Experimental determination of time lag due to phase shift on a flexible pipe conveying fluid

A finite element model of a flexible tube conveying fluid is developed in MATLAB^© based on the principle of virtual work, using a three-node isoparametric beam element. Finite element equations are formulated by applying Galerkin technique on the coupled equations of pipe conveying fluid. The present element developed is capable to model three-dimensional flexible tubes by including curved geometry, effects of damping, velocity and gyroscopic effects. The external excitation applied at the middle of the tube in the lateral direction produced a time lag between the lateral responses, which were measured at two equidistant points from the excitation point. This is due to the Coriolis effect, and the same is simulated using the developed code. An experiment, supported with a robust error analysis, is also conducted on a straight polyurethane tube conveying water, subjected to a sinusoidal excitation at the center between the clamped supports. The measured time responses are compared with the numerical values predicted by the code. The time lags for both cases are obtained from the temporal shift along the time axis, between the zero crossing points of the time–response curves. The results obtained agreed well. The code can be used to predict the time lag, which is correlated to the mass flow rate. The proposed method will help to design Coriolis mass flow meters for existing pipelines, without altering the system.     

On the vibrations of three-dimensional angled piping systems conveying fluid

The vibrations of three dimensional angled pipe systems conveying fluid are studied by using the finite element method. Extended Hamilton's principle is applied to derive the equations of motion. The characteristics matries consisting of inertia, stiffness, and Coriolis terms are derived by variational method, in which the effects of the internal flow velocity and pressure are considered. The change of dynamic characteristics of the piping system due to the variation of flow velocity, pressure and the geometry of the system is investigated. As a result, it can be found that the natural frequency of the system decreases generally as the flow velocity and pressure increase and that the tendency is more significant as the geometry of the system is similar to the straight pipe.     

Bending-torsional instability of a viscoelastic cantilevered pipe conveying pulsating fluid with an inclined terminal nozzle

In the present study, dynamic stability of a viscoelastic cantilevered pipe conveying fluid which fluctuates harmonically about a mean flow velocity is considered; while the fluid flow is exhausted through an inclined end nozzle. The Euler-Bernoulli beam theory is used to model the pipe and fluid flow effects are modelled as a distributed load along the pipe which contains the inertia, Coriolis, centrifugal and induced pulsating fluid flow forces. Moreover, the end nozzle is modelled as a follower force which couples bending vibrations with torsional ones. The extended Hamilton's principle and the Galerkin method are used to derive the bending-torsional equations of motion. The coupled equations of motion are solved using Runge-Kutta algorithm with adaptive time step and the instability boundary is determined using the Floquet theory. Numerical results present effects of some parameters such as fluid flow fluctuation, bending-to-torsional rigidity ratio, nozzle inclination angle, nozzle mass and viscoelastic material on the stability margin of the system and some conclusions are drawn.     

A finite element model for the analysis of flexible tube Coriolis mass flow meter using first order shear deformation shell theory

The numerical modelling of Coriolis Mass flow Meter (CFM) is essential for predicting its outcomes accurately in terms of sensitivity as well as exact mass flow rates. In the majority of mathematical and numerical modelling concerning the flexible structures, the authors neglect the dimensional and shape variation of the structure due to self-weight. The shell based on the First-order shear deformation shell theory (FSDST) is preferred in modelling shells compared to the beam model. The current work includes numerical modelling of CFM using eight noded isoparametric shell elements and twenty noded Acoustic fluid elements. The fluid energy describes as the potential, and the dynamic boundary condition is assumed utilising the displacement of structure and potential of the fluid. The fluid dynamic equation combining suitable numerical model, fluid-structure interaction module and cross-correlation technique helps to achieve the numerical modelling of CFM. The numerical model of CFM utilises the Newmark Beta method of numerical integration, and the response of two equidistant locations from the point of tube excitation is acquired. For the flexible tube conveying fluid, there exists sagging of tube due to the weight of tube and fluid. The Coriolis force and the external excitation force cause the fluid conveying tube to bend and twist, and as a result, the velocity responses picked from two equidistant points shows a difference in phase. The effect of sagging leads to a lower phase shift and time decay, and hence the sensitivity of the CFM is low for low pre-stretched flexible tubes. The pre-stretching of flexible tubes reduces the effect of sagging, facilitates to regain the cylindrical shape of the tube and increases the sensitivity of CFM. The result reveals that the shell element along with the three-dimensional acoustic fluid element provides the most accurate numerical model for the CFM and the change in sensitivity, as well as the change in mass flow measurements, can appropriately be analysed with the help of this numerical model. The amplitude of the velocity of the structure, measured from the two equidistant points, shows a difference. The severe variation in amplitude of velocity measured from two points is an implication of the out of plane deflection of the tube. For a CFM made up of metal tubes, the amplitude of velocity variation is minimal and ignored by the authors.     

Dynamic response of rotating flexible cantilever pipe conveying fluid with tip mass

In this study the vibration system is consisted of a rotating cantilever pipe conveying fluid and a tip mass. The equation of motion is derived by using the Lagrange's equation. Also, the equation of motion is derived applying a modeling method that employs hybrid deformation variables. The influences of the rotating angular velocity and the velocity of fluid flow on the dynamic behavior of a cantilever pipe are studied by the numerical method. The effects of a tip mass on the dynamic behavior of a rotating cantilever pipe are also studied. The influences of a tip mass, the velocity of fluid, the angular velocity of a cantilever pipe and the coupling of these factors on the dynamic behavior of a cantilever pipe are analytically clarified. The natural frequencies of a cantilever pipe conveying fluid are proportional to the angular velocity of the pipe and a tip mass in both axial direction and lateral direction.     

Transfer matrix modelling for the 3-dimensional vibration analysis of piping system containing fluid flow

For three dimensional vibration analysis of piping system containing fluid flow, a transfer matrix formulation is presented. The fluid velocity and pressure were considered, that coupled to longitudinal and flexural vibrations. Transfer matrices were derived from direct solutions of the differential equations of motion of pipe conveying fluids, and the variations of natural frequency with flow velocities for straight and curved pipes were investigated. The results were confirmed to the corrections of known data. The scheme of this study can be easily applied to the related fields, using small size personal computers with core memory about 200kbytes.     

Vibration and dynamic stability of pipes conveying fluid on elastic foundations

The paper deals with the vibration and dynamic stability of cantilevered pipes conveying fluid on elastic foundations. The relationship between the eigenvalue branches and corresponding unstable modes associated with the flutter of the pipe is thoroughly investigated. Governing equations of motion are derived from the extended Hamilton’s principle, and a numerical scheme using finite element methods is applied to obtain the discretized equations. The critical flow velocity and stability maps of the pipe are obtained for various elastic foundation para-meters, mass ratios of the pipe, and structural damping coefficients. Especially critical mass ratios, at which the transference of the eigenvalue branches related to flutter takes place, are precisely determined. Finally, the flutter configuration of the pipe at the critical flow velocities is drawn graphically at every twelfth period to define the order of the quasi-mode of flutter configuration.     

An aeroelastic model of Coriolis mass and density meters operating on aerated mixtures

A lumped parameter, aeroelastic model of Coriolis mass flow and density meters is presented which addresses the effects of compressibility and inhomogeneity introduced by aerated fluids. The model addresses U-tube Coriolis meters containing radially outward and inward flows of a aerated fluid. The mass flow rate and density of the fluid measured by the Coriolis meter are given by solution of an eigenvalue problem governing the dynamics of the aeroelastic system. Mass flow is determined by the phase lag between the displacement of the out-bound and in-bound tubes in the lowest frequency bending mode of the system. Fluid density is related to the natural frequency of this mode.The aerated fluid is assumed to be a well-mixed, dispersed bubbly flow in which the bubbles are small compared to the diameter of the tube. Under this assumption, the effects of compressibility can be incorporated using a lumped parameter model of the first acoustic cross mode of the tube. The effects of inhomogeneity introduced by the bubbles are incorporated using a lumped parameter model of a bubble in a oscillatory acceleration field contained in an viscous, incompressible fluid. The resulting aeroelastic equations of motion for the Coriolis meter show that the behavior of the system is influenced by non-dimensional parameters characterizing the aerated mixtures including reduced frequency, void fraction, and fluid viscosity parameters.The model is used to examine the effect of aeration for a range of parameters considered to be broadly representative of the commercially available Coriolis meters. Results show that aeration can significantly influence the aeroelastic behavior of Coriolis meters, but that, if appropriately considered, Coriolis meters can be used to provide accurate characterization of aerated fluids.     

主动液压激波作用下管道振动控制的运动分析与试验研究

为研究在主动液压激波作用下管道振动的动力学特性,建立流体的数学模型,设计出变频液压管网激振测试系统,用特征线法编程对激波作用下的有压脉动内流进行数值模拟。采用有限元法把管道简化为梁模型,建立考虑流固耦合的充液管道在激波作用下的振动方程,在保证特征线各断面与有限元节点重合的前提下,采用Newmark法编程将特征线法求得的流体各断面横向压力载荷施加到管道有限元的单元节点上,求得各断面处的动力响应。仿真结果表明,管道在轴向弹性支撑条件下,在激波作用下管道各断面压力和流速为简谐波,但两者呈反相关系。其横向各断面运动为简谐振动,振幅随系统压力的升高而升高,发现管道横向各断面振动波明显滞后于各断面对应的压力波,而轴向振动则由于弹簧与液体轴向力的耦合作用而出现较高的振动频率。数值模拟结果与试验结果基本比较吻合,揭示出流体动力学参数与管道振动之间的耦合关系,为激波作用下管道的二维振动特性及可控性研究提供了一些理论依据。     

Weight vector study of velocity profile effects in straight-tube Coriolis flowmeters employing different circumferential modes

The weight vector theory is applied to evaluate the effects of velocity profile on the sensitivity of Coriolis mass flowmeters employing different circumferential modes of a straight measuring tube. The study is limited to fully developed, axisymmetric steady flow. The measuring tube is modelled as a thin, elastic shell with clamped ends, and the vibrational fluid motion inside the vibrating tube is determined using potential flow theory. By applying a power series expansion with respect to the aspect ratio of the tube, the general form of the weight function is simplified, allowing (approximate) analytical evaluations of profile effects. The results show that large velocity profile effects are expected for the higher circumferential (shell-type) modes and rather lower (but generally also significant) velocity profile effects for the first circumferential (beam-type) mode.     

立体网状静电传感器空间灵敏特性研究

提出一种全新的立体网状静电传感器,以克服现有静电传感器对流场内速度分布无法测量的不足。通过建立立体网状静电传感器的三维仿真模型,利用有限元分析法获得了立体网状静电传感器的灵敏度空间分布规律,并对其动态灵敏度进行了分析。结果表明:立体网状静电传感器具有局部敏感特性,对于管内颗粒分布比较敏感。而且由于其极棒间相互交错的特性,可以探测到颗粒的运动方向;传感器的动态灵敏度可以反映带电颗粒在各个极棒间的空间位置及其速度,即带电颗粒距离极棒交织点越近,速度越快,动态灵敏度越大。最后通过重力输送颗粒实验,证明理论分析的结果与实验结果一致,从而为立体网状静电传感器实现颗粒流动参数测量提供了理论依据。     

悬臂输送管道流-固耦合动力学系统的直接解法

根据变分原理导出了输送管道(流—固耦合问题)自由振动的变分方程,采用直接解法求出了输送管道自由振动的固有频率和极限流速。     

Performance evaluation of an indigenously designed copper (U) tube Coriolis mass flow sensors

This paper describes an experimental study carried out for three different sets of U tube configurations made of copper for an indigenously developed setup of Coriolis mass flow sensor integrated with virtual instrumentation. The setup has been installed on an especially designed and constructed foundation using Passive Vibration Isolation technique. The location of sensors, drive frequency and L/d ratio were found to be influencing design parameters on the performance. In the present study water has been used as fluid for flow measurement. The setup was calibrated for the experimentally obtained result indicates a linear relationship between measured time lag and mass flow rate. The results also indicates that for the undertaken experimental conditions the case C is the optimum configuration from the point of view of minimum percentage of error in calibrated mass flow rate.     

Accurate mass flow and density of bubbly liquids using speed of sound augmented Coriolis technology

Speed of sound augmented Coriolis technology utilizes a process fluid sound speed measurement to improve the accuracy of Coriolis meters operating on bubbly liquids. This paper presents a theoretical development and experimental validation of speed of sound augmented Coriolis meters. The approach utilizes a process fluid sound speed measurement, based on a beam-forming interpretation of a pair of acoustic pressure transducers installed on either side of a Coriolis meter, to quantify, and mitigate, errors in the mass flow, density, and volumetric flow reported by two modern, dual bent-tube Coriolis meters operating on bubbly mixtures of air and water with gas void fractions ranging from 0% to 5%. By improving accuracy of Coriolis meters operating on bubbly liquids, speed of sound augmented Coriolis meters offer the potential to improve the utility of Coriolis meters on many existing applications and expand the application space of Coriolis meters to address additional multiphase measurement challenges.The sources of measurement errors in Coriolis meters operating on bubbly liquids have been well-characterized in the literature. In general, conventional Coriolis meters interpret the mass flow and density of the process fluid using calibrations developed for single-phase process fluids which are essentially incompressible and homogeneous. While these calibrations typically provide sufficient accuracy for single-phase flow applications, their use on bubbly liquids often results in significant errors in both the reported mass flow, density and volumetric flow. Utilizing a process fluid sound speed measurement and an empirically-informed aeroelastic model of bubbly flows in Coriolis meters, the methodology developed herein compensates the output of conventional Coriolis meters for the effects of entrained gas to provide accurate mass flow, density, volumetric flow, and gas void fraction of bubbly liquids.Data presented are limited to air and water mixtures. However, by influencing the effective bubble size through mixture flow velocity, the bubbly liquids tested exhibit decoupling characteristics which spanned theoretical limits from nearly fully-coupled to nearly fully-decoupled flows. Thus, from a non-dimensional parameter perspective, the data presented is representative of a broad range of bubbly liquids likely to be encountered in practice.     

Dynamic behavior of cracked pipe conveying fluid with moving mass based on timoshenko beam theory

In this paper we studied about the effect of the open crack and the moving mass on the dynamic behavior of simply supported pipe conveying fluid. The equation of motion is derived by using Lagrange’s equation and analyzed by numerical method. The crack section is represented by a local flexibility matrix connecting two undamaged pipe segments i.e. the crack is modeled as a rotational spring. The influences of the crack severity, the position of the crack, the moving mass and its velocity, the velocity of fluid, and the coupling of these factors on the vibration mode, the frequency, and the mid-span displacement of the simply supported pipe are depicted.     

Estimation of velocity profile effects in the shell-type Coriolis flowmeter using CFD simulations

This paper presents a study of the axisymmetric velocity profile effects in the shell-type Coriolis flowmeter with a second circumferential mode, which exploits the results of computational fluid dynamics simulations. Simulations were carried out for viscous fluid flow in the vibrating measuring tube, whose mode shape remained fixed during the transient simulation process. We observed time responses of the integral anti-symmetric fluid forces acting on the inner wall of the measuring tube. Their magnitudes and their relative variations with the mass flow rate of the fluid were used for the integral estimation of the velocity profile effect. Simulation results are presented for different fluid velocities through the measuring tube and show considerable loss of flowmeter’s sensitivity in the range of lower Reynolds numbers. The results are also compared with the weight vector estimations of the velocity profile effect and are evaluated for two different turbulent models.     

Influence of tip mass on dynamic behavior of cracked cantilever pipe conveying fluid with moving mass

In this paper, we studied about the effect of the open crack and a tip mass on the dynamic behavior of a cantilever pipe conveying fluid with a moving mass. The equation of motion is derived by using Lagrange’s equation and analyzed by numerical method. The cantilever pipe is modelled by the Euler-Bernoulli beam theory. The crack section is represented by a local flexibility matrix connecting two undamaged pipe segments. The influences of the crack, the moving mass, the tip mass and its moment of inertia, the velocity of fluid, and the coupling of these factors on the vibration mode, the frequency, and the tip-displacement of the cantilever pipe are analytically clarified.     

Diagnostic flow metering using ultrasound tomography

For an accurate flow metering without considering the influences of flow control devices such as valves and elbows in closed conduits, velocity distribution in the cross-sectional area must be integrated. However, most flow meters, including multi-path ultrasonic, electromagnetic or Coriolis mass flow meters, require assumptions on the fully-developed turbulent flows to calculate flow rates from physical quantities of their own concern. Therefore, a long straight pipe has been a necessary element for accurate flow metering because the straight pipe can reduce flow disturbances caused by flow control devices. To reduce costs due to the installation of long straight pipes, another flow metering technique is required. For example, flow rates can be estimated by integrating velocity distributions in the crosssection of conduits. In the present study, ultrasound tomography was used to find the velocity distribution in the cross-section of a closed conduit where flow was disturbed by a Coriolis mass flow meter or a butterfly valve. A commercial multi-path ultrasonic flow meter was installed in the pipeline to measure the line-averaged velocity distribution in the pipe flow. The ultrasonic flow meter was rotated 180° at intervals of 10° to construct line-averaged velocity distributions in Radon space. Flow images were reconstructed by using a backprojection algorithm (inverse Radon transform). Flow diagnostic parameters were defined by calculating statistical moments, i.e., average, standard deviation, skewness, and kurtosis, based on the normalized velocity distribution. The flow diagnostic parameters were applied to flow images to find whether the parameters could discern flow disturbances in the reconstructed velocity distribution.     

气力输送分支管路流量分配特性的研究

在水平T形分支管道中，用压缩空气作为输送气体，对不同粒径的砂石进行气力输送试验。分别通过试验和改进型BP神经网络预测两种方法对表观气速和分支管路流量控制阀开度变化时，固相在分支管路中的分配特性进行了研究。结果表明，随着表观气速减小和两分支管路流量控制阀开度差值变大，固相流量在两分支管中的分配产生较大差异。试验值和改进型BP网络预测值的对比结果表明，二者相互吻合较好，说明采用改进型的BP网络来模拟固相在分支管路中的分配特性适应性较好。     

粉煤灰管道气力输送特性的研究

对浓相粉煤灰气力输送系统三维流场进行了数值模拟,给出了流场特性,分析了颗粒对气相的影响,描述了颗粒沉降的具体过程。研究表明单位长度压力损失与速度平方、体积浓度成正比,而与管道长度无关;颗粒直径越大,压力损失就越小;颗粒的沉降与运动速度、颗粒直径和管道长度有关。