Time Domain Fluid Transmission Line Modelling using a Passivity Preserving Rational Approximation of the Frequency Dependent Transfer Matrix

Abstract

Flow and pressure transients in fluid transmission lines can be analysed starting from a modal approximation of the frequency domain irrational transfer matrix, relating pressure and flow rate at the line ends in Laplace transform. The obtained rational approximation can be converted in a state space representation and used in variable time step simulators to evaluate the influence of the line on fluid servosystems dynamics. Particular attention must be given to the causality, to the stability and to the energy passivity of the resulting line model.In this paper the application of a numerical approximation technique (Vector Fitting) to the frequency dependent transfer matrix describing the pipeline dynamics is proposed. The admittance matrix formulation is chosen, introducing an effective passivity enforcing technique, to ensure the energy passivity of the approximated matrix, thus preserving in the model the physical meaning of the real system.The rational approximation of the transfer matrix, combined with the passivity enforcement methodology, is applied to the study of the transient response of a single uniform line and of compound hydraulic line systems, showing the agreement between the simulation and the solution obtained with inverse fast Fourier transform.

A practical model for pressure probe system response estimation (with review of existing models)

“…Fluid transmission lines are now a standalone subject (e.g. [12][13][14]). Each continuous circular tube section is treated as a lossy transmission line, while connected volumes are treated as capacitive components.…”

Vector Fitting

“…The ability of VF to generate models compatible with transient simulations has also been exploited in the finite difference time domain (FDTD) method [57,60], the finite element time domain method [12,87], and the discontinuous Galerkin method [91]. Beyond electrical engineering, VF found countless applications in various domains, including acoustics [17,66], fluid dynamics [3,45,35], mechanical engineering [35,6] and in the thermal modeling of chemical batteries [44]. For a collection of VF applications and additional references, the Reader is referred to [35].…”

Pulsating characteristic of the hydraulic hybrid vehicle system with pipeline effect

“…Therefore, the simultaneous equations consisting of equations (12), (15), (18), (19), and (24) can be derived, respectively.…”

“…16,17 The most approximate calculation methods for pipeline are difficult to meet accuracy and simplicity. For example, Almondo and Sorli 18 proposed a numerical approximation technique of vector fitting to describe the frequency transfer matrix of pipeline. Taylor et al 19 present new approximations to frequencydependent friction for laminar and turbulent flow, and apply it to time domain simulation of modeling time-varying flow in hydraulic pipelines.…”

“…16,17 The most approximate calculation methods for pipeline are difficult to meet accuracy and simplicity. For example, Almondo and Sorli 18 proposed a numerical approximation technique of vector fitting to describe the frequency transfer matrix of pipeline. Taylor et al.…”