Abstract
An important application of radial pumps is their use under transient conditions in industrial and residential water distribution. In such applications, as the level of the suction and discharge reservoirs vary with time, the static head also varies. In this situation, the pump’s power consumption tends to increase and the overall efficiency to decrease, as the chance of the machine operating outside of the highest efficiency zone increases, causing significant economical impact on the process. Therefore, the primary objective of the present work is to obtain a novel analytical solution to the transfer of liquid between two reservoirs with free surface heights slowly varying with time. The system is supposed to evolve slowly and the friction factors are considered to be approximately constant. The model appears in the form of a nonlinear first-order ODE, relating the volumetric flow to time. The solution of this ODE yields time as a function of the flow rate, but the the problem is circumvented by use of the Lambert W function, and thus, an explicit dependence is obtained. This solution allows the identification and control of the factors influencing the total transfer time, helping in the selection of the most adequate pump for a particular application. The solution also helps to access the influence of the constant friction factor hypothesis on the dynamics of the process. Three illustrative examples presented suggest that the flow rate, volume transferred and total transfer time are only weakly dependent on the Darcy–Weisbach friction factor.
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Notes
Equipment ages for many reasons, the most common of which include erosion, corrosion and fatigue. The same apply to the piping system, as corrosion and accumulation of various elements around the internal surface of the pipe tends to increase its roughness.
Recently an explicit solution for the CW equation was developed using the real-valued Lambert W-function [10], but it is considered too complicated for everyday use.
And, consequently, a quadratic dependence between the volume transferred and the time.
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Acknowledgements
The authors would like to thank Professor Orestes Llanes Santiago, from Universidad Tecnológica de La Habana, for his useful suggestions and comments.
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Appendix
Appendix
The Moody chart, relating the friction factor to the Reynolds number and the surface roughness, for incompressible, steady, fully developed flows in circular pipes.
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Pellegrini, C.C., Zappi, G.A. & Vilalta-Alonso, G. An Analytical Solution for the Time-Dependent Flow in Simple Branch Hydraulic Systems with Centrifugal Pumps. Arab J Sci Eng 47, 16273–16287 (2022). https://doi.org/10.1007/s13369-022-06864-9
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DOI: https://doi.org/10.1007/s13369-022-06864-9