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Abstract
A 1D numerical model of a straight-tube Coriolis meter has been implemented and used to generate a simple and intuitive parametric relationship to predict sensitivity. This model is intended to aid the design of such meters and avoid the need to run a large number of time-consuming simulations. Three parameters were identified as being instrumental in determining the sensitivity of a meter: dimensionless bending stiffness (Σ̃), proximity to the Euler buckling limit (R̃) and the dimensionless sensor spacing (χ̃). Parametric relationships for sensitivity (dimensionless time-lag) and natural frequency were developed. These equations allow for the complete and rapid design of a straight-tube Coriolis meter with insignificant computational effort. The parametric model was validated against 11 experimental data sets, covering a range of flow conditions, tensions and materials. In all cases, the parametric model performed well, reporting a typical error of 2 to 5%.
... Таким образом, аналитические модели течения жидкости в кориолисовых расходомерах [1,9,[16][17][18][19][20][21][22] позволяют провести исследования в общем виде, получать соотношения, характеризующие поведение системы «расходомерная трубка -жидкость» при изменении ее параметров, например, установить зависимость влияния температуры на точность измерения массового расхода. Однако они применимы лишь для кориолисовых расходомеров с простой геометрической формой расходомерной трубки (U-образная трубка, прямая). ...
... Анализ возможностей аналитических моделей для решения задачи FSI. Аналитические модели для исследования течения жидкости в кориолисовых расходомерах использовали многие авторы [1,[16][17][18][19][20][21][22]. Cheesewright, Belhadj, Clark [16][17][18] исследовали влияние внешних вибраций работающего оборудования и пульсаций потока на точность измерения массового расхода жидкости и представили аналитические решения, позволяющие их учесть. ...
... Авторы [9,21,22] представили аналитические аппроксимации основных характеристик расходомера, в которых вариации собственной частоты и разности во времени использованы для измерения плотности жидкости и ее массового расхода. Kutin и Bajsic [9] разработали математическую модель, использующую метод Галеркина, на основе линейной суперпозиции нескольких модальных функций. ...
Background. The aim of the work is to implement numerical methods and algorithms in the form of a software package that allows for a parametric study of the "flow tube – liquid" system of a Coriolis flow meter, and to analyze the influence of the shape of the flow tube on the sensitivity of measurements of liquid mass flow parameters. The relevance is due to the need to identify the geometric design of the flow tube with the most acceptable characteristics when pumping highly paraffinic oils. Materials and methods. For the mathematical model, the finite element method and an iterative approach to solving systems of equations based on rigidly coupled coupling were used. A computational algorithm based on linear interpolation and an algorithm that automates the processing of data arrays and the calculation of average time and phase delays have been implemented. The structural and logical diagram of the software complex is presented and described. Results. In the course of computational experiments carried out to study the influence of the flow tube shape on the sensitivity of measurements of the parameters of liquid mass flow as a function of the average time delay using the developed hardware and software complex, it was revealed that the model of a Coriolis flow meter with a radius of curvature of the flow tube of 30° is recommended for use in oil and gas industry as a tube geometry that is less susceptible to waxing and at the same time has sufficient sensitivity for measurement. Conclusions. A software package has been developed and tested. A relationship has been identified between the geometric shape of the flow tube of a Coriolis flow meter and the measurement sensitivity of the device, which is characterized by the fact that at high values of the radius of curvature of the tube, the sensitivity increases compared to straight-tube designs.
This paper presents a new method to determine the external flow velocity using piezoelectric sensors. In this method, the piezoelectric sensor is molded as a composite bimorph beam including an aluminum layer, two piezoelectric layers as sensor layers, and a concentrated mass at the free end. The coupled electromechanical nonlinear differential equations governing the dynamic deformation of the bimorph beam related to the external fluid flow have been extracted from nonlinear theory based on Cosserat assumptions. After numerically solving the coupled nonlinear equations, the effect of various parameters on the voltage generated by the piezoelectric layers and its dependence on the fluid flow velocity is examined. The results indicate that as the fluid velocity increases, the voltage rises nonlinearly. Moreover, applying concentrated mass at the end of the bimorph beam leads the maximum voltage generated to increase by 4.8 times compared to that without concentrated mass. Considering the results, it can be concluded that the sensitivity of the proposed piezoelectric sensor to the fluid flow velocity is significant, and through using it, the fluid flow velocity within different types of channels can be determined effortlessly affordably.
The paper describes the calculated and experimental determination of the oscillation modes of a Coriolis flowmeter. To define the estimated parametric oscillation modes, we formed a model of the volumetric flowmeter finite element. The model allows the evaluation of the impact of changes in the meter size and the medium density on the flowmeter frequency. The results of the calculations of the finite element model were verified by modal tests of the flowmeter.
In this paper the influence of external vibrations on the measurement value of a Coriolis Mass-Flow Meter (CMFM) for low flows is investigated and quantified. Model results are compared with experimental results to improve the knowledge on how external vibrations affect the mass-flow measurement value. A flexible multi-body model is built and the working principle of a CMFM is explained. Some special properties of the model are evaluated to get insight in the dynamic behaviour of the CMFM. Using the model, the transfer functions between external vibrations (e.g. floor vibrations) and the flow error are derived. The external vibrations are characterised with a PSD. Integrating the squared transfer function times the PSD over the whole frequency range results in an RMS flow error estimate. In an experiment predefined vibrations are applied on the casing of the CMFM and the error is determined. The experimental results shows that the transfer functions and the estimated measurement error correspond with the model results.
The agreement between model and measurements implies that the influence of external vibrations on the measurement is fully understood. This result can be applied in two ways; firstly that the influence of any external vibration spectrum on the flow error can be estimated and secondly that the performance of different CMFM designs can be compared and optimised by shaping their respective transfer functions.
The vibration of a general plane tube with a flowing fluid, which is the measuring element in a Coriolis mass flow meter (CMF), is studied. The dynamic stiffness matrix method is used to model such a tube. The tube is divided into straight and circular elements. The elements dynamic stiffness matrices are derived from the equations of motion. By assembling the elements matrices into a global matrix the natural frequencies are obtained. The mode shapes are obtained by applying the boundary conditions at the supports and the compatibility conditions at the nodes. The effects of the flow velocity on the natural frequency and the relative phase difference are modeled. The method is applied to different tube shapes. The results are compared to the published data which reveals good agreement.
The aim of this paper is to discuss velocity profile effects in Coriolis flowmeters and to review related research work. The measurements made by Coriolis flowmeters are dependent upon the steady flow velocity distribution within them whenever certain features of the fluid vibrational fields are not uniform inside the measuring tube. This dependence is confirmed by simulation results on two straight tube configurations, one operating in a beam-type mode and the other in a shell-type mode. Findings to date and open questions regarding velocity profile effects in Coriolis flowmeters are discussed for both fully developed and disturbed inlet flow conditions.
The aim of this paper is to derive approximate, analytically expressed, theoretical characteristics for a straight, slender-tube Coriolis meter, which can be applied to any of its working modes. The mathematical model is based on the theories of the Euler beam and one-dimensional fluid flow, and includes the effects of axial force, added masses, damping and excitation. The analytical approximations are evaluated by applying a Taylor-series expansion to the solutions of the Galerkin method, which are considered as a superposition of the Euler-beam modal functions. On the basis of the obtained analytical expressions, the properties of the meter’s characteristics are discussed, with the emphasis being on particular nonidealities.
In Coriolis meter terminology, zero-shift is a general term describing the change in time-lag at zero-flow (zero-point) in response to any of a number of external factors. Zero-shift of a straight single-tube Coriolis meter has been investigated experimentally. The meter-tube was stainless-steel (2.66/2 × 300 mm) and has been tested across a range of flow rates (0–1 lpm, Red<104), pressures (1–4 barg) and pretensions (0 & 22 N). Zero-shift was shown to be principally dependent on the vibration-frequency. This was confirmed by operating the meter in both free and force-vibration modes. For zero-point to become a function of frequency the system must possess asymmetry, which may arise from the fluid or solid component. It was hypothesized that an asymmetric deformation field (i.e. a deviation from straightness) was the principal source of system asymmetry in this particular case. The deformation field was shown to be dynamic and therefore capable of modifying the zero-point even in fixed-frequency operation. The relationship between zero-point and frequency is complex owing to the many interconnected factors which can create asymmetry. However, it was shown that sensitivity of zero-point to frequency depends on the degree of asymmetry and particularly asymmetry of deformation-gradient (w0′).
The measuring tube is the core sensitive unit of the Coriolis mass flow sensor. Its design parameters directly influence natural frequency and sensitivity, such as shape and structure dimensions. In this study, we obtained under concentrated force the equivalent elastic coefficient of the measuring tube by adopting static analysis and calculating static deflection curves, including the respective U-shape, slightly curved, and straight tubes. We then obtained the resonant frequency from the second-order vibration equation. Additionally, the maximum sensitivity and position coordinates were obtained by calculating the torsional displacement curve of the measuring tube under the distribution of Coriolis force during a rated flow. Sensor models with different measuring tube shapes were designed by applying this theoretical analysis. Calibration tests for sensors were performed using a static gravimetric method. Theoretical analysis and test results show that the resonant frequency and sensitivity of the sensors calculated by applying static mechanical analysis and Coriolis distributing force align with the experimental results, thereby proving the validity of the theoretical method. Furthermore, the proposed method simultaneously obtained the relation curve of the measuring tube structure dimensions and natural frequency and sensitivity. It therefore provides theoretical evidence for the sensor design and detector installation position.
Coriolis mass flow meter (CFM) is used to measure the rate of mass flow through a pipe conveying fluid. In the present work, the Coriolis effect produced in the pipe due to a lateral excitation is modelled using the finite element (FE) method in MATLAB©. The coupled equation of motion for the fluid and pipe is converted to FE equations by applying Galerkin technique. The pipe conveying fluid is excited at its fundamental natural frequency. The time lag observed between symmetrically located measurement points which are equidistant from the point of excitation, is utilized to predict the mass flow rate. The results predicted by the present code is validated using the experimental, and numerical results published in the literature. The main contribution is the development of a FE model, using three node Timoshenko beam element to analyse the dynamics of fluid conveying pipes subjected to external excitation. The direction of the Coriolis force is perpendicular to the plane containing the velocity of flow vector and angular velocity vector of the pipe. Hence a three dimensional FE model is essential. This model can include curved geometry, damping, velocity and gyroscopic effects for three dimensional flexible tubes. The reduced integration used for overcoming shear locking in two node elements, will result in the formation of spurious modes leading to an incorrect prediction of natural frequencies and velocity. These modes will not occur while using three node elements. Influence of spatial as well as temporal discretisation on the time lag and frequency are also discussed. The sensitivity analysis shows that the time lag varies linearly with the mass flow rate.
Equations are more convenient than tables or graphical correlations in computer-aided design and operation. A single equation is derived that correlates the fluid-flow friction factor with all Reynolds numbers and all pipe-roughness ratios. It has been constructed from theoretical and correlating equations for the laminar, transition, and fully developed turbulent regimes of flow, using a general model developed by Churchill and Usagi. This equation not only reproduces the friction-factor plot but also avoids interpolation and provides unique values in the transiton region These values are, of course, subject to some uncertainty, because of the physical instability inherent in this region.
This paper starts from a brief revisit of key early published work so that an overview of modern Coriolis flowmeters can be provided based on a historical background. The paper, then, focuses on providing an updated review of Coriolis flow measurement technology over the past 20 years. Published research work and industrial Coriolis flowmeter design are both reviewed in details. It is the intention of this paper to provide a comprehensive review study of all important topics in the subject, which include interesting theoretical and experimental studies and innovative industrial developments and applications. The advances in fundamental understanding and technology development are clearly identified. Future directions in various areas together with some open questions are also outlined.
The behaviour of coriolis mass flowmeters (CMFs) with arbitrary pipe geometry is investigated in this paper regarding the effects of unsteady flow, geometry and damping. Timoshenko's Beam Theory, the Finite Element Method (FEM), Mathworks Matlab and Simulink are used for creating a suitable model. Various physical quantities such as coriolis, centrifugal and gravitational forces, mass inertia, mass flow, unsteady flow forces, damping and material parameters are considered to increase model precision. Stiffening effects due to fluid deflection are partly neglected in the model. The use of exact finite elements and model reduction via modal transformation results in fast and accurate numerical computation which corresponds well with experimental data. As a consequence, this approach allows further optimization of pipe geometries, actuation positions and concepts, actuation und flow control systems and signal processing concepts without co-simulation.
For steady flows, Coriolis flow meters accurately measure mass flow. To study the performance of Coriolis meters under transient flows, we measured the instantaneous and totalized flow determined by two Coriolis meters. The tests used a Transient Flow Facility (TFF) developed to generate transient flow, pressure, and temperature conditions similar to those that occur when a hydrogen powered vehicle is refueled. During simulated cascade fills, the TFF discharged 3 kg of helium in 3 minutes at flows between 10 g/s and 45 g/s through the Coriolis meters and the TFF's standard. The TFF's expanded uncertainty (95% confidence level) for totalized mass during this cascade fill was 0.45%. For the same simulated cascade fill, both Coriolis meters measured the instantaneous flow within the uncertainty of the TFF and measured totalized flow within the International Organization of Legal Metrology Recommendation 139 maximum permissible errors for meters in gaseous fuel dispensers (1.0%).
The aim of this paper is to develop an approximate analytical solution
for phase-shift (and thus mass flow) prediction along the length of the
measuring tube of a Coriolis mass-flowmeter. A single, straight
measuring tube is considered; added masses at the sensor and excitation
locations are included in the model, and thus in the equation of motion.
The measuring tube is excited harmonically by an electromagnetic driver.
Taking into account thermal effects, the equation of motion is derived
through use of the extended Hamilton's principle and constitutive
relations. The equation of motion is discretized into a set of ordinary
differential equations via Galerkin's technique. The method of multiple
timescales is applied to the set of resultant equations, and the
equations of order one and epsilon are obtained analytically for the
system at primary resonance. The solution of the equation of motion is
obtained by satisfying the solvability condition (making the solution of
order epsilon free of secular terms). The flow-related phase-shift in
the driver-induced tube vibration is measured at two symmetrically
located points on either side of the mid-length of the tube. The
analytical results for the phase-shift are compared to those obtained
numerically. The effect of system parameters on the measured phase-shift
is discussed. It is shown that the measured phase-shift depends on the
mass flow rate, of course, but it is also affected by the magnitude of
the added sensor mass and location, and the temperature change;
nevertheless, the factors investigated do not induce a zero phase-shift.
The non-linear equations of motion of a flexible pipe conveying unsteadily flowing fluid are derived from the continuity and momentum equations of unsteady flow. These partial differential equations are fully coupled through equilibrium of contact forces, the normal compatibility of velocity at the fluid- pipe interfaces, and the conservation of mass and momentum of the transient fluid. Poisson coupling between the pipe wall and fluid is also incorporated in the model. A combination of the finite difference method and the method of characteristics is employed to extract displacements, hydrodynamic pressure and flow velocities from the equations. A numerical example of a pipeline conveying fluid with a pulsating flow is given and discussed.
Theoretical investigations of a single, straight, vibrating, fluid-conveying pipe have resulted in simple analytical expressions for the approximate prediction of the spatial shift in vibration phase. The expressions have lead to hypotheses for real Coriolis flowmeters (CFMs). To test these, the flexural vibrations of two bent, parallel, non-fluid-conveying pipes are studied experimentally, employing an industrial CFM. Special attention has been paid on the phase shift in the case of zero mass flow, i.e. the zero shift, caused by various imperfections to the “perfect” CFM, i.e. non-uniform pipe damping and mass, and on ambient temperature changes. Experimental observations confirm the hypothesis that asymmetry in the axial distribution of damping will induce zero shifts similar to the phase shifts due to fluid flow. Axially symmetrically distributed damping was observed to influence phase shift at an order of magnitude smaller than the primary effect of mass flow, while small added mass and ambient temperature changes induced zero shifts two orders of magnitude smaller than the phase shifts due to mass flow. The order of magnitude of the induced zero shifts indicates that non-uniform damping, added mass as well as temperature changes could be causes contributing to a time-varying measured zero shift, as observed with some commercial CFMs. The conducted experimental tests of the theoretically based hypotheses have shown that simple mathematical models and approximate analysis allow general conclusions, that may provide a direct insight, and help increasing the benefit of time consuming numerical simulations and laboratory experiments.
Knowing the influence of fluid flow perturbations on the dynamic behavior of fluid-conveying pipes is of relevance, e.g., when exploiting flow-induced oscillations of pipes to determine the fluids mass flow or density, as done with Coriolis flow meters (CFM). This could be used in the attempts to improve accuracy, precision, and robustness of CFMs. A simple mathematical model of a fluid-conveying pipe is formulated and the effect of pulsating fluid flow is analyzed using a multiple time scaling perturbation analysis. The results are simple analytical predictions for the transverse pipe displacement and approximate axial shift in vibration phase. The analytical predictions are tested against pure numerical solution using representative examples, showing good agreement. Fluid pulsations are predicted not to influence CFM accuracy, since proper signal filtering is seen to allow the determination of the correct mean phase shift. Large amplitude motions, which could influence CFM robustness, do not appear to be induced by the investigated fluid pulsation. Pulsating fluid of the combination resonance type could, however, influence CFMs robustness, if induced pipe motions go unnoticed and uncontrolled during CFM operation by feedback control. The analytical predictions offer an immediate insight into how fluid pulsation affects phase shift, which is a quantity measured by CFMs to estimate the mass flow, and lead to hypotheses for more complex geometries, i.e. industrial CFMs. The validity of these hypotheses is suggested to be tested using laboratory experiments, or detailed computational models taking fluid–structure interaction into account.
The dynamic equilibrium matrix equation for a discretized pipe element containing flowing fluid is derived from the Lagrange principle, the Ritz method and consideration of the coupling between the pipe and fluid. The Eulerian approach and the concept of fictitious loads for kinematic correction are adopted for the analysis of geometrically non-linear vibration. The model is then deployed to investigate the vibratory behaviour of the pipe conveying fluid. The results for a long, simply supported, fluid-conveying pipe subjected to initial axial tensions are compared with experimentally obtained results and those from a linear vibration model.
In this paper, a spectral element model is developed for the uniform straight pipelines conveying internal unsteady fluid. Four coupled pipe-dynamics equations are derived first by using the Hamilton's principle and the principles of fluid mechanics. The transverse displacement, the axial displacement, the fluid pressure and the fluid velocity are all considered as the dependent variables. The coupled pipe-dynamics equations are then linearized about the steady-state values of the fluid pressure and velocity. As the final step, the spectral element model represented by the exact dynamic stiffness matrix, which is often called spectral element matrix, is formulated by using the frequency-domain solutions of the linearized pipe-dynamics equations. The fast Fourier transform (FFT)-based spectral dynamic analyses are conducted to evaluate the accuracy of the present spectral element model and also to investigate the structural dynamic characteristics and the internal fluid transients of an example pipeline system.
Experiments have been conducted on various configurations of Coriolis mass flowmeters, such as U-tube, S-tube, L-tube, and stair shaped tubes, with water as the working fluid. The sensitivity of the flowmeter has been measured for a wide range of parameters like position of sensors, geometry, material properties, etc. The results of this study indicate that the measured phase shift is proportional to the mass flow rate. The slope ratio (theoretical to measured) is observed to be constant for the range of parameters studies for U-tube flowmeters. Sensor positions for maximum sensitivity have been identified for S-tube and L-tube configurations.
Generally, current models of Coriolis flowmeters have a very limited dynamic response. This paper presents work which investigates the dynamic response of the meter flow tube alone (i.e. by-passing the flow transmitter) to flow pulsations. Because the shortest time for which an estimate of the phase difference can be achieved appears to be the period of one drive cycle, pulsations at frequencies somewhat lower than the meter drive frequency were investigated. Our existing signal processing algorithms were used to determine phase differences from sensor signals (or pseudo-signals) generated from three different sources: firstly, pseudo-data generated from our previously developed theoretical model of a straight tube meter response to pulsations; secondly, pseudo-data produced by finite element simulations of flow tubes with complex geometries, following validation against the straight tube theoretical results; finally, experimental data obtained by recording the internal sensor signals from eight commercially available flowmeters. Results from these three sources showed a high degree of consistency and confirmed the capability of meter flow tubes to reveal both the amplitude and reasonable definition of the waveform of flow pulsations. All meters tested experimentally showed a significant component of sensor signal noise at the Coriolis frequency. The results indicate that with appropriate flow tube design and improved meter drive and signal processing procedures, a high dynamic performance Coriolis meter is achievable.
CMFs are gaining a good reputation in the metering of liquid. However, because of some undesirable influences of the process parameters and fluid properties of the meter characteristics, it is recommended that the instrument be calibrated in situ when high accuracy must be achieved and/or the metering conditions (process parameters and fluid properties) differ noticeably from the calibration conditions. Field calibration is not always technically feasible and/or economically desirable. Therefore it is important to be able to predict the behaviour of the instrument. Taking the example of gas measurement, it is shown that straight pipe CMFs fulfil this requirement.
This paper presents some tests which have been carried out on a commercial model of Coriolis mass flowmeter under medium-pressure conditions (p > 15 bar), utilizing a primary standard flow calibration facility. The experimental results show that the fluid pressure affects the accuracy of the tested meter, which provides underestimated readings (i.e. negative errors). Finally, some diagrams which allow to correct the output signal (or the reading) provided by the tested meter (according to the operating fluid pressure) are also presented.
The current status of available work regarding the pressure effect on Coriolis mass flowmeters is reviewed, which shows significant improvement in the latest generation of Coriolis flowmeters. A theoretical method using the linear damping model is proposed to understand the pressure effect. This new method applied to Coriolis flow sensors provides intuitive insight into the flow-generated signal by studying undamped natural frequencies and mode shapes. Most importantly this method can be used to model virtually any shape and configuration of flow sensors as found in the practical design. It is found that when the pressure changes it alters the superimposed contribution and the mass flow measurement can deviate from the reference condition. Experimental results from both low and high pressure flow tests are reported, which are in general agreement with the theoretical prediction. Further specific work is finally suggested which may advance our understanding and improve the Coriolis mass flow measurement technology.
Coriolis flowmeters (CFM) are forced to vibrate by a periodic excitation usually applied midpipe through an electromagnetic actuator. From hands-on experience with industrial CFMs it appears, that the electromagnetic actuator has to be located as symmetric as possible. For CFM design and trouble-shooting it is of relevance to know how and if imperfections, related to the excitation location, influence the dynamic behavior of the vibrating fluid-conveying pipes employed in CFMs. A simple model of an imperfectly excited, simply supported, straight, single pipe CFM is investigated using a multiple time scaling perturbation analysis. The result is a simple analytical expression for the approximated phase shift, which offers a direct insight into how the location of the actuator influences the phase shift. It appears, that asymmetrical forcing combined with fluctuating pipe damping could be a factor contributing to lack of zero shift stability observed with some industrial CFMs. Tests of the approximated solution against results obtained by pure numerical analysis using Galerkin expansion show very good agreement. The effect of asymmetric detector positions is also investigated. Any asymmetry in the detectors position, e.g. due to manufacturing variations or improper handling of the CFM, induces a phase shift that leads to changes of the meter’s sensitivity, and could therefore result into erroneous measurements of the mass flow. This phase shift depends on the mass flow and does not contribute to a lacking zero-point stability. The validity of the hypotheses, which are assumed to be basically similar for more complicated geometries, e.g. bended and/or dual pipe CFMs, with or without multiple actuators, is suggested to be tested using laboratory experiments with purpose built non-ideal CFMs.
Explicit, but accurate formulas for the friction factor are given as a substitute for the more inconvenient implicit formulas which are at present considered to be the most accurate.
A simple analytical method is introduced to calculate the sensitivity of Coriolis mass flowmeter (CMF). The definition of the sensitivity is further developed based on the reciprocity principle, through employing two approaches: selecting the moment when the displacement of drive point is zero as the calculation moment; reducing the degree of high-order static indeterminacy by taking a half of the structure due to symmetry in structure and anti-symmetry in load. With these approaches, the method can be used to calculate sensitivities for the flowmeters with any shaped tubes and anywhere detected positions; thus, provides theory basis for tube shape design and detected positions determination. Detail analytical calculations for typical Straight-Circle-joint-shaped CMFs are illustrated. The method is validated on a published U-CMF and then, is further illustrated and experimentally validated through predicting the most sensitive detected point for a narrowed-U-CMF.
The paper aims at a detailed description of the physical background for the so-called Coriolis mass flow meter. It presents essentially an analysis of the (free) vibration modes of a fluid conveying straight pipe segment. Due to the inertial effects of the flowing fluid, mainly the Coriolis force, these modes deviate in shape (and in frequency) from those appearing in the absence of fluid motion. The effect of fluid inertia may, therefore, be exploited for the purpose of flow measurement. The analysis is performed under a simplifying approximation: The pipe is considered as a beam, the fluid as a moving string. This approximation leaves the fluid with only one degree of freedom, connected with its mean velocity, and eliminates an infinity of degrees of freedom of the pipe. Yet it keeps the essential features of the phenomenon. The equations describing the vibrations are derived variationally, with the constraint of a common vibration amplitude of both fluid and pipe. The Lagrange multiplier associated with the constraint gives the interaction force between pipe and fluid. The modes are determined by a perturbation procedure, wherein the small (perturbation) parameter is related to the fluid velocity. The analysis shows, as main result, how the time delay between the vibrations of two appropriately chosen points of the pipe may serve to determine the mass flow rate of the fluid. Other aspects of the problem, like the precise role of the Coriolis force, are considered. The possible improvement of the used approximation is discussed.
Resonant vibrations of a fluid-conveying pipe are investigated, with special consideration to axial shifts in vibration phase accompanying fluid flow and various imperfections. This is relevant for understanding elastic wave propagation in general, and for the design and trouble-shooting of phase-shift measuring devices such as Coriolis mass flowmeters in particular. Small imperfections related to elastic and dissipative support conditions are specifically addressed, but the suggested approach is readily applicable to other kinds of imperfection, e.g. non-uniform stiffness or mass, non-proportional damping, weak nonlinearity, and flow pulsation. A multiple time scaling perturbation analysis is employed for a simple model of an imperfect fluid-conveying pipe. This leads to simple analytical expressions for the approximate prediction of phase shift, providing direct insight into which imperfections affect phase shift, and in which manner. The analytical predictions are tested against results obtained by pure numerical analysis using a Galerkin expansion, showing very good agreement. For small imperfections the analytical predictions are thus comparable in accuracy to numerical simulation, but provide much more insight. This may aid in creating practically useful hypotheses that hold more generally for real systems of complex geometry, e.g. that asymmetry or non-proportionality in axial distribution of damping will induce phase shifts in a manner similar to that of fluid flow, while the symmetric part of damping as well as non-uniformity in mass or stiffness do not affect phase shift. The validity of such hypotheses can be tested using detailed fluid-structure interaction computer models or laboratory experiments.