Modelling pressure dynamics in a gas pipeline with an in-line sampling

   Gas pipelines, whether main, field, or urban, often operate in non-stationary modes. Changes in the operating modes of pumping stations, equipment start-up and shutdown, an in-line sampling or pumping and various factors are causes of instability of pressure, velocity, gas flow rate. Main pipelines are complex engineering systems with the pipeline itself being the main element. Fail-safety is the major factor in the operation of this system. Ensuring operational safety requires studying the movement regimes of the transported medium, particularly study of the dynamics of pressure during
start-up or shutdown and at specific extraction points.

   This article aims to build a mathematical model and study the pressure dynamics in a gas pipeline with a sampling point.

   The theory of non-stationary motion liquid in round pipes has been strongly developed in the works of I. A. Charnyj. These works consider a large complex of engineering tasks taking into account viscous properties of the transported medium and pipe resistance in the hydraulic approximation. In the article on the basis of I. A. Charnyj's researches on the motion of real liquid in circular pipes the equation in partial derivatives of hyperbolic type is compiled. The equation describes the unsteady pressure of a horizontal section of gas pipeline with an extraction point. Using the Dirac delta function allows the formulation of the problem in the form of a single equation. Pressures are set at the ends of a given section, and the initial velocity is related to the Dirac delta function. By applying the finite Fourier sine transform, the partial differential equation is transformed into an ordinary differential equation and solved. Solution of equation is vision of a solution to the initial task. Inverse transform formulas based on Fourier theory allowed us to proceed to the solution of this task. Explicit dependences for the dynamics of unsteady pressure are obtained. The qualitative analysis of the formulas indicates the wave motion of a medium during the initial phase of operation, transitioning into a stationary mode after a brief period. The duration of the transition period depends on factors such as the hydraulic resistance coefficient and the velocity of the transported medium. Coefficient of hydraulic resistance and the velocity of the transported medium are the main factors. An example is considered for a horizontal section under isothermal flow conditions. With assumed numerical parameters, the transition to a stationary state occurs approximately 17 minutes after the process begins, as illustrated in the graphs provided in the article. Engineers can use mathematical models of the liquid and gas motion through pipes in the design of pipelines, as well as in solving tasks that arise during their operation. These tasks include monitoring the condition of the pipeline system, optimizing the operation, accumulation capacity estimates and others.

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