Dynamic analysis of Coriolis flow meter using Timoshenko beam element

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 modeled 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.     

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.     

Numerical analysis of installation effects in Coriolis flowmeters: A case study of a short straight tube full-bore design

A fully coupled, partitioned, numerical model that accounts for fluid–structure interaction is applied for a study of installation effects in a straight-tube Coriolis flowmeter. Three flow disturbance elements positioned at different locations upstream of the measuring tube are considered in the study: a single elbow, closely coupled double elbows out-of-plane, and an orifice. The installation effects are estimated by comparing the mass-flow sensitivities obtained for the disturbed and the fully developed flow conditions in the measuring tube. The interpretation of the installation effects is given by analysing the anti-symmetric fluid forces in the measuring tube. The simulation results show that the magnitude of the installation effect in asymmetrically distorted flows varies for the different circumferential positions of the motion sensors. The sensitivity variations around the circumference of the measuring tube are less pronounced for measuring tubes with a higher circumferential stiffness.The predicted installation effects are also affected by changing the vibration direction of the tube with respect to the disturbance elements.     

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.     

Coriolis mass flow measurement of gas under normal conditions

This paper presents a new method of directly measuring the mass flow of gas using the well-known Coriolis principle, which has proved successful for mass flow measurement of liquids. The prototype consists of two U-shaped tubes, forming a device resembling very much a tuning fork, which is stimulated by electromagnetic actuators to perform autonomous bending oscillations. By this means the fluid is subjected to a radial velocity that, in combination with the axial velocity of the flow, induces harmonic Coriolis forces of the same frequency. This causes the U-shaped tube to perform torsional oscillations that superimpose on the bending oscillations. Both oscillations can be detected via electromagnetic transducers.

The amplitude of the torsional oscillation induced by the Coriolis forces is very small as the density of gas is very low. It can be amplified by tuning the eigenfrequencies of torsion and bending in a control loop. This results in an amplification of the torsional amplitude by a factor of 10², allowing the mass flow of gas to be measured under normal conditions.     

Motion induced signals of Coriolis flowmeters

Rotation induced Coriolis flowmeter output was observed in a robotic fluid dispensing test. This motion sensitivity of Coriolis flowmeters was confirmed to be universal for various noninertial reference frames. It was also confirmed that this motion sensitivity is a general aspect of many Coriolis flowmeter types. Qualitative tests revealed that the motion induced signals appear to be proportional to frame rotation rate and are strongly dependent on the flow tube oscillation velocity vectors.     

Calculation of the sensitivity of a straight tube Coriolis mass flowmeter with free ends

The weight vector theory for Coriolis mass flowmeters is applied to a simple theoretical meter configuration consisting of a single unsupported straight tube unattached to adjacent piping. The tube has free ends and vibrates in the fundamental mode. It is shown how the sensitivity of this meter depends in part on the interaction of the flow velocity profile with fluid vibrations occurring near the tube ends. This end effect is negative i.e. the meter reads lower than would be expected if end effects were ignored. On account of the end effect there is a predicted variation in sensitivity with Reynolds number (of the order 1% in a tube 25 diameters long) and this can be minimised by a certain choice of the sensor positions.     

内置扭带换热管三维流动与传热数值模拟

对自转扭带换热管内流体的运动进行了分析,根据流体在自转扭带管内的切向运动特点,提出将自转扭带等效虚拟于静止扭带的思路。建立内置螺旋扭带换热管流体流动的三维物理模型,采用大型CFD软件FLUENT6.0中的RNG k-ε模型对内置扭带换热管内的流动与传热进行了数值模拟,得到了内置扭带换热管流体流动的速度、压力、湍流强度场分布规律及传热特性。比较了静止、旋转及旋转等效虚拟静止扭带换热管的传热和阻力降特性,分析了不同螺距对强化传热和阻力降的影响。速度场的模拟值与激光测速仪试验值进行了比较,二者吻合较好。     

The performance characteristics of a micro-machined Coriolis flow meter: An evaluation by simulation

Micro-machined Coriolis meters will enable measurement of very low flow rates (0.1–500 g/h) and, potentially, ultra-low flows (0.1–100 mg/h). Application areas include the delivery of medical drug infusions to patients, and a wide variety of micro-fluidic devices. An evaluation of the performance of two prototype micro-machined flow-tubes of differing shapes is reported, based upon results obtained from a virtual Coriolis meter. This tool comprises a finite element modelling capability which simulates the meter flow-tube in motion, with the flow represented simply as a continuous string, i.e. 1-dimensional and frictionless, and the model allows the generation of pseudo-data at points on the tube corresponding to sensor locations. Application of signal processing algorithms then enables the representation of an indicated flow time history output by a Coriolis meter in response to a prescribed input flow. Results indicate that the devices investigated were all highly linear and that meter sensitivity is independent of fluid density. One flow-tube shape confers higher stiffness than the other and, for both tube shapes, increasing wall thickness increases tube stiffness at a greater rate than the tube mass. Higher stiffness results in reduced meter sensitivity, but increased drive frequency (hence, faster dynamic response). The spatial averaging resulting from the use of ‘distributed’ internal sensors inevitably yields meter sensitivity values that are lower than the potential maximum value that might be achieved by use of ‘point’ sensors; however there are practical reasons why this latter approach would not work. The dynamic response to a flow step is essentially the same as found for macro-Coriolis meters.     

管壳式换热器壳程流体流动与传热数值模拟

利用ANSYS参数化建模方法建立了管壳式换热器的参数化模型,在ANSYS FLUENT 13.0模拟软件中对管壳式换热器的壳程流体的流动与传热进行了数值模拟计算,得到了不同折流板间距及入口流速的情况下换热器壳程流体温度场、速度场和压力场,分析了折流板间距及入口流速对换热效率和流体诱导振动的影响,对换热器结构优化设计提出了改进措施。     

发夹式换热器管束流致振动数值模拟研究

运用计算流体力学方法,采用ANSYS CFX软件对发夹式换热器的壳侧流场进行了三维数值模拟。流场计算中采用多孔介质模型对管束区域进行简化,分析了壳侧流场的速度分布,结果表明:直管段部分的流体湍流强度大于弯管段,且外层管束所在区域为高流速区,受流体冲刷严重。结合流场信息,通过功率谱生成随机激振力,采用ABAQUS软件模拟计算了湍流激振下管束的振动响应,结果显示管束的面外均方根位移远大于面内位移,且弯管部分的振动位移最大。该研究结果可为发夹式换热器的性能分析和优化设计提供参考和依据。     

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.     

改进的浸入边界-晶格Boltzmann法的蠕动流分析

基于改进的浸入边界-晶格Boltzmann方法研究蠕动流问题,采用晶格Boltzmann法描述流场,用改进的浸入边界法实现管壁运动-流体流动之间的相互作用,将变形管壁的运动速度作为速度源引入晶格Boltzmann方程,代替了传统浸入边界-晶格Boltzmann法中固态变形力与流体速度之间的转换。分析了管道内蠕动流场的分布情况,研究了各相关参数如振幅比、频率、液体黏度以及波数对流量的影响,数值结果与已有的结果进行了对比,证实了本研究方法的合理性与有效性。     

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.     

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.     

The vibration of an artery-like tube conveying pulsatile fluid flow

A hybrid method for investigating pulsatile fluid flow in a long, thin, artery-like tube subjected to external excitations is presented. The non-linear partial differential equations governing the motion of the system, which incorporate the influence of circumferential strains, are solved by a combination of a finite element method, a finite difference method and a method of characteristics with interpolation. An initially axially stretched elastic tube conveying pulsating fluid, simply supported at both ends, is modelled to assess the effect of external harmonic excitation on the dynamic responses of the tube and the fluid flow. The results agree well with new experimental data. Comparison of the predicted results with those of a decoupled model demonstrates that it is necessary to consider the mechanism of fluid-structure interaction fully in the study of initially stretched cylindrical tubes conveying pulsatile fluid flow. An analysis of these coupling effects is presented for Womersley numbers alpha = 2.81 and 3.97 and a mean flow Reynolds number Re = 875.     

Velocity profile effects in Coriolis mass flowmeters: Recent findings and open questions

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.     

新型高效扭曲管双壳程换热器的研制

应用计算流体力学模拟了不同扭距下扭曲管的传热性能。通过实验室测试并比较扭曲管换热器和弓形折流板换热器的传热性能差异;介绍了扭曲管双壳程换热器的结构设计创新。通过工业试验产品在某厂的成功应用,验证了该新型高效扭曲管双壳程换热器的综合强化传热能力比传统折流板换热器提高30％～40％,节能效益显著,工业应用前景广阔。     

污垢对单直管科氏流量计灵敏度的影响及仿真分析

运用Euler梁理论,建立力学模型,研究管道污垢对单直管科氏流量计测量灵敏度的影响,并利用ansys进行仿真验证。结果表明:当单直管科氏流量计测量管内产生污垢时,其灵敏度将发生变化,并且灵敏度的变化取决于污垢质量和所在位置。     

Numerical analysis of installation effects in Coriolis flowmeters: Single and twin tube configurations

A fully coupled, partitioned, numerical model that accounts for fluid–structure interaction is applied for a study of the installation effects of Coriolis flowmeters. The modeled configurations include a single straight-tube full-bore flowmeter and two different twin tube flowmeters with straight and U-shaped measuring tubes. Three different flow disturbance elements positioned upstream of the flowmeter are considered in the study, as well as two different types of flow splitters in the case of the twin tube configurations. The installation effects are estimated by comparing the mass-flow sensitivities under the disturbed and fully developed flow conditions at the inlet of the flowmeter. For the modeled twin tube flowmeters they are found to be of the order of one-tenth of a per cent. These relatively small values of the installation effects are related to the presence of flow splitters and to the averaging of the motion of both measuring tubes in the twin tube configurations. Similarly, averaging the response from two sensor pairs instead of only a single sensor pair reduces the circumferential variations and the peak values of the installation effects for asymmetric flows in the single straight-tube flowmeter.