Moelling of Transit-Time Ultrasonic Flo Meters Uner Multi-phase Flo Conitions Matej Simura an Lars Duggen Mas Clausen Institute University of Southern Denmark Sønerborg, 64, Denmark Email: simura@mci.su.k Email: uggen@mci.su.k Benny Lassen DONG Energy Denmark Nils T. Basse Siemens A/S Flo Instruments Sønerborg, 64, Denmark Abstract A pseuospectral moel for transit time ultrasonic flometers uner multiphase flo conitions is presente. The metho solves first orer stress-velocity equations of elastoynamics, ith acoustic meia being moelle by setting shear moulus to zero. Aitional terms to account for the effect of the backgroun flo are inclue. Spatial erivatives are calculate by a Fourier collocation scheme alloing the use of the Fast Fourier transform. The metho is compare against analytical solutions an experimental measurements. Aitionally, a stuy of clamp-on an in-line ultrasonic flometers operating uner multiphase flo conitions is carrie out. I. INTRODUCTION Transit-time ultrasonic flometers (TTUF) is a family of evices that measure flui flo rate base on a ifference of times it takes an ultrasonic signal to cross the pipe hen propagating ith an against the flo. There are to main solutions available on the market. In-line flometers are manufacture as a spool-piece ith embee transucers that is mounte irectly into a pipeline. In clamp-on configuration, transucers are mounte from the outsie so the measurement oes not affect the flo. As a trae off the signal nees to penetrate through the pipe all. Moreover, floing meia can consist of to or more substances of very ifferent acoustic impeance so the ultrasonic beam transmits through an reflects from multiple soli-soli, soli-flui an fluiflui interfaces affecting the measurement signal significantly. Numerical simulations are a useful tool for analyzing an possibly accounting for such scenarios. There have been several approaches publishe on this matter such as the use of geometrical acoustics [1], [] hich, hoever o not account for iffraction. Finite ifference (FD) [3] an finite element methos (FEM) [4] employ ave theory through solving the linearize Euler equations (acoustic meia) or the equations of linear elasticity (soli meia). These require a relatively high number of points per minimum avelengths (PPMW) hich can be an issue in terms of computing resources in case of ultrasonic flometer simulations here the signal usually propagates over a istance of hunres of avelengths. Bezek et al. [5] presente an alternative approach here FEM is only use in the soli parts an a bounary integral metho is applie in the floing part assuming a homogeneous flui. In this paper e focus on the propagation of ultrasonic signals in to phase flo here the to substances have a high acoustic impeance mismatch. This is the case for flo measurement of ater/air bubbles mixture here the scattering from bubbles affects the measurement accuracy significantly. The metho evaluates spatial erivatives by a Fourier collocation scheme alloing the use of the Fast Fourier transform hile a finite ifference scheme (the thir orer Runge-Kutta [6]) is use to avance in time. This approach is sometimes calle a pseuospectral time omain (PSTD) [7] metho. After valiating the coe against analytical solutions, moels of in-line an clamp-on ultrasonic flometers operating uner multiphase flo conitions an stuy of measurement accuracy are presente. II. GOVERNING EQUATIONS In this paper, e assume a moifie system of first orer partial ifferential equations coupling particle isplacement velocity v i an stress σ ij escribing propagation of elastic aves in isotropic meia [8], [9] σ ij t ρ v i t ( v k vi = λδ ij + µ + v j x k x j ( σij λδ ij v k x k λ = σ ij x j ) x i ), (1) + f i ρv k v i x k ρv k v i x k, () here x i, i = 1,, 3 are Cartesian position coorinates,λ, µ, ρ an f i enote the Lamé elastic constant, shear moulus, mass ensity an boy force respectively. In these equations the Einstein summation convention is use an δ ij is the Kronecker Delta. Acoustic meia can be moelle by setting shear moulus µ to zero an v k is the backgroun flo velocity that can vary both in time an space but is zero in regions of elastic meia.
III. NUMERICAL METHOD The Fourier collocation metho is implemente in this ork to evaluate all erivatives ith respect to to imensional physical space. Consier an arbitrary interval of length l iscretize into N evenly istribute points ith step size x (l = xn). The spatial ifferentiation can be then approximate by [1] [.] x Re ( F 1 (ikf(.)) ), (3) here F an F 1 are the iscrete Fourier an inverse Fourier transform of a function f(x) respectively F (f(x)) (k) = ˆf(k) = x F 1 ( ˆf(k) ) (x) = f(x) = 1 xn an k is the vector of avenumbers N f(x j )e ikxj, j=1 N ˆf(k j )e ikjx, j=1 {[ N k =, N + 1,..., N 1] π [ xn N 1, N 1 + 1,..., N 1 1 ] π xn (4) if N is even if N is o. Using the Fourier collocation metho inherently imposes perioic bounary conitions. In orer to suppress aves leaving omain on one sie to instantly reappear on the opposite sie multi-axial perfectly matche layers(m-pml) [11] are implemente in this ork. Stability an efficiency of the metho is improve by implementing spatial staggering such that the stress an the velocity components are evaluate on gris shifte by half of the gri spacing x/. Temporal staggering is impossible ue to the fact that the equations (1),() are not interlace [1] because of the presence of the backgroun flo terms v k. In this stuy, e consier flui mixtures of very high acoustic impeance contrasts. As the Fourier collocation metho expresses the solution in basis of trigonometric functions it ill inherently evelop numerical issues for problems ith such strong iscontinuities ue to the Gibbs phenomenon. An alternative approach is therefore taken, here zero values on stress variables are irectly impose at each stage of the time integration hich is an acceptable approximation as long as the acoustic impeance of the host meium is several orers of magnitue higher than the acoustic impeance of the secon phase. This ay e avoi numerical issues that oul be associate ith a spectral ifferentiation of mass ensity. Finally, the equations are integrate in time ith an explicit thir orer Runge-Kutta scheme [6]. IV. TEST CASES In this section a comparison against an analytical solution is presente an the valiity of the above mentione approach of simulating high contrast agents is iscusse. The metho is then valiate against experimental measurements. source a y receiver Fig. 1. Cylinrical inclusion in a homogeneous meium. All imensions are in mm TABLE I PARAMETERS OF THE MODEL OF POINT SOURCE IN A HOMOGENEOUS MEDIUM WITH A CYLINDRICAL INCLUSION x 7 mm a.65 mm µ meium.5 1 9 Pa ρ meium 14.4 kg/m 3 µ inclusion 1.44 1 5 Pa ρ inclusion 1. kg/m 3 x = y.11 mm t 1.54 1 8 s A. Point source in a homogenenous meium ith a cylinrical inclusion We consier shear aves propagating in x y plane in a homogeneous elastic meium ith a cylinrical cavity of raius a centere at the origin. A point source is locate at x s =, y s = an the isplacement is recore at a point x r =, y r =. Dimensions are epicte in Figure 1 an all parameters tabulate in Table I here t represent the time integration step. The material properties are chosen such that shear ave velocities of the host meium an the inclusion match longituinal velocities of ater an air respectively. Consiering the high impeance mismatch, the above mentione approach of assigning zero values on the stress variables at gri points of the cavity as taken in the moel. The input signal is a burst of five sine pulses of frequency f c multiplie by the Hanning ino. The simulation results ere compare to an analytical solution that as evaluate by numerically convolving Green s function [13] an the comparison for ifferent input signal center frequencies f c is presente in Figure. The moel agrees exceptionally ell in the range of lo frequencies, but exhibits amplitue iscrepancies in the range of shorter avelengths. This is very likely ue to the fact that the cylinrical cavity is approximate on the rectangular gri. It is noteorthy that there seems to be a threshol for the amplitue error, being practically zero at f c =.5MHz, but alreay about 5% at.75mhz an 15% at 1MHz. The iscretization length (i.e. istance beteen gri points) in these simulations as chosen as.11mm. Hence the threshol of circle iscretization error is beteen 7 an 18 PPMW. There is, hoever, no significant change in the signal envelope,
TABLE II PARAMETERS OF THE MODEL AND THE EXPERIMENTAL MEASUREMENT SETUP 39.4 mm l 138.7 mm 16.1 mm a.5, 3, 4 mm λ ater.5 1 9 Pa ρ ater 14.4 kg/m 3 x = y.11 mm t 1.54 1 8 s V pp 1 V f c 1 MHz Particle isplacement uz [m] 1 11.5 1.5 1.5.5 1 1.5 Numerical moel.5.1.1.14.16.18...4.6.8 (a) f c =.5 MHz hich is essential for transit time ultrasonic flo measurement applications. B. Experimental measurements A measurement setup similar to the test case iscusse in the previous subsection as use to verify the moel experimentally. To ultrasonic transucers consisting of a piezoceramic crystal ith center frequency f c = 1 MHz couple to a half-avelength steel ino ere immerse in a aterbath irectly facing each other. Seale cyliners of ifferent raii a ere manufacture from aluminium foil an use as air phantoms. The foil thickness is eclare to be approximately. mm an is not inclue in the moel as it is about 75 times smaller than the expecte avelength of 1 MHz signal in ater. The measurement setup is schematically epicte in Figure 3 an parameters tabulate in Table II. An input signal burst of 5 sine cycles ith peak-to-peak voltage V pp as generate by the Agilent 331A Waveform Generator an the receive voltage as measure over a 5 Ω resistor connecte in series in the receiver circuit. Maesurements ere carrie out using Agilent MSO614A igital oscilloscope an are shon in Figure 4. To transfer from electric signal to stress-velocity variables an back a 1-D transucer moel evelope by Willatzen [16] as use. Some parameters of the moel such as properties of the piezoceramic material ere foun by optimizing the error beteen the measurement an the simulation result for the case of propagation in homogeneous meium. It shoul therefore come as a no surprise that the curves epicte in Figure 4a are in such a goo agreement. The remaining comparisons exhibit some ifferences in both envelope shape an magnitue. Once the phantom has been introuce, the amplitue mismatch is about 11% hich fits fairly ell ith the error seen in Figure c. For the bigger phantom iameters, the amplitues scale quite similarly, i.e. the scaling factors beteen cases II,III, an IV are 1.55, 1.57 for experimental ata an 1.36, 1.56 for our simulation ata, respectively. Hence, the ifference in amplitue is explaine by the bubble iscretization iscusse in the previous section. The ifference in envelope shape is very likely ue to the transucer moel that assumes the crystal operation in thickness moe. Once an inclusion is place in the sonic path the resulting pressure on the receiver plane is not Particle isplacement uz [m] Particle isplacement uz [m] Particle isplacement uz [m].5 1 11 1.5 1.5 1 Numerical moel 1.5.1.11.1.13.14.15.16.17.18.19 1 11 1.5 1.5.5 1 1.5 1 1 1 8 6 4 4 6 8 1 (b) f c =.5 MHz Numerical moel.1.11.1.13.14.15.16 (c) f c =.75 MHz Numerical moel.1.11.1.13.14 () f c = 1 MHz Fig.. Comparison of the experimental measurements an the simulation results
transmitter a receiver.4.. Simulation, max. voltage:.33 V.4.9.1.11.1.13.14.15.16 l Fig. 3. Cylinrical inclusion in a homogeneous meium. All imensions are in mm TABLE III PARAMETERS OF THE CLAMP-ON MEASUREMENT SETUP h 1 mm h p 3.7 mm h f 13.9 mm 15 mm 3.5 mm a.8 mm ϑ 45 o λ 4.7 1 9 Pa µ 1.53 1 9 Pa ρ 18 kg/m 3 λ p 1.1 1 11 Pa µ p 8.4 1 1 Pa ρ p 785 kg/m 3 λ f.5 1 9 Pa µ f Pa ρ f 14.4 kg/m 3 x = y.14 mm t 4.87 1 9 s f c.77 MHz t elay 1 µs uniform possibly giving rise to transverse aves in the steel ino an other than thickness moes in the piezo crystal. Such behaviour cannot be escribe by the one imensional transucer moel an is therefore very likely the reason for the envelope mismatch. V. FLOWMETER SIMULATIONS In this section a stuy on impact of the multiphase flo on the accuracy of clamp-on an in-line ultrasonic flometers is presente. The moel of a clamp-on flometer is epicte in Figure 5 shoing a system of layers that represent coupling ege, pipe, an the floing flui respectively. The moel parameters are tabulate in Table III. A cluster of four air bubbles is positione at the center of the pipe ith its rightmost bubble lying on the ieal acoustic path an a uniform flo profile ith velocity v x is assume insie the pipe. The in-line flometer is simulate by setting material properties of the coupling ege an the pipe to be the same as for the flui an the receiver position is shifte appropriately so that the transucers are acoustically facing each other. The air bubbles are again simulate by setting zero stresses at these gri points. The input signal is a burst of eight sine pulses of frequency f c multiplie by the Hanning ino..4.. Measurement, max. voltage:.33 V.4.9.1.11.1.13.14.15.16..1.1 (a) Case I: homogeneous meium Simulation, max. voltage:.19 V..9.1.11.1.13.14.15.16..1.1 Measurement, max. voltage:.17 V..9.1.11.1.13.14.15.16..1.1 (b) Case II: = 4.1 mm Simulation, max. voltage:.14 V..9.1.11.1.13.14.15.16..1.1 Measurement, max. voltage:.11 V..9.1.11.1.13.14.15.16.1.5.5 (c) Case III: = 6 mm Simulation, max. voltage:.9 V.1.9.1.11.1.13.14.15.16.1.5.5 Measurement, max. voltage:.7 V.1.9.1.11.1.13.14.15.16 () Case III: = 8 mm Fig. 4. Comparison of the experimental measurements an the simulation results
h h p h p h y transmitter ϑ x h f v x h f / Simulate measurement error [%].3.3195.319 bubbles of raius a receiver Fig. 5. Clamp-on flometer scheme Homogeneous meium Bubbly flo ege pipe flui pipe ege.3185 1 3 4 5 6 7 Flo velocity [m/s] Fig. 6. Comparison of clamp-on flometer simulate measurement error for homogeneous meium an bubbly flo Simulate measurement error [%].185.18.175.17.165 Homogeneous meium Bubbly flo.16 1 3 4 5 6 7 Flo velocity [m/s] Fig. 7. Comparison of in-line flometer simulate measurement error for homogeneous meium an bubbly flo For each backgroun flo velocity the propagation of the signal in the onstream irection is calculate. The bubbles are then shifte in x irection by a istance x elay = v x t elay here t elay represents a elay beteen the upstream an onstream measurement. The upstream propagation is then simulate an the transit time ifference is calculate by cross correlating up- an onstream receive signals an the measurement error is subsequently etermine using the stanar flo measurement equation [14] an compare to the case of a homogeneous meium as in the previous stuy [17]. The consiere bubble istribution ensures that the acoustic path is not clear of bubbles hen the upstream measurement is simulate. In case of a homogeneous floing meium, the clampon simulations shon in Figure 6 yiel more than 99.6% accuracy, hile the in-line simulations shon in Figure 7 are even more precise ith 99.8% accuracy. The reason for 5 4 3 5 15 1 5 the better performance of the in-line moel is foun in the absence of step changes in material parameters beteen the pipe all an ater causing Gibbs oscillations in the clampon case. For bubbly flo conitions our simulations preict higher measurement errors, if the stanar flo measurement equation is employe. This is because the transit time signal is calculate by cross-correlating up- an onstream receive signals. Such metho expects signals of the same envelope hich is a reasonable assumption in case of the homogeneous flui but fails once multiphase flo of high acoustic countrast appears. We see in Figure 6 that our simulations preict up to 4% error, slightly ecreasing for higher flo velocities. For the in-line configuration consiere in Figure 7, the error is much larger, an is much more ifficult to preict. The reason for the ifference beteen the to scenarios lies in the shape of the acoustic beam. The signal frequency f c is chosen such that a resonance in the pipe all is achieve [15] so the signal raiates into the flui over a larger surface, making the flometer less sensitive to small shifts in bubble position. This effect is not foun in in-line configurations, here the acoustic beam is of comparatively small cross section. VI. CONCLUSION A goal of this ork as to simulate an ultrasonic flometer operating uner multiphase flo conitions in meia of high acoustic impeance contrast. For that purpose a pseuospectral time omain metho as evelope here scattering from high contrast agents is approximate by irectly imposing zero stresses in the areas of the secon phase. The metho as valiate against analytical solutions an shoe a goo agreement here the main reason for possible mismatches arise from the rasterization of the inclusions. This hoever only affects the receive signal amplitue an not the envelope hich is more important in transit time flo measurement. The moel as use to emonstrate ho a simple cluster of bubbles can result in a complete measurement misreaing of the flo rate hen the algorithms esigne for the homogeneous meia are use. ACKNOWLEDGMENT The authors oul like to acknolege support from the Innovation Fun Denmark an from the DeIC National HPC Center for access to computational resources on Abacus.. They oul also like to thank Anrei-Alexanru Popa for his help ith the experimental measurements. REFERENCES [1] B. Iooss, C. Lhuillier an H. Jeanneau, Numerical simulation of transittime ultrasonic flometers: uncertainties ue to flo profile an flui turbulence, Ultrasonics 4 () 19-115. [] H. Koechner an A. Mellin, Numerical Simulation of Ultrasonic Flometers, Acustica-acta acustica 86 () 39-48. [3] B. Fornberg, High-orer finite ifferences an the pseuospectral metho on staggere gris, SIAM J. Numer. Anal. 7 (199) 94-918. [4] D. V. Mahaeva an R. C. Baker an J. Woohouse, Stuies of the Accuracy of Clamp-on Transit Time Ultrasonic Flometers, IMTC IEEE (May 8) 969-973.
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