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2Рис. 7. Зависимость режимов горения метановоздушной смеси (стехиометрия) от параметров
теплового источника зажигания
Полученные в численном эксперименте температуры зажигания метановоздушной смеси несколько выше определенных в аналитическом расчете, что объясняется учетом газодинамических процессов, которые ускоряют диссипацию тепловой энергии.
Выводы. Полученное аналитическое решение распределение температуры в тепловом слое позволило определить тепловой эффект реакции окисления метана вблизи источника зажигания и на этой основе показать сходимость численного метода с результатами аналитического решения в части выполнения критерия зажигания. Выполненный анализ точности вычислительного процесса позволяет применять численный метод в практических расчетах нахождения безопасных условий эксплуатации оборудования с учетем прогнозировании последствий аварийных ситуаций.
Список литературы
1. Варнатц Ю. Горение. Физические и химические аспекты, моделирование, эксперименты, образование загрязняющих веществ / Ю. Варнатц, У. Маас, Р. Дибл. Пер. с англ. Г.Л. Агафонова. Под ред. П.А. Власова .- М.: Физматлит, 2003.- 352 с.
2. Теория горения и взрыва / А. В. Тотай, О. Г. Казаков, Н. О. Радькова [и др.]; под ред. А. В. Тотая, О. Г. Казакова.- Москва: Изд. «Юрайт», 2013.- 296 с.
3. Чернай А. В. Математическое моделирование вынужденного воспламенения газовоздушной смеси при оценке безопасных условий ликвидации аварий / А. В. Чернай, Н. Н. Налисько // Науковий вюник НГУ.- 2016.- №5(155).- С. 106-114.
THE EVALUATION OF THE POSSIBILITY OF USING APPROXIMATE MODELS IN THE EVALUATION OF THE AVERAGE CHARACTERISTICS OF SCATTERING OF ELECTROMAGNETIC WAVES
Preobrazhenskiy A.P.
Professor of Voronezh institute of high technologies, doctor of technical sciences, Voronezh, Russia
Choporov O.N.
Professor of Voronezh state trchnical university, doctor of technical sciences, Voronezh, Russia
ABSTRACT
The paper considers the problem of scattering of electromagnetic waves on a hollow structure. The hollow structure of simple form and complex with the final loading are considered. To characterize these structures the integral equation method is used, the integral equation is solved on the base the method of moments. The dependence of the difference average RCS of hollow structures with complex load and its model from the tilt angle of back side for various apertures is shown.
Keywords: hollow structure, integral equation, scattering characteristics.
Modern scatterers of electromagnetic waves are characterized by the fact that they are in a very large number of cases have a complex structure. The analysis and design of such facilities must undertake to implement with the use of such models and methods, which provide a possible smaller mistakes [1, 2].
In the research and development of electrody-namic objects now more intensively computer aided design (CAD) are used. This gives you the opportunity to formulate and to solve various problems of the theory of diffraction of electromagnetic waves (EMW) in various structures with complex shapes.
When analyzing the possibilities of solving problems of diffraction of radio waves, as well as the design of facilities in some cases knowledge of the constraints on the average characteristics of the scattering is required.
In this paper, we analyze the two-dimensional model of scattering of electromagnetic waves. This is due to the fact that in large number of cases, the three-dimensional problem can be reduced to two-dimensional [3, 4].
The aim of this work to study the possibility of using approximate models to estimate the average characteristics of the scattering objects on the example of hollow organs and the development of proposals for the approximation of characteristics.
The results can be used to construct subsystems of CAD design objects with the required average characteristics of the scattering.
In currently available software systems and systems design solution is usually to split the space into cells in the numerical solution of differential or integral equations, for example, by the method of moments. But such a division does not account for the specific behavior of the currents on the surfaces (contours) of objects. The problem is solved without taking into account the possibility of dimension reduction of the resulting systems of linear equations, for example, by data to two-
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dimensional task, or the use of analytical solutions, a combination of several methods. All this, ultimately, significantly increases the calculation time of the characteristics of objects, as one of the main stages of design of complex diffractive structures and antennas.
When using a diffraction approach, the hollow structure is considered as a body of complex shape, in which the scattering of EMW occurs. The method of integral equations when considering hollow structures (and other objects), is a rather cumbersome method, often requiring large resources, like other numerical methods. However, if we consider structures that are bodies of revolution, the most successful is a combination of methods of integral equations and eigenfunc-tions. The main role is played by the angular or azi-muthal coordinate of the 9. For this coordinate the required fields, as when using the method of eigenfunctions, decomposed in Fourier series, and the field of individual harmonics because of the orthogonality be independent. This allows for each azimuthal harmonic to build a relatively simple integral equation which is solved numerically. This reduces the dimensionality of the electrodynamic problem is solved and reduced requirements for the amount of machine memory and computation time of the computer.
Integral equations for one body can be generalized for a system of bodies [6, 7]. Under the integration domain and the domain of variation of the observation point in this case should be understood the surface of not one, but together bodies.
Let's consider the scattering of electromagnetic effects for two-dimensional perfectly conducting hollow structure (Fig. 1).
This structure can have complex loading (Fig. 2). To estimate the average characteristics of this structure, you can use the model shown in Fig. 1.
L
<-►
Fig. 1. The geometry of two-dimensional hollow structure of a simple form, a is the aperture; L is the length of the structure; E is vector of the incident electromagnetic waves; 0 is the angle flat EMW.
Fig. 2. The geometry of two-dimensional hollow structure with a complex load. a — aperture; L — the length of the structure; a - the tilt angle; E is the vector of incident electromagnetic wave; 0- angle flat EMV
It is necessary to estimate sector angles counted and its model did not exceed 3 dB. Integral Fredholm
from the normal to the aperture of a cavity with a com- equation of the first kind for the density of the unknown
plex load, which applies the model to estimate the av- electric current in the case of E-polarization is of the
erage characteristics of the scattering. We have sought form [2, 7] : to ensure that the difference in average radar cross section (RCS) of a hollow structure with a complex load
^ • J j(t) • H0 [k • L0 (x, t)] 2 (t) + n'2 (t)dt = E0 (x;,
4 a
a<i<ß,
(1)
where Lo ( I,t) = V [ Ç( I ; - tft)] 2 + [ l( I ; - T(t)] Z -
the distance from the observation point to the point of integration, E^ (T) - the longitudinal component of the tension of the primary electric field at a point on the contour. The outline structure is given in parametric form:
x = ^(t),y = X\(t), a< t <J, a
'(t), l|'(t) - the first derivatives of the corresponding functions, k = 2 •% / À, À - the length of the incident electromagnetic wave.
The equation (1) was solved using the method of moments. The average RCS is calculated based on the following expression
i=0
N +1
(2)
where 6;) - is the value of RCS for the
viewing angle 6i.
In Fig.3 we can see the dependence of the difference A medium RCS of hollow structures with complex load and its model from the angle of a. We chose the
value of the length L=4 X . This value can be explained by the fact that when falling on the aperture of the cavity and further extending inside the hollow structure of the wave comes to the steady-state regime.
The dependence was approximated in the framework of the method of least squares for different values of the aperture a. The polynomial approximation was the following:
y(x) = b + b 'x + b ' X2 + b ' x3 + •• ■
Fig. 3. The dependence of the difference A average RCS of hollow structures with complex load and its model
from the angle of a for various apertures a.
The coefficients of the approximation are given in The approximation coefficients can be stored in
table 1. the CAD database and used in the calculations of scat-
The relative approximation error did not exceed tering characteristics of hollow structures. 1.5%.
Table1.
The ^ values of the coefficients of the approximating polynomial
b0 b1 b2
a=2X -0.698 0.224 -2.377X10"3
a=3X 0.668 0.159 -5.067X10"3
a=4X 0.135 6.786X10"4 9.289X10"3
Conclusion.The most important direction of development of modern CAD antenna feed, microwave devices and systems and diffraction structures is the expansion of the circle of problems solved with their help objectives, as well as an increase in the number of classes and varieties analyzed (projected) electrodynamic objects. Typically, software like CAD is based on the use of universal numerical methods for solving integral equations (finite elements, Galerkin). One of the features of object-oriented CAD systems is used in these analytical methods, which are optimal for solving certain classes of problems.
Thus, the investigated approach and the results obtained can be useful in the design of objects with the specified requirements for average characteristics of the scattering.
References
1. Preobrazhenskiy A.P. Predicting radar characteristics of objects in the wavelength band using results of measurement of scattering characteristics on discrete frequencies. Telecommunications and Radio Engineering. 2004. V. 62. № 9. pp. 843-850.
2. Preobrazhenskiy A.P. Estimation of possibilities of combined procedure for calculation of scattering cross section of two-dimensional perfectly conductive
cavities. Telecommunications and Radio Engineering. 2005. T. 63. № 3. C. 269-274.
3. Preobrazhenskiy A.P. Prediction of radar characteristics of a perfectly conducting cavity within a wavelength range. Telecommunications and Radio Engineering. 2007. V. 66. № 17. Pp. 1543-1548.
4. Lvovich I.Y., Lvovich Y.E., Preobrazhenskiy A.P., Choporov O.N. Optimization of electromagnetic scattering characteristics on the objects of complex shape based on the "ant" algorithm. Research Journal of Pharmaceutical, Biological and Chemical Sciences. 2016. T. 7. № 5. C. 990-998.
5. Lvovich I., Preobrazhensky A., Choporov O. The development of cad of information systems and software for diffractive structures. Information Technology Applications. 2016. № 1. C. 107-116.
6. Lvovich Ya., Preobrazhensky A., Choporov O. Modeling of scattering of electromagnetic waves on the base of multialternative optimization. Information Technology Applications. 2016. № 1. C. 117-125.
7. Lvovich I., Schindler F., Preobrazhensky A. The simulation of the scattering characteristics for cavities with complex shape. Current Issues of Science and Research in the Global World - Proceedings of the International Conference on Current Issues of Science and Research in the Global World 2015. C. 271-276.
