Abstract
In fluid power systems, the presence of undissolved air greatly influences the properties of the liquid-gas mixture. Even marginal amounts of undissolved air may drastically reduce the apparent bulk modulus of the mixture. In current state-of-the-art 1D simulation tools, the estimation of the apparent bulk modulus of the mixture is based on the assumption that both liquid and gas fractions act as springs. However, the so-called Rayleigh-Plesset equation frequently used for cavitation analysis shows that the gas bubbles should rather be regarded as non-linear mass-spring-damper systems, implicating a frequency-dependent stiffness of the gas phase. In the present paper, these dynamic effects are investigated by considering monodisperse as well as polydisperse mixtures. For the polydisperse case, a log-normal bubble size distribution is used. First, a frequency domain solution for the bubble dynamics is developed by linearizing the Rayleigh-Plesset equation. An expression of the mixture bulk modulus is derived, which is complex-valued and frequency-dependent. Based on the bulk modulus, a theoretical solution for the dynamics of a whole pipeline is developed by utilizing transmission line theory. It is shown that the dynamics of the bubbles leads to a significant shift of the system’s natural frequencies towards lower values — a phenomenon that needs to be accounted for during the design phase of a fluid power system. After the development of this analytical solution, by introducing a bubble dynamics source term, an established numerical scheme for 1D pipe simulation based on the method of characteristics is expanded. Finally, the newly developed numerical approach is compared with the analytical solution in order to determine its accuracy. The findings and simulation approaches in this work will enable fluid power system engineers to predict dynamic system behavior more precisely during early stages of system layout.
