Analytical Solutions of Water Hammer in Metal Pipes. Part I—Brief Theoretical Study

Urbanowicz, Kamil; Bergant, Anton; Stosiak, Michał; Lubecki, Marek

doi:10.1007/978-3-030-97822-8_7

Cited by 5 publications

(3 citation statements)

References 10 publications

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“…The initial verification of this model presented in recent papers [4,40,41] showed its enormous usefulness, especially in modeling water hammer occurring in water pipe flows (relatively low-viscosity water supply systems). Unfortunately, in typical hydraulic systems, i.e., wherever the working fluid is hydraulic oil (water hammer number 0.05 > Wh > 0.5 [42]), computational compliance was not sufficient.…”

Section: Introductionmentioning

confidence: 98%

About Inverse Laplace Transform of a Dynamic Viscosity Function

et al. 2022

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A dynamic viscosity function plays an important role in water hammer modeling. It is responsible for dispersion and decay of pressure and velocity histories. In this paper, a novel method for inverse Laplace transform of this complicated function being the square root of the ratio of Bessel functions of zero and second order is presented. The obtained time domain solutions are dependent on infinite exponential series and Calogero–Ahmed summation formulas. Both of these functions are based on zeros of Bessel functions. An analytical inverse will help in the near future to derive a complete analytical solution of this unsolved mathematical problem concerning the water hammer phenomenon. One can next present a simplified approximate form of this solution. It will allow us to correctly simulate water hammer events in large ranges of water hammer number, e.g., in oil–hydraulic systems. A complete analytical solution is essential to prevent pipeline failures while still designing the pipe network, as well as to monitor sensitive sections of hydraulic systems on a continuous basis (e.g., against possible overpressures, cavitation, and leaks that may occur). The presented solution has a high mathematical value because the inverse Laplace transforms of square roots from the ratios of other Bessel functions can be found in a similar way.

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Section: Introductionmentioning

confidence: 98%

About Inverse Laplace Transform of a Dynamic Viscosity Function

et al. 2022

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“…Holmboe [1] managed to find an analytical solution of pressure but in the Laplace domain 𝑝 ̃(𝑥, 𝑠) only. Recently [3][4][5], using Holmboe's solution, it was also possible to derive the missing XXV Fluid Mechanics Conference (FMC 2022) Journal of Physics: Conference Series 2367 (2022) 012026 IOP Publishing doi:10.1088/1742-6596/2367/1/012026 2 solutions in the Laplace domain describing the transforms of velocity 𝑣 ̃(𝑥, 𝑠) and wall shear stress 𝜏(𝑥, 𝑠) in transient laminar pipe flow:…”

Section: Introductionmentioning

confidence: 99%

“…In our first work [3] most promising models were compared and few novel analytical solutions useful for predicting velocity and pressure time histories were presented. In more recent papers the novel analytical solution for wall shear stress was derived and compared [4,5]. If one carefully examine transforms in the Laplace domain (2) it is clear that the time domain solutions (for pressure, velocity and wall shear stress) will be based on inverse Laplace transforms of the following functions: 𝕄 2 , 𝕄 and 𝑒 −𝑎𝑠𝕄 .…”

Section: Introductionmentioning

confidence: 99%

On the generalization of Calogero-Ahmed summation formulas

Urbanowicz

Stosiak

Bergant

2022

J. Phys.: Conf. Ser.

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The use of the Laplace transform gives the solution of water hammer equations in the frequency domain. The inverse transform of this solution over the years seemed impossible to derive, due to the significant complexity and the fact that the square root of the Bessel function was embedded in the argument of the resulting hyperbolic functions. In this work, we consider some generalizations that enable the determination of the modified Calogero-Ahmed infinite series. These generalizations will allow us in the near future (using the machine learning and artificial intelligence algorithms) a return to the time domain in a very wide range of the dynamic viscosity function, which plays the most important role in this complex fluid dynamic problem.

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