In fluid power networks, the presence of undissolved air greatly influences the properties of the liquid-gas mixture. Since the gas reduces the stiffness of the mixture, the wave propagation speed and – consequently – the system’s natural frequencies decrease. It can be shown that this effect is particularly pronounced for zones of low pressures (e.g. return lines or low-pressure systems). In these situations, even marginal amounts of undissolved air drastically reduce the apparent bulk modulus of the mixture.Current state-of-the-art 1D simulation tools assume that both liquid and gas fractions act as springs in order to calculate the apparent bulk modulus of the mixture. 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. Accordingly, the apparent bulk modulus and wave propagation speed change with frequency. Moreover, due to an external excitation by pumps and other components, individual bubbles may be excited with their natural frequency, which leads to a cancellation of pressure waves. In the present paper, these dynamic effects which also result in a shift of a system’s natural frequencies are investigated by considering monodisperse (one gas bubble size) as well as polydisperse (multiple bubble sizes) mixtures. For the polydisperse case, a log-normal bubble size distribution is used. Furthermore, a dilute mixture is assumed throughout this paper such that bubble-bubble interactions can be neglected. In the beginning of this work, a solution for the bubble dynamics in the frequency domain is presented using a linearized Rayleigh-Plesset equation. The Rayleigh-Plesset equation connects kinematic quantities like the bubble’s radius and its temporal derivatives with the pressure within and in the vicinity of the bubble. 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. This theoretical solution enables a thorough investigation of how the hydraulic pipe resonances are affected by the frequency-dependence of the bulk modulus. As a reference case, a pipe, which is closed on one end and excited by varying pressures on the other end, is investigated. By comparing the pipe resonance situations for mixtures with frequency-dependent as well as constant bulk moduli, the benefit of incorporating bubble dynamics within 1D simulations is emphasized. It is proven that the dynamics of the bubbles leads to a significant shift of the system’s natural frequencies towards lower values that needs to be accounted for during the layout of a fluid power system.After the development of this analytical solution, a numerical scheme for 1D pipe simulation is expanded such that bubble dynamics are taken into account. The numerical scheme utilizes the method of characteristics to transform the PDE system consisting of continuity and momentum equation into ordinary differential equations, which are solved numerically by a finite-difference scheme. By introducing a source term to the continuity equation that accounts for the dynamic effects of bubbles, the accuracy of the simulation of heterogeneous mixtures is increased and the profound effect on the system’s eigenfrequencies is evident. Finally, the newly developed numerical approach is compared with the analytical solution in order to determine its accuracy. The findings and implementation of the presented simulation approaches in this work will enable fluid power system engineers to predict dynamic system behavior during the early stages of system layout more precisely.

Consideration of Air Bubble Dynamics in 1D Hydraulic Pipeline Simulation – Source Term Development and Verification Utilizing Transmission Line Theory

Paper Proceedings: https://asmedigitalcollection.asme.org/FPST/FPMC2021/volume/85239

2021 ASME/BATH Symposium on Fluid Power and Motion Control (FPMC2021)

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