Numerical Simulations of Coriolis Flow Meters for Low Reynolds Number Flows

Transcription

1 MAPAN - Journal Numerical of Metrology Simulation Society of of Corioli India, Vol. Flow 26, Meter No. 3, 2011; for Low pp. Reynold Number Flow ORIGINAL ARTICLE Numerical Simulation of Corioli Flow Meter for Low Reynold Number Flow VIVEK KUMAR and MARTIN ANKLIN Reearch and Development, Endre+Hauer Flowtec AG, Kaegen Str. 7 Reinach (BL) CH-4132, Switzerland vivek.kumar@flowtec.endre.com [Received: ; Revied: ; Accepted: ] Abtract In proce indutrie Corioli ma flow meter (CMF) are widely employed for meauring ma flow. Quite often, epecially in the oil and ga (O&G) indutry, owing to fluid with high vicoitie, flow meaurement may lie in low Reynold number region. At low Reynold number (Re), a CMF reading may deviate under the influence of fluid-dynamic force. With the help of extenive Fluid- Structure-Interaction imulation (FSI), a detailed inight into phyical mechanim leading to thi deviation i provided. The main finding i that thi deviation i a function of the Reynold number and the effect can be explained by a periodic hear mechanim which interact with the ocillatory Corioli force and reduce the tube deflection. Experimental reult with and without a correction for thi effect are hown and compared with correponding numerical reult. If the low Reynold number effect were ignored, it would lead to error a large a 0.5% to 1% at Re = 800, however by meauring the Re and making correction, the effect i reduced to < 0.2%. 1. Introduction Although a Corioli ma flow meter (CMF) i independent of flow profile or intallation effect, it may be dependent on the Reynold number (Re) of the mean flow. With the help of the preent fluidtructure interaction (FSI) imulation, the mechanim reponible for the Reynold effect in CMF can be elucidated. It ha been hown that the Reynold effect induce a econdary flow in the ocillating tube of a CMF [1]. The ocillatory econdary flow lead to a change in the enitivity of the flow meter. The meter deviation can be corrected in-line in the flow meter provided the ma flow and ytem damping are known. In a CMF, the ma flow and tube damping information can be directly utilized to etimate the approximate Reynold number of the mean flow. Metrology Society of India, All right reerved Advance and development in the field of computational cience in the pat have led to extenive ue of numerical method in engineering. Unlike a few year ago, now computational fluid dynamic (CFD) and computational tructural mechanic (CSM) find plenty of application and interet in indutrie other than aero-pace and turbomachinery. The flow meaurement indutry i one uch example where application of thee numerical tool i helping to improve product quality and to find innovative olution. In many flow meaurement device, epecially a CMF, fluidtructure interaction (FSI), i.e. where CSM and CFD need to be coupled, related problem are often encountered and a complete undertanding of phyical phenomena occurring in device become vital. 225

2 Vivek Kumar and Martin Anklin The CMF are widely utilized in the proce indutry due to their high accuracy, reliability and direct meaurement of ma flow and fluid denity in continuou and batch procee. In a CMF, the phae difference between two enor point acro the center of the ocillating tube i directly proportional to the ma flow in the tube. The proportionality contant between ma flow and the phae difference i referred to a the calibration factor of the meter. The phae difference i meaured with an electro-magnetic inductive enor to calculate the ma flux and flow in the meter. A 2-inch CMF from Endre+Hauer (E+H) Flowtec AG i hown in Fig. 1. Under certain condition, a meter may deviate from the ideal linear behaviour depending on the proce condition in the meauring line. Thee diturbance are mainly due to coupled dynamic of fluid and tructure. Typical example of diturbance are: change in the compreibility of the fluid, preence of air-bubble, line preure and extremely low Re flow. Fig. 1. Proma F DN50, the two-inch Corioli ma flow meter of E+H Flowtec AG In the preent tudy, the focu i given to poible mechanim leading to deviation in meter reading from high to low Re region. The meaurement deviation at low Re ha ignificant importance in the metering of highly vicou fluid. Several laboratory and field meaurement with certain device clearly indicate that there can be a hift in the meter calibration 0.50 Uncorrected Fuel oil approx 250 cst (uncorrected) Fuel oil approx 200 cst (uncorrected) Fuel oil approx 150 cst (uncorrected) Crude approx. 8 cst (uncorrected) Deviation [%] Re Fig. 2(a) 226

3 Numerical Simulation of Corioli Flow Meter for Low Reynold Number Flow Fuel oil approx 250 cst (corrected) 0.50 Corrected Fuel oil approx 200 cst (corrected) Fuel oil approx 150 cst (corrected) Crude approx. 8 cst (corrected) Deviation [%] Re Fig. 2(b) Fig. 2. A hift in the meter reading with decreaing Re indicating the preence of a fluid dynamic phenomenon reponible behind the hift. (a) Meter deviation without any correction, (b) meter deviation after the Re correction for a 6-inch Proma device of Endre+Hauer factor at low Re. A typical deviation tart at =10000 [Eq. (9)] and the maximum deviation i approximately 0.5 % to 1 % at about Re = 800. The Re effect for a typical CMF i hown in Fig. 2. Both corrected and uncorrected experimental data are preented in the figure. The effect hown in Fig. 2(a) i explored with the help of numerical imulation. In the ma flow meter of E+H Flowtec AG, a compenation algorithm i implemented in the ignal proceing device in order to correct thi deviation inline. The meaurement data after the inline correction i preented in Fig. 2(b). Note that the compenation without any additional calibration work well. A far a CMF are concerned, there are a few attempt to imulate a CMF uing coupled FSI approach [2-4]. Kutin et al. [2-4] coupled the finiteelement Abacu and finite-volume Comet program mainly to invetigate flow profile effect in traighttube Corioli meter. According to Kutin et al., the preent effect i due to the change in axial flow profile due to the variation in the Re. They attributed the hift in meter reading with repect to decreaing Re to the change in axial flow profile from turbulent to laminar tranition. The preent numerical imulation ugget that a periodic or time-dependent mechanim ariing due to the interaction of ocillating inertial and ocillating hear force give rie to the hift in the meter reading of a CMF. The ratio of the two ocillating force i directly proportional to the Re of the mean flow 2. Mathematical Modeling Simulation of a Corioli meter involve the meh or boundary movement on both tructure and fluid ide. The movement of the ocillating tube i uperimpoed on the flowing fluid which in repone exert a force on the tructure due to it inertia. The inertial force of the fluid lead to a change in the ocillatory behaviour of the tructure that i recorded in term of diplacement. In thi ection, we briefly preent the governing equation and correponding general boundary/initial condition which we have utilized in the preent imulation. 227

4 Vivek Kumar and Martin Anklin 2.1. Fluid Domain For the fluid ide the governing equation are tranformed to Arbitrary Lagrangian-Eulerian (ALE) form in order to account for the convective fluxe which are due to the meh movement. The conervation equation of ma and momentum in integral form for the preent imulation are: d f dv f ( Ui ui ) dsi 0 dt ρ + ρ V = (1) S d dt S V ρ U dv + ρ ( U u ) U ds = f j S f i i j i ( τ + τ Pδ ) ds T ij ij ij i (2) where ρ f denote the denity of fluid, U j i the fluid velocity vector, u the velocity of the meh due to i T tructural motion, and are the vicou and τ ij turbulent part of the momentum tranport tenor, τ ij repectively, P repreent the fluid preure, S i denote the urface-area vector and V the volume of the control-volume. The turbulent hear-tre tenor i given by eddy-vicoity hypothei and the eddyvicoity wa modelled by the tandard k-ε turbulence model for turbulent flow. For the fluid domain, tandard inlet and outlet boundary condition (BC) were utilized where on the inlet patch fully-developed flow profile i precribed. Both inlet and outlet BC were kept away from the FSI-urface to minimize the influence of BC error on the imulation reult. For turbulent quantitie (i.e. k and ε) zero-gradient boundary condition were ued at the outlet. At the fluidtructure interface or ocillating tube-wall, the no-lip boundary wa pecified and an implicit meh motion wa impoed which wa provided by the CSM olver. A typical tructural mapped numerical meh ued in the FSI imulation i preented in Fig. 3. The height of the cell next to the ocillating wall wa determined on the bai of thickne of the Reynold and Stoke layer [6]. Ocillating pipe Fig. 3. A typical block-tructured numerical meh ued for FSI imulation of the fluid domain for a 2-inch Proma- F 228

5 Numerical Simulation of Corioli Flow Meter for Low Reynold Number Flow 2.2 Solid Domain In all computation in the preent work, the tructure wa aumed to be a linear elatic tructure and the differential form of equation of motion for a linear elatic tructure may be written a: 2 ρ φj τ ij = + ρ 2 bj t x where i (3) φ j repreent tructural diplacement vector, ρ the olid denity, t the time, b the body- j ν E λ = (1 + ν )(1 2 ν ) (6) For the tructural-ide boundary condition, the olid-tube wa alway kept fixed at both end. In order to imulate the tube exciter, a periodic or harmonic force wa applied at the center of tube only for the firt cycle. The frequency of the ocillating force wa et equal to the firt eigen-frequency of the pipe or the drive frequency of the meter. For pipe ocillation in the x-direction, a periodic force wa applied at the center in order to imulate the function of an exciter: force acting on a tructure. Here, tre tenor and can be written a: ij repreent the b b0 n fd t n j = [ in(2 ), 0, 0] if 20, [0, 0, 0] ele (7) j i k = + + xi x j xk ij ij (4) where µ and λ are Lame' coefficient and are related to Young' modulu of elaticity E and Poion ratio ν a follow: = E 2(1 + ) and (5) here t i the integration time tep, f d denote the drive frequency, and b 0 repreent the amplitude of the periodic force. The coupled imulation are performed until the amplitude of the ocillation goe below a certain value. The tructural domain and numerical meh i hown in Fig. 4. It may be een that the ditribution of cell on the tructural meh and fluid meh are not identical. The interface quantitie were tranferred acro the fluid-olid interface in each iteration. Fig. 4. The meh ued for the Corioli flow meter tructure (olid domain) 229

6 Vivek Kumar and Martin Anklin 2.3 CFD-FEM Coupling Approach The olution approach in an FSI computation i a critical factor for the convergence of taggered iteration. Before the tart of the tranient imulation, eparate initial computation were performed on the fluid and on the tructure ide. On the tructure ide, a modal (or eigen) analyi wa performed to find out the drive frequency (f d ) of the meter. On the fluid ide, a teady-tate imulation wa carried out to initialize the flow and preure field on the fluid ide. A a next tep, fluid and tructure field were coupled and a ingle teady-tate imulation wa performed to achieve reaonable initial field on the fluid and the tructure ide. The ize of the time-tep wa etimated by dividing each cycle of the pipe ocillation into 20- tep. Thi temporal reolution wa found to give reaonable prediction. For the FSI coupling, the following information i tranferred between the fluidtructure interface: U φ i i = U = φ i l i l FSI T FSI Fj = ( Pδ ij + τ ij + τ ij ) dsi (8) The above boundary condition et both kinematic and dynamic contraint for the FSI interface, FSI where F j denote the total force vector from the fluid olver to the tructural olver. On the other hand, tructural diplacement φ i were tranferred from the tructure to the fluid in order to fulfill kinematic contraint. The entire FSI imulation wa run over period or approximately 300 time-tep and the diplacement at two enor location were recorded at each time tep. For a typical cae, a computing time of approximately 60 CPU hour i required for fully converged olution for 15 period. Conequently the phae difference between two enor location wa found out with the help of a ignal proceing tool. 3. Reult and Dicuion 3.1 Reynold Number Effect A mentioned in an earlier ection, the calibration contant of a CMF meter may hift at low Reynold number. The Reynold number in the meauring tube i calculated a 4 m R e = n d t (9) where m i the ma flow, n t the number of meauring tube, µ the dynamic vicoity and d denote the inner diameter of the meauring tube. In the preent ection we try to elucidate the mechanim which i behind thi hift. With the help of coupled FSI imulation, the influence of Reynold number i imulated and numerical computed phae difference value are compared with the experimental data of the ame device. From Fig. 5 it may be noticed that the numerical imulation qualitatively upport the experimental obervation. Now the quetion arie: what exactly i behind thi hift in the meter repone? In order to undertand thi mechanim, following the Reynold decompoition, the fluid flow equation are plit [5] into two part: a teady component and an ocillating or firt harmonic component uch that U = U + u ' j j j ' P = P + p = + ij ij ij (10) where quantitie with an overbar are the mean or time independent quantitie and upercript ' indicate ocillating or periodic quantitie. The ocillating quantitie are only a function of drive frequency and time, and the influence of higher order harmonic e.g. from turbulence and tructure dynamic can be neglected for the cae of a Corioli meter. The higher order harmonic and fluctuation are anyway filtered out by the digital ignal proceing device and therefore in general do not contribute to the meter reading. By employing the above definition one may plit velocity, preure and hear-tre term in the Navier- Stoke equation into teady and ocillating part in order to derive equation for ocillating velocity field. Conequently, with the help of a few mathematical 230

7 Numerical Simulation of Corioli Flow Meter for Low Reynold Number Flow Re [ ] Fig. 5. A comparion between experimental and numerical imulation reult howing agreement in the hift in meter calibration factor in the low Re region for a Proma meter operation and neglecting the non-linear ocillating term, that i, the momentum balance for the ocillating flow in the differential form can be written a: ' ' ' u j u jui uiu j + + = t xi xi p' + ocillating Corioli force term ij ij xi xi ocillatinghear force term (11) With the help of pot-proceing tool, the ocillating Corioli force and correponding hear-rate are integrated over the tube cro-ection at enor location and plotted in Fig. 6 againt the Reynold number. It i intereting to note that both inertial force and hear in the tube cro-ection cloely follow the meter deviation. Therefore one may conclude that the hear ' rate u j / xi contribute to the change in the Corioli force and hence it alter the calibration factor a well. Furthermore, it can be eaily deduced that the ratio of ocilating Corioli to hear force in Eq. (11) i proportional to the Reynold number of the mean flow. The interaction of the ocillatory hear force with the inertial force in the meauring tube give rie to an ocillatory econdary flow. At a given time, thi econdary flow move in oppoite direction on either ide of the tube center and diappear at the center of the tube. The econdary flow at a Reynold number of 100 i hown in Fig. 7(a). At ufficiently high Reynold number thi econdary flow diappear a the ratio of hear force to Corioli force become negligibly mall. Better undertanding of the propoed mechanim behind the Reynold number effect i achieved with the help of FSI imulation. The phenomenon i briefly decribed with the help of a chematic diagram in Fig. 7. A hown in the figure, the Corioli force interacting with the hear force induce an aymmetric force in the meauring tube. The lower two panel illutrate that the Corioli force induce a hear layer indicating a econdary circulation in the cro-ection of the meauring tube. The Corioli force ha to overcome the hear force and part of the energy of the Corioli force i diipated in the econdary circulation and doe not contribute to the deflection of the tube. Thi explain why the meter reading i below the actual ma flow. The magnitude of the econdary circulation trongly decreae with higher Reynold number a the thickne of the hear-layer decreae exponentially with the increaing Reynold number. Conequently, the effect become inignificant above a certain Reynold number. A few other important quetion arie: i the Reynold effect dependent on tube form and ize and how do different meter behave in the low Reynold number regime? Unfortunately, there i no traight 231

8 Vivek Kumar and Martin Anklin deviation [%] ocillatory hear rate[^-1] Re [ ] deviation inertial force hear rate Fig. 6. A deviation in the meter reading with Reynold number and correponding volume-averaged Corioli or inertial force, ρ ' uu j i dv x i ' uj, and hear rate x i forward anwer to thee quetion. Thi iue i not yet fully explored in the preent tudy and may be a part of a future tudy. However, from the preent undertanding, we do not expect that the tube form directly influence the Reynold number effect. The mot important apect are the length and time-cale of the econdary flow and their ratio with the correponding inertial time and length cale. The length cale of the econdary flow i the thickne of the hear layer in the tube cro-ection and the tube diameter i the inertial length cale. 3.2 Real time Correction for the Low-Re Effect In order to meet accuracy requirement in low-re application, a Reynold number effect correction can be applied. Thi i adviable for meter that are ued in the low Reynold number range, i.e. with highly vicou product in cutody tranfer application [7]. If a Corioli meter ha the ability to etimate vicoity directly, compenation can be done on-line by on-board mean. The E+H Flowtec CMF meter can meaure ma flow and fluid vicoity in real time and hence the correction for the effect i performed directly in the electronic. Fluid vicoity i meaured with the help of damping of the ocillation. The damping can be etimated by meauring the drive current required to keep the meauring tube ocillating at a contant amplitude. The meter are operated either in lateral mode or in dual mode, that i lateral and torion mode. The damping of the latter mode i utilized to extract more accurate information on fluid vicoity than the former mode [8]. In thi tudy, meaurement with nine 10-inch Proma F meter are preented. Proma F DN250 meter were calibrated at the SPSE facility in outhern France with different vicoity hydrocarbon. Thi facility allow the calibration of flow meter with liquid hydrocarbon within the flow range of 1500 to 3000 m³/h with product having kinematic vicoitie of 0.5 to 500 mm²/. The reference flow i from a ball prover ytem with an overall 0.13 % uncertainty for ma flow and Reynold number meaurement. Figure 8 how that the low Re compenation method can be applied with good reult. The preent correction reduce the difference between the reference and the Corioli meter to ± 0.2 % without additional calibration. 232

9 Numerical Simulation of Corioli Flow Meter for Low Reynold Number Flow Fig. 7(a) Fig. 7(b) Fig. 7. (a) The computed ocillatory econdary flow in moving-frame of reference at the poition where the flow meter phae change enor i located, (b) a chematic repreentation of the ocillatory mechanim reponible for Reynold effect 233

10 Vivek Kumar and Martin Anklin Fuel oil approx 250 cst (corrected) 0.50 Corrected Fuel oil approx 200 cst (corrected) Fuel oil approx 150 cst (corrected) Crude approx. 8 cst (corrected) Deviation [%] Re Fig. 8(a) Fig. 8(b) Fig. 8. The repone of an E+H Flowtec AG Re-correction (a) for a 6-inch meter Proma F and (b) nine 10-inch Proma F meter at a flow calibration facility of SPSE in outhern France 234

11 Numerical Simulation of Corioli Flow Meter for Low Reynold Number Flow It i important to mention that thi correction i implanted in the ignal proceing device of E+H device and moreover the procedure i patented by E+H Flowtec. 4. Concluion In thi work, reult from coupled fluid-tructure numerical imulation mainly for low Reynold number are preented. With the help of thee imulation the fluid dynamic effect reponible for the meter deviation at low Reynold number i well undertood. A econdary ocillatory flow in the tube cro-ection, induced by the interaction of Corioli and hear force, give rie to a change in the calibration factor of the meter. The econdary flow i a function of the Reynold number of the mean flow. Since Corioli meter of E+H Flowtec can determine the Reynold number directly, a real time Reynold correction i implemented in the ignal proceing tool in order to compenate for the low Re effect. It ha been hown that the online compenation with the tandard procedure work well. In general, all Corioli device are ubjected to the Reynold number effect and for high accuracie the effect mut be corrected. In the low Re regime, the compenation algorithm i already implemented in the electronic of all E+H Corioli meter. Reference [1] V. Kumar, M. Anklin and B. Schwenter, Fluid- Structure Interaction (FSI) Simulation on the Senitivity of Corioli FlowMeter Under Low Reynold Number Flow, 15 th Flow Meaurement Conference FLOMEKO, Taipei Taiwan, (2010). [2] J. Kutin, J. Hemp, G. Bobovnik and I. Baji ć, Weight Vector Study of Velocity Profile Effect in Straight-Tube Corioli Flow Meter Employing Different Circumferential Mode, Flow Meaurement and Intrumentation, 16 (2005) [3] G. Bobovnik, N. Mole, J. Kutin, B. Stok and I. Baji, Coupled Finite-Volume/Finite Element Modeling of the Straight-Tube Corioli Flow Meter, Journal of Fluid and Structure, 20 (2005) [4] J. Kutin, G. Bobovnik, J. Hemp and I. Baji, Velocity Profile Effect in Corioli Ma Flow Meter: Recent Finding and Open Quetion, Flow Meaurement and Intrumentation, 17 (2006) [5] O. Reynold, On the Dynamical Theory of Incompreible Vicou Fluid and Determination of the Criterion, Philo. Tran. of R. Soc. London, Ser. A-186 (1895) [6] H. Schlichting and K. Gerten, Boundary Layer Theory, 8th Edition, Springer-Verlag, (2000). [7] V. Kumar, P. Tchabold and M. Anklin, Influence and Compenation of Proce Parameter on Corioli Meter with a View to Cutody Tranfer of Hydrocarbon Product, NEL 9th South Eat Aia Hydrocarbon Flow Meaurement Workhop, (2010). [8] A. Rieder, Corioli Ma Flow Meter with Direct Vicoity Meaurement, Pump and Compreor with Compreed Air and Vacuum Technology, (2010)