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An aeroelastic model of Coriolis mass and density meters operating on aerated mixtures
Flow Measurement and Instrumentation Flow Measurement and Instrumentation 18 (2007) 69–77 www.elsevier.com/locate/flowmeasinst
An aeroelastic model of Coriolis mass and density meters operating on aerated mixtures Daniel L. Gysling ∗ CiDRA Corporation, 50 Barnes Park North, Wallingford, CT 06492, USA Received 5 June 2006; received in revised form 24 October 2006; accepted 19 December 2006
Abstract 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. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Coriolis; Two-phase flow; Entrained gases; Aeroelastic; Reduced order model
1. Introduction Coriolis mass flow and density meters are considered the flow metering solution of choice for many precision flow applications. 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 $400 million annual sales on approximately 100,000 units. The success of Coriolis meters has been attributed to many factors, including accuracy, reliability, and ability to measure multiple process parameters, including mass flow and process fluid density. However, despite this long list of attributes, Coriolis meters have significant limitations regarding aerated liquids. Although the mass ∗ Tel.: +1 203 626 3343; fax: +1 203 294 4211.
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0955-5986/$ - see front matter doi:10.1016/j.flowmeasinst.2006.12.006
flow rate and process fluid density measurements determined by Coriolis meters are derived from independent physical principles, the accuracy of both are significantly degraded with the introduction of small amounts of entrained gases. This paper develops a lumped parameter aeroelastic model for Coriolis meters operating on aerated fluids. The model is used to examine the influence of several key, non-dimensional parameters that influence the effect on aeration on the mass flow and density measurement reported by commercially available Coriolis flow meters. 2. Coriolis mass and density measurement 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
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D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77 Table 1 Definition of non-dimensional parameters governing the equation of motion for the lumped parameter quasi-steady Coriolis meter Symbol
Description
Definition
α Γ Und τ y
Mass ratio Torsional spring parameter Radial flow velocity Non-dimensional time Non-dimensional temporal derivative of x
m fluid /m struct K torsion /K struct L U/ωstruct R t ωstruct dx/dτ
Fig. 1. Schematic of a U-tube Coriolis meter.
inertia associated with the tubes. The out-of-plane vibration of the fluid flowing within the tubes generates a Coriolis force that acts to damp the vibration of the tube with the radially outward flow, and feeds the vibration in the radially inward flow tube [3]. The equations of motion for the quasi-steady model depicted in Fig. 2 are given below. (Mstruct + Mfluid )
U dx1 d2 x1 + 2Mfluid + K struct x1 2 R dt dt
K torsion (x1 − x2 ) = 0 L d2 x2 U dx2 (Mstruct + Mfluid ) 2 − 2Mfluid + K struct x2 R dt dt K torsion (x2 − x1 ) = 0 (1) + L where M is the lumped parameter mass of the fluid or structure, K struct represents the lumped parameter transverse spring constant for each tube, and K torsion represents the torsion spring constant of the tube, L is the axial distance between the flow tubes, t is time, x is the transverse displacement of the tube, and subscripts 1 and 2 denote the outward-flowing and the inward-flowing tubes respectively. The first term in Eq. (1) represents the translation mass of the fluid and the structure; the second represents the Coriolis force acting on the fluid with radially velocity U at a radial position R. The last two terms represent the spring forces from the structural and torsion springs depicted in the schematic. The equations of motion of the above lumped parameter model, assuming solutions in the form of esτ where s is the complex frequency, can be expressed in non-dimensional form as +
Fig. 2. Typical section model of a U-tube Coriolis meter.
interaction of coupled fluid dynamic and structural dynamic systems, for example the static and dynamic responses of an aircraft under aerodynamic forces [1]. Coriolis flow meters rely on characterizing the aeroelastic response of fluid-filled, vibrating flow tubes to determine both the mass flow rate and process fluid density measurements. 3. Quasi-steady model Most Coriolis meters rely on quasi-steady models of the interaction between the structural and fluid dynamics within the meter to determine both mass flow and density. In a quasisteady model, the fluid within the flow tubes is assumed to be incompressible and homogeneous. The fluid essentially adds inertial terms associated with translation, and centrifugal and Coriolis acceleration to the vibrational dynamics of the flow tubes. Fig. 1 shows a schematic of a generic U-tube Coriolis meter. The Coriolis meter consists of two U-shaped flow tubes vibrating primarily in an out-of-plane bending mode. The dynamics of the typical section, shown as section AA, in Fig. 1 are considered to develop a lumped parameter, aeroelastic model of the Coriolis meter. Fig. 2 provides a schematic of a lumped parameter, quasi-steady model for the typical section dynamics. This model used herein is an extension of a similar model developed to address the effect of aeration on vibrating tube density meters in [2]. The model assumes symmetry about the centerline. The bending stiffness and inertia of each tube are modeled independently, with the tubes coupled structurally using a torsional spring to represent the torsional stiffness of the Utube. The translational inertia of the fluid adds linearly to the
2α − U −s 1 + α nd 1 0 0
−
1+Γ 1+α
0
−s
0
Γ 1+α
2α Und − s 1+α
0
1
Γ 1+α y1 0 x1 = 0. (2) 1 + Γ y2 − 1 + α x2 −s
The parameters governing the dynamic response of the model are defined in Table 1. Solving the fourth-order eigenvalue problem described provides a means to analytically link the fluid input parameters to the dynamic response of the Coriolis meter. Specifically, the Rossby number, Und , = U/(ω R), is closely related to fluid mass flow and influences the phase lag between the tubes in the first bending mode; and α, the fluid mass parameter, directly influences the natural frequency of the first bending mode.
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Fig. 3. Eigenvalues from the quasi-steady model of the Coriolis meter. Table 2 Definition of non-dimensional parameters governing the equation of motion for the lumped parameter quasi-steady Coriolis meter Symbol
Description
Value
α Γ Und
Mass ratio Torsional spring parameter Radial flow velocity
1.0 1.0 0–0.015
Fig. 4. Phase differences in outward- and inward-flowing tubes in the primary bending mode as a function of mass flow.
Table 3 Dimensional parameters defining the baseline vibrating tube density meter Parameter
Description
Value
f struct D R t ρfluid ρstruct m˙ U
Structural frequency of tubes Tube diameter Radial position of typical section Wall thickness Liquid density Tube density Mass flow Fluid velocity
100 Hz 2.0 in. 20 in. 0.060 in. 1000 kg/m3 8000 kg/m3 0–10 kg/s 0–5 m/s
Fig. 3 shows the eigenvalues for the quasi-steady model for a Coriolis meter with the parameters given in Table 2. The dimensional parameters for the representative Coriolis meter are given in Table 3. As shown, the quasi-steady model has two, lightly damped modes of oscillation. The locations of the eigenvalues do not change significantly with mass flow. The lower frequency mode is primarily associated with bending, and the higher frequency mode is that in which the tubes are predominantly in the out-of-phase torsional mode. Most commercial Coriolis meters leverage the information contained in the lower frequency bending mode to determine the mass flow rate and density of the fluid, namely the phase lag between the out-bound and in-bound tubes and the frequency of oscillation, respectively. Fig. 4 shows the phase lag between the two tubes as a function of mass flow. As shown, the phase lag scales directly with the mass flow through the Coriolis meter. The change in phase with mass flow rate is, however, relatively small, changing approximately 1 degree for a 10 kg/s change in mass flow with this Coriolis meter.
Fig. 5. Natural frequency of the primary bending mode as a function of the fluid density.
The normalized natural frequency of the primarily bending mode is shown as a function of the liquid density in Fig. 5. For this Coriolis meter with a structural frequency of 100 Hz operating on water, a 10% change is density corresponds to an approximately 2 Hz change in resonant frequency. 4. Aeroelastic model The model developed above illustrates the basic operating principles of Coriolis meters operating on the basis of a quasisteady model of the interaction between the fluid and the structure. The accuracy of the fluid measurements made by a Coriolis meter operating under this assumption is directly impacted by the validity of this assumption. In the following section, an aeroelastic model for a Coriolis meter is developed in which the quasi-steady assumption is relaxed, and the effects of fluid compressibility and inhomogeneity, typically associated with aerated liquids, are introduced.
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Fig. 6. Lump parameter model for the oscillation of a tube filled with a compressible fluid.
5. Fluid compressibility It is well known that most aerated liquids are significantly more compressible than non-aerated liquids. Compressibility of a fluid is directly related to the speed of sound and density of the fluid. Mixture density and sound speed can be related to component densities and sound speed through the following mixing rules which are applicable to single-phase and welldispersed mixtures and form the basis for speed of sound based entrained air measurement [4]. N X φi 1 = (3) 2 2 ρmix amix∞ i=1 ρi ai PN where ρmix = i=1 ρi φi and κmix is the mixture compressibility, and φi is the component volumetric phase fraction. Consistent with the above relations, introducing gases into liquids 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 causes the dynamics 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 be incorporated into a lumped parameter model of a vibrating tube as shown schematically in Fig. 6. The stiffness of the spring represents the compressibility of the fluid. As the compressibility approaches zero, the spring stiffness approaches infinity and the model becomes equivalent to one in which the mass of the fluid is directly lumped into the mass of the structure as in the model presented in Fig. 2. The natural frequency of the first transverse acoustic mode in a circular duct [5] can be used to estimate an appropriate spring constant for the model: s 1.84 1 K fluid amix = . (4) f = πD 2π m fluid
Fig. 7. Reduced frequency parameter as a function of fluid sound speed for three representative Coriolis meters.
κmix =
Note that this frequency corresponds to a wavelength of an acoustic oscillation of approximately two diameters, i.e., this transverse mode is closely related to a “half-wavelength” acoustic resonance of the tube. In characterizing aeroelastic systems, it is often convenient to define a reduced frequency parameter [1] to gauge the
Fig. 8. Reduced frequency parameter as a function of gas volume fraction for three representative Coriolis meters.
significance of the interaction between coupled dynamic systems. For a vibrating tube filled with fluid, a useful reduced frequency can be defined as a ratio of the natural frequency of the structural system to that of the fluid dynamic system: f red =
f struct D . amix
(5)
Here f struct 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. Strictly speaking, quasi-steady models are valid for systems in which the reduced frequency is small, i.e. negligible compared to unity. In these cases, models which neglect the compressibility of the fluid are likely to be sufficient. However, the effects of unsteadiness increase with increasing reduced frequency. Fig. 7 shows the reduced frequency as a function of fluid sound speed for three representative Coriolis meters. For a given Coriolis meter, mixture sound speed can have a dominant influence on changes in reduced frequency. Fig. 8 shows the reduced frequency plotted as a function of gas
D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77
Fig. 9. Lump parameter model for the inertia of an incompressible aerated fluid contained in a tube undergoing harmonic motion.
volume fraction of air in water at standard temperature and pressure for two-inch diameter tubes with structural natural frequencies ranging from 100 to 500 Hz. As shown, the reduced frequency is quite small for the non-aerated water. However, it builds rapidly with increasing gas volume fraction, indicating that the significance of compressibility increases with gas volume fraction. Small levels of gas volume fraction can result in significant reduced frequencies, with the impact of gas increasing in severity with tube natural frequency, and similarly, increasing with tube diameter. 6. Fluid inhomogeneity In additional to dramatically increasing the compressibility of the fluid, aeration also introduces inhomogeneity to the mixture. For flow regimes in which the gas is entrained in a liquid continuous flow field, the first-order effects of the aeration can be modeled using bubble theory. By considering the motion of an incompressible sphere of density of ρ0 contained in an inviscid, incompressible fluid with a density of ρ and set into motion by the fluid, previous authors [6] have shown that the velocity of the sphere is given by Vsphere =
3ρ Vfluid . ρ + 2ρ0
(6)
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 in an inviscid mixture. Application of this theory to bubbly flows in Coriolis meters is presented in greater detail in [7,8]. 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 the oscillation, resulting in a reduced apparent system inertia. Fig. 9 illustrates 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 gas volume fraction (often
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Fig. 10. Bubble to fluid velocity ratio in primary bending modes as a function of the gas damping ratio ζgas .
abbreviated as GVF) φ is connected across a fulcrum to a compensating mass of fluid with volume 2φ. The fulcrum is attached to the center of mass of the fluid of the remaining fluid within the tube. The effect of viscosity is modeled using a damper connected to restrict the motion of the gas bubble with respect to the rest of the liquid. The remaining volume of liquid in the tube cross section (1 − 3φ) is filled with an inviscid fluid. In the inviscid limit, the compensating mass of fluid (2φ) does not participate in the oscillation of the remaining fluid, and the velocity of the massless gas bubble approaches three times the velocity of the remaining fluid. The velocity ratio of the bubble and the remaining fluid is shown in Fig. 10 as a function of the gas damping ratio defined in Table 4. For effectively inviscid fluids (ζgas < ∼0.1), the effect of this relative motion is to reduce the effective inertia of the fluid inside the tube to (1–3φ) times that presented by a homogeneous fluid-filled tube. For high viscosity (ζgas > ∼1.0), the increased damping reduces the relative motion of the gas bubble and the liquid, and the effective inertia of the aerated fluid approaches 1 − φ. The effective inertia of an aerated, but incompressible, fluid oscillating within a tube predicted by this model is consistent with a potential theory model [7,8] in the limits of high and low viscosities. A first-principles estimate of an appropriate gas damping ratio can be obtained by assuming that the viscous drag on a sphere of radius a and velocity U is given by the Stokes law: Drag = 6πaµU.
(7)
Under this model, the gas damping constant for the flow of an aerated fluid of density ρ and viscosity µ for bubbles of radius a through a tube oscillating at ωs is given by ζgas =
9µ . 4ρa 2 ωstruct
(8)
7. Combined lumped parameter model Models were presented for the effects of aeration on vibrating tube density meters in which the effects of
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Table 4 Definition of non-dimensional parameters governing the equation of motion for the aeroelastic, lumped parameter model of a Coriolis meter with a compressible, aerated fluid Symbol
Description
φ Q ζfluid
Gas volume fraction Natural frequency ratio Critical damping ratio of fluid system Critical damping ratio of structural system Critical damping ratio of structural system
ζstruct ζgas bubble
Definition ωfluid /ωstruct bfluid /(2m fluid ωfluid )
Table 5 Parameters defining the baseline vibrating tube density meter Parameter
Description
Value
ζstruct
Critical damping ratio — structure
0.01
ζfluid
Critical damping ratio — fluid
0.01
ζgas
Critical damping ratio — gas
1.0
Q
Frequency ratio
D
Tube diameter
As determined by sound speed of air/water at STP and structural parameters 1.0 in.
bstruct /(2m struct ωstruct ) bgas /(2(2φ)m fluid ωstruct )
Fig. 11. Schematic of the lumped parameter model for a compressible, inhomogeneous, aerated fluid contained in a tube undergoing harmonic motion.
compressibility and inhomogeneity were addressed independently. Fig. 11 shows a schematic of a lumped parameter model that incorporates the effects of compressibility and inhomogeneity using the mechanism-specific models developed above. The purpose of this model is to illustrate trends and parametric dependencies associated with aeration; it is not intended as a quantitative predictive tool. The equations of motion of the above lumped parameter model, assuming solutions in the form of esτ where s is the complex frequency, can be expressed in non-dimensional form as y1 x1 y2 x2 y 3 A11 A12 x3 = {0} (9) A21 A22 q1 z1 q 2 z 2 q 3 z3 where A11 , A22 , A12 and A21 are given in Box I. The additional non-dimensional parameters that govern the dynamic response of the aeroelastic model compared to the quasi-steady model are defined in Table 4.
Fig. 12. Eigenvalues for the lumped parameter model of a representative lower tube frequency Coriolis meter.
Solving the twelfth-order eigenvalue problem described above provides a means to assess the influence of the various parameters on the ability of the Coriolis meter to measure the mass flow and density of a fluid under a variety of operating conditions. The eigenvalues for the standard Coriolis meter described in Tables 1–5 are shown in Fig. 12 as a function of gas volume fraction. Arrows indicate the movement of the eigenvalues as a function of gas volume fraction as the gas volume fraction is increased from 0.1% to 10%. The model shows four lightly damped modes, two corresponding to the modes present in the quasi-steady model, and two primarily associated with the acoustic modes within the tubes. At low gas volume fraction (0.1% in this case), the oscillatory frequency of the acoustic modes is well above that of the structural modes (∼60 times the structural frequency in this case). However, as the gas volume fraction is increased, the primarily acoustic modes move toward the primarily structural modes and compressibility of the fluid begins to influence the dynamics of the structural mode. Also, as the gas volume fraction increases, the natural frequency of the primarily bending mode is shown to increase as well, consistently with the quasi-steady effect of decreased mixture density.
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D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77
A11
s + 2ζ α Q + 2ζ struct fluid −1 2ζfluid Q − 1 − 3ϕ ≡ 0 0
1 + α Q2 + Γ
−2ζfluid α Q
−α Q 2
0
s
0
0
0
Q2
2ζgas (2ϕ) 1 − 3ϕ 0 2ζgas s+ + 2Und 2ϕ −1
−
Q2 1 − 3ϕ 0
A22
A12 = A21
−Γ 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
−
0
0
0
0
1 + α Q2 + Γ
−2ζfluid α Q
−α Q 2
0
s
0
0
0
Q2
2ζgas (2ϕ) 1 − 3ϕ 0 2ζgas s+ − 2Und 2ϕ −1
−
Q2 1 − 3ϕ 0
0 0 0 0 ≡ 0 0 0
2ζgas (2ϕ) 2ζfluid Q + + 2Und 1 − 3ϕ 1 − 3ϕ −1
1 − 3ϕ s −2ζgas 2ϕ 0
0 s + 2ζ α Q + 2ζ struct fluid −1 2ζfluid Q − 1 − 3ϕ ≡ 0 0
s+
0 0 0 0 0 0
s+
2ζgas (2ϕ) 2ζfluid Q + − 2Und 1 − 3ϕ 1 − 3ϕ −1
0
0
0
0
1 − 3ϕ s −2ζgas 2ϕ 0
−
0 0 0 0 0 s 0 0 0 0 0 s
0 0 0 0 0 0 Box I.
The movement of the eigenvalues also indicates that the damping of the primary bending mode increases with the gas volume fraction as well, consistently with commonly observed increase in drive gain required to maintain a constant amplitude oscillation in commercial Coriolis meters subjected to entrained gases. Fig. 13 shows the eigenvalues for a Coriolis meter operating at significantly higher tube frequencies; all the other parameters are identical. As shown, the locations of the eigenvalues of the various modes are in much closer proximity, indicative of a significantly higher level of aeroelastic interaction. Note that the higher structural frequency is reflected in the nondimensional frequency of the “acoustic modes”, approaching 10 in the limit of no gas and decreasing to approximately 1 at 10% φ. The lumped parameter model can be used to estimate the errors that would result from using a quasi-steady calibration (i.e. non-aerated fluid) on a meter operating in aerated fluids. Fig. 14 shows the mass flow observed based on a quasi-steady interpretation of the phase shift in the primary bending mode as a function of the gas volume fraction. As shown, the model predicts that the Coriolis meter operating with the lowest frequency tubes reports a mass flow measurement that is in good agreement with the actual mass flow over the range of gas volume fraction evaluated. However, for the higher frequency tubes, aeroelastic interactions associated with the increased compressibility and the inhomogeneity of the fluid result in significant errors
Fig. 13. Eigenvalues for the lumped parameter model of a representative higher tube frequency Coriolis meter.
between the actual and the interpreted mass flow rates. The “1GVF “and “1–3GVF” lines are shown for reference purposes. For the density measurement, the natural frequency of the primary tube bending mode predicted by the eigenvalue analysis is input into the frequency/density from the quasisteady, homogeneous model to determine the apparent density
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D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77
Fig. 16. Schematic of a speed of sound augmented Coriolis meter.
Fig. 15 shows similar results for the density measurement. The Coriolis meter with the lowest frequency tubes accurately reports the mixture density (1–φ). The higher frequency tube designs are shown to report significant errors with increasing gas volume fraction.
these models that gas volume fraction plays a dominant role, with several other parameters including the gas damping ζgas and reduced frequency also influencing performance. 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. Attempts to mitigate the effects of aeration on Coriolis measurements have proven to be challenging. Firstly, introducing aeration transforms the Coriolis meter from a well-understood device operating in quasi-steady parameter space into a device operating in a complex, non-homogeneous, multiphase parameter space. The impact of this complexity is further magnified by the inability of current Coriolis meters to accurately determine the relevant aeroelastic operating parameters, namely the inability to precisely determine the gas volume fraction or the reduced frequency of operation. This paper presents an approach in which a speed of sound measurement of the process fluid is integrated with a Coriolis meter to form a system with an enhanced ability to operate accurately on aerated fluids. A schematic of a speed of sound augmented Coriolis system is shown in Fig. 16. Introducing a real time, speed of sound measurement addresses the effects of aeration on multiple levels with the intent of enabling Coriolis meters to maintain mass flow and liquid density measurements in the presence of entrained air with accuracy approaching that for non-aerated liquids. 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 the gas volume fraction, are then utilized with analytically or empirically derived correction factors to improve the interpretation of the measured phase difference and natural frequency of the vibrating tubes in terms of the density of the aerated fluid.
8. Speed of sound augmented Coriolis meter
9. Discussion
The lumped parameter aeroelastic model presented above provides a first-principles basis for analyzing the effects of aeration on Coriolis based mass and density measurements. As shown, the model can be used to illustrate qualitatively the role of several non-dimensional parameters which govern the performance of the Coriolis meters. It can be concluded from
The intent of this work is to provide a unifying framework for assessing the impact of entrained gases on Coriolis meters under several simplifying assumptions. The most important of these assumptions is perhaps the assumption that the bubbles are well mixed within the liquid continuous phase. As a result, the model presented herein does not specifically address
Fig. 14. Interpreted non-dimensional mass flow as a function of gas volume fraction for three representative Coriolis meters.
Fig. 15. Apparent density as a function of gas volume fraction for three representative Coriolis meters.
of the fluid as follows. ρapparent
ρliq = α
f s2 2 f observed
! −1 .
(10)
D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77
slugging and/or stratified flows within Coriolis phenomena that can often occur in start-up and batching processes, but rather it is intended to address a class of applications in which relatively small amounts of gas (<∼10% by volume) are entrained in a liquid continuous mixture. These conditions are often encountered in the processing of viscous liquids, for which the entrained gases are both difficult to remove and challenging to accurate Coriolis based measurement. Examples of these applications include pulp slurries, heavy oil separation, cement, viscous consumer products, etc. The results of the analytical model presented herein suggest that Coriolis meters operating on sufficiently viscous fluids, with well-mixed entrained gases, at sufficiently low reduced frequencies, will continue to behave as quasi-steady devices and continue to accurately report mixture mass flow and mixture density. Furthermore, with knowledge of the gas volume fraction, the output of the Coriolis meter can be accurately interpreted in terms of liquid mass flow, liquid density, and liquid volumetric flow. These results are supported by experimental data presented in [4,9,10], addressing relatively low reduced frequency Coriolis meters operating on liquids with entrained gases with conditions consistent with those described above. The results from the analytical model further predict that the increases in reduced frequency of operation in the presence of entrained gases will result in errors in mixture mass flow and mixture density, even for well-mixed, sufficiently viscous, liquid continuous flows due the increasing importance of unsteady effects in the interaction the fluid and structural dynamics within the Coriolis meter. The author is unaware of any public domain data that experimentally categorizes relatively high reduced frequency Coriolis meters operating on liquids with entrained gases with conditions consistent with those described above. It should be noted that numerous experimental studies have been published in the literature on the response of Coriolis meters to two-phase flow [11] and [12]. The data appears to cover a wide range of flow conditions, flow regimes, and tube geometries. Errors in mass flow and density due to twophase flow are shown to vary with numerous “dimensional parameters” such as viscosity [12,13], tube size [12,13], and bubble size [12], as well as flow rate and flow regime [11, 13]. Although these studies provide insight into the potential adverse effects of general two-phase flows in Coriolis meters, the scope of these experimental studies, in general, exceeds the class of problems considered herein, namely, the application of Coriolis meters to “well-mixed” liquid continuous mixtures. 10. Summary A lumped parameter aeroelastic model for Coriolis meters operating in aerated fluid was developed. The model incorporates the effects of compressibility and inhomogeneity association with entrained gases. The model identifies several aeroelastic parameters that influence the relationship between
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quantities directly measured by the Coriolis meter and the interpreted fluid mass flow and density. The effect of aeration was evaluated using the model for a set of representative Coriolis meters spanning a range of tube natural frequencies. Results of the model show that the low frequency Coriolis meter analyzed was significantly less sensitive to errors associated with the aeroelastic effects of aeration than the higher frequency designs. A concept was presented for enhancing the performance of Coriolis meters on aerated fluids. The concept utilizes a speed of sound measurement of the process fluid integrated with a Coriolis meter. Measuring fluid sound speed enables the determination of real time aeroelastic operating parameters, which enables the meter to accurately interpret the directly measured phase difference and natural frequency of the tubes in terms of fluid mass flow and density in the presence of aeration. Acknowledgements The results presented in this work have benefited from several years of development efforts in flow measurement based on sonar technology. The authors gratefully acknowledge the efforts of the many colleagues and co-workers that have contributed to the results presented herein. References [1] Bisplinghoff RL, Ashley H, Halfman R. Aeroelasticity. Addison-Wesley Publishing; 1955. [2] Gysling DL, Banach T. Enhanced density measurement using speed-ofsound augmented Coriolis meters. Presented at ISA. Houston, TX; 2004. [3] Sultan G, Hemp J. Modeling of the Coriolis mass flowmeter. Journal of Sound and Vibration 1989;132(3):473–89. [4] Gysling DL, Loose DH. Sonar-based volumetric flow and entrained air measurement for pulp and paper applications. Presented at the TAPPI Spring Technical Conference. Paper 58-1. Chicago, IL; 2003. [5] Munjal ML. Acoustics of ducts and mufflers. John Wiley and Sons; 1987. [6] Landau LD, Lifshitz EM. Fluid mechanics, 2nd ed. Course of theoretical physics, section 11, vol. 6. Butterworth–Heinemann; 1987. [7] Hemp J, Yeung H, Kassi L. Coriolis meter in two phase conditions. Department of Process and Systems Engineering. Cranfield University; 2003. [8] Hemp J, Sultan G. On the theory and performance of Coriolis mass flowmeter. In: Proceedings of the international conference on mass flow measurement. IBC technical Services. London; 1989. [9] Volz R, Gysling D, Loose D. Accurate volumetric flow rate and densitybased watercut measurement in bubbly liquid hydrocarbon flow. Society of Petroleum Engineers SPE Paper 102962-PP. Presented at the 2006 Annual Technical Conference. San Antonio, TX; 2006. [10] Ward E. New class of meter solves old problem impacting well test accuracy. Society of Petroleum Engineers SPE Paper 100893. Presented at the 2006 Western Regional Conference. Anchorage, AK; 2006. [11] Lui RP, Fuent MJ, Henry MP, Duta MD. A neural network to correct mass flow errors caused by two phase flow in a digital Coriolis mass flow meter. Flow Measurement and Instrumentation 2001;12:53–63. [12] Seeger M. Coriolis flow measurement in two phase flow. IEE Computing and Control Engineering 2005. [13] Grumski JT, Bajura RA. Performance of a Coriolis-type mass flow meter in the measurement of two-phase (air–liquid) mixtures. ASME Fluids Engineering Division 1984;17.
An aeroelastic model of Coriolis mass and density meters operating on aerated mixtures Daniel L. Gysling ∗ CiDRA Corporation, 50 Barnes Park North, Wallingford, CT 06492, USA Received 5 June 2006; received in revised form 24 October 2006; accepted 19 December 2006
Abstract 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. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Coriolis; Two-phase flow; Entrained gases; Aeroelastic; Reduced order model
1. Introduction Coriolis mass flow and density meters are considered the flow metering solution of choice for many precision flow applications. 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 $400 million annual sales on approximately 100,000 units. The success of Coriolis meters has been attributed to many factors, including accuracy, reliability, and ability to measure multiple process parameters, including mass flow and process fluid density. However, despite this long list of attributes, Coriolis meters have significant limitations regarding aerated liquids. Although the mass ∗ Tel.: +1 203 626 3343; fax: +1 203 294 4211.
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0955-5986/$ - see front matter doi:10.1016/j.flowmeasinst.2006.12.006
flow rate and process fluid density measurements determined by Coriolis meters are derived from independent physical principles, the accuracy of both are significantly degraded with the introduction of small amounts of entrained gases. This paper develops a lumped parameter aeroelastic model for Coriolis meters operating on aerated fluids. The model is used to examine the influence of several key, non-dimensional parameters that influence the effect on aeration on the mass flow and density measurement reported by commercially available Coriolis flow meters. 2. Coriolis mass and density measurement 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
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D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77 Table 1 Definition of non-dimensional parameters governing the equation of motion for the lumped parameter quasi-steady Coriolis meter Symbol
Description
Definition
α Γ Und τ y
Mass ratio Torsional spring parameter Radial flow velocity Non-dimensional time Non-dimensional temporal derivative of x
m fluid /m struct K torsion /K struct L U/ωstruct R t ωstruct dx/dτ
Fig. 1. Schematic of a U-tube Coriolis meter.
inertia associated with the tubes. The out-of-plane vibration of the fluid flowing within the tubes generates a Coriolis force that acts to damp the vibration of the tube with the radially outward flow, and feeds the vibration in the radially inward flow tube [3]. The equations of motion for the quasi-steady model depicted in Fig. 2 are given below. (Mstruct + Mfluid )
U dx1 d2 x1 + 2Mfluid + K struct x1 2 R dt dt
K torsion (x1 − x2 ) = 0 L d2 x2 U dx2 (Mstruct + Mfluid ) 2 − 2Mfluid + K struct x2 R dt dt K torsion (x2 − x1 ) = 0 (1) + L where M is the lumped parameter mass of the fluid or structure, K struct represents the lumped parameter transverse spring constant for each tube, and K torsion represents the torsion spring constant of the tube, L is the axial distance between the flow tubes, t is time, x is the transverse displacement of the tube, and subscripts 1 and 2 denote the outward-flowing and the inward-flowing tubes respectively. The first term in Eq. (1) represents the translation mass of the fluid and the structure; the second represents the Coriolis force acting on the fluid with radially velocity U at a radial position R. The last two terms represent the spring forces from the structural and torsion springs depicted in the schematic. The equations of motion of the above lumped parameter model, assuming solutions in the form of esτ where s is the complex frequency, can be expressed in non-dimensional form as +
Fig. 2. Typical section model of a U-tube Coriolis meter.
interaction of coupled fluid dynamic and structural dynamic systems, for example the static and dynamic responses of an aircraft under aerodynamic forces [1]. Coriolis flow meters rely on characterizing the aeroelastic response of fluid-filled, vibrating flow tubes to determine both the mass flow rate and process fluid density measurements. 3. Quasi-steady model Most Coriolis meters rely on quasi-steady models of the interaction between the structural and fluid dynamics within the meter to determine both mass flow and density. In a quasisteady model, the fluid within the flow tubes is assumed to be incompressible and homogeneous. The fluid essentially adds inertial terms associated with translation, and centrifugal and Coriolis acceleration to the vibrational dynamics of the flow tubes. Fig. 1 shows a schematic of a generic U-tube Coriolis meter. The Coriolis meter consists of two U-shaped flow tubes vibrating primarily in an out-of-plane bending mode. The dynamics of the typical section, shown as section AA, in Fig. 1 are considered to develop a lumped parameter, aeroelastic model of the Coriolis meter. Fig. 2 provides a schematic of a lumped parameter, quasi-steady model for the typical section dynamics. This model used herein is an extension of a similar model developed to address the effect of aeration on vibrating tube density meters in [2]. The model assumes symmetry about the centerline. The bending stiffness and inertia of each tube are modeled independently, with the tubes coupled structurally using a torsional spring to represent the torsional stiffness of the Utube. The translational inertia of the fluid adds linearly to the
2α − U −s 1 + α nd 1 0 0
−
1+Γ 1+α
0
−s
0
Γ 1+α
2α Und − s 1+α
0
1
Γ 1+α y1 0 x1 = 0. (2) 1 + Γ y2 − 1 + α x2 −s
The parameters governing the dynamic response of the model are defined in Table 1. Solving the fourth-order eigenvalue problem described provides a means to analytically link the fluid input parameters to the dynamic response of the Coriolis meter. Specifically, the Rossby number, Und , = U/(ω R), is closely related to fluid mass flow and influences the phase lag between the tubes in the first bending mode; and α, the fluid mass parameter, directly influences the natural frequency of the first bending mode.
D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77
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Fig. 3. Eigenvalues from the quasi-steady model of the Coriolis meter. Table 2 Definition of non-dimensional parameters governing the equation of motion for the lumped parameter quasi-steady Coriolis meter Symbol
Description
Value
α Γ Und
Mass ratio Torsional spring parameter Radial flow velocity
1.0 1.0 0–0.015
Fig. 4. Phase differences in outward- and inward-flowing tubes in the primary bending mode as a function of mass flow.
Table 3 Dimensional parameters defining the baseline vibrating tube density meter Parameter
Description
Value
f struct D R t ρfluid ρstruct m˙ U
Structural frequency of tubes Tube diameter Radial position of typical section Wall thickness Liquid density Tube density Mass flow Fluid velocity
100 Hz 2.0 in. 20 in. 0.060 in. 1000 kg/m3 8000 kg/m3 0–10 kg/s 0–5 m/s
Fig. 3 shows the eigenvalues for the quasi-steady model for a Coriolis meter with the parameters given in Table 2. The dimensional parameters for the representative Coriolis meter are given in Table 3. As shown, the quasi-steady model has two, lightly damped modes of oscillation. The locations of the eigenvalues do not change significantly with mass flow. The lower frequency mode is primarily associated with bending, and the higher frequency mode is that in which the tubes are predominantly in the out-of-phase torsional mode. Most commercial Coriolis meters leverage the information contained in the lower frequency bending mode to determine the mass flow rate and density of the fluid, namely the phase lag between the out-bound and in-bound tubes and the frequency of oscillation, respectively. Fig. 4 shows the phase lag between the two tubes as a function of mass flow. As shown, the phase lag scales directly with the mass flow through the Coriolis meter. The change in phase with mass flow rate is, however, relatively small, changing approximately 1 degree for a 10 kg/s change in mass flow with this Coriolis meter.
Fig. 5. Natural frequency of the primary bending mode as a function of the fluid density.
The normalized natural frequency of the primarily bending mode is shown as a function of the liquid density in Fig. 5. For this Coriolis meter with a structural frequency of 100 Hz operating on water, a 10% change is density corresponds to an approximately 2 Hz change in resonant frequency. 4. Aeroelastic model The model developed above illustrates the basic operating principles of Coriolis meters operating on the basis of a quasisteady model of the interaction between the fluid and the structure. The accuracy of the fluid measurements made by a Coriolis meter operating under this assumption is directly impacted by the validity of this assumption. In the following section, an aeroelastic model for a Coriolis meter is developed in which the quasi-steady assumption is relaxed, and the effects of fluid compressibility and inhomogeneity, typically associated with aerated liquids, are introduced.
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Fig. 6. Lump parameter model for the oscillation of a tube filled with a compressible fluid.
5. Fluid compressibility It is well known that most aerated liquids are significantly more compressible than non-aerated liquids. Compressibility of a fluid is directly related to the speed of sound and density of the fluid. Mixture density and sound speed can be related to component densities and sound speed through the following mixing rules which are applicable to single-phase and welldispersed mixtures and form the basis for speed of sound based entrained air measurement [4]. N X φi 1 = (3) 2 2 ρmix amix∞ i=1 ρi ai PN where ρmix = i=1 ρi φi and κmix is the mixture compressibility, and φi is the component volumetric phase fraction. Consistent with the above relations, introducing gases into liquids 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 causes the dynamics 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 be incorporated into a lumped parameter model of a vibrating tube as shown schematically in Fig. 6. The stiffness of the spring represents the compressibility of the fluid. As the compressibility approaches zero, the spring stiffness approaches infinity and the model becomes equivalent to one in which the mass of the fluid is directly lumped into the mass of the structure as in the model presented in Fig. 2. The natural frequency of the first transverse acoustic mode in a circular duct [5] can be used to estimate an appropriate spring constant for the model: s 1.84 1 K fluid amix = . (4) f = πD 2π m fluid
Fig. 7. Reduced frequency parameter as a function of fluid sound speed for three representative Coriolis meters.
κmix =
Note that this frequency corresponds to a wavelength of an acoustic oscillation of approximately two diameters, i.e., this transverse mode is closely related to a “half-wavelength” acoustic resonance of the tube. In characterizing aeroelastic systems, it is often convenient to define a reduced frequency parameter [1] to gauge the
Fig. 8. Reduced frequency parameter as a function of gas volume fraction for three representative Coriolis meters.
significance of the interaction between coupled dynamic systems. For a vibrating tube filled with fluid, a useful reduced frequency can be defined as a ratio of the natural frequency of the structural system to that of the fluid dynamic system: f red =
f struct D . amix
(5)
Here f struct 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. Strictly speaking, quasi-steady models are valid for systems in which the reduced frequency is small, i.e. negligible compared to unity. In these cases, models which neglect the compressibility of the fluid are likely to be sufficient. However, the effects of unsteadiness increase with increasing reduced frequency. Fig. 7 shows the reduced frequency as a function of fluid sound speed for three representative Coriolis meters. For a given Coriolis meter, mixture sound speed can have a dominant influence on changes in reduced frequency. Fig. 8 shows the reduced frequency plotted as a function of gas
D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77
Fig. 9. Lump parameter model for the inertia of an incompressible aerated fluid contained in a tube undergoing harmonic motion.
volume fraction of air in water at standard temperature and pressure for two-inch diameter tubes with structural natural frequencies ranging from 100 to 500 Hz. As shown, the reduced frequency is quite small for the non-aerated water. However, it builds rapidly with increasing gas volume fraction, indicating that the significance of compressibility increases with gas volume fraction. Small levels of gas volume fraction can result in significant reduced frequencies, with the impact of gas increasing in severity with tube natural frequency, and similarly, increasing with tube diameter. 6. Fluid inhomogeneity In additional to dramatically increasing the compressibility of the fluid, aeration also introduces inhomogeneity to the mixture. For flow regimes in which the gas is entrained in a liquid continuous flow field, the first-order effects of the aeration can be modeled using bubble theory. By considering the motion of an incompressible sphere of density of ρ0 contained in an inviscid, incompressible fluid with a density of ρ and set into motion by the fluid, previous authors [6] have shown that the velocity of the sphere is given by Vsphere =
3ρ Vfluid . ρ + 2ρ0
(6)
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 in an inviscid mixture. Application of this theory to bubbly flows in Coriolis meters is presented in greater detail in [7,8]. 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 the oscillation, resulting in a reduced apparent system inertia. Fig. 9 illustrates 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 gas volume fraction (often
73
Fig. 10. Bubble to fluid velocity ratio in primary bending modes as a function of the gas damping ratio ζgas .
abbreviated as GVF) φ is connected across a fulcrum to a compensating mass of fluid with volume 2φ. The fulcrum is attached to the center of mass of the fluid of the remaining fluid within the tube. The effect of viscosity is modeled using a damper connected to restrict the motion of the gas bubble with respect to the rest of the liquid. The remaining volume of liquid in the tube cross section (1 − 3φ) is filled with an inviscid fluid. In the inviscid limit, the compensating mass of fluid (2φ) does not participate in the oscillation of the remaining fluid, and the velocity of the massless gas bubble approaches three times the velocity of the remaining fluid. The velocity ratio of the bubble and the remaining fluid is shown in Fig. 10 as a function of the gas damping ratio defined in Table 4. For effectively inviscid fluids (ζgas < ∼0.1), the effect of this relative motion is to reduce the effective inertia of the fluid inside the tube to (1–3φ) times that presented by a homogeneous fluid-filled tube. For high viscosity (ζgas > ∼1.0), the increased damping reduces the relative motion of the gas bubble and the liquid, and the effective inertia of the aerated fluid approaches 1 − φ. The effective inertia of an aerated, but incompressible, fluid oscillating within a tube predicted by this model is consistent with a potential theory model [7,8] in the limits of high and low viscosities. A first-principles estimate of an appropriate gas damping ratio can be obtained by assuming that the viscous drag on a sphere of radius a and velocity U is given by the Stokes law: Drag = 6πaµU.
(7)
Under this model, the gas damping constant for the flow of an aerated fluid of density ρ and viscosity µ for bubbles of radius a through a tube oscillating at ωs is given by ζgas =
9µ . 4ρa 2 ωstruct
(8)
7. Combined lumped parameter model Models were presented for the effects of aeration on vibrating tube density meters in which the effects of
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D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77
Table 4 Definition of non-dimensional parameters governing the equation of motion for the aeroelastic, lumped parameter model of a Coriolis meter with a compressible, aerated fluid Symbol
Description
φ Q ζfluid
Gas volume fraction Natural frequency ratio Critical damping ratio of fluid system Critical damping ratio of structural system Critical damping ratio of structural system
ζstruct ζgas bubble
Definition ωfluid /ωstruct bfluid /(2m fluid ωfluid )
Table 5 Parameters defining the baseline vibrating tube density meter Parameter
Description
Value
ζstruct
Critical damping ratio — structure
0.01
ζfluid
Critical damping ratio — fluid
0.01
ζgas
Critical damping ratio — gas
1.0
Q
Frequency ratio
D
Tube diameter
As determined by sound speed of air/water at STP and structural parameters 1.0 in.
bstruct /(2m struct ωstruct ) bgas /(2(2φ)m fluid ωstruct )
Fig. 11. Schematic of the lumped parameter model for a compressible, inhomogeneous, aerated fluid contained in a tube undergoing harmonic motion.
compressibility and inhomogeneity were addressed independently. Fig. 11 shows a schematic of a lumped parameter model that incorporates the effects of compressibility and inhomogeneity using the mechanism-specific models developed above. The purpose of this model is to illustrate trends and parametric dependencies associated with aeration; it is not intended as a quantitative predictive tool. The equations of motion of the above lumped parameter model, assuming solutions in the form of esτ where s is the complex frequency, can be expressed in non-dimensional form as y1 x1 y2 x2 y 3 A11 A12 x3 = {0} (9) A21 A22 q1 z1 q 2 z 2 q 3 z3 where A11 , A22 , A12 and A21 are given in Box I. The additional non-dimensional parameters that govern the dynamic response of the aeroelastic model compared to the quasi-steady model are defined in Table 4.
Fig. 12. Eigenvalues for the lumped parameter model of a representative lower tube frequency Coriolis meter.
Solving the twelfth-order eigenvalue problem described above provides a means to assess the influence of the various parameters on the ability of the Coriolis meter to measure the mass flow and density of a fluid under a variety of operating conditions. The eigenvalues for the standard Coriolis meter described in Tables 1–5 are shown in Fig. 12 as a function of gas volume fraction. Arrows indicate the movement of the eigenvalues as a function of gas volume fraction as the gas volume fraction is increased from 0.1% to 10%. The model shows four lightly damped modes, two corresponding to the modes present in the quasi-steady model, and two primarily associated with the acoustic modes within the tubes. At low gas volume fraction (0.1% in this case), the oscillatory frequency of the acoustic modes is well above that of the structural modes (∼60 times the structural frequency in this case). However, as the gas volume fraction is increased, the primarily acoustic modes move toward the primarily structural modes and compressibility of the fluid begins to influence the dynamics of the structural mode. Also, as the gas volume fraction increases, the natural frequency of the primarily bending mode is shown to increase as well, consistently with the quasi-steady effect of decreased mixture density.
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D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77
A11
s + 2ζ α Q + 2ζ struct fluid −1 2ζfluid Q − 1 − 3ϕ ≡ 0 0
1 + α Q2 + Γ
−2ζfluid α Q
−α Q 2
0
s
0
0
0
Q2
2ζgas (2ϕ) 1 − 3ϕ 0 2ζgas s+ + 2Und 2ϕ −1
−
Q2 1 − 3ϕ 0
A22
A12 = A21
−Γ 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
−
0
0
0
0
1 + α Q2 + Γ
−2ζfluid α Q
−α Q 2
0
s
0
0
0
Q2
2ζgas (2ϕ) 1 − 3ϕ 0 2ζgas s+ − 2Und 2ϕ −1
−
Q2 1 − 3ϕ 0
0 0 0 0 ≡ 0 0 0
2ζgas (2ϕ) 2ζfluid Q + + 2Und 1 − 3ϕ 1 − 3ϕ −1
1 − 3ϕ s −2ζgas 2ϕ 0
0 s + 2ζ α Q + 2ζ struct fluid −1 2ζfluid Q − 1 − 3ϕ ≡ 0 0
s+
0 0 0 0 0 0
s+
2ζgas (2ϕ) 2ζfluid Q + − 2Und 1 − 3ϕ 1 − 3ϕ −1
0
0
0
0
1 − 3ϕ s −2ζgas 2ϕ 0
−
0 0 0 0 0 s 0 0 0 0 0 s
0 0 0 0 0 0 Box I.
The movement of the eigenvalues also indicates that the damping of the primary bending mode increases with the gas volume fraction as well, consistently with commonly observed increase in drive gain required to maintain a constant amplitude oscillation in commercial Coriolis meters subjected to entrained gases. Fig. 13 shows the eigenvalues for a Coriolis meter operating at significantly higher tube frequencies; all the other parameters are identical. As shown, the locations of the eigenvalues of the various modes are in much closer proximity, indicative of a significantly higher level of aeroelastic interaction. Note that the higher structural frequency is reflected in the nondimensional frequency of the “acoustic modes”, approaching 10 in the limit of no gas and decreasing to approximately 1 at 10% φ. The lumped parameter model can be used to estimate the errors that would result from using a quasi-steady calibration (i.e. non-aerated fluid) on a meter operating in aerated fluids. Fig. 14 shows the mass flow observed based on a quasi-steady interpretation of the phase shift in the primary bending mode as a function of the gas volume fraction. As shown, the model predicts that the Coriolis meter operating with the lowest frequency tubes reports a mass flow measurement that is in good agreement with the actual mass flow over the range of gas volume fraction evaluated. However, for the higher frequency tubes, aeroelastic interactions associated with the increased compressibility and the inhomogeneity of the fluid result in significant errors
Fig. 13. Eigenvalues for the lumped parameter model of a representative higher tube frequency Coriolis meter.
between the actual and the interpreted mass flow rates. The “1GVF “and “1–3GVF” lines are shown for reference purposes. For the density measurement, the natural frequency of the primary tube bending mode predicted by the eigenvalue analysis is input into the frequency/density from the quasisteady, homogeneous model to determine the apparent density
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D.L. Gysling / Flow Measurement and Instrumentation 18 (2007) 69–77
Fig. 16. Schematic of a speed of sound augmented Coriolis meter.
Fig. 15 shows similar results for the density measurement. The Coriolis meter with the lowest frequency tubes accurately reports the mixture density (1–φ). The higher frequency tube designs are shown to report significant errors with increasing gas volume fraction.
these models that gas volume fraction plays a dominant role, with several other parameters including the gas damping ζgas and reduced frequency also influencing performance. 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. Attempts to mitigate the effects of aeration on Coriolis measurements have proven to be challenging. Firstly, introducing aeration transforms the Coriolis meter from a well-understood device operating in quasi-steady parameter space into a device operating in a complex, non-homogeneous, multiphase parameter space. The impact of this complexity is further magnified by the inability of current Coriolis meters to accurately determine the relevant aeroelastic operating parameters, namely the inability to precisely determine the gas volume fraction or the reduced frequency of operation. This paper presents an approach in which a speed of sound measurement of the process fluid is integrated with a Coriolis meter to form a system with an enhanced ability to operate accurately on aerated fluids. A schematic of a speed of sound augmented Coriolis system is shown in Fig. 16. Introducing a real time, speed of sound measurement addresses the effects of aeration on multiple levels with the intent of enabling Coriolis meters to maintain mass flow and liquid density measurements in the presence of entrained air with accuracy approaching that for non-aerated liquids. 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 the gas volume fraction, are then utilized with analytically or empirically derived correction factors to improve the interpretation of the measured phase difference and natural frequency of the vibrating tubes in terms of the density of the aerated fluid.
8. Speed of sound augmented Coriolis meter
9. Discussion
The lumped parameter aeroelastic model presented above provides a first-principles basis for analyzing the effects of aeration on Coriolis based mass and density measurements. As shown, the model can be used to illustrate qualitatively the role of several non-dimensional parameters which govern the performance of the Coriolis meters. It can be concluded from
The intent of this work is to provide a unifying framework for assessing the impact of entrained gases on Coriolis meters under several simplifying assumptions. The most important of these assumptions is perhaps the assumption that the bubbles are well mixed within the liquid continuous phase. As a result, the model presented herein does not specifically address
Fig. 14. Interpreted non-dimensional mass flow as a function of gas volume fraction for three representative Coriolis meters.
Fig. 15. Apparent density as a function of gas volume fraction for three representative Coriolis meters.
of the fluid as follows. ρapparent
ρliq = α
f s2 2 f observed
! −1 .
(10)
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slugging and/or stratified flows within Coriolis phenomena that can often occur in start-up and batching processes, but rather it is intended to address a class of applications in which relatively small amounts of gas (<∼10% by volume) are entrained in a liquid continuous mixture. These conditions are often encountered in the processing of viscous liquids, for which the entrained gases are both difficult to remove and challenging to accurate Coriolis based measurement. Examples of these applications include pulp slurries, heavy oil separation, cement, viscous consumer products, etc. The results of the analytical model presented herein suggest that Coriolis meters operating on sufficiently viscous fluids, with well-mixed entrained gases, at sufficiently low reduced frequencies, will continue to behave as quasi-steady devices and continue to accurately report mixture mass flow and mixture density. Furthermore, with knowledge of the gas volume fraction, the output of the Coriolis meter can be accurately interpreted in terms of liquid mass flow, liquid density, and liquid volumetric flow. These results are supported by experimental data presented in [4,9,10], addressing relatively low reduced frequency Coriolis meters operating on liquids with entrained gases with conditions consistent with those described above. The results from the analytical model further predict that the increases in reduced frequency of operation in the presence of entrained gases will result in errors in mixture mass flow and mixture density, even for well-mixed, sufficiently viscous, liquid continuous flows due the increasing importance of unsteady effects in the interaction the fluid and structural dynamics within the Coriolis meter. The author is unaware of any public domain data that experimentally categorizes relatively high reduced frequency Coriolis meters operating on liquids with entrained gases with conditions consistent with those described above. It should be noted that numerous experimental studies have been published in the literature on the response of Coriolis meters to two-phase flow [11] and [12]. The data appears to cover a wide range of flow conditions, flow regimes, and tube geometries. Errors in mass flow and density due to twophase flow are shown to vary with numerous “dimensional parameters” such as viscosity [12,13], tube size [12,13], and bubble size [12], as well as flow rate and flow regime [11, 13]. Although these studies provide insight into the potential adverse effects of general two-phase flows in Coriolis meters, the scope of these experimental studies, in general, exceeds the class of problems considered herein, namely, the application of Coriolis meters to “well-mixed” liquid continuous mixtures. 10. Summary A lumped parameter aeroelastic model for Coriolis meters operating in aerated fluid was developed. The model incorporates the effects of compressibility and inhomogeneity association with entrained gases. The model identifies several aeroelastic parameters that influence the relationship between
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quantities directly measured by the Coriolis meter and the interpreted fluid mass flow and density. The effect of aeration was evaluated using the model for a set of representative Coriolis meters spanning a range of tube natural frequencies. Results of the model show that the low frequency Coriolis meter analyzed was significantly less sensitive to errors associated with the aeroelastic effects of aeration than the higher frequency designs. A concept was presented for enhancing the performance of Coriolis meters on aerated fluids. The concept utilizes a speed of sound measurement of the process fluid integrated with a Coriolis meter. Measuring fluid sound speed enables the determination of real time aeroelastic operating parameters, which enables the meter to accurately interpret the directly measured phase difference and natural frequency of the tubes in terms of fluid mass flow and density in the presence of aeration. Acknowledgements The results presented in this work have benefited from several years of development efforts in flow measurement based on sonar technology. The authors gratefully acknowledge the efforts of the many colleagues and co-workers that have contributed to the results presented herein. References [1] Bisplinghoff RL, Ashley H, Halfman R. Aeroelasticity. Addison-Wesley Publishing; 1955. [2] Gysling DL, Banach T. Enhanced density measurement using speed-ofsound augmented Coriolis meters. Presented at ISA. Houston, TX; 2004. [3] Sultan G, Hemp J. Modeling of the Coriolis mass flowmeter. Journal of Sound and Vibration 1989;132(3):473–89. [4] Gysling DL, Loose DH. Sonar-based volumetric flow and entrained air measurement for pulp and paper applications. Presented at the TAPPI Spring Technical Conference. Paper 58-1. Chicago, IL; 2003. [5] Munjal ML. Acoustics of ducts and mufflers. John Wiley and Sons; 1987. [6] Landau LD, Lifshitz EM. Fluid mechanics, 2nd ed. Course of theoretical physics, section 11, vol. 6. Butterworth–Heinemann; 1987. [7] Hemp J, Yeung H, Kassi L. Coriolis meter in two phase conditions. Department of Process and Systems Engineering. Cranfield University; 2003. [8] Hemp J, Sultan G. On the theory and performance of Coriolis mass flowmeter. In: Proceedings of the international conference on mass flow measurement. IBC technical Services. London; 1989. [9] Volz R, Gysling D, Loose D. Accurate volumetric flow rate and densitybased watercut measurement in bubbly liquid hydrocarbon flow. Society of Petroleum Engineers SPE Paper 102962-PP. Presented at the 2006 Annual Technical Conference. San Antonio, TX; 2006. [10] Ward E. New class of meter solves old problem impacting well test accuracy. Society of Petroleum Engineers SPE Paper 100893. Presented at the 2006 Western Regional Conference. Anchorage, AK; 2006. [11] Lui RP, Fuent MJ, Henry MP, Duta MD. A neural network to correct mass flow errors caused by two phase flow in a digital Coriolis mass flow meter. Flow Measurement and Instrumentation 2001;12:53–63. [12] Seeger M. Coriolis flow measurement in two phase flow. IEE Computing and Control Engineering 2005. [13] Grumski JT, Bajura RA. Performance of a Coriolis-type mass flow meter in the measurement of two-phase (air–liquid) mixtures. ASME Fluids Engineering Division 1984;17.