Release:
2023. Vol. 9. № 2 (34)About the authors:
Ilya F. Chuprov, Dr. Sci. (Tech.), Professor, Department of Higher Mathematics, Ukhta State Technical University, Ukhta, Russia; ichuprov@ugtu.netAbstract:
In this work, based on the fundamental research of I. A. Charny on the unsteady movement of a real fluid in pipes, an equation that describes the pressure dynamics in a complex section of a gas pipeline was compiled. The use of the Dirac impulse function made it possible to describe the dynamics of unsteady pressure in a single equation in the case of point pumping and injection at given points. Linearization of the model by averaging the velocity of the gas made it possible to bring the equation to a hyperbolic form. If we neglect the forces of inertia compared to the forces of resistance, then the mathematical model will represent a partial differential equation of the second order parabolic type. The dynamics of pressure at specific points of pumping and injection under boundary conditions of the second kind (rates are given) was obtained using the finite cosine Fourier transform. The working formula that allows you to determine the pressure at any point at a given point in time is a trigonometric series. The series are rapidly convergent, so there is no difficulty in working with them. Particular cases are considered (without pumping and injection). The fulfillment of the boundary conditions is checked. It is easy to rebuild the working formula for pumping and / or injection at several points in a given area. Calculations of the mass flow and average velocity are made. In practice, the pressure is usually set at the start and end points of the area under study. In this case, it becomes possible to go to the costs (to derivatives) at the ends of the section under consideration.Keywords:
References:
Altimul, A. D. (1970). Hydraulic resistance. Nedra. [In Russian]
Arsenin, V. Ya. (1984). Methods of mathematical physics and special functions. Nauka. [In Russian]
Baykov, V. A., & Zhiber, A. V. (2012). Equations of mathematical physics. Izhevsk Institute of Computer Science. [In Russian]
Bobrovskiy, S. A., Shcherbakov, S. G., & Guseyn-zade, M. A. (1972). Movement of gas in gas pipelines with line extraction. Nauka. [In Russian]
Bobrovskiy, S. A., Shcherbakov, S. G., & Yakovlev, E. I. (1976). Pipeline transport of gas. Nauka. [In Russian]
Gelfand, I. M., & Shilov, G. E. (1957). Generalized functions and actions on them. Fizmatgiz. [In Russian]
Godunov, S. K. (1991). Equations of mathematical physics. Nauka. [In Russian]
Gradshteyn, I. S., & Ryzhik, I. M. (2011). Tables of integrals, series and products. BKhV-Peterburg. [In Russian]
Grachev, V. V., Guseyn-zade, M. A., Ksenz, B. I., & Yakovlev, E. I. (1982). Complex piping systems. Nedra. [In Russian]
GSSSD 160–93. (1993). Tables of standard reference data. Natural gas rated. Density, compressibility factor, entalpy, entropy, isobaric specific heat, velocity of sound, index of adiabate and coefficient of volume thermal expansion in the temperature range 230...430 К and pressure range 0.1...12 MPa. Russian Federation Committee for Standardization, Metrology and Certification. [In Russian]
Ditkin, A. V., & Prudnikov, A. P. (1974). Integral transformations and operational calculus. Nauka. [In Russian]
Kuvakina, A. O., & Chuprov, I. F. (2022). Gas pressure in the pipeline with extraction or pumping.
Petroleum Engineering, 20(3), 176–181. https://doi.org/10.17122/ngdelo-2022-3-176-181
[In Russian]
Mirzadzhanzade, A. Kh., & Guseynzade, M. A. (1971). Solving problems of oil and gas field mechanics. Nedra. [In Russian]
Piskunov, N. S. (2001). Differential and integral calculus: a textbook for universities. In 2 vols. Vol. 1. Integral-Press. [In Russian]
Rabinovich, E. Z. (1980). Hydraulics. Nedra. [In Russian]
Sedov, L. I. (1978). Continuum mechanics. Nauka. [In Russian]
Tikhonov, A. N., & Samarskiy, A. A. (1972). Equations of mathematical physics. Nauka. [In Russian]
Charnyy, I. A. (1975). Unsteady motion of a real fluid in pipes. Nedra. [In Russian]
Chuprov, I. F., Sekutov, V. V., & Parmuzina, M. S. (2021). Dynamics of gas pressure in a gas pipeline with a in-line sampling and pumping. Problems of Gathering, Treatment and Transportation of Oil and Oil Products, (3), 67–75. https://doi.org/10.17122/ntj-oil-2021-3-67-75 [In Russian]
Shirokov, Yu. M. (1979). Algebra of one-dimensional generalized functions. Theoretical and Mathematical Physics, 39(3), 471–477. https://doi.org/10.1007/BF01017992
