In the modern environmental situation, modeling atmospheric air pollution is an urgent problem. The modeling of the state of atmospheric air quality is considered using various mathematical approaches that describe physical and chemical processes that are modeled depending on the type of pollution, emission parameters, meteorological, topographic and other conditions affecting the dispersion of pollutants. The key requirements for air pollution models are given. The stages of construction and classification of air pollution models are considered. One of the types of air pollution models are models based on a mathematical description of the physical processes occurring in the atmosphere. Similar are the models built on the basis of solving the equation of turbulent diffusion. Solutions of the equation to describe the phenomenon of transport and diffusion of a pollutant for the “ball”, “torch”, “box” and “finite-difference” models are considered. The advantages and disadvantages of these models are described. The software implementation of the “torch” model is described.
air pollution
modeling
"clew"
turbulent diffusion equations
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In the modern environmental situation, modeling atmospheric air pollution is an urgent problem.
The development of computing capabilities makes it possible to use mathematical modeling tools to study complex physical and chemical processes such as atmospheric diffusion, transformation of pollutants in the atmosphere, processes of leaching and deposition of impurities, etc., taking into account meteorological and topographic conditions.
The atmospheric air pollution model must meet the following basic requirements: the necessary forecast resolution in space and time; take into account weather conditions and the state of the troposphere and the earth's surface at points of contact, types of pollution sources; increasing model accuracy as the amount of information increases or its quality improves.
The stages of constructing a model of atmospheric air pollution are presented in Fig. 1.
The result of the simulation is the distribution of the concentration of harmful substances in space and time.
The content of the modeling problem statement can be to obtain either an operational forecast or long-term planning. An operational forecast is considered to be for a time from 30 minutes to one day. Other sources consider other forecasting periods: express or operational, assuming a time of 1-2 hours, short-term for a time from 12 hours to 1-2 days, long-term - from 3 days to 2-3 weeks, long-term - from 1 month to several years .
The presence of different approaches to modeling processes occurring in the atmosphere is due to the lack of a generalizing physical and mathematical model that takes into account all parameters of atmospheric diffusion phenomena. The choice of modeling approach depends on the formulation of the problem and determines the quality of the model and the accuracy of the forecast.
Rice. 1. Stages of constructing an air pollution model
When modeling atmospheric air pollution, it is necessary to take into account the type and time of forecasting, determine the class of sources of atmospheric air pollution - point, linear, area, etc., as well as the territorial location of pollution sources.
The classification of approaches to modeling processes occurring in the atmosphere is shown in Fig. 2.
One of the types of air pollution models are models based on a mathematical description of the physical processes occurring in the atmosphere. Similar are the models built on the basis of solving the equation of turbulent diffusion (Fig. 3).
In these models, the physical phenomena of transport and diffusion of pollutants in atmospheric air are described by the equation
where C is the concentration of the pollutant, are the coefficients of turbulent diffusion, is the vector of the averaged air velocity field; QC is a source of pollution.
To mathematically formulate the problem of solving equation (1), it is necessary to specify initial and boundary conditions, the choice of which is determined by the type of pollution source and surface characteristics.
It is possible to obtain a solution to equation (1) only under certain assumptions and restrictions, or using numerical methods.
Rice. 2. Classification of air pollution models
Rice. 3. Models based on solving the turbulent diffusion equation
Assuming in equation (1) the absence of distribution of pollutant particles with air flows, the heterogeneity of the atmosphere, and also assuming that the source of pollution is located outside the area, we obtain the equation
(2)
The fundamental solution to this equation is a Gaussian curve and is used in the tangle and plume models.
The tangle model assumes that the source of pollution acts instantaneously. The transfer of pollutant emissions under the influence of wind is represented in a moving coordinate system.
The “ball” model has the following form:
where x, y, z are the coordinates of the center of the “ball”, which determine the trajectory of its movement; u, v, w - average values of wind speeds in directions x, y, z at time t; σ x, σ y, σ z - standard deviations of the “ball” sizes in the x, y, z directions, respectively; Q is the amount of pollutant emitted by the source at time t.
The “ball” model has some disadvantages, such as the need for numerous measurements of wind speeds in the x, y, z directions, the difficulty of identifying the parameters of a ball of pollutants (height of the center, size deviations in directions), and the complexity of software implementation.
Let's consider the “torch” model. In this model, it is assumed that the source is point and acts continuously.
The “flare” model is used in the case of the release of pollutants from point sources of different heights; the temperature and nature of the emissions are not taken into account.
The torch model has the following form:
where C(x, y, z, H) is the distribution of concentration along coordinates x, y, z, Q is the rate of release of the pollutant; u - average wind speed; σ y (x), σ z (x) - standard deviations of the “torch” dimensions in the horizontal and vertical directions for a given x, H = h + Dh - effective height of the torch; h - pipe height; Dh is the rise of the torch due to its buoyancy.
When considering the model, we will take into account the following assumptions:
Within the area under consideration, weather conditions are uniform and do not change over time;
Chemical reactions do not occur with the pollutant;
The pollutant is not absorbed by the surface;
The surface in the area under consideration is flat.
The “flare” model is relatively simple and allows one to calculate the concentrations of pollutants based on a limited number of parameters that are determined experimentally, which is its main advantage. As research experience shows, this model can be used in 70% of meteorological situations.
The box model is used to approximate pollutant levels from large surface areas.
This model looks like
where l is the width of the “box”, h is the height, C is the average concentration at the rear (in the direction of the wind) wall of the “box”; u is the average wind speed through the “box”.
When using numerical methods to solve the diffusion equation, “finite-difference” models are obtained. Models obtained in this way do not depend on the parameters of the sources, environment, or boundary conditions.
The main disadvantage of these models is the difficulty of determining their stability and accuracy, as well as the high probability of calculation errors.
This paper examines the software implementation of the “torch” model. The program is written in C++ in the Borland C++ Builder 6.0 development environment.
The menu of the “Atmospheric Air Pollution Model” program consists of three items: File, Calculation, Help. The contents of menu items are shown in Fig. 4. The program allows you to both load calculation parameters from a file and enter them from the keyboard. Detailed instructions for using the program are also provided.
The main window of the program consists of three areas for filling in parameters and one for displaying the calculated results. The upper left area contains fields for entering atmospheric parameters: wind speed and direction. On the right is an area for entering parameters of pollution sources. When the program starts, the “Source number” input field is set to “1”. Next, you should fill in the fields for the coordinates of the source, pollution speed, pipe height and flame height. When you click the “Save” button, the parameters of the current source are saved, the values in the input fields are reset, and the “Source Number” field is automatically changed to the next number value.
Rice. 4. Contents of menu items
Rice. 5. Main window
In the lower left area there are fields for entering the coordinates of the measuring point. After filling in all the data for each source, click on the “Calculate” button.
At the bottom of the main window there is a field for displaying results. This field accumulates the values of calculated pollutant concentrations for each measurement point. The results of the program can be saved to a text file. This file contains the results for each measurement point: the entered atmospheric parameters, the number of pollution sources and their parameters in accordance with the serial number, as well as the coordinates of the measurement point.
The input file for loading parameters must contain the following data in a given order: wind speed, wind direction, measurement point coordinates in three directions, number of sources and for each source, respectively, the number of the current source, source coordinates in three directions, pollution speed, pipe height, height torch.
The main program window with filled-in input fields and calculated results for five measurement points is shown in Fig. 5.
This paper examines various models of the distribution of pollutants that describe the state of atmospheric air using various mathematical approaches that take into account types of pollution, emission parameters, meteorological, topographic and other conditions affecting the dispersion of pollutants. The key requirements for air pollution models are given. The stages of construction and classification of air pollution models are considered.
The “torch” model is implemented in software. The developed program provides the ability to calculate the concentration of pollutants at the measurement point. The results obtained from the simulation were confirmed experimentally.
In the future, it is planned to create an automated system that will allow both operational forecasting of the level of atmospheric air pollution and long-term planning.
Bibliographic link
Khashirova T.Yu., Akbasheva G.A., Shakova O.A., Akbasheva E.A. MODELING OF ATMOSPHERIC AIR POLLUTION // Fundamental Research. – 2017. – No. 8-2. – P. 325-330;URL: http://fundamental-research.ru/ru/article/view?id=41669 (access date: 02/01/2020). We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"
Let us consider the biosphere processes of the spread of pollution from single industrial sources, paying special attention to the study of sanitary and hygienic situations due to particularly dangerous pollution conditions.
In the general case, the change in average concentration values U is described by the equation
where the x and y axes are located in the horizontal plane; z axis - vertical; t - time; V, P, W - components of the average speed of movement of impurities relative to the direction of the x, y, z axes; - horizontal and vertical components of the exchange coefficient; - coefficient that determines the change in concentration due to the transformation of impurities.
However, air pollution in the city in the case of a non-inversion state of the air basin may be insignificant and does not require special methods to protect the population.
Another situation arises due to unpleasant meteorological conditions (temperature inversions with light winds and calm weather). Taking into account unpleasant meteorological conditions is one of the poorly studied issues.
During the occurrence of inversions, the air temperature in the surface layer increases rather than falls, as in the case of persistent thermal stratification of the atmosphere. Mixing occurs weakly, and the lower part of the inversion layer plays the role of a screen, from which the torch of pollutants is partially or completely reflected, and in the ground layer the concentration of harmful impurities increases to values that are dangerous to the health and life of people.
Theoretical models for calculating atmospheric air pollution do not reflect the entire set of factors that influence pollution from an industrial source in extreme situations, but are only approximate models that require complex additional studies (theoretical and experimental) to determine the model coefficients and process parameters if they are used on practice. Extreme conditions due to pollution, which arise during surface inversions in the atmosphere and the absence of turbulent exchange, are described by a special case of the general diffusion equation. However, it is these conditions that are the most dangerous for human health and should be the subject of hygienic forecasts when planning the location of industrial enterprise zones.
To achieve this goal, it becomes necessary to create forecast equations based on the principles of self-organization, which have the following advantages:
The structure of the forecast equation and the coefficients of the algorithm models are found from field observations of the concentration of pollutants under appropriate conditions, which ensures a significant refinement of the model;
Theoretical information about the class of operators is used, and the final calculation formulas in the form of final operators are simple and make it possible to designate the sanitary and hygienic zones of enterprises.
According to this technique, theoretical models in the form of differential operators and their semi-imperial analogues are first determined using observational data, and then their adequacy is checked when calculating concentrations with data that are not involved in identification.
The theoretical model for the propagation of impurities from a single source is the diffusion equation in cylindrical coordinates:
In the case of a single point source, taking into account in the most general form, equation (3.2) has the form:
where M is the mass of the ejection per unit time; r - distance from the source; z - vertical distance; - angle of rotation relative to the axis; - functions:
As can be seen from equation (3.3), the source of pollution is located at point r=0 at height H. At a point other than r=0, the equation has the form:
Let's make a cut along the line of maximum contamination along the torch at a height:
and the diffusion equation (3.3) becomes one-dimensional:
Note that the functions, in the general case, are also functions of the source height H, i.e.; ; .
The structure of equation (3.7) is the starting point for identifying difference analogues - models of atmospheric pollution from industrial sources.
Field observations of industrial emissions were used to construct equations for the distribution of individual ingredients, and these formed the basis for practical testing of the models.
Synthesis of the equation for predicting the maximum level of dust pollution:
To approximate functions, we used the following expressions:
where are linear functions.
We write the derivatives in the form of the corresponding difference:
Then the structure of the difference operator must be found in the class of linear operators F:
where is the concentration of the pollutant at the i point; - distance beyond the radius from the origin to the i - point.
Based on research data in different cities of Ukraine, continuous pollution observation curves were approximated. Using a combinatorial algorithm, a model was obtained:
Where; ; - dust concentration (maximum value at i point).
Thus, the method for determining the quality of atmospheric air in a city consists of calculating the concentration of a pollutant until the concentration reaches the maximum permissible value for a given substance.
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Novozhilov Artem Sergeevich. Mathematical models of interaction between pollution and the environment: Dis. ...cand. physics and mathematics Sciences: 05.13.18 Moscow, 2002 84 p. RSL OD, 61:02-1/855-4
Introduction
1. Conceptual model of the interaction of pollution with the environment 12
1.1. Single release of pollutants into the environment 12
1.2. Behavior of the destruction curve during multiple releases 13
1.3. Numerical simulation of multiple release 16
1.4. General notes 18
2. Differential model of interaction between pollution and the environment 20
2.1. Atmospheric diffusion model 20
2.2. Differential model of interaction of pollution with the environment at point 22
2.3. Qualitative study of a differential mathematical model 24
2.3.1. Replacing variables 24
2.3.2. Physical meaning of parameters 25
2.3.3. Stationary points of the system under study 26
2.3.4. Parametric portrait 27
2.3.5. Bifurcations of equilibrium positions 29
2.4. Modification of the functional model of the impact of nature
for pollution 31
2.5. Possible modifications of model 33
2.5.1. Taking into account the Ollie effect 33
2.5.2. Modification of the pollution source power function 35
2.6. Preliminary findings 36
2.7. System pollution - environment in the presence of a periodic source of pollution 37
3. Distributed mathematical model of pollution interaction
with the environment 45
3.1. Problem formulation 45
3.2. Model on a plane 46
3.3. 3D model 47
3.4. Numerical solution of distributed models 48
3.5. Simulation modeling of the interaction of pollution with the environment 50
3.5.1. Mathematical model on plane 50
3.5.2. 3D model 52
3.5.3. Remarks 53
4. Identification of parameters of a mathematical model of interaction of pollution with the environment 54
4.1. Mathematical model 54
4.2. Model 55 Analytical Record
4.3. Observational data 58
4.3.1. Brief description of the ecological and geographical conditions of the Kola Peninsula region and the Severonickel plant 59
4.3.2. Ecological and geographical characteristics of the region of the Southern Urals and the Karabash copper smelter 61
4.3.3. Data on pollution levels and biomass density in the studied regions 62
4.4. Algorithm for solving the problem of identifying parameters of a mathematical
models of interaction between pollution and the environment 67
4.4.1. Final formulation of the mathematical model 67
4.4.2. Supporting Results 68
4.4.3. Problem statement and solution algorithm 71
4.5. Results and analysis of the results obtained 72
4.5.1. Parameter estimates 72
4.5.2. Analysis of the results 74
CONCLUSION 80
LITERATURE 81
Introduction to the work
Relevance of the topic. Anthropogenic impact, increasing urbanization, and the development of industry and agriculture have posed the task of developing and applying a set of measures to prevent environmental degradation and stabilize the state of the biosphere. This led to the separation from ecology - a science whose subject is the concept of an ecosystem as an integral, evolutionarily formed formation - into a field dealing with the study and protection of the environment (environmental science) - the theoretical basis of human behavior in an industrial society in nature.
Despite the fact that ecology is a biological discipline, solving complex, multidimensional dynamic problems of description, forecasting, optimal use and rational design of various ecological systems requires a quantitative and systematic approach, the implementation of which is unthinkable without the widespread use of mathematical models and computers. As J. Hutchinson (1965) emphasized, it is impossible to write about the ecology of populations without the use of mathematics. To date, a significant number of different mathematical models of ecological systems of any level have been developed - gene, individual, population. In the science of environmental protection, mathematical models are also used (Marchuk, 1982; Marchuk, Kondratyev, 1992).
Since experiment and observation are most consistent with knowledge only when they are conceived and implemented on the basis of scientific theory, it should be recognized that one of the most fruitful methods is the method of mathematical modeling.
In accordance with the ideology of mathematical modeling, in order to adequately describe the processes occurring in the environment, it is necessary to identify the key factors that have a major influence on the processes being studied. There is no doubt that pollution has a negative impact on the environment. It is also known that vegetation absorbs and processes pollution to a certain extent. It is natural to raise the question about the importance of taking into account the impact of the environment on pollution when formulating certain mathematical models that describe the dynamics of biomass in the presence of pollution.
Considering the pollution-environment system from the point of view of mathematical modeling, it is first necessary to identify the specific characteristics of the object under study, the variety of connections between elements, their different qualities and subordination. For this reason, the first object of study should be recognized as a separate system - an industrial enterprise - a specific ecosystem. In this case, the process of interaction between pollution and the environment is clearly expressed, which simplifies the analysis of the adequacy of the mathematical model, and, on the other hand, such a system is not an exception to the rule. Examples include the Severonickel plant and the Karabash copper smelting plant discussed in this work, and, in addition, the Pechenganikel plant, the Guzum metallurgical plant in Sweden, and the metallurgical plant in Sudbury (Canada).
The degree of development of the problem. Starting from the fundamental works of V. Volterra at the beginning of the 20th century (Volterra, 1926) to the present day, the subject of mathematical biology - the study of biological systems by the method of mathematical modeling - has turned into a difficult-to-see conglomerate of ideas and approaches, using all the capabilities of modern mathematics (Mshtu, 1996 ; Bazykin, 1985; Gimmelfarb A.A., 1974; Karev, Berezovskaya, 2000; Odum, 1975; Riznichenko, Rubin, 1993; Smith, 1976; Fedorov, Gilmanov, 1980 and many others).
The question of the mathematical description of forest phytocenoses can be considered as an integral part of mathematical biology. By now this section is also well developed. Models for describing forest growth dynamics can be divided into two categories. The first describe forests as a whole (continuous approach), considering, in principle, the entire thin film of green cover as one large tree. This approach was developed, for example, in the following works (Toorming, 1980; Kuml, Oya, 1984; Rosenberg, 1984). The second approach is to describe a forest ecosystem as a community of discrete elements with internal connections (Rachko, 1979;BotkinataI., 1972).
Considering that the topic of this work is related to the spread of pollution, we note that this issue is a well-studied area of knowledge. However, the main problem studied by many scientists is the problem of short-term forecasting of the spread of pollution (Berland, 1985). There are numerous models for describing the spread of pollution in the presence of different climatic conditions, fog, smog, different types of underlying surfaces, and various terrain (Berland, 1975,1985; Gudarian, 1979; Atmospheric turbulence and modeling of the spread of impurities, 1985).
Since the main task of any environmental protection measures is the issue of environmental regulation of the impact on the ecosystem, we note that although the theoretical aspects of this task have been formulated (Israel, 1984), in practice this question remains open. Currently, we only have values for maximum permissible concentrations (MPC) for human protection. The next step should be the establishment of EPDC - environmentally maximum permissible concentrations that protect the ecosystem from anthropogenic impact (Impact of metallurgical production on forest ecosystems of the Kola Peninsula, 1995).
Observations show (Bui Ta Long, 1999) that pollution dynamics and forest ecosystem dynamics are highly correlated, so a natural step would be to try to combine the two well-researched applications of mathematical modeling into one system. Many mathematical models take into account the impact of pollution on the environment. The impact of pollution on humanity was included as an integral block of the models of “World Dynamics” by J. Forrester (Forrester, 1978) and “The Limits to Growth” by D. Meadows (Meadows at a]., 1972) when constructing global models for studying the processes of economic development of the world. A number of models examine the dynamics of wildlife in the presence of pollution (Tarko et al., 1987). However, the factor of the cleansing effect of nature on pollution is considered for the first time when constructing mathematical models. The correlation between pollution concentration and biomass density was studied by ecologists using statistical methods (Impact of metallurgical production on forest ecosystems of the Kola Peninsula, 1995; Comprehensive assessment of technogenic impact on the ecosystems of the southern taiga, 1992; Butusov, Stepanov, 2000, 2001).
Goal of the work. The purpose of this work is to create mathematical models of the interaction of pollution with the environment and to assess the adequacy of a distributed mathematical model of the interaction of pollution with the environment based on environmental monitoring data. To achieve this goal, the following tasks were solved:
An analysis of the conceptual model of the interaction of pollution with the environment was carried out, identifying possible scenarios for the behavior of the closed system pollution - environment.
Based on the analysis of the conceptual model, a number of mathematical models are proposed that are described by autonomous systems of ordinary differential equations (models localized at a point). A qualitative study of differential models was carried out, including an analysis of the behavior of systems with bifurcation values of parameters. A qualitative correspondence between the proposed differential models and the conceptual model of the interaction of pollution with the environment has been established.
A mathematical model of the interaction of pollution with the environment in the presence of a periodic source of pollution is considered. A solution has been found to the problem of controlling a source of pollution in the presence of a critical condition for the survival of living nature.
Distributed mathematical models described by systems of semilinear differential equations of parabolic type are proposed. An algorithm for numerical solution of the recorded models is formulated. Examples of the dynamics of interaction between pollution and living nature are given.
Based on environmental monitoring data, the problem of identifying (obtaining numerical estimates of model parameters) a distributed mathematical model of the interaction of pollution with the environment has been studied. An algorithm for solving the identification problem is proposed as a search for the minimum of a functional connecting the solution of the mathematical model and observational data.
Scientific novelty of the results
1. For the first time, a number of mathematical models (systems of differential equations) have been proposed to describe the dynamics of the interaction of pollution with the environment, the distinctive feature of which is the presence in them of terms that describe the influence of vegetation cover on the concentration of pollution. In this work, a program has been developed and implemented to carry out simulation modeling of the interaction of pollution with the environment.
Based on a computational experiment using the proposed mathematical model, estimates of the values of the parameters of the mathematical model were obtained and an analysis of the adequacy of the model under consideration to the dynamics of a real ecosystem was carried out,
Based on simulation modeling of the proposed mathematical model, estimates of maximum permissible pollution concentrations for the regions of the Kola Peninsula (Severonnkel plant) and the Southern Urals (Karabash copper smelter) are given.
The reliability of the scientific provisions of the conclusions and recommendations is justified by the use of mathematical evidence, proven simulation modeling methodology, comparability of the results of analytical and computer calculations with available empirical data and expert assessments of specialists.
The practical significance of the work lies in the study and analysis of the proposed mathematical models of the interaction of pollution with the environment, taking into account the ability of vegetation to absorb and process harmful impurities. As an integral part of the work, results are presented on identifying the parameters of a mathematical model of interaction based on environmental monitoring data in the regions of the Kola Peninsula and the Southern Urals and obtaining estimates of maximum permissible pollution concentrations in the regions under consideration.
Proposals for defense:
Mathematical analysis of a conceptual model of the interaction of pollution with the environment.
Formulation and analysis of mathematical models of interaction of pollution with the environment, described by autonomous systems of ordinary differential equations,
Solving the problem of controlling a periodic source of pollution.
Formulation and numerical solution of distributed mathematical models of interaction of pollution with the environment, described by systems of semi-linear equations of parabolic type.
Identification of parameters of a distributed mathematical model of interaction between pollution and the environment based on environmental monitoring data.
Assessment of environmentally maximum permissible pollution concentrations for the regions of the Russian Federation considered in the work.
Approbation of work. The results of the dissertation were presented at the international conference “Control of Oscillations and Chaos” (“COC"OO”), St. Petersburg, July 2000; discussed at a scientific seminar at the Institute of Mathematics and Electronics, Moscow, 2001, a scientific seminar at the Institute of Problems mechanics, Moscow, 2001.
Various parts of the work were reported and discussed at different times at research seminars at Moscow State University, MIIT, in 1999-2001.
Publications. The main provisions of the dissertation were published in the works:
Bratus A.S., Mescherin A.S., Novozhilov A.S. Mathematical models of interaction between pollution and the environment II Bulletin of Moscow State University, ser. 15, Computational Mathematics and Cybernetics, No. 1, 200] pp. 23-28. Bratus A., Mescherin A. and Novozhilov A. Mathematical Models of Interaction between Pollutant and Environment It Proc. of the conference "Control of Oscillations and Chaos", July, St. Petersburg, Russia, 2000, vol. 3, pp. 569 - 572.
Novozhilov A.S. Identification of the parameters of one dynamic system that models the interaction of pollution with the environment II Izvestiya RAS, ser. Theory and control systems, No. 3, 2002.
Structure of the dissertation. The dissertation consists of an introduction, four chapters, a conclusion and a list of references. The volume of work includes 84 pages of text, 26 drawings, 5 tables. The list of cited literature includes 67 titles (59 Russian and 8 English).
The introduction substantiates the relevance of the topic, assesses the degree of development of the problem, formulates the goals and objectives of the work, shows the scientific and practical value of the research conducted, and indicates the provisions of the dissertation to be defended.
The subject of the first chapter is the conceptual model of the interaction of pollution with the environment, proposed by R.G. Khleboprosom (Hlebopros, Fet, 1999). A qualitative analysis of the model under consideration as a one-dimensional discrete mapping is provided, three main scenarios of ecosystem dynamics within the framework of this model are shown, analytical dependencies are given that describe the dynamics of interaction, on the basis of which the process of multiple pollution emissions is numerically simulated.
In the second chapter, assumptions are formulated on the basis of which a system of autonomous differential equations is written that describes the interaction of pollution with the environment. In accordance with the systems approach in ecology, the ecosystem is viewed as a black box. From the variety of external factors, only the factor (considered, in accordance with V. Shelford’s law of tolerance, as limiting (Fedorov, Gilmanov, 1980)) of the impact of polluting emissions from an industrial enterprise on the environment is selected. Using the qualitative theory of differential equations, an analysis of phase flows was carried out for various parameter values and a qualitative correspondence of the differential model was established at the point of the conceptual model of the interaction of pollution with the environment. A number of modifications of the differential model are proposed, based on well-studied systems of the Lotka-Volterra type (Ollee effect, use of trophic functions). A mathematical model of interaction in the presence of a periodic source of pollution has been considered and studied numerically and analytically, and a sufficient condition for the survival of nature within the framework of the model under consideration has been found.
The subject of the third chapter is further complication and modification of the mathematical model of interaction. Based on natural considerations about the heterogeneity of the distribution of pollution concentration and biomass density in space, mathematical models are proposed, described by systems of semilinear parabolic equations that take into account the spatial distribution of pollution and biomass. A diagram of the numerical solution of the studied models is presented and, on the basis of simulation modeling, the processes of interaction of pollution with the environment are considered.
The fourth chapter has practical significance. From the spectrum of mathematical models under consideration, a specific system of partial differential equations is selected. Using statistical data from environmental monitoring of the regions of the Kola Peninsula (Severonickel plant) and the Southern Urals (Karabash copper smelter), a solution algorithm was developed and the problem of identifying (estimating the numerical values of parameters) of the mathematical model was solved. A comparative analysis of observational data and simulation results was carried out. Estimates of maximum permissible pollution levels for the regions under consideration were obtained. The limits of applicability of a specific mathematical model of the interaction of pollution with the environment have been established.
Gratitude. The author expresses sincere gratitude to Professor, Doctor of Physical and Mathematical Sciences A.S. Bratus, who proposed the topic of the dissertation, supported this work and provided assistance to the author in solving many problems. The author also expresses gratitude to the employee of the Center for Problems of Ecology and Forest Productivity of the Russian Academy of Sciences, Butusov O.B., who provided the author with material on environmental monitoring of various regions of our country and repeatedly discussed the results of the work.
This work was partially supported by a grant from the Russian Foundation for Basic Research No. 98 - 01 - 00483.
One-time release of pollutants into the environment
In almost any case, the first step in constructing a mathematical model is the description of one or another biological, environmental, physical, etc. system in terms of a conceptual model that reflects the main qualitative aspects of the nature of the behavior of a given system. The construction of a conceptual model is based on data and statements from experts in a particular subject area. Let's consider a conceptual model of the interaction of pollution with the environment (Khlebopros, Fet, 1999).
Let there be a point source of pollution (for example, a pipe of a metallurgical enterprise). At some initial point in time, an instantaneous release of a pollutant into the environment occurs. It is natural to assume that there is an interaction between nature and pollution. After a certain fixed period of time T, the concentration of pollution will decrease, since natural dissipation of pollution occurs and part of the pollution is processed and absorbed by nature. In other words, the functional relationship between the ejected pollution concentration and the remaining concentration after T time units is described by a certain curve that lies below the bisector of the first coordinate angle. This dependence (destruction curve) was obtained experimentally by ecologists and has the form shown in Fig. ІЛ.
The value Γ is chosen for natural reasons of clarity, since if we take a very short period of time, then the destruction curve will simply be the bisector of the first coordinate angle (how much is thrown out, so much is left); if T is large, then the destruction curve will approach the x-axis (after a long period of time, the pollution concentration will become close to zero).
In Fig. 1.1, the value є indicates a constant background of pollution. The shape of the destruction curve is due to the fact that up to a certain concentration x0 the environment actively reacts with pollution, greatly influencing the concentration, and at point x0 saturation occurs and a threshold effect occurs. This effect is confirmed experimentally for almost all harmful substances (Comprehensive assessment of technogenic impact on the ecosystems of the southern taiga, 1992). For example, forests can even process heavy metals, such as lead, while low concentrations of pollution not only do not negatively affect the density of biomass, but also act as catalysts for growth in some way.
The destruction curve can be considered as a one-dimensional discrete mapping xk+l = f(xk), which has one fixed point. In this case, this fixed point is a global attractor: no matter how large the release of a pollutant into the environment, after a finite time the concentration of pollution will decrease to the natural background value.
Atmospheric diffusion model
It is known that in general, the spatial and temporal change in the concentration of any pollutant u(t, x, y, z) can be described by the following partial differential equation (Berland, 1985): where u = u(t, x, y, z) - concentration of the pollutant, x, y, z - spatial Cartesian coordinates, t - time, v(yx,vy,v2) components of the average speed of movement of the pollutant and, accordingly, in the direction of the x, y, z axes (wind contribution to the movement of the pollutant), Kx, Ky,Kz - molecular diffusion coefficients, R-R(u,(,xty,z) - changes due to atmospheric turbulence, emission, dissipation and movement. Note that the components of the wind vector can be functions of time, diffusion coefficients can be functions of time and spatial coordinates
The R function can be represented as follows:
R = E(t, x, y, z) + P(u) - w, (u) - w2 (u) ,
where E(t,x,y,z) is the characteristic function of pollutant emission sources, P(i)
An operator describing the physical and chemical transformations of a pollutant, w u)
The rate of leaching of the pollutant by precipitation, w2 (u) is the rate of dry deposition.
Since in the future we will be dealing with a point source of a pollutant located at a point with coordinates x0, ua and at a height H, then
the characteristic function of emission sources can be specified using the Dirac delta function (Tikhonov and Samarsky, 1977; Berland 1975,1985):
(/, x, yt z) - a6(x - x0, y - y0, z - #),0 t oo,
where a is the power of the pollution source, (xt,y0,R) are the coordinates of the source.
The remaining terms allow for many different descriptions depending on the type of pollutant and the underlying surface, however, in this particular case, since we are considering a generalized pollutant, it is possible to limit ourselves to a linear dependence with a certain proportionality coefficient g:
P(u) - №, (u) - w2 (u) = -gu, g 0 ,
which indicates that sedimentation, leaching and self-disintegration of the pollutant is constantly occurring.
Equation (2.1) is a second-order partial differential equation of parabolic type, so it is necessary to set initial and boundary conditions. Assuming the existence of an initial pollution distribution, we can write
“(O, x, y, z) = w0 (x, y, z) .
Based on natural considerations that at a considerable distance from the source of pollution the concentration of the pollutant should tend to zero, we set the boundary conditions:
u(t,x,y,z) - 0 for \x\ - yes, \y\ - x ,z - yes, t 0 .
Finally, it is necessary to set a boundary condition at z = 0. Here too
considerable choice is possible (Berland, 1985). For example, if the underlying surface is water, which mostly absorbs the pollutant, then the necessary boundary condition will look like u(t,x,y,0) - 0.
Pollutants usually interact weakly with the soil surface. Once on the soil surface, pollutants do not accumulate on it, but are carried back into the atmosphere with turbulent eddies. If it is believed that the average turbulent flow at the earth's surface is small, then
di Kz - = G at z - 0.0 t yes.
22. In the general case, the boundary condition on the underlying surface is formulated taking into account the possibility of absorption and reflection of the pollutant. Some authors (Monin and Krasitsky, 1985) suggested setting this boundary condition in the form:
Zi Kz--pu= at z = 0.0 o. dz
In order to simplify the model, let us consider averaging the pollutant concentration over height, in other words, we will exclude the third coordinate from consideration. Taking into account the above, the mathematical model of the spread of a pollutant in space R1 (on a plane) will be a mixed problem
di „. . di di „ d2i „ d2i
u(0,x,y) = u(x,y) . (2.2)
u(t,x,y) = 0, for \x\- x ,\y\- co,t 0
In problem (2.2), it is assumed that the diffusion coefficients and components of the wind vector are constant quantities. All parameters included in problem (2.2), except for the components of the wind vector, are considered non-negative.
2.2. Differential model of interaction of pollution with the environment at a point
The patterns of behavior that take place in the conceptual model of the interaction of pollution with wildlife (Chapter 1) underlie the formulation of a mathematical model described by ordinary differential equations.
Let us consider equation (2.1), assuming that the process is localized at some point in space. Then we can write the ordinary differential equation
u = a-gu, w(0) = w0, (2.3)
where a is the generalized power taking into account wind and diffusion, m0 is the initial concentration of pollution.
Equation (2.3) has a solution
u(t) = - + (u0--)e ,
from which it is clear that u(t) -» - at t co. As one would expect, the concentration of pollution with a constant source tends to a certain limit,
the corresponding moment when the power of the source is balanced by the process
self-disintegration.
Let us now assume that pollution is in constant interaction
with the environment, and the environment has a cleansing effect on
pollution. We will consider the pollution-nature system as closed.
Based on these assumptions and assuming that and is the pollution concentration, v is the biomass density, we can write down a system of ordinary differential
equations:
lv = 0 v)-iK«,v)
where /(u, v) 0 is a function of the influence of the environment on pollution, p(v) is a function that describes the behavior of biomass density in the absence of pollution, t//(u,v) 0 is a function of the influence of pollution on the environment.
The behavior of the environment in the absence of pollution will be described by the usual logistic equation:
V(v) = rv(\-), (2.5)
where r is the rate of exponential growth at v « K, K is the potential capacity of the ecosystem, determined by external factors: soil fertility, competition, etc. The solution to the logistic equation (2.5) with the initial condition v(0) = vu is the function
W0= -. v(t)- K at /- «.
Note that, despite the fact that there is a quadratic term in equation (2.5), the solution cannot go to infinity in a finite time, since we consider (2.5) as a mathematical model of biomass dynamics, and therefore v0 0 .
For simplicity, we take bilinear relationships as models of interaction between pollution and wildlife:
f(u,v) = cuv y/(u, V) - duv
Taking into account (2.4) - (2.6), the simplest dynamic model of the interaction of pollution with the environment, described by a system of nonlinear ordinary differential equations, has the form:
and - a - gu - cuv
where all parameters are assumed to be non-negative. Considering (2.7) as a mathematical model of the interaction of pollution with the environment, it is necessary to consider only non-negative solutions (2.7), that is, phase points with coordinates (u,v)eRl - ((u,v) : and 0,v 0).
Model (2.7) is a Lotka-Volterra type system for two competing “species”: pollution and wildlife. The only difference is that the growth pattern in the first equation has no biological, “living” meaning.
class3 Distributed mathematical model of pollution interaction
with the environment class3
Problem formulation
From the point of view of any practical applications, it is clear that it is not enough to study the proposed mathematical model as a system concentrated at a fixed point. In the theory of mathematical modeling, models naturally appear where either the parameters or the phase coordinates themselves are functions of not only time, but also spatial coordinates. In many cases, the parameters are perturbed randomly. For the most part, such a generalization leads to mathematical models described either by a single equation or by a system of partial differential equations - an infinite-dimensional dynamical system.
In the particular case under consideration, it is natural to assume that the spatial distribution of pollution concentration and biomass density is heterogeneous, that is, pollution and biomass are functions of spatial coordinates:
v = v(x, y, Z, i) We consider the source of pollution to be a point source; the mathematical model for it will be the Dirac delta function. If there are n sources of pollution, then the source function is the sum of the delta functions:
E(xty,h) = Y,at S(x-xi y-yi,h hi),i \...n,
where o, is the power of the i-th source of pollution, (x y h are the coordinates of the i-th source of pollution.
If the set of coordinates of the pollution source is infinite, then the delta function from this set should be included in the equation - for example, if the set of coordinates of the pollution source is described by the equation y-ax + b, then it is necessary to consider the term S(y -ax-b) (this, for example, may correspond to a motorway).
Mathematical model
The experience of the development of natural science in general and ecology in particular indicates that observations and experiments contribute to knowledge to the greatest extent only when they are conceived and implemented on the basis of scientific theory. In the exact natural sciences, to which modern ecology is increasingly striving, models are a very effective form of expressing theoretical concepts, and one of the most fruitful methods is the method of modeling, that is, constructing, testing, studying models and interpreting the results obtained with their help.
The essence of the modeling method is that, along with the system (original), which we denote J", its model is considered, which is some other system - J, which is an image (similarity) of the original y0 under the modeling display (similarity correspondence) /: where the brackets indicate that / is a partially defined mapping, that is, not all features of the composition and structure of the original are displayed by the model. It is usually advisable to represent / as a composition of two mappings - coarsening and homomorphic. Depending on the nature of coarsening and the degree of aggregation (model capabilities in a certain sense, correctly reflect the original) for the same original you can obtain several different models. One of the advantages of the modeling method is the ability to build models with a “convenient” implementation (characteristic of “how and what the model is made of” (Poletaev, 1966) ), because a successful choice of implementation makes the study of the model incomparably easier than the study of the original, and at the same time allows one to preserve the essential features of its composition, structure and functioning.
Two types of iconic (ideal) models are of greatest importance for ecology: conceptual and mathematical models. The conceptual model of the interaction of pollution with the environment was discussed in Chapter 1, and various mathematical models were discussed in Chapters 2 and 3, for the purposes of this one. Chapter - comparison of modeling results with observational data - it is necessary to select a specific mathematical model from those discussed above, using an adequate coarsening mapping that, if possible, simplifies the model as much as possible.
To obtain information about the spatial variability of the concentrations of harmful substances in the air and, based on experimental data, to create a map of air pollution, it is necessary to systematically carry out air sampling at regular grid nodes with a step of no more than 2 km. Such a task is practically impossible. Therefore, to construct concentration fields, methods of mathematical modeling of the processes of dispersion of impurities in atmospheric air are used, implemented on a computer. Mathematical modeling assumes the availability of reliable data on meteorological features and emission parameters. The applicability of models to real conditions is verified using data from network or specially organized observations. Calculated concentrations should match those observed at sampling points.
The model can be any algorithmic or analog system that allows one to simulate the processes of dispersion of impurities in atmospheric air.
In our country, the model of Professor M.E. is most widespread. Berlyanda. In accordance with this model, the degree of atmospheric air pollution by emissions of harmful substances from continuously operating sources is determined by the highest calculated value of a single ground concentration of harmful substances (C m), which is established at a certain distance (x m) from the place of release under unfavorable meteorological conditions, when the wind speed reaches a dangerous value (V m), and intense turbulent exchange occurs in the surface layer. The model makes it possible to calculate the field of one-time maximum concentrations of impurities at ground level for emissions from a single source and a group of sources, for heated and cold emissions, and also makes it possible to simultaneously take into account the effect of heterogeneous sources and calculate the total air pollution from a combination of emissions from stationary and mobile sources.
The algorithm and procedure for calculating fields of maximum concentrations are set out in the "Methodology for calculating concentrations in the atmospheric air of harmful substances contained in emissions from enterprises. OND - 86" and in the corresponding instructions for calculation programs.
As a result of the calculations carried out on a computer, the following results are obtained:
- · maximum concentrations of impurities at the nodes of the computational grid, mg/m 3 ;
- · maximum surface concentrations (C m) and the distances at which they are reached (x m) for sources of emissions of harmful substances;
- · share of the contribution of the main sources of emissions at the nodes of the computational grid;
- · maps of atmospheric air pollution (in fractions of MPC mr);
- · printout of input data on sources of pollution, meteorological parameters, physical and geographical features of the area;
- · list of sources that make the greatest contribution to the level of air pollution;
- · other data.
Due to the high saturation of cities with sources of pollution, the level of air pollution in them, as a rule, is significantly higher than in the suburbs and, even more so, in rural areas. In certain periods unfavorable for the dispersion of emissions, the concentrations of harmful substances can greatly increase relative to the average and background urban pollution. The frequency and duration of periods of high atmospheric air pollution will depend on the regime of emissions of harmful substances (one-time, emergency, etc.), as well as on the nature and duration of weather conditions that contribute to an increase in the concentration of impurities in the ground layer of air.
In order to avoid increasing levels of atmospheric air pollution under meteorological conditions unfavorable for the dispersion of harmful substances, it is necessary to predict and take into account these conditions. Currently, factors have been established that determine changes in the concentrations of harmful substances in the atmospheric air when meteorological conditions change.
Forecasts of adverse weather conditions can be made for the city as a whole, or for groups of sources or individual sources. There are usually three main types of sources: high with hot (warm) emissions, high with cold emissions and low.
In addition to complexes of adverse weather conditions, the following can be added:
- - For high sources with hot (warm) emissions:
- · the height of the mixing layer is less than 500 m, but greater than the effective height of the source;
- · the wind speed at the height of the source is close to dangerous wind speed;
- · presence of fog and wind speed more than 2 m/s.
- - For high sources with cold emissions: presence of fog and calm.
- - For low emission sources: a combination of calm and surface inversion.
It should also be borne in mind that when impurities are transferred to densely built areas or in difficult terrain, concentrations can increase several times.
To characterize air pollution in the city as a whole, i.e. for background characteristics, parameter P is used as a generalized indicator:
where N is the number of observations of the concentration of impurities in the city during one day at all stationary posts; M is the number of observations during the same day with an increased impurity concentration (q), exceeding the average seasonal value (qI ss) by more than 1.5 times (q > 1.5 qI ss).
The P parameter is calculated for each day both for individual impurities and for all of them together. This parameter is a relative characteristic, and its value is determined mainly by meteorological factors that influence the state of atmospheric air throughout the city.
The use of parameter P in forecasting as a characteristic of air pollution for the city as a whole (predictant) provides for the identification of three groups of air pollution, determined by the characteristics given in Table. 1
In order to prevent extremely high levels of pollution, a subgroup of gradations with P > 0.5 is distinguished from the first group, the repeatability of which is 1 - 2%.
The methodology for predicting the likely increase in the concentrations of harmful substances in the atmospheric air of a city involves the use of a predictive air pollution scheme, which is developed for each city based on the experience of many years of monitoring the state of its atmosphere. Let's consider the general principles of constructing predictive schemes.
Forecast schemes for air pollution in the city should be developed for each season of the year and each half of the day separately. With a sliding air sampling schedule, the first half of the day includes sampling times at 7, 10 and 13 hours, and the second - at 15, 18 and 21 hours. With three-time sampling, the first half of the day includes sampling times at 7 and 13 h, and for the second - at 13 and 19 h.
Meteorological predictors for the first half of the day are taken for a period of 6 hours, and radio sounding data for a period of 3 hours. For the second half of the day, meteorological elements for a period of 15 hours are taken as predictors. The characteristics of meteorological conditions and predictors, as well as their procedure for using in forecasts, are described in detail in "Guidelines for forecasting air pollution in cities."
Operational forecasting of atmospheric air pollution is carried out with the aim of short-term reduction of emissions of harmful substances into the atmospheric air during periods of unfavorable meteorological conditions.
Usually, two types of forecasts of atmospheric air pollution for the city are compiled: preliminary (a day in advance) and updated (6 to 8 hours in advance, including in the morning for the current day, in the afternoon for the evening and at night).
UDC 004.942
ON THE. Solyanik, V.A. Kushnikov
MATHEMATICAL MODELING OF THE PROCESS OF ATMOSPHERIC AIR POLLUTION IN THE ZONE OF INFLUENCE OF INDUSTRIAL ENTERPRISES
Models and algorithms for information software for environmental monitoring in the zone of influence of industrial enterprises are presented. Models of atmospheric dispersion are considered with the aim of their optimization and further application in the developed information and software complex. A mathematical model based on the Gaussian equation is used as the main model of atmospheric dispersion.
Mathematical modeling, environmental monitoring, atmospheric air, Gaussian distribution of concentrations, automated control system, source of pollution, industrial complex.
N.A. Solyanik, V.A. Kushnikov
THE MATHEMATICAL SIMULATION OF AIR POLLUTION IN INDUSTRIAL ZONE OF INFLUENCE
The paper presents models and algorithms for information-software of the ecological monitoring in a zone of the industrial enterprises’ influence. We consider models of an atmospheric dispersion with the goal of their optimization and the further application in a developed information-program complex. As the basic model of the atmospheric dispersion the mathematical model on the basis of Gauss equation is applied.
Mathematical modeling, environmental monitoring, air, concentrations Gaussian distribution, automated control system, the source of pollution, industrial complex.
In the context of intensification of economic activity and an increase in the number of regularly operating industrial facilities on the territory of the Russian Federation, assessing the negative impact on the environment from the industrial complex is becoming increasingly important. At the same time, the most dangerous is air pollution in the zone of influence of industrial enterprises.
Environmental monitoring in large industrial centers of the Russian Federation is not carried out effectively enough. For example, due to the fact that the city of Saratov is a large industrial center located in an area with complex terrain and having a satellite city of Engels, it is necessary to increase the number of monitoring posts for monitoring the state of atmospheric air, which will require significant material costs.
There are also alternative methods for obtaining up-to-date information about the level of air pollution, for example, aerospace monitoring of atmospheric air. But their use, as well as the construction of additional observation posts, is associated with significant material investments.
In this regard, the task of mathematical modeling of the processes of distribution of pollutants in the atmospheric air in the zone of influence of industrial enterprises is relevant. Modeling is a more cost-effective alternative to the use of stationary observation posts and aerospace monitoring of the air basin. At the same time, the use of mathematical models of the distribution of impurities in atmospheric air will significantly increase the efficiency of obtaining results.
It is necessary to develop a set of mathematical models designed for environmental monitoring of atmospheric air in the zone of influence of industrial enterprises.
These mathematical models are focused on use as part of an automated system for controlling the process of environmental pollution in the zone of influence of industrial enterprises; in this regard, there is a need to consider the most common procedures for controlling the qualitative composition of the air basin.
Firstly, timely receipt of information on the level of concentration of pollutants makes it possible to identify sources whose influence significantly increases the health risk of the population at receptor points. At the same time, by modeling the process of atmospheric air pollution by an intruder source, we can change the input parameters of the control object, such as the emission power, the height of the source (pipe), in order to minimize the concentration level. This will make it possible to formulate requirements for the source of pollution, the implementation of which will reduce the level of its negative impact on the environment to a minimum. In addition, it becomes possible to simulate various types of weather conditions. This will allow the relevant services to more clearly develop rules regulating the level of emissions in accordance with adverse meteorological conditions for each source of pollution.
Let us consider the main physical processes, the mathematical modeling of which will be used to solve the problem.
The mathematical model is based on dependencies that make it possible to calculate the distribution of impurities in the atmospheric air from a pollution source, taking into account the parameters of the source and the environment. At the same time, most authors consider two large classes of models: models based on the Gaussian concentration distribution and transport models, which are based on the turbulent diffusion equation. Let us dwell in more detail on Gaussian models (Fig. 1).
The subject of modeling is the processes of distribution of pollutants in the atmospheric air in the zone of influence of industrial enterprises.
The input parameters of the model include:
H is the effective height of the torch rise, expressed in meters and characterizing the initial rise of the impurity. The work provides an overview of the basic formulas for calculating N;
Q - power or
intensity of the emission source, expressed in g/s and characterizing the amount of substance released by the source at time t.
Model disturbances
characterized by the following
parameters:
K - atmospheric stability class. There are 6 classes of stability of the surface layer of air,
symbolically designated through the first 6 letters of the English alphabet (from A to B). Each of the classes corresponds to certain values of wind speed and, degree of insolation and time of day;
I is the wind speed at height H, expressed in m/s;
Ф - wind direction, expressed through the angle of inclination to the base coordinate system.
The output of the model is the level of pollutant concentration C(xy,z) at a point in space (xy^), expressed in μg/m3.
Rice. 1. Operating principle of the model for the distribution of impurities in atmospheric air based on the Gaussian distribution of concentrations
sustainability
atmosphere
Outrage
i- speed
κ - wind direction (expressed through the angle of inclination to the base coordinate system)
N- effective
Inputs torch lift height Mathematical model C(x,y^) - concentration y X -O co
(^- power of the pollutant emission source at a point in space (x/y/g)
Rice. 2. Input and output parameters of the mathematical model
In the model under consideration, the direction of the wind coincides with the direction of the OX axis; the origin of coordinates is considered to be the base of the source (for example, the base of a pipe). There are a number of Gaussian models that differ in the way they specify the dispersion of the propagation of impurities in the corresponding directions. Below is a general view of the non-stationary Gaussian model of the distribution of impurities in atmospheric air:
(27G)3 2STxSTu(72
((x-w)2 S---I)2’ (g + H I2
V x e U e 2 " + e
A simulation system for modeling the distribution of impurities in atmospheric air was developed (Fig. 3), designed to calculate the level of impurity concentration at all points in space x, y, z. The system allows you to calculate the level of pollutant concentration with predetermined input parameters, as well as monitor changes concentration values depending on changes in one or another parameter. At the same time, it is possible to calculate the average concentration level under conditions where the values of the input parameters change over time.
Rice. 3. Modeling algorithm and functional specification of a simulation system for modeling the distribution of impurities in atmospheric air
Simulation algorithm:
1. At the initial stage, the base coordinate system is set, as well as the number of steps of changes in input parameters over time.
3. The next step generates wind speed and direction values, as well as atmospheric stability classes.
5. The obtained result is “overlaid” on the base coordinate system, after which, depending on the size of the generated arrays of input variables, steps 3 to 5 are iteratively repeated.
6. At the last step, the average value of the concentration level is calculated
pollutant at all points in space x, y, z and visualization is carried out
result.
The output of the mathematical model contains a three-dimensional array containing the values of the pollutant concentration level at all points in space x, y, z. The obtained values are used to construct graphs,
characterizing the level of pollutant concentration at different distances from the source, including a graph of the surface of the impurity plume from the source (Fig. 4), as well as various types of graphs in the form of isolines (Fig. 5).
Rice. 4. Visualization of simulation results for various input parameters and disturbances
Rice. 5. Graphs of the level of pollutant concentration in isolines (x-axis - coordinates in the wind direction X, ordinate axis - coordinates perpendicular to the wind direction Y)
The results obtained confirm the possibility of using expression (1) when modeling the processes of distribution of pollutants in the atmospheric air in the zone of influence of industrial enterprises.
LITERATURE
1. Solyanik N.A. Information system for forecasting the state of atmospheric air in Saratov / N.A. Solyanik, V.A. Kushnikov, N.S. Pryakhina // Environmental problems of industrial cities: collection of articles. scientific tr. Saratov: SSTU, 2005. pp. 153-156.
2. GOST 17.2.3.01-86 “Rules for monitoring air quality in populated areas.” M.: Publishing house of standards, 1986. 26 p.
3. Berlyand M.E. Forecast and regulation of atmospheric pollution / M.E. Berland. L.: Gidrometeoizdat, 1985. 272 p.
emissions in the information and analytical system of environmental services of a large city: textbook. allowance / S.S. Zamai, O.E. Yakubailik. Krasnoyarsk: KSU, 1998. 109 p. Solyanik Nikolay Aleksandrovich -
graduate student of the Department of Information Graduate Student of the Department
systems in the humanitarian field" of "Information Systems in Humanities"
Saratov State of Saratov State Technical University
technical university
Kushnikov Vadim Alekseevich -
Professor, Doctor of Technical Sciences, Head of the Department of Information Systems in the Humanities, Saratov State Technical University
Kushnikov Vadim Alekseyevich -
Professor, Doctor of Technical Sciences, Head of the Department of “Information Systems in Humanities” of Saratov State Technical University
