01 March 2005
Measurement at speed of sound
New method using coriolis can gain aerated liquid accuracy
By Daniel L. Gysling
Coriolis mass flow and density meters are the solution of choice for precision flow applications these days. Since their introduction to the mainstream flow metering community in the early 1980's, Coriolis meters have grown into one of largest and fastest growing market segments, representing roughly $400 million annual sales on 100,000 units. The reasons for the success of Coriolis meters comes from many factors, including its accuracy, reliability, and ability to measure multiple process parameters, including mass flow and process fluid density. However, despite this long list of attributes, Coriolis meters have significant limitations regarding aerated liquids.
Although the mass flow rate and process fluid density measurements determined by Coriolis meters derive from independent physical principles, the accuracy of both significantly degrade with the introduction of small amounts of entrained gases. That is why there is a new method to improve the accuracy of vibrating tube-based density measurements of aerated liquids.
Although the specific design parameters of Coriolis meters are many and varied, all Coriolis meters are essentially aeroelastic devices. Aeroelasticity is a term developed in the aeronautical sciences that describes the study of dynamic interaction of coupled fluid dynamic and structural dynamic systems. Coriolis flow meters rely on characterizing the aeroelastic response of fluid-filled, vibrating flow tubes to determine the mass flow rate and process fluid density measurements.
The physical principle used to determine process fluid density in a Coriolis meter is similar to that used in vibrating tube density meters. In these devices, the density of the process fluid is determined by relating the natural frequency of a fluid-filled tube to the density of the process fluid.
Introducing fluid to the tube changes the natural frequency of the oscillation. Under a quasi-steady and homogeneous model of the fluid, the primary effect of the fluid is to mass-load the tubes. The fluid typically has a negligible effect on the stiffness of the system. Thus, within the framework of this model, the mass of the fluid adds directly to the mass of the structure.
The mass of the fluid in the tube is proportional to fluid density, and therefore, the natural frequency decreases with increasing fluid density as described below:
β is a calibrated constant related to the geometry and vibratory characteristic of the vibrating tube.
Rearranging, the algebraic relation between the measured natural frequency of the vibrating tube and the density of the fluid within the tube can be as follows.
Defining the ratio between the effective mass of the fluid to that of the structure as a, the natural frequency of the fluid loaded tubes is:
This basic framework provides an accurate means to determine process fluid density under most operating conditions. However, some of the fundamental assumptions regarding the interaction of the fluid and the structure can deteriorate under different operating conditions.
Most aerated liquids are significantly more compressible than nonaerated liquids. Compressibility of a fluid directly relates to the speed of sound and density of the fluid. Mixture density and sound speed can relate to component densities and sound speed through the following mixing rules that are applicable to single phase and well-dispersed mixtures and form the basis for speed-of-sound-based entrained air measurement.
Is the mixture compressibility.
Is the component volumetric phase fraction
Consistent with the above relations, introducing air into water dramatically increased the compressibility of the mixture. For instance, at ambient pressure, air is approximately 25,000 times more compressible than water. Thus, adding 1% entrained air increases the compressibility of the mixture by a factor of 250. Conceptually, this increase in compressibility introduces dynamic effects that cause the dynamic of behavior of the aerated mixture within the oscillating tube to differ from that of the essentially incompressible single-phase fluid.
The effect of compressibility of the fluid can incorporate into a lumped parameter model of a vibrating tube. The stiffness of the spring represents the compressibility of the fluid. As the compressibility approaches zero, the spring stiffness approaches infinity.
As before, the effective mass of the fluid is proportional to the density of the fluid and the geometry of the flow tube. The natural frequency of the first transverse acoustic mode in a circular duct can estimate an appropriate spring constant for the model.
Note this frequency corresponds to a wavelength of an acoustic oscillation of two diameters, i.e., this transverse mode closely relates to a "half wavelength" acoustic resonance of the tube. For low levels of entrained air the frequency of the first transverse acoustic mode is quite high if you compare it to the typical structural resonant frequencies of Coriolis meters of 100 Hz, however, the resonant acoustic frequency decreases rapidly with increased levels of entrained air.
In characterizing aeroelastic systems, it is often convenient to define a reduced frequency parameter to gauge the significance of the interaction between coupled dynamic systems.
For a vibrating tube filled with fluid, a reduced frequency can be a ratio of the natural frequency of the structural system to that of the fluid dynamic system.
Where fstruct is the natural frequency of the tubes in vacuum, D is the diameter of the tubes, and amix is the sound speed of the process fluid.
For this application, as the reduced frequency becomes negligible compared to 1, the system approaches quasi-steady operation. In these cases, models, which neglect the compressibility of the fluid are likely to be suitable. However, the effects of unsteadiness increase with increasing reduced frequency. For a given Coriolis meter, mixture sound speed has the dominant influence of changes in reduced frequency. When considering Coriolis meters of varying design parameters, increases in tube natural frequency or tube diameter will increase the effects of unsteadiness for a given level of aeration.
In addition to dramatically increasing the compressibility of the fluid, aeration introduces inhomogeneity to the fluid. For flow regimes where the entrained gas is in a liquid-continuous flow field, the first–order effects of the aeration can model using the bubble theory. By considering the motion of an incompressible sphere of density of r0 contained in an inviscid, incompressible fluid with a density of r and set into motion by the fluid, the velocity of the sphere is:
For most entrained gases in liquids, the density of the sphere is orders of magnitude below that of the liquid and the velocity of bubble approaches three times that of the fluid.
Considering this result in the context of the motion of a sphere in a cross section of a vibrating tube, the increased motion of the sphere compared to the remaining fluid must result in a portion of the remaining fluid having a reduced level of participation in oscillation, resulting in a reduced, apparent system inertia.
For a given Coriolis meter, the level of aeration has a dominant effect on the difference between actual and apparent mixture density. However, other parameters identified by the lumped parameter model also play important roles. For example, the damping parameter associated with the movement of the gas bubble relative to the fluid within the tube, ζgas is an important parameter governing the response of the system to aeration.
For ζgas approaching zero, the apparent density approaches 1-3Γ (i.e., the meter under reports the density of the aerated mixture by 2Γ. However, as the ζgas increases, the apparent density approaches the actual fluid density of 1-Γ.
Speed of sound
Although simplified models may provide some insight into the influence of various parameters, quantitative models remain elusive due to the inherent complexity of multiphase, unsteady fluid dynamics. Furthermore, the difficulty associated with correcting for the effects of aeration in the interpreted density of the liquid compounds not only by the transformation of the Coriolis meter from a well understood device operating in homogeneous, quasi-steady parameter space into a device operating in a complex, non-homogeneous, unsteady operation space, but also by the inability of current Coriolis meters to precisely determine the amount of aeration present in the process mixture.
A speed-of-sound measurement of the process fluid can integrate with the natural frequency measurement of a vibrating tube density meter to form a system with an enhanced ability to operate accurately in aerated fluids. Introducing a real time, speed-of-sound measurement addresses the effects of aeration on multiple levels with the intent to enable vibrating-tube-based density measurement to continue to report liquid density in the presence of entrained air with accuracy approaching that for a non-aerated liquid. Firstly, by measuring the process sound speed with process pressure, the aeration level of the process fluid can be determined with high accuracy on a real time basis. Secondly, the real time measurements of sound speed, and the derived measurement of gas volume fraction, then utilize with empirically derived correction factors to improve the interpretation of the measured natural frequency of the vibrating tubes in terms of the density of the aerated fluid.
Thirdly, the combined knowledge of aerated mixture density and aerated mixture sound speed, enable the determination of the nonaerated liquid component density, providing improved compositional information. Note liquids phase includes pure liquids, mixtures of liquids, as well as liquid/solids mixtures.
We built a facility to evaluate the performance of Coriolis meters on aerated water. The facility uses a mag meter operating on single phase water as a reference flow rate and the sonar-based meter to monitor the gas volume fraction of the aerated mixtures.
One assumption was the density of the liquid component of the aerated liquid, i.e. the water, was constant. Several Coriolis meters of various designs and manufactures were tested. One example showed apparent density measured by a Coriolis meter with 1-inch diameter tubes with a structural resonant frequency of 100Hz. Data recorded over flow rates ranging from 100-200 gpm and Coriolis inlet pressures of 16 to 26 psi. During the experiment, the apparent density of the Coriolis meter is highly correlated to the gas volume fraction.
Another example showed the apparent density measured by the Coriolis meter with 1-inch diameter tubes with a structural resonant frequency of ~300Hz. Data recorded over a similar range of flow rate and inlet pressures as the previous meter. As with the other meter tested, the apparent density of the Coriolis meter is highly correlated to the gas volume fraction as measured by the sonar-based meter. DT
Behind the byline
Daniel L. Gysling, Ph.D. is the chief technology officer of CiDRA Corp. Gysling holds several patents in fluid dynamic devices. He holds a Ph.D. in aeronautics and astronautics from MIT, a M.S. degree in aeronautics and astronautics from MIT, and a B.S. in aerospace engineering from Pennsylvania State University. Visit www.cidra.com for more information.
A Look at lump model
A lumped parameter model for the effects of inhomogeneity in the oscillation of an aerated-liquid-filled tube. In this model, a gas bubble of volume fraction Γ connects across a fulcrum to a compensating mass of fluid with volume 2Γ. The fulcrum rigidly connects to the outer tube. A user can model the effects of viscosity using a damper connected to restrict the motion of the gas bubble with respect to the rest of the liquid and the tube itself.
The remaining volume of liquid in the tube cross section (1-3Γ) fills with an inviscid fluid. In the inviscid limit, the compensating mass of fluid (2Γ) does not participate in the oscillations, and the velocity of the mass-less gas bubble becomes three times the velocity of the tube. The effect of this relative motion is to reduce the effective inertia of the fluid inside the tube to (1-3Γ) times that presented by a homogeneous fluid-filled the tube. In the limit of high viscosity, the increased damping constant minimizes the relative motion between the gas bubble and the liquid, and the effective inertia of the aerated fluid approaches 1-Γ. The effective inertia predicted by this model of an aerated, but incompressible, fluid oscillating within a tube agrees with those presented by in the limits of high and low viscosities.
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