Integrodifferential approaches to frequency analysis and control design for compressible fluid flow in a pipeline element

Abstract

In this study, modelling, frequency analysis, and optimization of control processes are considered for the fluid flow in pipeline systems. A mathematical model of controlled pipeline elements with distributed parameters is proposed to describe the dynamical behaviour of compressible fluid which is transported in a long rigid tube. By exploiting specific functions representing cross-sectional forces and effective displacements as well as linear approximations of fluidic resistances, the original problem with non-uniform parameters is reduced to a partial differential equation (PDE) system with constant coefficients and homogeneous initial and boundary conditions. Three numerical approaches are applied to an efficient analysis of natural vibrations and reliable control-oriented modelling of pipeline elements. The conventional Galerkin method is compared with the method of integrodifferential relations based on a weak formulation of the constitutive laws. In the latter approach, the original initial-boundary value problem is reduced to the minimization of an error functional which provides explicit energy estimates of the solution quality. A novel projection approach is implemented on the basis of the Petrov-Galerkin method combined with the method of integrodifferential relations. This technique benefits from the advantages of the above-mentioned projection and variational approaches, namely sufficient numerical stability, a lower differential order, and an explicit quality estimation. Numerical optimization procedures, making use of a modified finite element technique, are proposed to obtain a feedforward control strategy for changing the pressure and mass flow inside the pipeline system to a desired operating state. At this given finite point of time, residual elastic oscillations inside the pipeline are minimized. Numerical results, obtained for ideal as well as viscous fluid models, are analysed and discussed.

An unscented transformation approach to stochastic analysis of measurement uncertainty in magnet resonance imaging with applications in engineering

“…In such cases, the admissibility of the replacement of complex, commonly nonlinear, models consisting of coupled partial differential equations for heat and mass transfer has to be validated. This replacement is possible if sufficiently accurate formulations can be found in terms of lumped-parameter, reduced-order models, which are typically derived by finite volume, finite difference, or finite element approaches (Rauh et al, 2011;Kostin et al, 2014). As such, MRI results may help in optimizing future control scenarios, where a trade-off between high modeling accuracy and a minimum amount of numerical effort is required.…”

Dynamic stability of deformable elements of one class of aeroelastic constructions

“…Among the last studies on the dynamics and stability of the pipelines and its elements during the flow inside them of liquid or gas flow should be noted the next studies [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and many others. Among the works of the authors of this article by the investigations of the dynamics and stability of elastic bodies interacting with the gas flow, note the monographs and articles [17][18][19][20][21][22][23][24].…”