΠ΄ΡΠΊΠ°ΡΠ° Π½Π° ΠΈΠ·Π»Π°Π·Ρ
ΠΈΠ· Π»ΠΎΠΆΠΈΡΡΠ°. Π Π°Π·ΠΌΠ°ΡΡΠ°Π½ ΡΠ΅ ΡΡΠΈΡΠ°Ρ Π²Π΅Π»ΠΈΠΊΠΎΠ³ Π±ΡΠΎΡΠ° ΡΠ°Π΄Π½ΠΈΡ
ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΠ°ΡΠ° ΠΏΡΠΎΡΠ΅ΡΠ°
ΠΎΠ΄ΡΡΠΌΠΏΠΎΡΠ°Π²Π°ΡΠ°, ΠΊΠ°ΠΎ ΡΡΠΎ ΡΡ: ΠΌΠ΅ΡΡΠΎ ΡΠ½ΠΎΡΠ΅ΡΠ° ΠΈ Π΄ΠΈΡΡΡΠΈΠ±ΡΡΠΈΡΠ° ΡΠ΅ΡΡΠΈΡΠ° ΡΠΎΡΠ±Π΅Π½ΡΠ°,
ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΡΠΊΠ° ΠΈΡΡΠΎΡΠΈΡΠ° ΠΈ Π²ΡΠ΅ΠΌΠ΅ Π±ΠΎΡΠ°Π²ΠΊΠ° ΡΠ΅ΡΡΠΈΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ, Π»ΠΎΠΊΠ°Π»Π½Π° ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ° Π³Π°ΡΠ°
Ρ Π»ΠΎΠΆΠΈΡΡΡ, ΠΌΠΎΠ»Π°ΡΠ½ΠΈ ΠΎΠ΄Π½ΠΎΡ ΠΊΠ°Π»ΡΠΈΡΡΠΌΠ° ΠΈ ΡΡΠΌΠΏΠΎΡΠ°, Π»ΠΎΠΊΠ°Π»Π½Π° ΠΊΠΎΠ½ΡΠ΅Π½ΡΡΠ°ΡΠΈΡΠ° ΠΎΠΊΡΠΈΠ΄Π°
ΡΡΠΌΠΏΠΎΡΠ° ΠΈ ΠΊΠΈΡΠ΅ΠΎΠ½ΠΈΠΊΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ, ΠΈΡΠ΄. ΠΠ·Π²Π΅Π΄Π΅Π½ΠΈ ΡΡ Π·Π°ΠΊΡΡΡΡΠΈ ΠΎ ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈΠΌΠ° ΡΠ½ΠΎΡΠ΅ΡΠ°
ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΠ΅ ΠΏΠ°ΡΠ½ΠΎΠ³ ΠΊΠΎΡΠ»Π°, ΠΊΠ°ΠΎ ΠΈ ΠΏΡΠΎΠ½Π°Π»Π°ΠΆΠ΅ΡΡ ΠΎΠΏΡΠΈΠΌΠ°Π»Π½ΠΎΠ³ Π½Π°ΡΠΈΠ½Π°
ΡΠ½ΠΎΡΠ΅ΡΠ° Ρ Π·Π°Π²ΠΈΡΠ½ΠΎΡΡΠΈ ΠΎΠ΄ ΡΠ°Π΄Π½ΠΎΠ³ ΡΠ΅ΠΆΠΈΠΌΠ° ΠΊΠΎΡΠ»Π°.
Π Π°Π·Π²ΠΈΡΠ΅Π½ΠΈ ΡΠΎΡΡΠ²Π΅Ρ ΡΠ΅ ΠΎΠΏΡΠ΅ΠΌΡΠ΅Π½ ΠΊΠΎΡΠΈΡΠ½ΠΈΡΠΊΠΈΠΌ ΠΈΠ½ΡΠ΅ΡΡΠ΅ΡΡΠΎΠΌ ΠΊΠΎΡΠΈ ΠΎΠΌΠΎΠ³ΡΡΠ°Π²Π° ΡΠ΅Π΄Π½ΠΎΡΡΠ°Π²Π½ΠΎ
Π·Π°Π΄Π°Π²Π°ΡΠ΅ ΡΠ»Π°Π·Π½ΠΈΡ
ΠΏΠΎΠ΄Π°ΡΠ°ΠΊΠ° Π·Π° ΠΏΡΠ΅Π΄ΠΌΠ΅ΡΠ½ΠΎ Π»ΠΎΠΆΠΈΡΡΠ΅, ΡΡΠΎ ΠΎΠ»Π°ΠΊΡΠ°Π²Π° Π°Π½Π°Π»ΠΈΠ·Π΅, Π° ΠΎΠΌΠΎΠ³ΡΡΠ°Π²Π°
ΠΈ ΠΈΠ½ΠΆΠ΅ΡΠ΅ΡΡΠΊΠΎΠΌ ΠΊΠ°Π΄ΡΡ ΠΎΠ»Π°ΠΊΡΠ°Π½ ΡΠ°Π΄ ΡΠ° ΡΠΎΡΡΠ²Π΅ΡΠΎΠΌ, Π° Ρ ΡΠΈΡΡ ΡΠ°Π³Π»Π΅Π΄Π°Π²Π°ΡΠ° ΠΏΡΠ΅Π΄ΠΌΠ΅ΡΠ½ΠΎΠ³
ΠΏΡΠΎΡΠ΅ΡΠ° ΠΊΠ°ΠΎ ΠΌΠΎΠ³ΡΡΠ΅ ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ ΠΈ ΡΠ΅Π½Π΅ ΠΏΠΎΡΠ΅Π½ΡΠΈΡΠ°Π»Π½Π΅ ΠΏΡΠΈΠΌΠ΅Π½Π΅ Π½Π° Π»ΠΎΠΆΠΈΡΡΠΈΠΌΠ° ΠΏΠ°ΡΠ½ΠΈΡ
ΠΊΠΎΡΠ»ΠΎΠ²Π°
Modelling of flue gas desulfurization process by sorbent injection into the pulverized coal-fired utility boiler furnace.
ΠΠΊΠΎΠ»ΠΎΡΠΊΠΈ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠΈ ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΠΏΡΠ΅ΡΠ²Π°ΡΠ°ΡΠ° Π΅Π½Π΅ΡΠ³ΠΈΡΠ΅ ΡΠ°Π΄ΡΠΆΠ°Π½Π΅ Ρ ΡΠ³ΡΡ Ρ Π΅Π»Π΅ΠΊΡΡΠΈΡΠ½Ρ Π΅Π½Π΅ΡΠ³ΠΈΡΡ ΡΡ ΠΎΠ΄ ΠΈΠ·ΡΠ·Π΅ΡΠ½ΠΎΠ³ Π·Π½Π°ΡΠ°ΡΠ° ΠΈ ΠΏΠΎΡΠ²Π΅ΡΡΡΠ΅ ΠΈΠΌ ΡΠ΅ ΠΏΠΎΡΠ΅Π±Π½Π° ΠΏΠ°ΠΆΡΠ°. ΠΡΠ΅ ΠΏΡΠ΅Π΄ΡΠ·ΠΈΠΌΠ°ΡΠ° ΠΊΠΎΠ½ΠΊΡΠ΅ΡΠ½ΠΈΡ
ΠΌΠ΅ΡΠ° Π½Π° ΡΠ½Π°ΠΏΡΠ΅ΡΠ΅ΡΡ ΠΏΠΎΡΡΠΎΡΠ΅ΡΠΈΡ
ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠ°, ΠΈΠ»ΠΈ ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΠΏΠ»Π°Π½ΠΈΡΠ°ΡΠ° ΠΈ ΠΈΠ·Π³ΡΠ°Π΄ΡΠ΅ Π½ΠΎΠ²ΠΈΡ
ΠΏΠΎΡΡΠ΅Π±Π½ΠΎ ΡΠ΅ ΠΈΠ·Π²Π΅ΡΡΠΈ Π΄Π΅ΡΠ°ΡΠ½Π΅ Π°Π½Π°Π»ΠΈΠ·Π΅, ΠΊΠ°ΠΊΠΎ ΡΠ΅Ρ
Π½ΠΎ-Π΅ΠΊΠΎΠ½ΠΎΠΌΡΠΊΠ΅, ΡΠ°ΠΊΠΎ ΠΈ Π°Π½Π°Π»ΠΈΠ·Π΅ ΡΡΠΈΡΠ°ΡΠ° Π½Π° ΠΆΠΈΠ²ΠΎΡΠ½Ρ ΡΡΠ΅Π΄ΠΈΠ½Ρ. Π‘ΡΠ΅Π΄ΠΈΠ½ΠΎΠΌ ΠΏΡΠΎΡΠ»ΠΎΠ³ Π²Π΅ΠΊΠ° ΠΎΡΠΏΠΎΡΠ΅ΠΎ ΡΠ΅ ΡΠ±ΡΠ·Π°Π½ ΡΠ°Π·Π²ΠΎΡ ΡΠ°ΡΡΠ½Π°ΡΠ°, ΡΠ· ΠΈΡΡΠΎΠ²ΡΠ΅ΠΌΠ΅Π½ΠΎ ΠΏΠΎΡΠ΅ΡΡΠΈΡΠ΅ΡΠ΅ ΠΈ Π΄ΠΎΡΡΡΠΏΠ½ΠΎΡΡ ΠΊΡΠ°ΡΡΠ΅ΠΌ ΠΊΠΎΡΠΈΡΠ½ΠΈΠΊΡ, a 21. Π²Π΅ΠΊ ΡΠ΅ ΡΡΠΎΠ»Π΅ΡΠ΅ ΠΊΠΎΡΠ΅ ΡΠ΅ ΠΎΠ±Π΅Π»Π΅ΠΆΠΈΡΠΈ ΠΈ Π²Π΅Ρ ΠΎΠ±Π΅Π»Π΅ΠΆΠ°Π²Π°ΡΡ Π·Π½Π°ΡΠ°ΡΠ½Π΅ ΠΏΡΠΎΠΌΠ΅Π½Π΅ Ρ ΡΡΡΡΠΊΡΡΡΠΈ ΡΠ°ΡΡΠ½Π°ΡΠ°, ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈΠΌΠ° ΠΈ ΡΠΏΠΎΡΡΠ΅Π±ΠΈ...Environmental problems during energy conversion from coal into electric power are of great importance and must be addressed as such. Before undertaking measures to improve existing utility boilers, or during planning and building new plants, detailed analysis are required, considering both techno-economic and the environmental issues. During the middle of the last century a rapid development of computers started, and at the same time computers became affordable and available to the end user..
Modelling of flue gas desulfurization process by sorbent injection into the pulverized coal-fired utility boiler furnace
Environmental problems during energy conversion from coal into electric power are of great
importance and must be addressed as such. Before undertaking measures to improve existing
utility boilers, or during planning and building new plants, detailed analysis are required,
considering both techno-economic and the environmental issues. During the middle of the last
century a rapid development of computers started, and at the same time computers became
affordable and available to the end user. Thus, the 21st century becomes the era that will be
marked by significant changes in computer structure, possibilities and use. Advances in
computer development allowed for improvement of the computational methods in mechanical
engineering and in other fields as well. Process control and plant design with the aid of
computers are becoming everyday task and allow dealing with engineering problems that
have previously been unsolvable and required empirical approach.
One of the major contributors to environmental pollution is the emission of pollutants from
large stationary sources, that is, more precisely, from the pulverized coal powered utility
steam boilers. The subject of research in the dissertation is numerical modelling of complex
processes in utility boiler furnace during direct injection of pulverized calcium-based sorbent
(limestone, or lime) into the furnace for sulfur oxides reduction, with the model development,
as well as numerical analysis and optimization of the processes as the primary goals. Process
is well known in theory, however, as it can be found in the literature, the sorbent behavior
during the furnace sorbent injection is still not understood enough, and thus on the full-scale
plants the efficiency of the process significantly varies. Problems and the causes of significant
drops in efficiency can be attributed to the poor process control. Numerical modeling allows
for investigation of furnace behavior during various configurations of the sorbent injection
process, before any changes are made at the plant itself, which is of primary importance
during analysis and decision making about directions of the changes and upgrades of the
existing plants, and can give good ideas about the design of the new plants.
Developed software for three-dimensional furnace calculation includes differential model of
flow and heat transfer processes, combustion reactions model, nitrogen oxides formation and
destruction reactions model, and two selected and optimized models of sorbent particle
reactions with sulfur oxides from furnace gasses, applied within the comprehensive model of
furnace processes. A k-Ξ΅ model is used for turbulence modeling, while the radiative heat
exchange is modelled by using the six fluxes model. Two-phase gas-particle turbulent flow is
modeled with Euler-Lagrangian approach. Interaction between gas phase and particles is
treated by PSI-Cell method, with transport equations for gas phase having source terms that
takes into account the particles influence.
Significance of development and application of such a software for calculations is mostly
notable in possibility to perceive and analyze processes inside of the furnace which cannot be
analyzed and (the entire system cannot be) predicted by other means. Understanding the
behavior of the boiler furnace during certain operation regimes, with the use of various fuels,
as well as under modifications such as the furnace sorbent injection is of great importance,
and represents a prerequisite for achieving efficient, reliable and environmentally friendly
boiler operation with compromises between the three, important but to some extent opposed
conditions.
Particular attention is devoted to the modeling of pulverized sorbent furnace injection,
regarding that a primary goal is investigation of possibility to reduce sulfur oxides emission
by means of direct sorbent injection into the boiler furnace. Problem is approached through
several phases, starting with the analysis of selected models of calcination, sintering and
sulfation reactions, their stability and behavior in two-dimensional simulated reactors with
focus on comparison with available experimental results in order to validate the models
implementation. In further study, models are implemented in three-dimensional numerical
code for simulation of in-furnace processes, with particular interest to observe, beside the
sorbent influence on sulfur oxides content, the influence it has on the furnace exiting gas
temperature and other relevant process parameters in the furnace.
During the research, a complex numerical study of the furnace sorbent injection possibilities
and accompanying phenomena was performed. Sorbent injection was simulated through the
burner tiers, and through the special injection ports above the burner tiers, individually and in
combination. Process was analyzed for several fuels with different heating values and varied
sulfur content, and various the impacts of different operation regimes and combustion
configurations on the gaseous combustion products at the furnace exit were shown. Influence
of wide range of desulfurization process parameters was considered, such as: sorbent injection
position and particle distribution, particle temperature history and residence time, local gas
temperature within the furnace, calcium β sulfur molar ratio, local sulfur oxides concentration,
local oxygen concentration, etc. Conclusions were drawn considering possibilities for direct
sorbent injection into the pulverized coal fired boiler furnace, as well as suggestions were
given on optimal furnace sorbent injection configuration, depending on the boiler operation
parameters.
The developed software includes a user interface for easier data input for the case-study boiler
furnace, allowing for easier boiler analysis, and provides engineering staff with a tool for an
efficient software control, with the purpose of considering and analyzing better the furnace
sorbent injection technology and its potential applications in the utility boiler furnaces.ΠΠΊΠΎΠ»ΠΎΡΠΊΠΈ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠΈ ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΠΏΡΠ΅ΡΠ²Π°ΡΠ°ΡΠ° Π΅Π½Π΅ΡΠ³ΠΈΡΠ΅ ΡΠ°Π΄ΡΠΆΠ°Π½Π΅ Ρ ΡΠ³ΡΡ Ρ Π΅Π»Π΅ΠΊΡΡΠΈΡΠ½Ρ
Π΅Π½Π΅ΡΠ³ΠΈΡΡ ΡΡ ΠΎΠ΄ ΠΈΠ·ΡΠ·Π΅ΡΠ½ΠΎΠ³ Π·Π½Π°ΡΠ°ΡΠ° ΠΈ ΠΏΠΎΡΠ²Π΅ΡΡΡΠ΅ ΠΈΠΌ ΡΠ΅ ΠΏΠΎΡΠ΅Π±Π½Π° ΠΏΠ°ΠΆΡΠ°. ΠΡΠ΅ ΠΏΡΠ΅Π΄ΡΠ·ΠΈΠΌΠ°ΡΠ°
ΠΊΠΎΠ½ΠΊΡΠ΅ΡΠ½ΠΈΡ
ΠΌΠ΅ΡΠ° Π½Π° ΡΠ½Π°ΠΏΡΠ΅ΡΠ΅ΡΡ ΠΏΠΎΡΡΠΎΡΠ΅ΡΠΈΡ
ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠ°, ΠΈΠ»ΠΈ ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΠΏΠ»Π°Π½ΠΈΡΠ°ΡΠ° ΠΈ
ΠΈΠ·Π³ΡΠ°Π΄ΡΠ΅ Π½ΠΎΠ²ΠΈΡ
ΠΏΠΎΡΡΠ΅Π±Π½ΠΎ ΡΠ΅ ΠΈΠ·Π²Π΅ΡΡΠΈ Π΄Π΅ΡΠ°ΡΠ½Π΅ Π°Π½Π°Π»ΠΈΠ·Π΅, ΠΊΠ°ΠΊΠΎ ΡΠ΅Ρ
Π½ΠΎ-Π΅ΠΊΠΎΠ½ΠΎΠΌΡΠΊΠ΅, ΡΠ°ΠΊΠΎ ΠΈ
Π°Π½Π°Π»ΠΈΠ·Π΅ ΡΡΠΈΡΠ°ΡΠ° Π½Π° ΠΆΠΈΠ²ΠΎΡΠ½Ρ ΡΡΠ΅Π΄ΠΈΠ½Ρ. Π‘ΡΠ΅Π΄ΠΈΠ½ΠΎΠΌ ΠΏΡΠΎΡΠ»ΠΎΠ³ Π²Π΅ΠΊΠ° ΠΎΡΠΏΠΎΡΠ΅ΠΎ ΡΠ΅ ΡΠ±ΡΠ·Π°Π½ ΡΠ°Π·Π²ΠΎΡ
ΡΠ°ΡΡΠ½Π°ΡΠ°, ΡΠ· ΠΈΡΡΠΎΠ²ΡΠ΅ΠΌΠ΅Π½ΠΎ ΠΏΠΎΡΠ΅ΡΡΠΈΡΠ΅ΡΠ΅ ΠΈ Π΄ΠΎΡΡΡΠΏΠ½ΠΎΡΡ ΠΊΡΠ°ΡΡΠ΅ΠΌ ΠΊΠΎΡΠΈΡΠ½ΠΈΠΊΡ, a 21. Π²Π΅ΠΊ ΡΠ΅
ΡΡΠΎΠ»Π΅ΡΠ΅ ΠΊΠΎΡΠ΅ ΡΠ΅ ΠΎΠ±Π΅Π»Π΅ΠΆΠΈΡΠΈ ΠΈ Π²Π΅Ρ ΠΎΠ±Π΅Π»Π΅ΠΆΠ°Π²Π°ΡΡ Π·Π½Π°ΡΠ°ΡΠ½Π΅ ΠΏΡΠΎΠΌΠ΅Π½Π΅ Ρ ΡΡΡΡΠΊΡΡΡΠΈ ΡΠ°ΡΡΠ½Π°ΡΠ°,
ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈΠΌΠ° ΠΈ ΡΠΏΠΎΡΡΠ΅Π±ΠΈ. ΠΠ°ΠΏΡΠ΅Π΄Π°ΠΊ Ρ ΡΠ°Π·Π²ΠΎΡΡ ΡΠ°ΡΡΠ½Π°ΡΠ° ΠΎΠΌΠΎΠ³ΡΡΠΈΠΎ ΡΠ΅ ΡΠ°Π·Π²ΠΎΡ Π½ΠΎΠ²ΠΈΡ
ΠΏΡΠΎΡΠ°ΡΡΠ½ΡΠΊΠΈΡ
ΠΌΠ΅ΡΠΎΠ΄Π° Ρ ΠΌΠ°ΡΠΈΠ½ΡΡΠ²Ρ ΠΊΠ°ΠΎ ΠΈ Π΄ΡΡΠ³ΠΈΠΌ ΠΎΠ±Π»Π°ΡΡΠΈΠΌΠ°. ΠΠΎΡΠ΅ΡΠ΅ ΠΏΡΠΎΡΠ΅ΡΠ° ΠΈ ΠΏΡΠΎΡΠ΅ΠΊΡΠΎΠ²Π°ΡΠ΅ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠ° ΡΠ· ΠΏΡΠΈΠΌΠ΅Π½Ρ ΡΠ°ΡΡΠ½Π°ΡΠ° ΠΏΠΎΡΡΠ°ΡΡ Π½Π°ΡΠ° ΡΠ²Π°ΠΊΠΎΠ΄Π½Π΅Π²Π½ΠΈΡΠ° Ρ ΠΊΠΎΡΠΎΡ ΡΠ΅ ΠΌΠΎΠ³ΡΡΠ΅ ΡΠ΅ΡΠΈΡΠΈ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ΅ ΠΊΠΎΡΠΈ ΡΡ ΡΠ°Π½ΠΈΡΠ΅ Π±ΠΈΠ»ΠΈ Π½Π΅ΡΠ΅ΡΠΈΠ²ΠΈ ΠΈ ΠΏΡΠΈΡΡΡΠΏΠ°Π»ΠΎ ΠΈΠΌ ΡΠ΅ ΠΈΡΠΊΡΡΡΠΈΠ²ΠΎ
Π΅ΠΌΠΏΠΈΡΠΈΡΡΠΊΠΈ.
ΠΠ΅Π΄Π°Π½ ΠΎΠ΄ ΠΎΠ²ΠΈΡ
ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΡΠ΅ΡΡΠ΅ ΠΈ ΠΏΡΠΎΠ±Π»Π΅ΠΌ Π΅ΠΌΠΈΡΠΈΡΠ΅ ΡΡΠ΅ΡΠ½ΠΈΡ
ΡΠ΅Π΄ΠΈΡΠ΅ΡΠ° ΠΈΠ· ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΈΡ
ΠΈΠ·Π²ΠΎΡΠ° Π²Π΅Π»ΠΈΠΊΠΈΡ
ΠΊΠ°ΠΏΠ°ΡΠΈΡΠ΅ΡΠ°, ΠΎΠ΄Π½ΠΎΡΠ½ΠΎ, Ρ Π½Π°ΡΠ΅ΠΌ ΠΊΠΎΠ½ΠΊΡΠ΅ΡΠ½ΠΎΠΌ ΡΠ»ΡΡΠ°ΡΡ ΠΈΠ· Π΅Π½Π΅ΡΠ³Π΅ΡΡΠΊΠΈΡ
ΠΏΠ°ΡΠ½ΠΈΡ
ΠΊΠΎΡΠ»ΠΎΠ²Π° Π½Π° ΡΠ³ΡΠ΅Π½ΠΈ ΠΏΡΠ°Ρ
. ΠΡΠ΅Π΄ΠΌΠ΅Ρ ΠΏΡΠΎΡΡΠ°Π²Π°ΡΠ° Ρ ΠΎΠ²ΠΎΡ Π΄ΠΈΡΠ΅ΡΡΠ°ΡΠΈΡΠΈ ΡΠ΅ΡΡΠ΅
Π½ΡΠΌΠ΅ΡΠΈΡΠΊΠΎ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ°ΡΠ΅ ΡΠ»ΠΎΠΆΠ΅Π½ΠΈΡ
ΠΏΡΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ ΠΏΠ°ΡΠ½ΠΎΠ³ ΠΊΠΎΡΠ»Π° ΠΏΡΠΈ ΡΠ½ΠΎΡΠ΅ΡΡ
ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Π½Π° Π±Π°Π·ΠΈ ΠΊΠ°Π»ΡΠΈΡΡΠΌΠ° (ΠΊΡΠ΅ΡΡΠ°ΠΊΠ°, ΠΈΠ»ΠΈ ΠΊΡΠ΅ΡΠ°) Π΄ΠΈΡΠ΅ΠΊΡΠ½ΠΎ Ρ Π»ΠΎΠΆΠΈΡΡΠ΅ ΡΠ°Π΄ΠΈ
ΡΠΌΠ°ΡΠ΅ΡΠ° Π΅ΠΌΠΈΡΠΈΡΠ΅ ΠΎΠΊΡΠΈΠ΄Π° ΡΡΠΌΠΏΠΎΡΠ°, Π° ΡΠΈΡΠ΅Π²ΠΈ Π΄ΠΈΡΠ΅ΡΡΠ°ΡΠΈΡΠ΅ ΡΡ ΡΠ°Π·Π²ΠΎΡ ΠΌΠΎΠ΄Π΅Π»Π°, ΠΊΠ°ΠΎ ΠΈ
Π½ΡΠΌΠ΅ΡΠΈΡΠΊΠ° Π°Π½Π°Π»ΠΈΠ·Π° ΠΈ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΡΠ° ΠΎΠ²ΠΈΡ
ΠΏΡΠΎΡΠ΅ΡΠ°. ΠΡΠΎΡΠ΅Ρ ΡΠ΅ ΠΏΠΎΠ·Π½Π°Ρ, Π°Π»ΠΈ, ΠΊΠ°ΠΎ ΡΡΠΎ ΡΠ΅
ΠΌΠΎΠΆΠ΅ ΠΏΡΠΎΠ½Π°ΡΠΈ Ρ Π»ΠΈΡΠ΅ΡΠ°ΡΡΡΠΈ, ΠΏΠΎΠ½Π°ΡΠ°ΡΠ΅ ΡΠΎΡΠ±Π΅Π½ΡΠ° ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΡΠ½ΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΠ΅ ΡΠ΅ ΠΈ
Π΄Π°ΡΠ΅ Π½Π΅Π΄ΠΎΠ²ΠΎΡΠ½ΠΎ ΠΏΠΎΠ·Π½Π°Ρ ΠΏΡΠΎΡΠ΅Ρ, ΠΈ Π½Π° ΡΡΠ²Π°ΡΠ½ΠΈΠΌ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠΈΠΌΠ° Π΅ΡΠΈΠΊΠ°ΡΠ½ΠΎΡΡ ΠΏΡΠΎΡΠ΅ΡΠ°
Π·Π½Π°ΡΠ°ΡΠ½ΠΎ Π²Π°ΡΠΈΡΠ° ΠΈΠ·ΠΌΠ΅ΡΡ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠ° ΠΈΡΡΠ΅ ΠΈΠ»ΠΈ ΡΠ»ΠΈΡΠ½Π΅ ΡΠ½Π°Π³Π΅. ΠΡΠΎΠ±Π»Π΅ΠΌΠ΅ ΠΈ ΡΠ·ΡΠΎΠΊΠ΅ Π·Π½Π°ΡΠ°ΡΠ½ΠΈΡ
ΡΠ°Π·Π»ΠΈΠΊΠ° Ρ Π΅ΡΠΈΠΊΠ°ΡΠ½ΠΎΡΡΠΈ ΠΌΠΎΠ³ΡΡΠ΅ ΡΠ΅ ΡΡΠ°ΠΆΠΈΡΠΈ Ρ Π»ΠΎΡΠ΅ΠΌ Π²ΠΎΡΠ΅ΡΡ ΠΏΡΠΎΡΠ΅ΡΠ°. ΠΡΠΌΠ΅ΡΠΈΡΠΊΠΎ
ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ°ΡΠ΅ Π½Π°ΠΌ ΠΎΠΌΠΎΠ³ΡΡΠ°Π²Π° Π΄Π° ΠΈΡΠΏΠΈΡΠ°ΠΌΠΎ ΠΏΠΎΠ½Π°ΡΠ°ΡΠ΅ Π»ΠΎΠΆΠΈΡΡΠ° ΠΏΡΠΈΠ»ΠΈΠΊΠΎΠΌ ΡΠ°Π·Π»ΠΈΡΠΈΡΠΈΡ
ΠΎΡΠ³Π°Π½ΠΈΠ·Π°ΡΠΈΡΠ° ΠΏΡΠΎΡΠ΅ΡΠ° Π²Π΅Π·Π°Π½ΠΈΡ
Π·Π° ΡΠ½ΠΎΡΠ΅ΡΠ΅ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΠ΅, ΠΏΡΠ΅ Π±ΠΈΠ»ΠΎ ΠΊΠ°ΠΊΠ²ΠΈΡ
ΠΈΠ·ΠΌΠ΅Π½Π°
Π½Π° ΠΏΠΎΡΡΠΎΡΠ΅ΡΠ΅ΠΌ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΡ, ΡΡΠΎ ΡΠ΅ ΠΎΠ΄ ΠΈΠ·ΡΠ·Π΅ΡΠ½ΠΎΠ³ Π·Π½Π°ΡΠ°ΡΠ° ΠΏΡΠΈ Π°Π½Π°Π»ΠΈΠ·Π°ΠΌΠ° ΠΈ ΠΎΠ΄Π»ΡΡΠΈΠ²Π°ΡΡ ΠΎ
ΠΏΡΠ°Π²ΡΠΈΠΌΠ° Ρ ΠΊΠΎΡΠΈΠΌΠ° ΡΡΠ΅Π±Π° Π²ΡΡΠΈΡΠΈ ΠΈΠ·ΠΌΠ΅Π½Π΅ Π½Π° ΠΏΠΎΡΡΠΎΡΠ΅ΡΠΈΠΌ ΠΏΠΎΡΡΡΠΎΡΠ΅ΡΠΈΠΌΠ°, ΠΎΠ΄Π½ΠΎΡΠ½ΠΎ Π΄Π°ΡΠΈ
ΡΠΌΠ΅ΡΠ½ΠΈΡΠ΅ ΠΏΡΠΈ Π΄ΠΈΠ·Π°ΡΠ½Ρ Π½ΠΎΠ²ΠΈΡ
.
Π Π°Π·Π²ΠΈΡΠ΅Π½ ΡΠ΅ ΡΠΎΡΡΠ²Π΅Ρ Π·Π° ΠΏΡΠΎΡΠ°ΡΡΠ½ ΠΏΡΠΎΡΠ΅ΡΠ° Ρ ΡΡΠΎΠ΄ΠΈΠΌΠ΅Π½Π·ΠΈΠΎΠ½Π°Π»Π½ΠΎΠΌ ΠΊΠΎΡΠ»ΠΎΠ²ΡΠΊΠΎΠΌ Π»ΠΎΠΆΠΈΡΡΡ
ΠΊΠΎΡΠΈ ΡΠΊΡΡΡΡΡΠ΅ Π΄ΠΈΡΠ΅ΡΠ΅Π½ΡΠΈΡΠ°Π»Π½ΠΈ ΠΌΠΎΠ΄Π΅Π» ΡΡΡΡΡΠ½ΠΎΡΠ΅ΡΠΌΠΈΡΠΊΠΈΡ
ΠΏΡΠΎΡΠ΅ΡΠ°, ΠΌΠΎΠ΄Π΅Π» ΡΠ΅Π°ΠΊΡΠΈΡΠ°
ΡΠ°Π³ΠΎΡΠ΅Π²Π°ΡΠ°, ΡΠ΅Π°ΠΊΡΠΈΡΠ° Π½Π°ΡΡΠ°ΡΠ°ΡΠ° ΠΈ Π΄Π΅ΡΡΡΡΠΊΡΠΈΡΠ΅ ΠΎΠΊΡΠΈΠ΄Π° Π°Π·ΠΎΡΠ°, ΠΈ Π΄Π²Π° ΠΎΠ΄Π°Π±ΡΠ°Π½Π° ΠΈ
ΠΎΠΏΡΠΈΠΌΠΈΠ·ΠΎΠ²Π°Π½Π° ΠΌΠΎΠ΄Π΅Π»Π° ΡΠ΅Π°ΠΊΡΠΈΡΠ° ΡΠ΅ΡΡΠΈΡΠ° ΡΠΎΡΠ±Π΅Π½ΡΠ° ΡΠ° ΠΎΠΊΡΠΈΠ΄ΠΈΠΌΠ° ΡΡΠΌΠΏΠΎΡΠ° ΠΈΠ· Π»ΠΎΠΆΠΈΡΠ½ΠΈΡ
Π³Π°ΡΠΎΠ²Π°, ΠΏΡΠΈΠΌΠ΅ΡΠ΅Π½Π° Ρ ΡΠ»ΠΎΠΆΠ΅Π½ΠΎΠΌ ΠΌΠΎΠ΄Π΅Π»Ρ ΠΏΡΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ. Π£ΠΏΠΎΡΡΠ΅Π±ΡΠ°Π²Π° ΡΠ΅ οΏ½ β οΏ½
ΠΌΠΎΠ΄Π΅Π» ΡΡΡΠ±ΡΠ»Π΅Π½ΡΠΈΡΠ΅, Π΄ΠΎΠΊ ΡΠ΅ Π·Π° ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ°ΡΠ΅ ΡΠ°Π΄ΠΈΡΠ°ΡΠΈΠΎΠ½Π΅ ΡΠ°Π·ΠΌΠ΅Π½Π΅ ΡΠΎΠΏΠ»ΠΎΡΠ΅ ΠΊΠΎΡΠΈΡΡΠΈ ΠΌΠΎΠ΄Π΅Π»
ΡΠ΅ΡΡ ΡΠ»ΡΠΊΡΠ΅Π²Π°. ΠΠ²ΠΎΡΠ°Π·Π½ΠΈ Π³Π°Ρ-ΡΠ΅ΡΡΠΈΡΠ΅ ΡΡΡΠ±ΡΠ»Π΅Π½ΡΠ½ΠΈ ΡΠΎΠΊ ΡΠ΅ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ° ΠΏΡΠΈΠΌΠ΅Π½ΠΎΠΌ EulerLagrange-ΠΎΠ²ΠΎΠ³ ΠΏΠΎΡΡΡΠΏΠΊΠ°. ΠΠ½ΡΠ΅ΡΠ°ΠΊΡΠΈΡΠ° ΠΈΠ·ΠΌΠ΅ΡΡ Π³Π°ΡΠΎΠ²ΠΈΡΠ΅ ΡΠ°Π·Π΅ ΠΈ ΡΠ΅ΡΡΠΈΡΠ° ΡΠ΅ ΡΡΠ΅ΡΠΈΡΠ°
ΠΏΠΎΠΌΠΎΡΡ PSI-Cell ΠΌΠ΅ΡΠΎΠ΄Π΅, ΠΎΠ΄Π½ΠΎΡΠ½ΠΎ Ρ ΡΡΠ°Π½ΡΠΏΠΎΡΡΠ½ΠΈΠΌ ΡΠ΅Π΄Π½Π°ΡΠΈΠ½Π°ΠΌΠ° Π·Π° Π³Π°ΡΠ½Ρ ΡΠ°Π·Ρ ΠΏΠΎΡΡΠΎΡΠ΅
ΠΈΠ·Π²ΠΎΡΠ½ΠΈ ΡΠ»Π°Π½ΠΎΠ²ΠΈ ΠΊΠΎΡΠΈΠΌΠ° ΡΠ΅ ΡΠ·ΠΈΠΌΠ° Ρ ΠΎΠ±Π·ΠΈΡ ΡΡΠΈΡΠ°Ρ ΡΠ΅ΡΡΠΈΡΠ°.
ΠΠ½Π°ΡΠ°Ρ ΡΠ°Π·Π²ΠΎΡΠ° ΠΈ ΠΏΡΠΈΠΌΠ΅Π½Π΅ ΠΎΠ²Π°ΠΊΠ²ΠΎΠ³ ΡΠΎΡΡΠ²Π΅ΡΠ° Π·Π° ΠΏΡΠΎΡΠ°ΡΡΠ½ ΡΠ΅ ΠΎΠ³Π»Π΅Π΄Π° Ρ ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈ
ΡΠ°Π³Π»Π΅Π΄Π°Π²Π°ΡΠ° ΠΈ Π°Π½Π°Π»ΠΈΠ·Π΅ ΠΏΡΠΎΡΠ΅ΡΠ° ΡΠ½ΡΡΠ°Ρ Π»ΠΎΠΆΠΈΡΡΠ° ΠΊΠΎΡΠ΅ Π½Π° Π΄ΡΡΠ³ΠΈ Π½Π°ΡΠΈΠ½ Π½ΠΈΡΠ΅ ΠΌΠΎΠ³ΡΡΠ΅
Π°Π½Π°Π»ΠΈΠ·ΠΈΡΠ°ΡΠΈ Π½ΠΈΡΠΈ ΠΏΡΠ΅Π΄Π²ΠΈΠ΄Π΅ΡΠΈ ΠΏΠΎΠ½Π°ΡΠ°ΡΠ΅ ΡΠΈΡΡΠ΅ΠΌΠ° Π΄ΡΡΠ³ΠΈΠΌ ΡΠ΅Π΄Π½ΠΎΡΡΠ°Π²Π½ΠΈΡΠΈΠΌ ΠΌΠ΅ΡΠΎΠ΄Π°ΠΌΠ°.
ΠΠΎΠ·Π½Π°Π²Π°ΡΠ΅ ΠΏΠΎΠ½Π°ΡΠ°ΡΠ° ΠΊΠΎΡΠ»ΠΎΠ²ΡΠΊΠΎΠ³ Π»ΠΎΠΆΠΈΡΡΠ° ΠΏΡΠΈ ΠΎΠ΄ΡΠ΅ΡΠ΅Π½ΠΈΠΌ ΡΠ°Π΄Π½ΠΈΠΌ ΡΠ΅ΠΆΠΈΠΌΠΈΠΌΠ°, ΡΠ·
ΡΠΏΠΎΡΡΠ΅Π±Ρ ΡΠ°Π·Π»ΠΈΡΠΈΡΠΈΡ
Π³ΠΎΡΠΈΠ²Π°, ΠΊΠ°ΠΎ ΠΈ ΠΏΡΠΈ ΠΌΠΎΠ΄ΠΈΡΠΈΠΊΠ°ΡΠΈΡΠ°ΠΌΠ° ΠΏΠΎΠΏΡΡ ΡΠ½ΠΎΡΠ΅ΡΠ° ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ
Π»ΠΎΠΆΠΈΡΡΠ΅ ΡΠ΅ ΠΎΠ΄ ΠΈΠ·ΡΠ·Π΅ΡΠ½ΠΎΠ³ Π·Π½Π°ΡΠ°ΡΠ°, ΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²ΡΠ° ΠΏΡΠ΅Π΄ΡΡΠ»ΠΎΠ² Π·Π° ΠΏΠΎΡΡΠΈΠ·Π°ΡΠ΅ Π΅ΡΠΈΠΊΠ°ΡΠ½ΠΎΠ³,
ΠΏΠΎΡΠ·Π΄Π°Π½ΠΎΠ³ ΠΈ Π΅ΠΊΠΎΠ»ΠΎΡΠΊΠΈ ΠΏΡΠΈΡ
Π²Π°ΡΡΠΈΠ²ΠΎΠ³ ΡΠ°Π΄Π° ΡΠ· ΠΊΠΎΠΌΠΏΡΠΎΠΌΠΈΡΠ΅ ΠΊΠΎΡΠΈ ΠΈΠ· ΡΠ° ΡΡΠΈ Π±ΠΈΡΠ½Π°, Π°Π»ΠΈ
Π΄ΠΎΠ½Π΅ΠΊΠ»Π΅ ΡΡΠΏΡΠΎΡΡΡΠ°Π²ΡΠ΅Π½Π° Π·Π°Ρ
ΡΠ΅Π²Π° ΠΏΡΠΎΠΈΠ·ΠΈΠ»Π°Π·Π΅.
ΠΠ²Π΄Π΅ ΡΠ΅ ΠΏΠΎΡΠ΅Π±Π½Π° ΠΏΠ°ΠΆΡΠ° ΠΏΠΎΡΠ²Π΅ΡΠ΅Π½Π° ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠ°ΡΡ ΡΠ½ΠΎΡΠ΅ΡΠ° ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ
Π»ΠΎΠΆΠΈΡΡΠ΅ ΠΏΠ°ΡΠ½ΠΎΠ³ ΠΊΠΎΡΠ»Π°, Ρ ΠΎΠ±Π·ΠΈΡΠΎΠΌ Π΄Π° ΡΠ΅ Π³Π»Π°Π²Π½ΠΈ ΡΠΈΡ ΠΏΡΠΎΠ²Π΅ΡΠ° ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈ ΡΠΌΠ°ΡΠ΅ΡΠ°
Π΅ΠΌΠΈΡΠΈΡΠ΅ ΠΎΠΊΡΠΈΠ΄Π° ΡΡΠΌΠΏΠΎΡΠ° ΠΏΠΎΠΌΠΎΡΡ Π΄ΠΈΡΠ΅ΠΊΡΠ½ΠΎΠ³ ΡΠ½ΠΎΡΠ΅ΡΠ° ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΠ΅.
Π Π΅ΡΠ°Π²Π°ΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΠΏΡΠΈΡΡΡΠΏΡΠ΅Π½ΠΎ ΡΠ΅ ΠΊΡΠΎΠ· Π΅ΡΠ°ΠΏΠ΅, ΠΏΠΎΡΠ΅Π² ΠΎΠ΄ ΠΏΡΠΎΠ²Π΅ΡΠ΅ ΠΈΠΌΠΏΠ»Π΅ΠΌΠ΅Π½ΡΠ°ΡΠΈΡΠ΅
ΠΎΠ΄Π°Π±ΡΠ°Π½ΠΈΡ
ΠΌΠΎΠ΄Π΅Π»Π° ΡΠ΅Π°ΠΊΡΠΈΡΠ° ΠΊΠ°Π»ΡΠΈΠ½Π°ΡΠΈΡΠ΅, ΡΠΈΠ½ΡΠ΅ΡΠΎΠ²Π°ΡΠ° ΠΈ ΡΡΠ»ΡΠ°ΡΠΈΠ·Π°ΡΠΈΡΠ΅ ΡΠ΅ΡΡΠΈΡΠ΅
ΡΠΎΡΠ±Π΅Π½ΡΠ°, ΠΏΡΠΎΠ²Π΅ΡΠ΅ ΡΠΈΡ
ΠΎΠ²Π΅ ΡΡΠ°Π±ΠΈΠ»Π½ΠΎΡΡΠΈ ΠΈ ΠΏΠΎΠ½Π°ΡΠ°ΡΠ° Ρ Π΄Π²ΠΎΠ΄ΠΈΠΌΠ΅Π½Π·ΠΈΠΎΠ½Π°Π»Π½ΠΈΠΌ ΠΊΠ°Π½Π°Π»ΠΈΠΌΠ°
ΠΊΠΎΡΠΈΠΌΠ° ΡΠ΅ ΡΠΈΠΌΡΠ»ΠΈΡΠ°ΡΡ ΡΠ΅Π°ΠΊΡΠΎΡΠΈ ΠΈ Ρ ΠΊΠΎΡΠΈΠΌΠ° ΡΠ΅ ΠΏΠΎΡΠ΅Π±Π½ΠΎ ΠΏΠΎΡΠ²Π΅ΡΠ΅Π½Π° ΠΏΠ°ΠΆΡΠ° ΠΏΠΎΡΠ΅ΡΠ΅ΡΡ
ΡΠ΅Π·ΡΠ»ΡΠ°ΡΠ° ΡΠ° Π΄ΠΎΡΡΡΠΏΠ½ΠΈΠΌ Π΅ΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½ΡΠ°Π»Π½ΠΈΠΌ ΡΠ΅Π·ΡΠ»ΡΠ°ΡΠΈΠΌΠ° ΡΠ°Π΄ΠΈ Π²Π°Π»ΠΈΠ΄Π°ΡΠΈΡΠ΅ ΠΌΠΎΠ΄Π΅Π»Π°.
ΠΠ°Π΄Π°ΡΠ΅ ΡΡ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΏΡΠΈΠΌΠ΅ΡΠ΅Π½ΠΈ Ρ ΡΡΠΎΠ΄ΠΈΠΌΠ΅Π½Π·ΠΈΠΎΠ½Π°Π»Π½ΠΎΠΌ Π½ΡΠΌΠ΅ΡΠΈΡΠΊΠΎΠΌ ΠΊΠΎΠ΄Ρ Π·Π° ΡΠΈΠΌΡΠ»Π°ΡΠΈΡΡ
ΠΏΡΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ ΠΏΠ°ΡΠ½ΠΎΠ³ ΠΊΠΎΡΠ»Π° ΠΈ ΡΡ ΡΠ΅ ΠΏΠΎΡΠ΅Π±Π½ΠΎ ΠΈΠ½ΡΠ΅ΡΠ΅ΡΠ°Π½ΡΠ½ΠΎ Π±ΠΈΠ»ΠΎ ΠΏΠΎΡΠΌΠ°ΡΡΠ°ΡΠΈ, ΠΏΠΎΡΠ΅Π΄
ΡΡΠΈΡΠ°ΡΠ° ΡΠΎΡΠ±Π΅Π½ΡΠ° Π½Π° ΡΠ°Π΄ΡΠΆΠ°Ρ ΠΎΠΊΡΠΈΠ΄Π° ΡΡΠΌΠΏΠΎΡΠ°, ΠΈ ΡΡΠΈΡΠ°Ρ Π½Π° ΠΈΠ·Π»Π°Π·Π½Π΅ ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ΅ ΠΈ Π΄ΡΡΠ³Π΅
ΡΠ΅Π»Π΅Π²Π°Π½ΡΠ½Π΅ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠ΅ ΠΏΡΠΎΡΠ΅ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ.
Π£ ΡΠΎΠΊΡ ΠΈΡΡΡΠ°ΠΆΠΈΠ²Π°ΡΠ° ΠΈΠ·Π²Π΅Π΄Π΅Π½Π° ΡΠ΅ ΠΎΠ±ΠΈΠΌΠ½Π° Π½ΡΠΌΠ΅ΡΠΈΡΠΊΠ° Π°Π½Π°Π»ΠΈΠ·Π° ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈ ΡΠ½ΠΎΡΠ΅ΡΠ°
ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΠ΅ ΠΈ ΠΏΡΠ°ΡΠ΅ΡΠΈΡ
ΠΏΠΎΡΠ°Π²Π°. Π‘ΠΈΠΌΡΠ»ΠΈΡΠ°Π½ΠΎ ΡΠ΅ ΡΠ½ΠΎΡΠ΅ΡΠ΅ ΠΊΡΠΎΠ· Π΅ΡΠ°ΠΆΠ΅
Π³ΠΎΡΠΈΠΎΠ½ΠΈΡΠΊΠΈΡ
ΠΏΠ°ΠΊΠ΅ΡΠ°, ΠΊΠ°ΠΎ ΠΈ ΠΊΡΠΎΠ· ΠΏΠΎΡΠ΅Π±Π½Π΅ ΠΎΡΠ²ΠΎΡΠ΅ ΠΈΠ·Π½Π°Π΄ Π³ΠΎΡΠΈΠΎΠ½ΠΈΡΠΊΠΈΡ
ΠΏΠ°ΠΊΠ΅ΡΠ°, ΠΏΠΎΡΠ΅Π΄ΠΈΠ½Π°ΡΠ½ΠΎ
ΠΈ Ρ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΈΡΠΈ. ΠΠ½Π°Π»ΠΈΠ·ΠΈΡΠ°Π½Π΅ ΡΡ ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈ ΠΏΡΠΎΡΠ΅ΡΠ° ΡΠ° Π²ΠΈΡΠ΅ Π³ΠΎΡΠΈΠ²Π°, ΡΠ° ΡΠ°Π·Π»ΠΈΡΠΈΡΠΈΠΌ
ΡΠΎΠΏΠ»ΠΎΡΠ½ΠΈΠΌ ΠΌΠΎΡΠΈΠΌΠ° ΠΈ ΡΠ°Π΄ΡΠΆΠ°ΡΠΈΠΌΠ° ΡΡΠΌΠΏΠΎΡΠ° ΠΈ ΠΏΡΠΈΠΊΠ°Π·Π°Π½ΠΈ ΡΡ ΡΡΠΈΡΠ°ΡΠΈ ΠΊΠΎΡΠ΅ ΡΠ°Π·Π»ΠΈΡΠΈΡΠΈ ΡΠ°Π΄Π½ΠΈ
ΡΠ΅ΠΆΠΈΠΌΠΈ ΠΈ ΠΊΠΎΠ½ΡΠΈΠ³ΡΡΠ°ΡΠΈΡΠ΅ ΡΠ°Π³ΠΎΡΠ΅Π²Π°ΡΠ° ΠΈΠΌΠ°ΡΡ Π½Π° ΡΠ°Π΄ΡΠΆΠ°ΡΠ΅ Π³Π°ΡΠΎΠ²ΠΈΡΠΈΡ
ΠΏΡΠΎΠ΄ΡΠΊΠ°ΡΠ° Π½Π° ΠΈΠ·Π»Π°Π·Ρ
ΠΈΠ· Π»ΠΎΠΆΠΈΡΡΠ°. Π Π°Π·ΠΌΠ°ΡΡΠ°Π½ ΡΠ΅ ΡΡΠΈΡΠ°Ρ Π²Π΅Π»ΠΈΠΊΠΎΠ³ Π±ΡΠΎΡΠ° ΡΠ°Π΄Π½ΠΈΡ
ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΠ°ΡΠ° ΠΏΡΠΎΡΠ΅ΡΠ°
ΠΎΠ΄ΡΡΠΌΠΏΠΎΡΠ°Π²Π°ΡΠ°, ΠΊΠ°ΠΎ ΡΡΠΎ ΡΡ: ΠΌΠ΅ΡΡΠΎ ΡΠ½ΠΎΡΠ΅ΡΠ° ΠΈ Π΄ΠΈΡΡΡΠΈΠ±ΡΡΠΈΡΠ° ΡΠ΅ΡΡΠΈΡΠ° ΡΠΎΡΠ±Π΅Π½ΡΠ°,
ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΡΠΊΠ° ΠΈΡΡΠΎΡΠΈΡΠ° ΠΈ Π²ΡΠ΅ΠΌΠ΅ Π±ΠΎΡΠ°Π²ΠΊΠ° ΡΠ΅ΡΡΠΈΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ, Π»ΠΎΠΊΠ°Π»Π½Π° ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ° Π³Π°ΡΠ°
Ρ Π»ΠΎΠΆΠΈΡΡΡ, ΠΌΠΎΠ»Π°ΡΠ½ΠΈ ΠΎΠ΄Π½ΠΎΡ ΠΊΠ°Π»ΡΠΈΡΡΠΌΠ° ΠΈ ΡΡΠΌΠΏΠΎΡΠ°, Π»ΠΎΠΊΠ°Π»Π½Π° ΠΊΠΎΠ½ΡΠ΅Π½ΡΡΠ°ΡΠΈΡΠ° ΠΎΠΊΡΠΈΠ΄Π°
ΡΡΠΌΠΏΠΎΡΠ° ΠΈ ΠΊΠΈΡΠ΅ΠΎΠ½ΠΈΠΊΠ° Ρ Π»ΠΎΠΆΠΈΡΡΡ, ΠΈΡΠ΄. ΠΠ·Π²Π΅Π΄Π΅Π½ΠΈ ΡΡ Π·Π°ΠΊΡΡΡΡΠΈ ΠΎ ΠΌΠΎΠ³ΡΡΠ½ΠΎΡΡΠΈΠΌΠ° ΡΠ½ΠΎΡΠ΅ΡΠ°
ΡΠΏΡΠ°ΡΠ΅Π½ΠΎΠ³ ΡΠΎΡΠ±Π΅Π½ΡΠ° Ρ Π»ΠΎΠΆΠΈΡΡΠ΅ ΠΏΠ°ΡΠ½ΠΎΠ³ ΠΊΠΎΡΠ»Π°, ΠΊΠ°ΠΎ ΠΈ ΠΏΡΠΎΠ½Π°Π»Π°ΠΆΠ΅ΡΡ ΠΎΠΏΡΠΈΠΌΠ°Π»Π½ΠΎΠ³ Π½Π°ΡΠΈΠ½Π°
ΡΠ½ΠΎΡΠ΅ΡΠ° Ρ Π·Π°Π²ΠΈΡΠ½ΠΎΡΡΠΈ ΠΎΠ΄ ΡΠ°Π΄Π½ΠΎΠ³ ΡΠ΅ΠΆΠΈΠΌΠ° ΠΊΠΎΡΠ»Π°.
Π Π°Π·Π²ΠΈΡΠ΅Π½ΠΈ ΡΠΎΡΡΠ²Π΅Ρ ΡΠ΅ ΠΎΠΏΡΠ΅ΠΌΡΠ΅Π½ ΠΊΠΎΡΠΈΡΠ½ΠΈΡΠΊΠΈΠΌ ΠΈΠ½ΡΠ΅ΡΡΠ΅ΡΡΠΎΠΌ ΠΊΠΎΡΠΈ ΠΎΠΌΠΎΠ³ΡΡΠ°Π²Π° ΡΠ΅Π΄Π½ΠΎΡΡΠ°Π²Π½ΠΎ
Π·Π°Π΄Π°Π²Π°ΡΠ΅ ΡΠ»Π°Π·Π½ΠΈΡ
ΠΏΠΎΠ΄Π°ΡΠ°ΠΊΠ° Π·Π° ΠΏΡΠ΅Π΄ΠΌΠ΅ΡΠ½ΠΎ Π»ΠΎΠΆΠΈΡΡΠ΅, ΡΡΠΎ ΠΎΠ»Π°ΠΊΡΠ°Π²Π° Π°Π½Π°Π»ΠΈΠ·Π΅, Π° ΠΎΠΌΠΎΠ³ΡΡΠ°Π²Π°
ΠΈ ΠΈΠ½ΠΆΠ΅ΡΠ΅ΡΡΠΊΠΎΠΌ ΠΊΠ°Π΄ΡΡ ΠΎΠ»Π°ΠΊΡΠ°Π½ ΡΠ°Π΄ ΡΠ° ΡΠΎΡΡΠ²Π΅ΡΠΎΠΌ, Π° Ρ ΡΠΈΡΡ ΡΠ°Π³Π»Π΅Π΄Π°Π²Π°ΡΠ° ΠΏΡΠ΅Π΄ΠΌΠ΅ΡΠ½ΠΎΠ³
ΠΏΡΠΎΡΠ΅ΡΠ° ΠΊΠ°ΠΎ ΠΌΠΎΠ³ΡΡΠ΅ ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ ΠΈ ΡΠ΅Π½Π΅ ΠΏΠΎΡΠ΅Π½ΡΠΈΡΠ°Π»Π½Π΅ ΠΏΡΠΈΠΌΠ΅Π½Π΅ Π½Π° Π»ΠΎΠΆΠΈΡΡΠΈΠΌΠ° ΠΏΠ°ΡΠ½ΠΈΡ
ΠΊΠΎΡΠ»ΠΎΠ²Π°