**1. Introduction**

*1.1. Gaseous and Vaporous Cavitation*

Cavitation is one of the natural phenomena whose thorough understanding should be a scientific priority. Among others, it takes place when gas is released from the liquid. It occurs in hydraulic systems (water supply, hydropower, heating, cooling, etc.) in which the flow (forced by the pressure gradient) takes place through pressurized pipes. There are two types of cavitation: gaseous and vaporous [1–3].

More dangerous is vaporous cavitation, which occurs when the pressure drops to the saturated vapor pressure. This type of cavitation is rapidly changing, as it only takes place during the duration of the reduced pressure. In the literature, there is a group of mathematical models based on this type of cavitation, the so-called discrete vapor cavity models (DVCM) [1–3]. In the event of a water hammer, the reduction of pressure to the

**Citation:** Urbanowicz, K.; Bergant, A.; Kodura, A.; Kubrak, M.; Malesi ´nska, A.; Bury, P.; Stosiak, M. Modeling Transient Pipe Flow in Plastic Pipes with Modified Discrete Bubble Cavitation Model. *Energies* **2021**, *14*, 6756. https://doi.org/ 10.3390/en14206756

Academic Editor: Dmitry Eskin

Received: 11 September 2021 Accepted: 15 October 2021 Published: 17 October 2021

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saturated vapor pressure takes a relatively short time, which is followed by an implosion of the resulting vapor regions. The implosion is accompanied by large local increases in the velocity of the liquid, because the cavitation space must rapidly be filled with liquid at the time of pressure increase (above the vapor pressure). The impact of the liquid against the walls of the pipe (as well as the walls of other elements of the systems: valves, turbines, pumps, flow meters, etc.) results in cavitation erosion in the long term. Irreversible losses appear in the material of the walls of the pipes and other elements of the system. Sections in which such erosion takes place are systematically weakened in terms of strength, and it is in these places that leaks or, in extreme cases, complete damage of the structure can occur. Cavitation also leads to a reduction of the efficiency of hydraulic systems, contributing to the deterioration of the operation of energy-saving systems in hydraulic drives [4].

The second type of cavitation, namely gaseous cavitation, is a slowly changing phenomenon occurring in systems with unsteady flows (dynamic, rapid changes of velocity and pressure) or large pressure drops along the length of the system. Each liquid dissolves a certain amount of air (possibly a different gas). In water systems (water supply networks), the average amount of dissolved air is about 2%. In oils, on the other hand, the amount of dissolved air can reach up to about 10%. Hence, the influence of this type of cavitation is much more noticeable in oil-hydraulic systems than in water supply systems. Interestingly, during water hammer, such cavitation areas, due to the large time necessary for desorption and absorption, are beneficial. Their presence causes a faster damping of dynamic waveforms, as the "air bags" emitted by their action resemble local air–liquid shock absorbers. The influence of this type of cavitation is still poorly understood both experimentally and theoretically. There is a group of models called discrete gas cavity models (DGCM) [1,2], which take into account the influence of free gas in a simplified way.

The type of pipe material also significantly affects the intensity and timing of transient phenomena [5]. The flows in metal pipes with vapor–gas cavitation areas are well recognized and described. However, if we look at plastic pipes, which are now starting to displace metal pipes (especially in water supply systems), the researchers have mostly used the two basic cavitation models, i.e., the DVCM and the DGCM. Apart from these two models, alternative models have been developed, including a revised version of the DVCM model proposed by Adamkowski [6,7] as well as a model based on two-phase flow equations that can be called a discrete bubble cavity model (DBCM), which was developed by Shu [8]. Shu's model does not generate the unrealistic pressure spikes due to flow discontinuity at each computational section [8] that have been found in DVCM simulations. In DGCM, it is difficult to assign the physical amount of free air at computational sections along the pipeline. The model that is based on two-phase pipe flow equations is in principle more realistic than the model that is based on single-phase pipe flow equations with cavities lumped at computational sections. However, the aforementioned discrete Adamkowski cavity (DACM) and DBCM models have not been previously used to model transient cavitating flow in plastic pipes. The main objective of this paper is to present a novel DBCM that will enable the simulation of transient cavitating flows in plastic pipes.
