Inverse Problem of Pipeline Transport of Weakly-Compressible Fluids
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 6, November, 2020
INVERSE PROBLEM OF PIPELINE TRANSPORT OF WEAKLY-COMPRESSIBLE FLUIDS Kh. M. Gamzaev
UDC 532.546:519.6
Nonstationary one-dimensional flow of a weakly-compressible fluid in a pipeline is considered. The flow is described by a nonlinear system of two partial differential equations for the fluid flow rate and pressure in the pipeline. An inverse problem on determination of the fluid pressure and flow rate at the beginning of the pipeline needed for the passage of the assigned quantity of fluid in the pipeline at a certain pressure at the pipeline end was posed and solved. To solve the above problem, a method of nonlocal perturbation of boundary conditions has been developed, according to which the initial problem is split at each discrete moment into two successively solvable problems: a boundary-value inverse problem for a differential-difference equation of second order for the fluid flow rate and a direct differential-difference problem for pressure. A computational algorithm was suggested for solving a system of difference equations, and a formula was obtained for approximate determination of the fluid flow rate at the beginning of the pipeline. Based on this algorithm, numerical experiments for model problems were carried out. Keywords: pipeline transport, weakly-compressible fluid, nonstationary flow, boundary-value inverse problem, differential-difference problem. Introduction. At the present time, for transporting various fluids (water, oil, oil products), pipelines of various dimensions are used, beginning from the smallest ones, used in laboratories and control-measuring apparatuses, up to main ones. Usually, in designing a pipeline the fluid flow rate in it is assigned; it determines the pipeline efficiency and the positions of its beginning and end. One of the main tasks here is the determination of the pressure drop along the pipeline length needed for the passage of a given quantity of fluid through it. In practice, in solving this problem, use is made of the assumption according to which the fluid motion in the pipeline is stationary and, proceeding from this assumption, the Darcy–Weisbach formula is used in calculations [1–3]:
ΔP = λ
ρu 2 l. 2d
(1)
It should be noted that it was possible to justify this formula and to obtain an explicit expression for the coefficient of hydraulic resistance of the pipeline only for a homogeneous incompressible stationary laminar fluid flow obeying the corresponding rheological laws. However, as the practice of the fluid transport through pipelines shows, the start-up or stopping of a pipeline, switching-in or switching-off of the pumping-over station, the beginning or stopping of the fluid takeoffs, and other technological operations lead to the appearance of nonstationary fluid flow in the pipeline. In this connection, for the pipeline conveyance of fluids, of great importance is the investigation of nonstationary flow of compressible fluid in the pipeline with the aim of determining
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