Fluid-structure coupling analysis and simulation of viscosity effect. on Coriolis mass flowmeter

Transcription

1 APCOM & ISCM th December, 2013, Singapore luid-tructure coupling analyi and imulation of vicoity effect on Corioli ma flowmeter *Luo Rongmo, and Wu Jian National Metrology Centre, A*STAR, 1 Science Park Drive, Singapore *Correponding author: luo_rongmo@nmc.a-tar.edu.g Abtract Corioli ma flowmeter (CM) are increaingly ued in the oil and ga indutry with feature of directly meauring ma flow rate. The performance of CM influenced by on-line vicoity till need further tudy. A computational fluid dynamic model of U-hape CM wa developed. The imulation reult were evaluated in term of the natural frequency of the vibrating ytem and the correponding phae difference between the motion of the ening point ymmetrically located on the meauring tube. The imulation were conducted on comparion between water and vicou liquid with flow rate panning the laminar and turbulent region. The effect of vicoity on ma flow meaurement by CM are dicued in detail. The finding in the imulation can be ued for further compenation of deviation due to vicoity effect. Keyword: Corioli ma flowmeter, luid-tructure interaction, High vicoity, Deviation, Ma flow rate 1. Introduction Corioli ma-flow meaurement i ued in a huge range of indutry ector, including pharmaceutical, chemical and petrochemical, oil and ga, and food. Although a Corioli ma flow meter (CM) i independent of flow profile or intallation effect, it may be dependent on the Reynold number (Re) of the mean flow. 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 factor at vicou liquid (Henry, Tomb et al. 2006, Kumar and Anklin 2011). 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 CM, fluid-tructure interaction (SI), i.e. where computational tructural mechanic (CSM) and computational fluid dynamic (CD) need to be coupled, related problem are often encountered and a complete undertanding of phyical phenomena occurring in device become vital (Bobovnik, Mole et al. 2005, Mole, Bobovnik et al. 2008). A far a CM are concerned, there are a few attempt to imulate a CM uing coupled SI approach (Mole, Bobovnik et al. 2008, Kumar and Anklin 2011). In thi paper, a three-dimenional coupled fluid-tructure numerical model of a U-hape Corioli flowmeter i preented. The excitation force ha been complemented to addre properly the forced vibration condition of CM, and then the CD model i employed for the CM operating under inflow fluid condition with different vicoity. Reult from coupled fluid-tructure numerical imulation mainly for varied vicoity are preented. With the help of thee imulation the fluid dynamic effect reponible for the meter deviation at different vicoity can be undertood, and better reemblance can be achieved between the numerical model imulation and the true operation of the CM. 1

2 2. MATHEMATICAL MODELS The fluid-conveying meauring enor tube in the Corioli ma flowmeter i maintained to vibrate periodically at it natural frequency under impulively forced vibration condition (reonance). Ma flow i uually meaured a the time or phae difference between the motion of two ening point (S 1 and S 2 ) on the tube, which are poitioned ymmetrically along the tube length. However, the ditortion of ymmetry of the no-flow drive mode i reulted from the interaction between the motion of the tube and the fluid flow due to the CM inertial force field, where the traight meauring tube i clamped at both end and vibrating at it firt natural frequency. Thi ection preent the governing equation and correponding general boundary/initial condition which we have ued in the preent imulation. 2.1 luid domain The conervation equation of ma and momentum are written in the integral form for the threedimenional patial ditribution and time range ( t 0) of fluid flow a d ( S) d0 t V V n (1) t V d V ( V V ) nd σ nd f d (2) S where movement of fluid flow with the denity (, ) and the velocity V in the domain ( x ) are influenced by the motion of a urrounding boundary velocity v, Γ denote the urface-area vector. The vector f ( x, t ) in the momentum equation (2) i the volume force acting inide the domain, and ( x, t ) i the reulting tenor. The repective boundary condition can be written a V ( xt, ) V, inflow x inflow, p (, ) x t poutlow, x outflow, m V v, x () tube t, V ( xt, ) 0, in () out x tube t, x () tube t, (3) where Vinflow i the inflow fluid velocity, i the fluid boundary, p outflow i the abolute fluid preure at fluid outflow, and v i the velocity of the meauring tube urface. 2.2 Structure domain The conervation of momentum principle i utilized for the three-dimenional patial ditribution ( x ) and time evolution ( t 0 ) of the tructural repone, where the repective equation of motion can be derived by Hamilton variation principle, t2 ( WP Wk) dt 0 (4) t1 where Wp and WK are the total potential energy and the total kinetic energy of the moving olid tructure, repectively. The detailed expreion for them are defined a (Mole, Bobovnik et al. 2008), 1 W : d d W P u r (5) P P 2 1 ( v v ) d (6) k 2 S 2

3 where the urface traction P acting upon the moving hell boundary through the repective diplacement field u, and the concentrated force () t at point P, where the forced vibration i generated. and are the train and the tre tenor in the hell tructure, and r P i the poition vector of point P where the force i applied. i the tructure material denity and v i the tructure velocity field. At t 0, the initial velocity and acceleration field, v ( x,0) and a ( x,0) mut be given, and ( x,0) x( x,0) 0. Att 0, u u ( xt, ) 0, x tube, u σ n P, x () tube t, () t ( (),0,0) t, x xp (7) or the tructural-ide boundary condition, the ening tube wa fixed at both end. In order to imulate the tube exciter, a periodic or harmonic force wa applied at the center of tube point P. The firt frequency of the ening tube (the drive frequency of the meter) i equal to the frequency of the ocillating force. A periodic force wa applied at the centre Point P to ocillate the pipe in the x- direction, () t ( 0 in(2 nfdt),0,0) (8) where t i the integration time tep, f d denote the drive frequency, and 0 repreent the amplitude of the periodic force. 2.3 Method of analyi The preent imulation ue the ANSYS Workbench framework employing DeignModeler, SIMULATION, ANSYS, CX Meh and CX olver. The pipe i created in DeignModeler, with the tructural and fluid domain repreenting the tube wall and the fluid inide the tube. SIMULATION and CX Meh are ued to meh the olid and fluid domain, repectively. To determine the invetigated tube natural frequency, a modal analyi of the olid domain i conducted in ANSYS. The determined natural frequency i ued to calculate the excitation force at point P. And then a dynamic repone analyi for the olid domain i performed baed on the linear elatic theory, where the deformation of the ene tube are aumed to be mall. And it can determine the time tep for SI analyi. To obtain the initial condition for the tranient fluid analyi, a teady tate analyi i carried out in CX over the fluid domain. The two teady tate analye are employed for the tranient analyi of the fluid domain. The SI imulation are carried out within CX (ANSYS v13.0). During the information tranferring, both the kinematic and dynamic contraint are et for the SI interface, V( x, tn) V ( x, tn) u ( xt, n) u ( xt, n) SI SI j ( pij ij) di (9) 3

4 SI where j denote the total force vector from the CX olver to the tructural olver, and ij include the vicou and turbulent part of the momentum tenor. On the other hand, tructural diplacement u ( x, tn ) are tranferred from the tructure to the fluid in order to fulfill kinematic contraint. 3. RESULTS AND DISCUSSION The analyzed U-hape CM, with geometry a preented in ig. 1(a), i characterized by the length of pan L=0.4 m and height of CM H=0.38 m, and a cro-ection geometry which i determined by the internal diameter D= m and wall thickne δ=1/40d. The ditance between the ymmetrically poitioned ening point S 1 and S 2 i equal to L=0.36 m. The pecification ued for the imulation in thi work are hown in Table 1. A the tructure ha multiple degree of freedom, the tructure will vibrate in a different manner at different natural frequencie without any application of external force. Looking at all the deformation of the repective frequencie, in order for reonance to occur, the driver ha to impoe a driving force at the natural frequency fd Hz, a hown in ig. 1(b). a) 4

5 b) igure 1. a) Senor point, and geometry of the U-hape CM, and b) modal analyi for natural frequency at Hz. Table 1. Input parameter for modeling and imulation Specification Dimenion and Value Internal diameter of tube (D) m Thickne of tube 1/40D Material Stainle teel 316 Poion ratio 0.3 Young modulu 1.93E+11 Pa Denity of tainle teel 316 tube 8000kg/m 3 Vicoity of liquid fluid cst Denity of water 998.2kg/ m 3 Teted fluid velocitie 2m/ - 20m/ Number of time tep 100 Through the imulation, the time hift can be gained between the two enor point S 1 and S 2, a hown in ig 2. The blue curve plot the diplacement at monitor point 1, where the excitation force i applied. The red curve plot the diplacement at the enor point S 1 which i near the inlet. The green curve plot the meh diplacement at the enor point 2 which i near the outlet. 5

6 igure 2. Example reult of time difference between enor S 1 and S 2. The developed SI model are employed to invetigate the effect on fluid vicoity on the performance of the U-hape CM, including vicoitie 50 cst, 100 cst, 180 cst and 500 cst. The reult of time hift δt between enor S 1 and S 2 for the variou vicoitie are hown in ig 3. The time hift increae with the nominal flow velocity, and the drift from the reult of water (µ=1 cst) of time hift become obviou at high velocity. 6

7 igure 3. Time hift between enor S 1 and S 2 againt fluid vicoity. igure 4. Deviation to water metering of phae difference for different fluid vicoitie. 7

8 Since the ma flow rate m of a CM can be calculated by m K ft, where K f i the meter factor, and t i the time hift between the two enor S 1 and S 2, the deviation of metering vicou liquid can be calculated baed on water a benchmark, K ftvicou liquid K ftwater tvicou liquid twater Deviation 100% 100% (10) K ftwater twater The imulation reult have been plotted in ig 4 for the relation between the metering deviation and Reynold number Re Dv /. It i found that when the flow i laminar or tranient flow ( Re 4000 ), the deviation i fluctuating, while the deviation will be comparably flat when the flow i turbulent. It i known that fluid with different vicoity may have different damping factor (Kumar and Anklin 2011). When the ocillation of the tructure domain experience damping during fluid flow, the driver have to excite additional force to compenate for the amplitude lo caued by the fluid damping. Since the damping affect the natural frequency of the flow tube, the meter factor K f change, the change directly affect the ma flow rate. The effect of the reduction in natural frequency i caued by the interaction between the fluid and tructure dynamic in the CM. In the future compenation modeling, the damping effect hall be conidered. 4. CONCLUSION In thi paper, a computational fluid dynamic model of U-hape CM wa developed to invetigate the performance of CM influenced by on-line vicoity. The imulation reult were evaluated in term of the natural frequency of the vibrating ytem and the correponding time difference between the motion of the ening point ymmetrically located on the meauring tube. The imulation of water and vicou liquid were compared over flow rate panning the laminar and turbulent region. The effect of vicoity on CM performance have been dicued in detail. The finding in the imulation can be ued for further compenation of deviation due to vicoity effect for oil and ga indutry. Reference Bobovnik, G., N. Mole, J. Kutin, B. Stok and I. Bajic (2005). Coupled finite-volume/finite-element modelling of the traight-tube Corioli flowmeter. Journal of luid and Structure, 20, pp Henry, M., M. Tomb, M. Duta,. Zhou, R. Mercado,. Kenyery, J. Shen, M. Morle, C. Garcia and R. Langanan (2006). Two-phae flow metering of heavy oil uing a Corioli ma flow meter: A cae tudy. low Mea. Intrum., 17, pp Kumar, V. and M. Anklin (2011). Numerical imulation of Corioli flow meter for low Reynold number flow. Mapan, 26, pp Mole, N., G. Bobovnik, J. Kutin, B. Stok and I. Bajic (2008). An improved three-dimenional coupled fluid-tructure model for Corioli flowmeter. Journal of luid and Structure, 24, pp