Numerical Simulations of Coriolis Flow Meters for Low ...metrologyindia.org/26c/6-Vivek-Kumar.pdfNumerical Simulations of Coriolis Flow Meters for Low Reynolds Number Flows factor

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Numerical Simulations of Coriolis Flow Meters for Low Reynolds Number Flows

Numerical Simulations of Coriolis Flow Meters forLow Reynolds Number Flows

VIVEK KUMAR and MARTIN ANKLIN

Research and Development, Endress+Hauser Flowtec AG, Kaegen Str. 7Reinach (BL) CH-4132, Switzerland

e-mail: [email protected]

[Received: 14.01.2011 ; Revised: 28.03.2011 ; Accepted: 30.06.2011]

AbstractIn process industries Coriolis mass flow meters (CMFs) are widely employed for measuring mass flows.Quite often, especially in the oil and gas (O&G) industry, owing to fluids with high viscosities, flowmeasurements may lie in low Reynolds number regions. At low Reynolds numbers (Re), a CMFreading may deviate under the influence of fluid-dynamic forces. With the help of extensive Fluid-Structure-Interaction simulations (FSI), a detailed insight into physical mechanisms leading to thisdeviation is provided. The main finding is that this deviation is a function of the Reynolds number andthe effect can be explained by a periodic shear mechanism which interacts with the oscillatory Coriolisforce and reduces the tube deflection. Experimental results with and without a correction for this effectare shown and compared with corresponding numerical results. If the low Reynolds number effect wereignored, it would lead to errors as large as 0.5% to 1% at Re = 800, however by measuring the Re andmaking corrections, the effect is reduced to < 0.2%.

© Metrology Society of India, All rights reserved 2011.

1. Introduction

Although a Coriolis mass flow meter (CMF) isindependent of flow profile or installation effects, itmay be dependent on the Reynolds number (Re) ofthe mean flow. With the help of the present fluid-structure interaction (FSI) simulations, the mechanismresponsible for the Reynolds effect in CMFs can beelucidated. It has been shown that the Reynolds effectinduces a secondary flow in the oscillating tubes of aCMF [1]. The oscillatory secondary flow leads to achange in the sensitivity of the flow meter. The meterdeviation can be corrected in-line in the flow meterprovided the mass flow and system damping areknown. In a CMF, the mass flow and tube dampinginformation can be directly utilized to estimate theapproximate Reynolds number of the mean flow.

MAPAN - Journal of Metrology Society of India, Vol. 26, No. 3, 2011; pp. 225-235ORIGINAL ARTICLE

Advances and developments in the field ofcomputational sciences in the past have led toextensive use of numerical methods in engineering.Unlike a few years ago, now computational fluiddynamics (CFD) and computational structuralmechanics (CSM) find plenty of applications andinterests in industries other than aero-space andturbomachinery. The flow measurement industry isone such example where application of thesenumerical tools is helping to improve product qualityand to find innovative solutions. In many flowmeasurement devices, especially a CMF, fluid-structure interaction (FSI), i.e. where CSM and CFDneed to be coupled, related problems are oftenencountered and a complete understanding ofphysical phenomena occurring in devices becomesvital.

Vivek Kumar and Martin Anklin

226

The CMFs are widely utilized in the processindustry due to their high accuracy, reliability anddirect measurement of mass flow and fluid density incontinuous and batch processes. In a CMF, the phasedifference between two sensor points across the centerof the oscillating tube is directly proportional to themass flow in the tube. The proportionality constantbetween mass flow and the phase difference is referredto as the calibration factor of the meter. The phasedifference is measured with an electro-magneticinductive sensor to calculate the mass flux and flowin the meter. A 2-inch CMF from Endress+Hauser(E+H) Flowtec AG is shown in Fig. 1. Under certainconditions, a meter may deviate from the ideal linearbehaviour depending on the process conditions inthe measuring line. These disturbances are mainlydue to coupled dynamics of fluid and structure.Typical examples of disturbances are: change in thecompressibility of the fluid, presence of air-bubbles,line pressure and extremely low Re flows.

In the present study, the focus is given to possiblemechanisms leading to deviation in meter readingfrom high to low Re regions. The measurementdeviation at low Re has significant importance in themetering of highly viscous fluids. Several laboratoryand field measurements with certain devices clearlyindicate that there can be a shift in the meter calibration

Fig. 1. Promass F DN50, the two-inch Coriolis massflow meter of E+H Flowtec AG

Uncorrected

-0.50

0.00

0.50

1000 10000 100000 1000000Re

Dev

iatio

n [%

]

Fuel oil approx 250 cSt (uncorrected)



Crude approx. 8 cSt (uncorrected)

Fig. 2(a)

227


factor at low Re. A typical deviation starts at =10000[Eq. (9)] and the maximum deviation is approximately0.5 % to 1 % at about Re = 800. The Re effect for atypical CMF is shown in Fig. 2. Both corrected anduncorrected experimental data are presented in thefigure.

The effect shown in Fig. 2(a) is explored with thehelp of numerical simulations. In the mass flow metersof E+H Flowtec AG, a compensation algorithm isimplemented in the signal processing device in orderto correct this deviation inline. The measurement dataafter the inline correction is presented in Fig. 2(b). Notethat the compensation without any additionalcalibration works well.

As far as CMFs are concerned, there are a fewattempts to simulate a CMF using coupled FSIapproach [2-4]. Kutin et al. [2-4] coupled the finite-element Abacus and finite-volume Comet programmainly to investigate flow profile effects in straight-tube Coriolis meters. According to Kutin et al., thepresent effect is due to the change in axial flow profile

superimposed on the flowing fluid which in responseexerts a force on the structure due to its inertia. Theinertial force of the fluid leads to a change in theoscillatory behaviour of the structure that is recordedin terms of displacements. In this section, we brieflypresent the governing equations and correspondinggeneral boundary/initial conditions which we haveutilized in the present simulations.

due to the variation in the Re. They attributed theshift in meter readings with respect to decreasing Reto the change in axial flow profiles from turbulent tolaminar transitions.

The present numerical simulations suggest thata periodic or time-dependent mechanism arising dueto the interaction of oscillating inertial and oscillatingshear forces give rise to the shift in the meter readingof a CMF. The ratio of the two oscillating forces isdirectly proportional to the Re of the mean flow

2. Mathematical Modeling

Simulation of a Coriolis meter involves the meshor boundary movement on both structure and fluidsides. The movement of the oscillating tubes is

Corrected

-0.50

0.00

0.50

1000 10000 100000 1000000Re

Dev

iatio

n [%

]Fuel oil approx 250 cSt (corrected)

Fuel oil approx 200 cSt (corrected)


Crude approx. 8 cSt (corrected)

Fig. 2(b)

Fig. 2. A shift in the meter reading with decreasing Re indicating the presence of a fluid dynamic phenomenonresponsible behind the shift. (a) Meter deviation without any correction, (b) meter deviation after the

Re correction for a 6-inch Promass device of Endress+Hauser


228

2.1. Fluid Domain

For the fluid side the governing equations aretransformed to Arbitrary Lagrangian-Eulerian (ALE)form in order to account for the convective fluxeswhich are due to the mesh movement. Theconservation equations of mass and momentum inintegral form for the present simulations are:

sf f i i iV S

( ) 0d dV U u dSdt

ρ ρ∆ ∆

+ − =∫ ∫ (1)

sf j f i i j iV S

Tij ij ij iS

( )

( )

d U dV U u U dSdt

P dS

ρ ρ

τ τ δ

∆ ∆

∆

+ − =

+ −

∫ ∫

∫ (2)

where ρf denotes the density of fluid, Uj is the fluidvelocity vector, s

iu the velocity of the mesh due to

structural motion, and are the viscous andturbulent part of the momentum transport tensors,

respectively, P represents the fluid pressure, Si denotesthe surface-area vector and V the volume of thecontrol-volume. The turbulent shear-stress tensor isgiven by eddy-viscosity hypothesis and the eddy-viscosity was modelled by the standard k-ε turbulencemodel for turbulent flows.

For the fluid domain, standard inlet and outletboundary conditions (BCs) were utilized where onthe inlet patch fully-developed flow profile isprescribed. Both inlet and outlet BCs were kept awayfrom the FSI-surface to minimize the influence of BCserrors on the simulation results. For turbulentquantities (i.e. k and ε) zero-gradient boundaryconditions were used at the outlet. At the fluid-structure interface or oscillating tube-wall, the no-slipboundary was specified and an implicit mesh motionwas imposed which was provided by the CSM solver.A typical structural mapped numerical mesh used inthe FSI simulations is presented in Fig. 3. The heightof the cell next to the oscillating wall was determinedon the basis of thickness of the Reynolds and Stokeslayers [6].

Fig. 3. A typical block-structured numerical mesh used for FSI simulations of thefluid domain for a 2-inch Promass- F

ijτTijτ

Oscillating pipes

229


2.2 Solid Domain

In all computations in the present work, thestructure was assumed to be a linear elastic structureand the differential form of equation of motion for alinear elastic structure may be written as:

2 ss j ij

s j2i

bt x

ρ φ τρ

∂ ∂= +

∂ ∂ (3)

where jφ represents structural displacement

vector, the solid density, t the time, the body-

force acting on a structure. Here, represents thestress tensor and can be written as:

(4)

where µs and λs are Lame's coefficient and are relatedto Young's modulus of elasticity E and Poisson ratioνs as follows:

(5)

and

ss

s s(1 )(1 2 )Eν

λν ν

=+ − (6)

For the structural-side boundary conditions, thesolid-tube was always kept fixed at both ends. In orderto simulate the tube exciter, a periodic or harmonicforce was applied at the center of tube only for the firstcycle. The frequency of the oscillating force was setequal to the first eigen-frequency of the pipe or thedrive frequency of the meter. For pipe oscillations inthe x-direction, a periodic force was applied at thecenter in order to simulate the function of an exciter:

(7)

here t∆ is the integration time step, fd denotes thedrive frequency, and b0 represents the amplitude ofthe periodic force. The coupled simulations areperformed until the amplitude of the oscillations goesbelow a certain value.

The structural domain and numerical mesh isshown in Fig. 4. It may be seen that the distribution ofcell on the structural mesh and fluid mesh are notidentical. The interface quantities were transferredacross the fluid-solid interface in each iteration.

Fig. 4. The mesh used for the Coriolis flow meter structure (solid domain)

sρ jb

sij

js kiij s s ij

i j kx x x ∂ ∂∂

= + + ∂ ∂ ∂

ss2(1 )

E

=

+

0 dj

[ sin(2 ),0,0] if 20,[0,0,0] else

∆ ≤=

b n f t nb


230

2.3 CFD-FEM Coupling Approach

The solution approach in an FSI computation is acritical factor for the convergence of staggerediterations. Before the start of the transient simulations,separate initial computations were performed on thefluid and on the structure side. On the structure side,a modal (or eigen) analysis was performed to find outthe drive frequency (fd) of the meter. On the fluid side,a steady-state simulation was carried out to initializethe flow and pressure fields on the fluid side. As anext step, fluid and structure fields were coupled anda single steady-state simulation was performed toachieve reasonable initial fields on the fluid and thestructure side. The size of the time-step was estimatedby dividing each cycle of the pipe oscillations into 20-steps. This temporal resolution was found to givereasonable predictions. For the FSI coupling, thefollowing information is transferred between the fluid-structure interface:

The above boundary conditions set both kinematicand dynamic constraints for the FSI interface,where FSI

jF denotes the total force vector from the fluidsolver to the structural solver. On the other hand,structural displacements φi were transferred from thestructure to the fluid in order to fulfill kinematicconstraints.

The entire FSI simulation was run over 12-15periods or approximately 300 time-steps and thedisplacements at two sensor locations were recordedat each time step. For a typical case, a computing timeof approximately 60 CPU hours is required for fullyconverged solution for 15 periods. Consequently thephase difference between two sensor locations wasfound out with the help of a signal processing tool.

3. Results and Discussion

3.1 Reynolds Number Effect

As mentioned in an earlier section, the calibration

constant of a CMF meter may shift at low Reynoldsnumber. The Reynolds number in the measuring tubeis calculated as

(9)

where is the mass flow, nt the number of measuringtubes, µ the dynamic viscosity and d denotes the innerdiameter of the measuring tube.

In the present section we try to elucidate themechanism which is behind this shift. With the helpof coupled FSI simulations, the influence of Reynoldsnumber is simulated and numerical computed phasedifference values are compared with the experimentaldata of the same device. From Fig. 5 it may be noticedthat the numerical simulations qualitatively supportthe experimental observations.

Now the question arises: what exactly is behindthis shift in the meter response? In order to understandthis mechanism, following the Reynoldsdecomposition, the fluid flow equations are split [5]into two parts: a steady component and an oscillatingor first harmonic component such that

(10)

where quantities with an overbar are the mean or timeindependent quantities and superscript “ ' “ indicatesoscillating or periodic quantities. The oscillatingquantities are only a function of drive frequency andtime, and the influence of higher order harmonics e.g.from turbulence and structure dynamics can beneglected for the case of a Coriolis meter. The higherorder harmonics and fluctuations are anyway filteredout by the digital signal processing device andtherefore in general do not contribute to the meterreadings.

By employing the above definitions one may splitvelocity, pressure and shear-stress terms in the Navier-Stokes equations into steady and oscillating parts inorder to derive equations for oscillating velocity fields.Consequently, with the help of a few mathematical

i is l

i i ls

ISFTISFj ij ij ij i( )

U U

F P dS

φ φ

δ τ τ

=

=

= + +∫ (8)

t

4e =

mR

n d

m

'j j j

'

ij ij ij

U U u

P P p

= +

= +

= +

231


operations and neglecting the non-linear oscillatingterm, that is , the momentum balance for the oscillatingflow in the differential form can be written as:

With the help of post-processing tools, the oscillatingCoriolis force and corresponding shear-rate areintegrated over the tube cross-section at sensor locationand plotted in Fig. 6 against the Reynolds number. Itis interesting to note that both inertial force and shearin the tube cross-section closely follow the meterdeviation. Therefore one may conclude that the shear

rate 'ju∂ / ix∂ contributes to the change in the Coriolis

force and hence it alters the calibration factor as well.Furthermore, it can be easily deduced that the ratio ofoscilating Coriolis to shear force in Eq. (11) isproportional to the Reynolds number of the mean flow.

The interaction of the oscillatory shear force withthe inertial force in the measuring tube gives rise to anoscillatory secondary flow. At a given time, thissecondary flow moves in opposite directions on eitherside of the tube center and disappears at the center of

the tube. The secondary flow at a Reynolds number of100 is shown in Fig. 7(a). At sufficiently high Reynoldsnumber this secondary flow disappears as the ratioof shear force to Coriolis force becomes negligiblysmall.

Better understanding of the proposed mechanismbehind the Reynolds number effect is achieved withthe help of FSI simulation. The phenomenon is brieflydescribed with the help of a schematic diagram inFig. 7. As shown in the figure, the Coriolis forceinteracting with the shear force induces anasymmetric force in the measuring tube. The lowertwo panels illustrate that the Coriolis force induces ashear layer indicating a secondary circulation in thecross-section of the measuring tube. The Coriolis forcehas to overcome the shear force and part of the energyof the Coriolis force is dissipated in the secondarycirculation and does not contribute to the deflectionof the tube. This explains why the meter reading isbelow the actual mass flow. The magnitude of thesecondary circulation strongly decreases with higherReynolds numbers as the thickness of the shear-layerdecreases exponentially with the increasing Reynoldsnumber. Consequently, the effect becomes insigni-ficant above a certain Reynolds number.

A few other important questions arise: is theReynolds effect dependent on tube form and size andhow do different meters behave in the low Reynoldsnumber regime? Unfortunately, there is no straight

Fig. 5. A comparison between experimental and numerical simulation results showing agreement in the shiftin meter calibration factor in the low Re region for a Promass meter

(11)

Re [ ]

' ' 'j j i i j

i i

oscillating Coriolis force term

ijij

i i

oscillating shear force term

'

∂ ∂ ∂+ + = ∂ ∂ ∂

∂∂− +

∂ ∂

u u U u Ut x x

px x


232

forward answer to these questions. This issue is notyet fully explored in the present study and may be apart of a future study. However, from the presentunderstanding, we do not expect that the tube formdirectly influences the Reynolds number effect. Themost important aspects are the length and time-scalesof the secondary flow and their ratio with thecorresponding inertial time and length scales. Thelength scale of the secondary flow is the thickness ofthe shear layer in the tube cross-section and the tubediameter is the inertial length scale.

3.2 Real time Correction for the Low-Re Effect

In order to meet accuracy requirements in low-Reapplications, a Reynolds number effect correction canbe applied. This is advisable for meters that are usedin the low Reynolds number range, i.e. with highlyviscous products in custody transfer applications [7].

If a Coriolis meter has the ability to estimateviscosity directly, compensation can be done on-lineby on-board means. The E+H Flowtec CMF meterscan measure mass flow and fluid viscosity in real timeand hence the correction for the effect is performed

directly in the electronics. Fluid viscosity is measuredwith the help of damping of the oscillations. Thedamping can be estimated by measuring the drivecurrent required to keep the measuring tubesoscillating at a constant amplitude. The meters areoperated either in lateral mode or in dual mode, thatis lateral and torsion mode. The damping of the lattermode is utilized to extract more accurate informationon fluid viscosity than the former mode [8].

In this study, measurements with nine 10-inchPromass F meters are presented. Promass F DN250meters were calibrated at the SPSE facility in southernFrance with different viscosity hydrocarbons. Thisfacility allows the calibration of flow meters withliquid hydrocarbons within the flow range of 1500 to3000 m³/h with products having kinematic viscositiesof 0.5 to 500 mm²/s. The reference flow is from a ballprover system with an overall 0.13 % uncertainty formass flow and Reynolds number measurements.Figure 8 shows that the low Re compensation methodscan be applied with good results. The presentcorrection reduces the difference between the referenceand the Coriolis meter to ± 0.2 % without additionalcalibrations.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

100 1000 10000 100000 1000000Re [ ]

devi

atio

n [%

]

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

oscil

lato

ry sh

ear r

ate[

s^-1

]

deviation inertial force shear rate

Fig. 6. A deviation in the meter reading with Reynolds number and corresponding volume-averaged Coriolis

or inertial force, i

'j

i

∂

∂∫u U

dVx

ρ , and shear rate 'j

i

ux

∂

∂

233


Fig. 7(a)

Fig. 7. (a) The computed oscillatory secondary flow in moving-frame of reference at the position where the flowmeter phase change sensor is located, (b) a schematic representation of the oscillatory mechanism responsible for

Reynolds effect

Fig. 7(b)


234

Corrected

-0.50

0.00

0.50

1000 10000 100000 1000000Re

Dev

iatio

n [%

]




Crude approx. 8 cSt (corrected)

Fig. 8(a)

Fig. 8(b)

Fig. 8. The response of an E+H Flowtec AG Re-correction (a) for a 6-inch meter Promass F and (b) nine 10-inchPromass F meters at a flow calibration facility of SPSE in southern France

235


It is important to mention that this correction isimplanted in the signal processing device of E+Hdevices and moreover the procedure is patented byE+H Flowtec.

4. Conclusion

In this work, results from coupled fluid-structurenumerical simulations mainly for low Reynoldsnumbers are presented. With the help of thesesimulations the fluid dynamic effect responsible forthe meter deviation at low Reynolds numbers is wellunderstood. A secondary oscillatory flow in the tubecross-section, induced by the interaction of Coriolisand shear forces, gives rise to a change in thecalibration factor of the meter. The secondary flow isa function of the Reynolds number of the mean flow.

Since Coriolis meters of E+H Flowtec candetermine the Reynolds number directly, a real timeReynolds correction is implemented in the signalprocessing tool in order to compensate for the low Reeffect. It has been shown that the online compensationwith the standard procedure works well. In general,all Coriolis devices are subjected to the Reynoldsnumber effect and for high accuracies the effect mustbe corrected. In the low Re regime, the compensationalgorithm is already implemented in the electronicsof all E+H Coriolis meters.

References

[1] V. Kumar, M. Anklin and B. Schwenter, Fluid-Structure Interaction (FSI) Simulations on theSensitivity of Coriolis FlowMeter Under LowReynolds Number Flows, 15th FlowMeasurement Conference FLOMEKO, TaipeiTaiwan, (2010).

[2] J. Kutin, J. Hemp, G. Bobovnik and I. Bajsi ć ,Weight Vector Study of Velocity Profile Effectsin Straight-Tube Coriolis Flow MetersEmploying Different Circumferential Modes,Flow Measurement and Instrumentation, 16(2005) 375-385.

[3] G. Bobovnik, N. Mole, J. Kutin, B. Stok and I.Bajsi , Coupled Finite-Volume/Finite ElementModeling of the Straight-Tube Coriolis FlowMeter, Journal of Fluid and Structures, 20 (2005)785-800.

[4] J. Kutin, G. Bobovnik, J. Hemp and I. Bajsi ,Velocity Profile Effects in Coriolis Mass FlowMeters: Recent Findings and Open Questions,Flow Measurement and Instrumentation, 17(2006) 349-358.

[5] O. Reynolds, On the Dynamical Theory ofIncompressible Viscous Fluids and Deter-mination of the Criterion, Philos. Trans. of R.Soc. London, Ser. A-186 (1895)123-164.

[6] H. Schlichting and K. Gersten, Boundary LayerTheory, 8th Edition, Springer-Verlag, (2000).

[7] V. Kumar, P. Tschabold and M. Anklin,Influence and Compensation of ProcessParameters on Coriolis Meters with a View toCustody Transfer of Hydrocarbon Products,NEL 9th South East Asia Hydrocarbon FlowMeasurement Workshop, (2010).

[8] A. Rieder, Coriolis Mass Flow Meter with DirectViscosity Measurement, Pumps andCompressors with Compressed Air andVacuum Technology, (2010) 52-55.

Numerical Simulations of Coriolis Flow Meters for Low ...metrologyindia.org/26c/6-Vivek-Kumar.pdfNumerical Simulations of Coriolis Flow Meters for Low Reynolds Number Flows factor

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