Completed:
Student of group VT26-5
Sadovsky M.V.
Checked:
Belolipetsky V.M.
Krasnoyarsk ’ 370 years
Introduction:
When studying any phenomenon, a qualitative description of the problem is first obtained. At the modeling stage, the qualitative representation turns into a quantitative one. At this stage, the functional dependencies between the variables for each solution and the input data and output data of the system are determined. Building models is an informal procedure and very much depends on the experience of the researcher, it always relies on certain experimental material. The model must correctly reflect the phenomena, but this is not enough - it must be convenient for use. Therefore, the degree of detail of the model, the form of its presentation depend on the study.
The study and formalization of experimental material is not the only way to build a mathematical model. An important role is played by obtaining models that describe particular phenomena from more general models. Today, mathematical modeling is used in various fields of knowledge; many principles and approaches have been developed that are of a fairly general nature.
The main task of scientific analysis is to single out real movements from the set of mentally permissible ones, to formulate the principles of their selection. Here the term "movement" is used in a broad sense - changes in general, any interaction of material objects. In various fields of knowledge, the principles of selection of movements are different. It is customary to distinguish three levels of organization of matter: inanimate, living and thinking. At the lowest level - inanimate matter - the basic principles of selection are the laws of conservation of matter, momentum, energy, etc. Any modeling begins with the choice of the main (phase) variables, with the help of which the conservation laws are written.
Conservation laws do not single out a single solution and do not exhaust all selection principles. Various conditions (restrictions) are very important: boundary, initial, etc.
At the level of living matter, all the principles of selection of motions, which are valid for non-living matter, retain their force. Therefore, here, too, the modeling process begins with the recording of conservation laws. However, the main variables are already different.
The advantages of mathematical models are that they are precise and abstract, conveying information in a logically unambiguous manner. Models are accurate because they allow predictions to be made that can be compared with real data by setting up an experiment or making the necessary observations.
Models are abstract, since the symbolic logic of mathematics extracts those and only those elements that are important for the deductive logic of reasoning, excluding all extraneous meanings.
The disadvantages of mathematical models often lie in the complexity of the mathematical apparatus. Difficulties arise in translating the results from the language of mathematics into the language of real life. Perhaps the biggest drawback of the mathematical model is related to the distortion that can be introduced into the problem itself by stubbornly defending a particular model, even if in reality it does not correspond to the facts, as well as the difficulties that sometimes arise when it is necessary to abandon a model that has turned out to be unpromising. . Mathematical modeling is such a fascinating activity that it is very easy for a “modeler” to move away from reality and get carried away with the application of mathematical languages to abstract phenomena. That is why it should be remembered that modeling in applied mathematics is only one of the stages of a broad research strategy.
The book is devoted to the problems of environmental pollution during accidents of industrial enterprises and objects of various profiles and is mainly of an overview reference character.
The dynamics of emergency turbulent emissions in the presence of atmospheric diffusion, the nature of the expansion of turbulent jet streams, their resistance in the blowing wind, the evolution of emissions in the real atmosphere in the presence of inversion delay layers are studied.
Possible accidents with emissions of pollutants and toxic substances into the atmosphere in gaseous, liquid or solid phases are classified and analyzed, and emergency risk factors are given.
Accidents associated with emissions of toxicants into the atmosphere are considered, mathematical models of emergency emissions are described. It is shown that the whole variety of anthropogenic sources of atmospheric air pollution during accidents can be conditionally divided into separate classes according to the type of emissions that have arisen and the nature of the movement of their matter. Fires, explosions and toxic emissions are considered as pollution sources. These sources, depending on the specifics of the supply of the working fluid to the surrounding space, form atmospheric emissions in the form of solid or liquid particles falling on the earth's surface, jets, terms and clubs, spills, evaporation volumes and heat columns. The environmental hazards of emissions during accidents and in everyday life are considered.
The book contains large illustrative material in the form of tables, graphs, figures and photographs, which helps the reader to understand the issues under discussion. It is addressed to a wide range of people whose occupation is mainly related to environmental issues: engineers, scientists, students and all those who are interested in environmental and environmental topics.
Book:
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At present, the efforts of scientists from all over the world have created a single fund of models of processes occurring in animate and inanimate nature. These models are usually based on a small number of fundamental principles that tie together the diverse facts and ideas of the natural sciences. Each model in this fund occupies a certain place, the limits of its applicability and connection with other models are established. The presence of such a fund of models gives confidence to researchers when using them in practical activities - after all, each of these models, thanks to connections with other models, relies not so much on a specific verification of itself, but on the entire practical experience of mankind. For each specific object in this fund, you can choose the most appropriate model or modify it from similar models.
With regard to the problems of environmental protection, the development of theories of the emergence and transformation of pollutants in natural environments, which has manifested itself in the presence of a grandiose fund of natural processes, on the one hand, determines the high efficiency of the use of mathematical models and methods in engineering practice, and on the other hand, gives researchers a single picture of the surrounding world.
In general, the basis for a constructive approach to the problem of human interaction with nature is provided by modeling (in particular, mathematical modeling) in combination with purposeful experimental studies. Pollution of natural environments is one of the most typical manifestations of such an interaction.
The set of factors that must be taken into account in models is at the intersection of a number of research programs implemented within the geosciences. The complex nature of such programs and the presence of complex direct and feedback links between hydrometeorological processes, pollution of natural environments, and the biosphere actively stimulate the development of theoretical foundations and the systematic organization of mathematical models. At this higher level, the system organization operates with the "simplest" models as with elementary objects.
With regard to mathematical modeling of the processes of occurrence and development in the atmosphere of accidental emissions of pollutants and toxic substances, we will proceed from models of physical processes. These include models of hydrothermodynamics of the atmosphere of various space-time scales, as well as models of the transfer and transformation of impurities, various methods of parameterization, etc. There are quite a lot of similar developments in the literature. Their physical meaning and differences between them depend on the specific problem setting. In any case, as applied to solving the problem by numerical simulation methods, one proceeds from the concepts of state functions and parameters.
For convenience and brevity of presentation, we use the operator form. Let us denote the vector state function as
Its components include fields of hydrometeorological elements and concentrations of pollutants.
We denote the parameter vector
The parameters are the coefficients of the equations, the parameters of the area of integration D t of the grid area D h t , the area of placement of observational systems D m t , the initial values of the functions of state, distribution and power of heat sources, moisture and other impurities and components.
In operator form, the mathematical model of the described process has the following form:
Nonlinear differential operator of matrix structure acting on sets of functions
;
Q(D t) - space of state functions satisfying boundary conditions;
R(D t) - area of acceptable parameter values;
B is a diagonal matrix in which all or part of the elements can be zeros;
Sources;
Where D is the area of change of spatial variables;
Time interval t.
The operator in relation (1.1)
It is determined by the equations of hydrothermodynamics of the atmosphere - soil - water system, the transfer and transformation of impurities, as well as the conditions at the interfaces.
Boundary and initial conditions are written for the specific physical content of the model.
In particular, for the mathematical model of the transfer of impurities in the atmosphere, which is included in equation (1.1) as an integral part, we obtain the equation
This model takes into account the processes of possible transformation of substances, turbulent exchange and exchange processes between natural media: water, air and soil.
In relation (1.2):
impurity concentration;
Velocity vector with u,v,w components in spatial direction
Respectively;
AND? - turbulence coefficients in horizontal (x 1 ,x 2) and vertical (x 3 = z) directions;
the index s marks the operators acting in horizontal directions;
Impurity transformation operators;
Sources of impurities (at the same time sources of natural and anthropogenic origin are taken into account).
Note that operations with a vector
They are implemented component by component, i.e. equation (1.2) is a system of n partial differential equations. Operator
Generally non-linear. It determines the rate of change in concentrations c i due to chemical and photochemical reactions. The velocities of the vertical movement of particles (settlement or ascent) are taken into account by the function w. Impurities are multicomponent, the number of components is the input parameter of the model. In practice, the model parameter is determined by the amount of chemicals involved in the reactions.
The model is supplemented with initial and boundary conditions:
R 1 and R 2 are some operators;
Sources and sinks of impurities at the upper and lower boundaries of region D.
For the global model, conditions for the periodicity of all functions on the surface of the sphere are set, and for models on a limited area, conditions for the concentration fields on the lateral boundaries of the region D t .
The processes of interaction of impurities with the underlying surface, including exchange processes between air, water, soil and vegetation, are described by the operator
Moreover, the concentration vector
is included in the vector function of the state of the system as a whole, and the coefficients of equations (1.2) and boundary conditions (1.4), (1.5), as well as the initial conditions (1.3), source functions
And the rate constants of gas-phase reactions in the operator
Included in the parameter vector.
Note that in computational models, an extended concept of parameters is used, including not only the numerical values of some quantities, but also algorithms for their calculation. Then the parameters include reaction schemes, algorithms for calculating radiative heat fluxes, turbulent exchange coefficients, as well as coefficients in models of the interaction of air masses with the underlying surface.
The development of the approaches presented here for the construction of discrete analogs of models and computational algorithms applies variational principles, the use of which provides qualitatively new information about the behavior of a mathematical model.
Obviously, in the process of numerical simulation, the meaning inherent in the initial formulations of the problem should not be lost, and the results of calculations should correspond to the actual processes.
When solving practical problems, there is always an acute problem of setting input parameters and initial data, information about which, as a rule, is fragmentary and incomplete. Therefore, the use of multidimensional and multicomponent models, while creating the illusion of a detailed consideration of the process, is not capable of producing results whose accuracy exceeds the accuracy of the initial setting parameters. Each mathematical model can only be considered valid when the reliability of the results of its use has been assessed.
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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 and software support for environmental monitoring in the zone of influence of industrial enterprises are presented. Atmospheric dispersion models are considered with the aim of their optimization and further application in the developed information and software complex. As the main model of atmospheric dispersion, a mathematical model based on the Gauss equation is used.
Mathematical modeling, environmental monitoring, atmospheric air, Gaussian distribution of concentrations, automated control system, pollution source, 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 the 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 the intensification of economic activity and an increase in the number of regularly functioning industrial facilities in the territory of the Russian Federation, the assessment of 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. So, for example, due to the fact that the city of Saratov is a large industrial center located on a territory with a difficult terrain and having a satellite city of Engels, it is necessary to increase the number of air monitoring posts, which will require significant material costs.
There are also alternative methods for obtaining up-to-date information on 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 problem of mathematical modeling of the processes of the spread 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 fixed observation posts and aerospace monitoring of the air basin. At the same time, the use of mathematical models of the spread of impurities in the atmospheric air will significantly increase the efficiency of obtaining the result.
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 control system for the process of environmental pollution in the zone of influence of industrial enterprises, in connection with this, it becomes necessary to consider the most common procedures for managing 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, the influence of which significantly increases the risk to the health of the population of receptor points. At the same time, by simulating the process of atmospheric air pollution by a violating 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, in the implementation of which the level of its negative impact on the environment will be minimized. In addition, it becomes possible to simulate various types of weather conditions. This will allow the relevant authorities to more clearly develop rules governing 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 in solving the problem.
The basis of the mathematical model is dependencies that allow calculating the distribution of impurities in the atmospheric air from the source of pollution, 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 distribution of concentrations and transport models based on the turbulent diffusion equation. Let us dwell in more detail on the 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 flare, expressed in meters and characterizing the initial rise of the admixture. The paper gives an overview of the basic formulas for calculating H;
Q - power or
the intensity of the release source, expressed in g/s and characterizing the amount of substance emitted by the source at time t.
Model Perturbations
are characterized by the following
parameters:
K - class of stability of the atmosphere. 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, the degree of insolation and time of day;
I - 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. The principle of operation of the model for the distribution of impurities in atmospheric air based on the Gaussian distribution of concentrations
sustainability
atmosphere
perturbations
i - speed
q> - wind direction (expressed in terms of the angle of inclination to the base coordinate system)
H- effective
Entrances Flare lift height Mathematical model C(x,y^) - concentration y X -O co
(^- power of the source of pollutant emission at a point in space (х/у/г)
Rice. 2. Input and output parameters of the mathematical model
In the model under consideration, the wind direction coincides with the direction of the OX axis, the origin of coordinates is the base of the source (for example, the base of the pipe). There are a number of Gaussian models that differ in the way they set the variance of the propagation of impurities in the respective directions. Below is a general view of the non-stationary Gaussian model for the propagation of impurities in atmospheric air:
(27G)3 2STxSTu(72
((x-w)2 C---I)2' (r + H I2
V x e Y e 2 "+ e
A simulation system was developed for modeling the spread of impurities in atmospheric air (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 track concentration values depending on the change of 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 the simulation system for modeling the spread 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 the input parameters over time.
3. At the next step, wind speed and direction values are generated, as well as atmospheric stability classes.
5. The result obtained is “imposed” on the base coordinate system, after which, depending on the dimension 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 of space x, y, z and visualization is carried out
result.
At the output of the mathematical model, there is 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 plot 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 parameters of inputs and disturbances
Rice. Fig. 5. Graphs of the level of pollutant concentration in isolines (abscissa axis - coordinates along the wind direction X, ordinate axis - coordinates perpendicular to the wind direction Y)
The results obtained confirm the possibility of using expression (1) in modeling the processes of the spread 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 // Ecological problems of industrial cities: Sat. scientific tr. Saratov: SGTU, 2005, pp. 153-156.
2. GOST 17.2.3.01-86 "Rules for air quality control in settlements". 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: KGU, 1998. 109 p. Solyanik Nikolay Aleksandrovich - Solyanik Nikolay Aleksandrovich -
Graduate Student of the Department
systems in the humanities" of "Information Systems in Humanities"
Saratov State Technical University of Saratov
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
Consider the biospheric processes of the spread of pollution from single industrial sources, paying special attention to the study of sanitary and hygienic situations due to especially dangerous pollution conditions.
In the general case, the change in the average values of the concentration U is described by the equation
where the x and y axes are placed in the horizontal plane; z-axis - vertically; t - time; V, P, W - components of the average speed of movement of impurities relative to the direction of the axes x, y, z; - 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 an inversion-free state of the air basin can be insignificant and does not require special methods to protect the population.
Another situation arises due to unpleasant meteorological conditions (temperature inversions during light winds and calm weather). Accounting for unpleasant meteorological conditions is one of the little-studied issues.
During the occurrence of inversions, the air temperature in the surface layer rises, and does not fall, as in the case of persistent thermal stratification of the atmosphere. Mixing is weak, and the lower part of the inversion layer plays the role of a screen, from which a torch of pollutants is partially or completely reflected, and the concentration of harmful impurities in the surface layer increases to values dangerous to human health and life.
Theoretical models for calculating atmospheric air pollution do not reflect the entire set of factors that affect 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 occur 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 precisely such conditions that are the most dangerous for human health and should be the object of hygienic forecasts in the case of planning the location of zones of industrial enterprises.
To achieve this goal, it becomes necessary to create prediction 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 the data of field observations of pollutant concentrations under appropriate conditions, which provides 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 analogs are first determined using observational data, and then their adequacy is checked when calculating concentrations with data that do not take part 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 of time; r is the distance from the source; z - vertical distance; - angle of rotation relative to the axis; - functions:
As can be seen from equation (3.3), the pollution source is located at the 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 pollution along the flame 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 emissions from industrial enterprises were used to construct equations for the distribution of individual ingredients, and they form the basis for the practical verification of models.
Synthesis of the equation for predicting the maximum level of dust pollution:
To approximate the functions, the following expressions were used:
where are linear functions.
We write the derivatives as 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.
According to research data in different cities of Ukraine, continuous curves of pollution observations were approximated. Following the combinatorial algorithm, the following 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 in calculating the concentration of a pollutant until the concentration reaches the maximum permissible values for a given substance.
1In the conditions of the modern ecological situation, modeling of 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 that affect the dispersion of pollutants. The key requirements for models of atmospheric air pollution are given. The stages of construction and classification of atmospheric air pollution models are considered. One of the types of atmospheric air pollution models are models based on a mathematical description of the physical processes occurring in the atmosphere. Models built on the basis of solving the turbulent diffusion equation are similar. Solutions of the equation for describing the phenomenon of transfer and diffusion of a pollutant for the models of "coil", "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
1. Egorov A.F., Savitskaya T.V. Safety management of chemical production based on new information technologies. - M.: Chemistry, Kolos, 2006. - 416 p.
2. Baranova M.E., Gavrilov A.S. Methods for computational monitoring of atmospheric pollution in megacities // Natural and technical sciences. - M .: LLC Publishing House Sputnik +, 2008. - No. 4. - P. 221–225.
3. Plotnikova L.V. Ecological management of the quality of the urban environment in highly urbanized areas. - M .: Publishing house of the Association of construction universities, 2008. - 239 p.
4. Tsyplakova E.G., Potapov A.I. Otsenka sostoyaniya i upravlenie kachestva otmosfernogo vozduha: uchebnoe posobie [Assessment of the state and management of atmospheric air quality: a textbook]. - St. Petersburg: Nestor-History, 2012. - 580 p.
5. Tyurikov B.M., Shkrabak R.V., Tyurikova Yu.B. Modeling the processes of distribution of pollutants in the air of working areas of industrial sites of agricultural enterprises / B.M. Tyurikov, R.V. Shkrabak, Yu.B. Tyurikova // Bulletin of the Saratov State Agrarian University. - 2009. - No. 10. - P. 58–64.
6. Modeling the spread of pollutants in the atmosphere on the basis of the “torch” model / Kondrakov O.V. [and others] // Bulletin of the Tambov University. - 2011. - T. 16, No. 1. - S. 196-198.
In the conditions of the modern ecological situation, modeling of atmospheric air pollution is an urgent problem.
The development of computer technology capabilities makes it possible to use mathematical modeling tools to study such complex physical and chemical processes as atmospheric diffusion, transformations of pollutants in the atmosphere, processes of leaching and sedimentation of impurities, etc., taking into account meteorological and topographic conditions.
The atmospheric air pollution model must meet the following basic requirements: the required resolution of the forecast in space and time; take into account weather conditions and the state of the troposphere and the earth's surface at the points of contact, types of pollution sources; increasing the accuracy of the model as the amount of information increases or its quality improves.
The stages of building a model of atmospheric air pollution are shown in fig. one.
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 either an operational forecast or long-term planning. The forecast for time from 30 minutes to one day is considered operational. Other sources consider other forecasting periods: express or operational, assuming a time of 1-2 hours, short-term for time from 12 hours to 1-2 days, long-term - from 3 days to 2-3 weeks, promising - from 1 month to several years .
The presence of various approaches to modeling the processes occurring in the atmosphere is due to the lack of a generalized physical and mathematical model that takes into account all the parameters of atmospheric diffusion phenomena. The choice of approach to modeling depends on the problem statement and determines the quality of the model and the accuracy of the forecast.
Rice. 1. Stages of building a model of atmospheric air pollution
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, areal, 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 atmospheric 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 turbulent diffusion equation (Fig. 3).
In these models, the physical phenomena of the transfer and diffusion of a pollutant in the 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 velocity field of the air; QC is the source of pollution.
For the mathematical formulation of the problem of solving equation (1), it is necessary to set the 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 the solution of the turbulent diffusion equation
Assuming in equation (1) the absence of the distribution of pollutant particles with air flows, the heterogeneity of the atmosphere, and also assuming that the source of pollution is outside the area, we obtain the equation
(2)
The fundamental solution to this equation is a Gaussian curve and is used in the "coil" and "torch" models.
The coil model assumes that the pollution source acts instantaneously. The transfer of pollutant emissions under the influence of wind is represented in a moving coordinate system.
The tangle model has the following form:
where x, y, z are the coordinates of the center of the "coil", which determine the trajectory of its movement; u, v, w - average wind speeds in directions x, y, z at time t; σ x , σ y , σ z - standard deviations of the size of the "coil" in the directions x, y, z, respectively; Q is the amount of pollutant emitted by the source at time t.
The “coil” model has some disadvantages, such as the need for numerous measurements of wind speeds in the x, y, z directions, the difficulty in identifying the pollutant coil parameters (center height, size deviations in directions), and the complexity of software implementation.
Consider the "torch" model. In this model, it is assumed that the source is point and acts continuously.
The "torch" model is used in the case of pollutant emissions from point sources of different altitudes, the temperature and nature of the emissions are not taken into account.
The flame model looks like this:
where C(x, y, z, H) - concentration distribution along x, y, z coordinates, Q - pollutant release rate; u - average wind speed; σ y (x), σ z (x) - standard deviations of the dimensions of the "torch" in the horizontal and vertical directions for a given x, H = h + Dh - effective height of the torch; h - pipe height; Dh - the rise of the flame 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 with the pollutant do not occur;
The pollutant is not absorbed by the surface;
The area under consideration is flat.
The “torch” model is relatively simple and allows calculating pollutant concentrations from a limited number of parameters that are determined experimentally, which is its main advantage. As research experience shows, this model can be applied in 70% of meteorological situations.
The box model is used to approximate pollutant levels from large surface sources.
This model has the form
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 for solving the diffusion equations, "finite-difference" models are obtained. The models obtained in this way do not depend on the parameters of the sources, medium, and boundary conditions.
The main disadvantage of these models is the difficulty in determining their stability and accuracy, as well as the high probability of calculation errors.
This paper discusses the software implementation of the "torch" model. The program was written in C++ in the Borland C++ Builder 6.0 development environment.
The program menu "Atmospheric air pollution model" consists of three items: File, Calculation, Help. The contents of the menu items are shown in fig. 4. The program allows both loading calculation parameters from a file and entering them from the keyboard. It also provides detailed instructions for working with the program.
The main window of the program consists of three areas for filling in the 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 the parameters of pollution sources. When the program starts, the value "1" is set in the "Source number" input field. Next, fill in the fields for source coordinates, pollution rate, pipe height and flare height. Pressing the "Save" button saves the parameters of the current source, resets the values in the input fields and automatically changes the "Source number" field to the next value of the number.
Rice. 4. Contents of menu items
Rice. 5. Main window
In the lower left area there are fields for entering the coordinates of the measurement 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 the results. This field accumulates the values of the 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 the parameters must contain the following data in the given order: wind speed, wind direction, measuring 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 velocity, pipe height, height torches.
The main window of the program with filled input fields and calculated results for five measurement points is shown in fig. 5.
This paper considers various models of the distribution of pollutants that describe the state of atmospheric air using various mathematical approaches that take into account the types of pollution, emission parameters, meteorological, topographic and other conditions that affect the dispersion of pollutants. The key requirements for models of atmospheric air pollution are given. The stages of construction and classification of atmospheric air pollution models are considered.
The model of "torch" is programmatically implemented. The developed program provides an opportunity to calculate the concentration of pollutants at the measurement point. The results obtained in the simulation are confirmed experimentally.
In the future, it is planned to create an automated system that allows 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. - S. 325-330;URL: http://fundamental-research.ru/ru/article/view?id=41669 (date of access: 02/01/2020). We bring to your attention the journals published by the publishing house "Academy of Natural History"
