PRESSURE AND TEMPERATURE EFFECTS FOR ORMEN LANGE ULTRASONIC GAS FLOW METERS

Transcription

1 25 th Internatonal North Sea Flow Measurement Workshop, Gardermoen, Norway, October 2007 PRESSURE AND TEMPERATURE EFFECTS FOR ORMEN LANGE ULTRASONIC GAS FLOW METERS Per Lunde 1,2, Kjell-Evnd Frøysa 1 and Trond Folkestad 3 1) Chrstan Mchelsen Research AS (CMR), P.O.Box 6031 Posttermnalen, N-5892 Bergen, Norway. 2) Unversty of Bergen, Department of Physcs and Technology, Allégaten 55, N-5007 Bergen, Norway. 3) StatolHydro ASA, Sandslveen 90, P.O.Box 7190, N-5020 Bergen, Norway. ABSTRACT Ultrasonc gas flow meters (USMs) may be nfluenced by pressure and temperature n several ways. Change of the meter body's cross-sectonal area (the "ppe bore") nfluences drectly on the amount of gas flowng through the meter. Change of the ultrasonc path geometry (.e. change of the nclnaton angles and lateral chord postons, caused by e.g. meter body dameter change and change of the orentaton of the ultrasonc transducer ports) nfluences on the transt tmes and the numercal ntegraton method of the meter. Change of the Reynolds number nfluences on the ntegraton method. Change of the length of the ultrasonc transducer ports nfluences on the acoustc path lengths, and thus on the transt tmes. Lkewse, change of the length of the ultrasonc transducers nfluences on the acoustc path lengths, and thus on the transt tmes. In addton, changes of the transducer propertes such as the drectvty, nfluences on the dffracton correcton, and thus on the transt tmes. Some of these ssues are addressed to some extent n current draft standards for such meters, such as the AGA-9 (1998) report, and the ISO/CD (August 2007). Other of these effects have not been descrbed or treated n the lterature. In the present paper, pressure and temperature effects have been nvestgated for 18" Elster-Instromet Q-Sonc 5 ultrasonc flow meters (USMs) to be operated n the Ormen Lange fscal meterng system at Nyhamna n Møre and Romsdal, Norway, from October Pressure and temperature changes from flow calbraton (Westerbork, at 63 barg and 7 o C) to feld operaton (Ormen Lange, nomnally at 230 barg and 40 o C) condtons are evaluated. The effects addressed are changes related to (a) the meter's cross-sectonal area, (b) the ultrasonc path geometry (nclnaton angles and lateral chord postons), (c) length expanson of the ultrasonc transducer ports, (d) length expanson/compresson of the ultrasonc transducers, and (e) Reynolds number correcton. The varous effects (a)-(e) contrbutng to the measurement error are dscussed and quantfed. Investgatons are made usng a combnaton of analytcal modelng and fnte element numercal modelng of the meter body and the ultrasonc transducers, combned wth a model for USM numercal ntegraton relevant for the Q-Sonc 5 multpath ultrasonc flow meter n queston. It s shown that for the Ormen Lange applcaton, nvestgaton and evaluaton of all of the factors (a)-(e) mentoned above have been necessary to evaluate the effect of pressure and temperature on the meter. Expressons for pressure and temperature effects on ultrasonc flow meters proposed n ISO/CD do not appear to be preferred for the Ormen Lange fscal meterng system. The study shows that pressure and temperature affects the Q-Sonc 5 by about 0.26 % n the Ormen Lange applcaton. If ths systematc measurement error s not corrected for, the Q-Sonc 5 wll underestmate the volumetrc flow rate by the same amount. Sgnfcant economc values are nvolved. Two correcton factors are thus proposed for the Q-Sonc 5 n ths applcaton: (1) one "nomnal P&T correcton factor" (accountng for by far the largest part of the correcton, about 0.26 %), and (2) an "nstantaneous P&T correcton factor" (accountng for small devatons n pressure and temperature from nomnal to actual Ormen Lange condtons), whch s typcally an order of magntude smaller than the nomnal P&T correcton factor. The correcton factors and the ndvdual contrbutors to these are dscussed and quantfed. 1. INTRODUCTION From October 2007 fve 18" Elster-Instromet Q-Sonc ultrasonc gas flow meters wll be operated at a land based fscal gas meterng staton at Nyhamna, Møre and Romsdal, Norway, for export of

2 2 gas through the 1200 km Langeled ppelne, to an mport meterng staton n Easngton, UK, bult by Statol, cf. Fg.1. The Ormen Lange export staton at Nyhamna was constructed and bult by Norsk Hydro, and wll be operated by Shell. The producton lfe of Ormen Lange s estmated to 50 years. The nomnal flow rate of the Ormen Lange export meterng staton s 70 mllon Sm 3 /day, or 25 bllon Sm 3 /year. At an assumed sales prce of 2 NOK/Sm 3 ths corresponds tentatvely to 140 mllon NOK/day, or 50 bllon NOK/year. An assumed systematc measurement error of only 0.3 % (as an example), would correspond to about NOK/day, or about 153 mllon NOK/year, for such a tentatve sales prce. Flow calbraton of the flow meters have been made at the Westerbork laboratory n the Netherlands, at temperature and pressure condtons of 7 o C og 63 barg, respectvely, wth two meters n seres nstalled n a "long ppe", and wth flow condtoner upstream of the meters. The hgh pressures n queston at the Ormen Lange meterng staton, 230 barg nomnal, have rased the queston whether correcton for pressure and temperature effects on the ultrasonc meters wll be needed, relatve to the 63 barg pressure used under flow calbraton at Westerbork. Pressure and temperature effects on the ultrasonc meters relates to factors such as e.g 1. Change of the meter's cross-sectonal area, Change of the ultrasonc path geometry (nclnaton angles and lateral chord postons), Change of the length of the ultrasonc transducer ports, Change of the length of the ultrasonc transducers, Change of the Reynolds number. The nfluence of these factors are addressed here, on bass of the results gven n [1]. 2. SPECIFICATIONS The Ormen Lange meterng staton conssts of 3 parallel meter runs, wth n total 5 ultrasonc flow meters, cf. Fg. 1: 2 parallel runs, each wth two 18" ultrasonc flow meters n seres, 1 parallel run wth one 18" ultrasonc flow meter, for backup measurement, Flow condtoner wll be used (DN450 Laws type 316SS or Duplex Materal), Elster-Instromet Q-Sonc 5 ultrasonc gas flow meters [2]. Table 1 gves varous parameters of the USM, and Table 2 other specfcatons for the study. 1 In general, the propertes of an ultrasonc flow meter wll also depend on the pressure and temperature propertes of the ultrasonc transducers used n the meter that s, the electrcal and acoustcal propertes of the transducers, whch for a large part determne the sgnal form, etc.). These are factors whch relate to the tme detecton of the meter (the sgnal processng). There s no evaluaton of such factors n the present study, snce these types of effects f the transducers functon as they should are not consdered to be very sgnfcant (several decades smaller than the other effects) n a 18" meter wth reflectng paths. (However, f pressure and temperature cause effects such as perod error, transducer error or defect, etc., that would of course be serous and sgnfcant.) Pressure test certfcates for the K10 transducers used n Q-Sonc 5 gven n [19] show that the transducers have survved pressure testng to 620 bar, n water at room temperature.

3 3 Table 1. Specfcatons of the ultrasonc flow meters used n the Ormen Lange meterng staton. Parameter Property Condtons Materal type Steel (Duplex) Length 1800 mm (at assumed 20 o C, 1 atm.) Outer dameter, OD mm (at assumed 20 o C, 1 atm.) Inner dameter, ID (366.5 ± 0.25) mm (at assumed 20 o C, 1 atm.) Inner radus, R mm (at assumed 20 o C, 1 atm.) Wall thckness, w mm (at assumed 20 o C, 1 atm.) w/r (at assumed 20 o C, 1 atm.) Young s modulus, Y MPa Posson s rato, σ 0.3 Coeff. of lnear thermal expanson, α K -1 (ASME) Table 2. Specfcatons for the study. Parameter Westerbork Ormen Lange meterng staton flow calbraton condtons (lne condtons, nomnal) Gas Dry natural gas Dred natural gas a) Pressurel P 63 barg 230 barg (desgn) Temperature 7 o C 40 o C (desgn) Vscosty Pa-s Pa-s Densty kg/m kg/m 3 Meterng confguraton 2 USMs n seres, wth upstream flow condtoner 2 USMs n seres, wth upstream flow condtoner Flow velocty m/s Volumetrc flow rate 70 MSm 3 /d (=> flow velocty = m/s per run) Reynolds number, Re a) The gas composton s known, but has not been necessary to specfy for the present study. (a) (b) Fg. 1. (a) Photograph of the Ormen Lange fscal gas meterng staton, under factory acceptance test (FAT) n Athens, Greece, (b) Sketch of the Ormen Lange transport system, wth fscal meterng statons at Nyhamna (Norway) and Easngton (UK).

4 4 3. MULTIPATH ULTRASONIC GAS FLOW METERS 3.1 USM functonal relatonshp In ultrasonc transt tme flow meters wth reflectng and/or non-reflectng paths, the volumetrc flow rate (at lne condtons) s gven as [3-5] q = πr, va = wv, 2 USM v A N = 1 v 2 R y ( t t ) = ( Nrefl, + 1), (1) t1 t2 sn 2φ where (cf. Fg. 2), R s the nner radus of the USM meter body; v A s the axal volume flow velocty (at lne condtons); N s the number of acoustc paths; s the path number; w s the ntegraton weght factor for path no. ; v s the average axal flow velocty along path no. (.e. the lne ntegral along the path); y s the lateral dstance from the ppe center (lateral chord poston) for path no. ; L s the nterrogaton length for path no. ; φ s the nclnaton angle (relatve to the ppe axs) of path no. ; t 1 and t 2 are the measured transt tmes for upstream and downstream sound propagaton of path no. ; and N refl, s the number of wall reflectons for path no. (N refl, = 0, 1 or 2 n current USMs), = 1,, N. z Recevng transducer Recevng cables & electroncs Pulse detecton x L p 2R y φ Sgnal generator Transmttng cables & electroncs Transmttng transducer TOP VIEW FRONT VIEW Fg. 2. Schematc llustraton of a sngle path n a multpath ultrasonc transt tme flow meter wth non-reflectng paths (for downstream sound propagaton). (Left: centre path example (y = 0); Rght: path at lateral chord poston y.) 3.2 Pressure and temperature nfluences on USMs Ultrasonc gas flow meters may be nfluenced by pressure and temperature n several ways, cf. Table 3. Change of the meter body's cross-sectonal area (the "ppe bore") nfluences drectly on the amount of gas flowng through the meter. Change of the ultrasonc path geometry (.e. change of the nclnaton angles and lateral chord postons, caused by e.g. meter body dameter change and change of the orentaton of the ultrasonc transducer ports) nfluences on the transt tmes and the numercal ntegraton method of the meter. Change of the Reynolds number nfluences on the ntegraton method. Change of the length of the ultrasonc transducer ports nfluences on the acoustc path lengths, and thus on the transt tmes. Lkewse, change of the length of the ultrasonc transducers nfluences on the acoustc path lengths, and thus on the transt tmes.

5 5 Table 3. Drect and ndrect pressure and temperature nfluences on USMs. Drect P&T effect Indrect P&T effect A Change of the meter body cross-sectonal area Affects amount of gas flowng through the flow meter B Change of the ultrasonc path geometry (changed nclnaton angles and lateral chord postons, caused by dameter change & changed transducer port orentaton) Affects acoustc path lengths and thus transt tmes. Influences on the numercal ntegraton method. C Change of the length of the ultrasonc transducer ports Affects acoustc path lengths and thus transt tmes. D Change of the length of the ultrasonc transducers Affects acoustc path lengths and thus transt tmes. E Change of the Reynolds number Influences on the numercal ntegraton method. 3.3 Elster-Instromet Q-Sonc 5 The Q-Sonc 5 ultrasonc meter employs 5 acoustc paths. Three of these paths are sngle reflecton paths. These paths are denoted path 1, 3 and 5, cf. Fg. 3. These paths are centre paths. Ths means that n the sde vew of Fg. 3, these 3 paths are represented by the three straght lnes gong through the centre of the ppe (y = 0). The nclnaton angle of these paths s typcally 70 for the meters n queston at Ormen Lange. The two remanng paths are double reflectng paths. These paths are denoted path 2 and 4, cf. Fg. 3. These paths propagate at a lateral dstance, y, of 0.5 R. In the sde vew of Fg. 3b, each of these paths s represented as a trangle. The nclnaton angle of these paths s typcally 60 for the meters n queston at Ormen Lange. (a) (c) Fg. 3. Meter body and path geometry of the Elster-Instromet Q-Sonc 5 ultrasonc gas flow meter (after [2].) In the present work, pressure and temperature effects for each of the fve acoustc paths are frst analyzed ndvdually. Thereafter, values for the pressure and temperature effect on the volumetrc flow rate are found through ntegraton over the fve acoustc paths, usng the ntegraton weght factors, w, = 1,, 5, cf. Eq. (1) and e.g. [22]. The ntegraton weghts of the Q-Sonc 5 have not been avalable for the present study. A tentatve set of ntegraton weght factors, w, = 1,, 5, has thus been worked out for the Q-Sonc 5. Ths gves one ntegraton weght factor for each of the three sngle-bouncng paths (centre paths), and a second one for each of the two double-bouncng paths (swrl paths). Ths set s the best estmate that has been possble to obtan for the present work, and has been used n the analyss presented here [1]. (b)

6 6 4. P RESSURE AND TEMPERATURE INFLUENCES ON THE METER CROSS-SECTIONAL AREA AND ULTRASONIC PATH GEOMETRY The present secton addresses changes n the USM's cross-sectonal area (dameter) and ultrasonc path geometry (nclnaton angles and lateral chord postons) caused by changes n pressure and temperature, and the consequences of such changes for the measurement accuracy. In partcular, ths relates to (a) changes n the Q-Sonc 5 ultrasonc meter's cross-sectonal area and ts path geometry, from factory ("dry calbraton") to Westerbork (flow calbraton) condtons, (b) changes n the Q-Sonc 5 cross-sectonal area and ts path geometry, from Westerbork (flow calbraton) to Ormen Lange (operatng) condtons, (c) the effect of these changes on the Q-Sonc 5 measurement uncertanty at Ormen Lange (operatng) condtons. In Secton 4.1, smplfed analytcal models for pressure and temperature expanson / contracton are dscussed and used for descrbng (a) - (c). In Secton 4.2, fnte element numercal modellng (FEM) s used, as a more accurate approach. Calculaton results are gven n Secton Smplfed analyss Analytcal models generally represent smplfed descrptons of pressure and temperature effects on USM cross-sectonal area and ultrasonc path geometry, both wth respect to meter geometry and valdty ranges, but may be useful for certan purposes, dependng on ther accuracy. Varous models used n the lterature and nternatonal standards to correct for pressure and temperature expanson of USMs are dscussed Analytcal Model A At a temperature T and pressure P, the meter body radus (R), the lateral chord postons (y ), and the nclnaton angles (φ ), can be shown to be approxmately gven by [5,1] R KT K PR 0, KT K P y0 y, 1 tan( φ 0) φ tan, = 1,, N (2) * 1 (1 β β )( K P 1) where subscrpt 0 s used to denote the respectve geometrcal quantty at "dry calbraton" condtons,.e. R 0, y 0 and φ 0. The correcton factors for the nner radus of the meter body due to dmensonal changes caused by temperature and pressure changes relatve to dry calbraton condtons, are gven as (cf. e.g. [6,7,8,10]) K K 1 + αδ, ΔT dry T-T dry, (3) T T dry 1 + βδ, ΔP dry P-P dry, (4) P P dry respectvely, where P dry and T dry are the pressure and temperature at dry calbraton condtons, e.g. P dry = 1 atm. and T dry = 20 o C. α s the coeffcent of lnear thermal expanson of the meter body materal. β and β* are the radal and axal lnear pressure expanson coeffcents for the

7 7 meter body, respectvely (cf. Secton 4.1.3). K P and K T are here referred to as the radal pressure and temperature correcton factors for the USM meter body, respectvely 2. Eqs. (2)-(4) are referred to as the "analytcal model A", and apples to all nclnaton angles Analytcal Model B For USMs where all nclnaton angles are equal to ±45 o,.e, φ 0 = ±45 o, = 1,, N, q USM can - from Eqs. (1)-(4) - be wrtten as [5,1] q USM qusm, 0 Ctsm C psm, (5) where C tsm 3 = K = (1 + αδt ) 1+ 3αΔT 3 T dry 3 3 dry, C psm = K P = (1 + βδpdry ) 1+ 3βΔPdry, (6) are the volumetrc thermal and pressure correcton factors of the USM meter body 3, and q USM, 0 s gven by Eqs. (1)-(2), wth the "dry calbraton" quanttes R 0, y 0, L 0, x 0 and φ 0 nserted nstead of the quanttes R, y, L, x and φ, = 1,, N. Eqs. (5)-(6) are referred to as the "analytcal model B". For USMs wth nclnaton angles equal to ±45 o, thus, the analytcal model B s equvalent to the analytcal model A. For other nclnaton angles t represents an approxmaton to the more accurate analytcal model A [5]. Ths s the case for Q-Sonc 5, whch employ nclnaton angles of 60 o and 70 o. Eq. (D.28) n [5] gves the relatve error by usng ths approxmaton. It turns out that for moderate pressure devatons Δ Pdry (a few tens of bars), the errors made by usng analytcal model B may be neglected, and that ths model may be used for nclnaton angles n the range of relevance for current USMs, 40 o to 60 o. However, for larger pressure devatons, and especally for nclnaton angles approachng 60 o, the error ntroduced by usng Eqs. (5)-(6) ncreases. The man advantage of analytcal model B over A lays n the fact that n model B, the P and T correctons of the geometrcal quanttes of the meter body can be separated from the basc USM functonal relatonshp and put outsde of the summng over paths, as llustrated by Eq. (6). Consequently, snce the analytcal model A s not easly applcable for use n the "nstantaneous correcton factor" to be mplemented at the flow computer level, Eqs. (5)-(6) s the model proposed for the "nstantaneous correcton factor" descrbed n Secton 7.2, used for relatvely small pressure changes only (a few bar) Coeffcents of lnear pressure expanson The radal and axal lnear pressure expanson coeffcents β and β* nvolved n the analytcal models A and B depend on the type of support provded for the meter body nstallaton (.e. the 2 The radal pressure and temperature correcton factors for the USM meter body, K P and K T, should not be confused wth the correspondng volumetrc pressure and temperature correcton factors of the meter body, C psm and C tsm, cf. e.g. Eqs. (6). 3 For the correcton factor of the meter body, a notaton s used accordng to common flow meterng termnology, where subsrpts t, p, s and m refer to temperature, pressure, steel and meter, respectvely, cf. e.g. [6-8].

8 8 model used for the meter body pressure expanson / contracton). For thn-walled cylndrcal and sotropc elastc meter bodys, β and β* are related by [1] σ = 0.3 for the cylndrcal ppe secton model (ends free), * β 0 β for the nfnte-length cylndrcal ppe model (ends clamped), (7) 1 2σ = σ for the cylndrcal tank model (ends capped), where the values gven for β * β apply to steel (σ = 0.3). Wth respect to K T and K P, there seems to be general agreement n the lterature that temperature and pressure expanson/contracton can be descrbed by expressons such as Eqs. (3)-(4). However, there s wdely vared practce wth respect to whch model s used for the coeffcent of lnear radal pressure expanson, β. Table 4 gves dfferent models n use for β, and a dscusson of these s gven n [1]. Note that all models n use represent smplfcatons. Table 4. Models used by USM manufacturers, standards, etc. for lnear pressure expanson of the nner radus of the USM meter body (sotropc materal assumed), under unform nternal pressure. Reference / USM manufacturer AGA-9 (1998) [10], Roark (2001), p. 592 [11] Danel Industres (2001) [20,5] FMC Technologes (2001) [21], [5], Roark (2001), p. 593 [11] Instromet (2001) [12], [5] ISO/CD (2007) [3] ISO/CD (2007) [3] Models for the coeffcent of lnear radal pressure expanson, β USM meter body assumptons R 0 Cylndrcal ppe secton model β = (ends free) wy Thn wall, w < R 0 / ( R0 + w ) + 0.4R Cylndrcal tank model 0 β = (ppe wth ends capped) 2 2 Y ( R0 + w ) R0 Thck wall R Steel materal (σ = 0.3) 0 ( β 0.85 for w << R 0 ) wy R 0 σ β = 1 wy 2 R0 ( β = 0.85 for σ = 0.3 (steel)) wy No P or T correcton used. Pressure expanson analyss based on: R0 β = 0.5 wy 1 3 D 0 R β = = wY wy 1 7D0 7 R0 R0 β = = = wY 6 wy wy Cylndrcal tank model (ppe wth ends capped) Thn wall, w < R 0 /10 Infntely long ppe model (ends clamped, no axal dsplacement) Radal expanson taken to be = 0.5 radal expanson for ends-free model Thn wall, w < R 0 /10 Flanged-n meter body Thn wall, w < R 0 /10 (?) Welded-n meter body Thn wall, w < R 0 /10 (?) The cylndrcal ppe secton model (ends free) [11] (1 st row of Table 4) apples to a fntelength ppe secton wth free ends and does not account for flanges, bends, etc. It s consdered to be relevant for thn-walled meter bodes mounted n ppe sectons where ends can move relatvely freely (note that accordng to the FEM analyss of Secton 4.2, axal dsplacements are n the submm range), e.g. wth U-bend as part of the ppe secton. Calculatons usng ths model are confrmed relatvely well by the FEM calculatons descrbed n Secton 4.2 (whch do account for flanged meter bodes), cf. Secton 4.3. For the Ormen Lange meterng staton, ths model for β s

9 9 consdered to be the most relevant of the analytcal models gven n Table 4, and consequently used here (cf. Secton 7.2). The two expressons proposed n ISO/CD [3], gven n the latter two rows of Table 4, and clamed to cover "flanged-n meter body" (5 th row) and "welded-n meter body" (6 th row), respectvely, gve 50 % smaller and 17 % hgher radal dsplacement than the ends-free model (1 st row). These models are not found to be very relevant for use n correcton factors for the Ormen Lange fscal meterng staton, bascally for two reasons: (a) no documentaton or references for the β expressons has been gven n ISO/CD (.e. no traceablty), and (b) the expressons gven n ISO/CD are not confrmed by the FEM calculatons, cf. Secton Fnte element modellng (FEM) analyss The analytcal models A and B descrbed n Secton 4.1 represent smplfed descrptons, accountng for "average" effects only, and are n general not able to account very precsely for the effects of P&T on the meter body. To analyze such effects n more detal and more accurately (ncludng effects of flange thckness, wall thckness, the resultng form of the meter body (e.g. ppe bulgng), nfluence of the transducer ports and ther locaton, dsplacement of the transducer ports, precse calculaton of the ultrasonc path lengths, etc.), a numercal fnte element model (FEM) s needed. Thus, as a second and consderably more accurate step to analyze pressure and temperature effects on the Q-Sonc 5 ultrasonc meter, a FEM approach was used. The fnte element mesh used for FEM calculatons of the Q-Sonc 5 meter body s shown n Fg. 4a. Dmensonal changes caused by temperature and pressure changes, at any poston of the meter body, are calculated usng FEM. Detals are gven n [1]. Wth respect to boundary condtons, the model of the meter body s fxed as follows, cf. Fg. 4: In vertcal drecton (x-drecton) n two ponts correspondng to bolts n 3 o clock and 9 o clock postons. In axal drecton (y-drecton) n the bolt's crcle dameter. In transveral drecton (z-drecton) n one pont correspondng to a bolt n 6 o clock poston. Ths means that the centre of the ppe n prncple does not move n vertcal and transversal drecton, and that no constrant loads wll appear. (a) (b) Fg. 4. (a) Fnte element grd used for FEM analyss of P & T effects on the meter body (spoolpece) of the Q- Sonc 5 ultrasonc gas flow meter. (b) Sketch of ppe secton accounted for n the FEM analyss.

10 10 Due to the U-bend and the header at the opposte sde of the USM n the Ormen Lange meterng staton (cf. Fg.1), t s assumed that the meter body can expand freely n the axal drecton (as for the ends-free analytcal model for β chosen n Secton 4.1.3) (note that axal dsplacements are n the sub-mm range). Flanges and the 10 transducer mountng ports of the Q-Sonc 5 meter body are all accounted for, as well as axal forces on the meter body flanges caused by the assocated 18" ppe secton n whch the Q-Sonc 5 s mounted, cf. Fg. 4 [1]. The materal data used for the calculatons are gven n Secton 2. The FEM calculatons gve the change of poston for every pont of the meter body. Thus, the change of dameter n the horzontal and vertcal drectons and changes wth respect to the transducer ports (.e. nclnaton angles and drectonal orentaton of the port (rotaton, etc.)), are calculated. Ths ncludes the rotaton around the vertcal, axal and transversal axes, of the back plane of the transducer ports. The results of the FEM calculatons are used as nput data to calculaton of the pressure and temperature effects on the ndvdual acoustc paths, and the pressure and temperature effects on the USM volumetrc flow rate measurement, through ntegraton over the 5 acoustc paths of the USM, usng the tentatve Q-Sonc 5 weght factors dscussed n Secton 3.3 [1]. 4.3 Calculaton results Frst, consder the accuracy of the dfferent models for radal lnear pressure expanson coeffcent β gven n Table 4. Table 5 provdes a comparson of 4 of the dfferent β models gven n Table 4, compared to the results of the FEM analyss 4. It appears from the pressure effect results gven n the table that the cylndrcal ppe secton model (ends free) (1 st row of Table 4) gves the best approxmaton to the FEM results. A relatvely good agreement s found between the FEM calculatons and ths β model, for both cases. The flanged-n and welded-n models proposed n ISO/CD [3] represent another type of correcton than found to be recommended here, and s not used for the Ormen Lange fscal meterng staton, Table 5. Calculated change n USM meter body radus, ΔR [mm], due to pressure and temperature expanson/contracton, for 4 dfferent models for β gven n Table 4, compared to the results of the FEM analyss. Two cases are consdered: (a) "dry calbraton" to flow calbraton (Westerbork) condtons, and (b) flow calbraton (Westerbork) to operatng (Ormen Lange) condtons. Flow calbraton (Westerbork) 7 o C, 63 barg Operaton (Ormen Lange) 40 o C, 230 barg Temperature Pressure P & T Temperature Pressure P & T effect effect effect effect effect effect ΔT = -13 o C ΔT = 0 o C ΔT = -13 o C ΔT = 33 o C ΔT = 0 o C ΔT = 33 o C ΔP = 0 bar ΔP = 63 bar ΔP = 63 bar ΔP = 0 bar ΔP = 167 bar ΔP = 167 bar Cylndrcal ppe secton β model (ends free) [AGA-9, 1998] [Roark, 2001, p.592] Cylndrcal tank β model (ends capped) [Roark, 2001, p.593] Flanged-n meter body β model [ISO 17089] Welded-n meter body β model [ISO 17089] FEM In the FEM analyss of pressure effects, the calculated dameter change s dfferent n the vertcal and horzontal drectons, due to the asymmetrc dstrbuton of the transducer ports (all located at the upper half of the meter body, cf. Fg. 4a). In the FEM results of Table 5, the vertcal and horzontal dameter changes have been averaged.

11 11 Next, consder the error n the volumetrc flow rate as measured by the USM caused by pressure and temperature effects on the meter dameter and the ultrasonc path geometry. For the change from flow calbraton (Westerbork) to feld operaton (Ormen Lange) condtons, there s a sgnfcant systematc shft n the volumetrc flow rate due to pressure and temperature effects, rangng approxmately from about 0.24 (double reflectng paths) to about 0.27 % (sngle reflecton paths) for the varous acoustc paths [1]. Integrated over the 5 acoustc paths the effect s calculated to %, cf. Fg. 5. It s also seen from Fg. 5 that the effect of the pressure effect solated s calculated to % and the temperature effect solated s calculated to %, whch act n the same drecton snce both the temperature and the pressure ncreases from Westerbork to Ormen Lange condtons. If not corrected for, the Q-Sonc 5 wll thus underestmate the volumetrc flow rate. For the analytcal model approaches, the correspondng results become % for the analytcal model A (0.125 % both for temperature and pressure effects (2) Flow calbraton cond. (Westerbork) Operaton (Ormen Lange) solated), and % for the more 7 C / 63 barg 40 N 2 C / 230 barg 2 L Dfference: 33 ( t1 t2 ) C / 167 bar q USM = π R w = 1 2 x t 1 t 2 smplfed analytcal model B, 7 C, 63 barg 40 C, 230 barg vald at nclnaton angles of 45 (Westerbork) (Ormen Lange) (0.10 % for the pressure effect Rel. dff Path no. R²L²/X [%] solated and % for the temperature effect solated) [1] The dfference between the results Integrated from the analytcal model A and Rel. dff Rel. dff the FEM results are therefore only Path no. R²L²/X [%] Path no. R²L²/X [%] %, and the dfference between the FEM results and the Integrated results from the smplfed Integrated C, 230 barg analytcal model B s %. Fg. 5. Effect of pressure and temperature on the measured volumetrc flow rate for the Elster-Instromet Q-Sonc 5 ultrasonc gas flow meter, calculated from the FEM analyss results, for Westerbork (flow calbraton) to Ormen Lange (operatonal). condtons (After [1].) Ths result s of hgh nterest n the sense that the FEM results may be used for the correcton from Westerbork condtons to nomnal Pressure change Temperature change Ormen Lange condtons (the "nomnal PT correcton factor"), and the analytcal model B may be used for the remanng correcton from nomnal to the actual Ormen Lange lne condtons (the "nstantaneous PT correcton factor"), cf. Secton P RESSURE AND TEMPERATURE INFLUENCES ON TRANSDUCER PORTS, TRANSDUCER LENGTH AND ACOUSTIC PATH LENGTH Changes n pressure and temperature nfluence on the length of the transducer ports n whch the ultrasonc transducers are mounted, the length of the ultrasonc transducers themselves, and consequently on the length of the acoustc paths, and thus the measured transt tmes. The present secton addresses these length changes, and the consequences for the measurement error. In partcular, ths relates to

12 12 (a) length changes from Westerbork (flow calbraton) to Ormen Lange (operatng) condtons, (b) the effect of these changes on the Q-Sonc 5 measurement uncertanty at Ormen Lange (operatng) condtons. 5.1 Expanson / compresson of the transducer ports Changes n pressure and temperature wll nduce expanson/compresson of the transducer ports and thus changes n the length of the ports, whch nfluences on the acoustc path lengths, and thus on the transt tme measurements. A detal study of the pressure and temperature nduced expanson / compresson of the transducer ports of the Q-Sonc 5 ultrasonc meter s gven n the followng, for the Ormen Lange applcaton. The analyss s based on the fnte element modellng (FEM) numercal calculatons descrbed n Secton 4.2. Fg. 4a gves the fnte element mesh used for modelng of the Q-Sonc 5 meter body wth transducer ports, and Fg. 6 a detal drawng of a transducer port of ths meter. In the FEM calculatons, each transducer port s assumed to be covered wth a pressure tght and rgd plate (nstead of the transducer flange), so that pressure-nduced length changes of the port can be descrbed, from Westerbork (flow calbraton) to Ormen Lange (operatng) condtons. Actual and realstc length changes are calculated for each of the 10 ports n the 5 acoustc paths. The calculated length changes of the transducer ports range from mm to mm, dependng on the path no., cf. Table 6 [1]. To llustrate the basc analyss and ndcate the level of sgnfcance, consder an average length change of about, say, 0.12 mm, as a smplfed and prelmnary approach. If the length change s not corrected for, the (solated) measurement error due to port length changes becomes, approxmately [1], Δv ΔL mm 2 = = 0.06%, (8) v L 780mm where L s the nterrogaton length of path no., = 1,, N, and pressure and temperature effects. Δ L s the change of L due to However, note that when the two effects of the changed transducer ports and the changed transducer length are combned, the nfluence on the Q-Sonc 5 volumetrc flow rate measurement n the Ormen Lange applcaton becomes dfferent from Eq. (8), cf. Secton Expanson / compresson of the ultrasonc transducers Fg. 6. Detal drawng of a transducer port of the Q-Sonc 5 meter body, as accounted for n the FEM analyss of the meter body. Fg. 7 shows photographs of an Q-Sonc 5 ultrasonc transducer of the K2 type used n the Ormen Lange applcaton. Changes n pressure and temperature wll nduce expanson/compresson of

13 13 the transducers, whch nfluences on the acoustc path lengths, and thus on the transt tme measurements. An extract of a detal study of the expanson / compresson of the Q-Sonc 5 ultrasonc transducer n the Ormen Lange applcaton s gven n the followng. The analyss s based on fnte element (FEM) numercal calculatons of pressure and temperature nduced length changes of the transducer [1]. In partcular, ths relates to (a) changes of the transducer length, from Westerbork (flow calbraton) to Ormen Lange (operatng) condtons, and (b) the effect of these changes on the Q-Sonc 5 measurement uncertanty at Ormen Lange (operatng) condtons. (a) (b) Fg. 7. Photographs of an Elster-Instromet Q-Sonc 5 ultrasonc transducer of the K2 type used n the Ormen Lange applcaton. Left: sde vew; Rght: front vew. (After [1].) The FEM analyss s based on nformaton provded by Elster-Instromet [12], n addton to "qualfed guess" on some of the constructon detals of the transducer. It s known that the actve acoustc part of the transducer (the pezoelectrc element, etc., n the front of the transducer) s pressure equalzed, that the pressure barrer s located behnd ths regon, and that mportant parts of the transducer nteror are made up of epoxy [12]. However, the nformaton provded on the constructon detals of the transducer was n general nsuffcent to do a precse FEM analyss of the pressure and temperature expanson/compresson of the transducer. Thus, some assumptons had to be made n relaton to constructonal detals, materals used, and materal propertes. The calculatons gven below are based on the best possble "qualfed guess" of the constructonal detals of the transducer one could establsh. A fnte element model mesh of the transducer was thus developed, and fnte element calculatons made, n two steps: () changes of the transducer length, from factory ("dry calbraton") to Westerbork (flow calbraton) condtons, and () changes of the transducer length, from factory ("dry calbraton") to Ormen Lange (operatng) condtons. These are then combned to evaluate the changes of the transducer length, from Westerbork (flow calbraton) to Ormen Lange (operatng) condtons. In short, the FEM calculatons ndcate that a pressure ncrease from 63 to 230 barg leads to a change n the transducer length of mm (compresson) [1]. A temperature ncrease from 7 to 40 o C leads to a calculated change n the transducer length of mm (expanson) [1]. Some further detals are gven n [1]. The combned effect of the calculated pressure nduced

14 14 compresson of mm and the calculated temperature nduced expanson of mm becomes [ ] mm = mm (expanson). Thus, the FEM results ndcate that the temperature effect domnates and that the transducer ncreases n length by mm from Westerbork to Ormen Lange condtons. It should be emphaszed that, from the above dscusson on the FEM analyss of the transducer, t s evdent that due to the uncertantes n relaton to some of the constructonal detals and the materals used, the calculated change n transducer length ( mm) s assocated wth some uncertanty (whch s dffcult to estmate, however). If ths length change s not corrected for, the measurement error due to transducer expanson becomes, approxmately [1] (by notng that a transducer expanson corresponds to negatve Δ L ), Δv ΔL 2 ( ) mm 2 = = %. (9) v L 780mm However, note that when the two effects of the changed transducer ports and the changed transducer length are combned, the nfluence on the Q-Sonc 5 volumetrc flow rate measurement n the Ormen Lange applcaton becomes dfferent than predcted by Eq. (9), cf. Secton Combned effect, n relaton to change of acoustc path length The results of Sectons 5.1 ndcate that pressure and temperature effects on the transducer ports wll lead to ncreased acoustc path length, n the range from mm to mm, dependng on the path no. On the other hand, the results of Sectons 5.2 ndcate that pressure and temperature effects on the transducers themselves wll lead to a decreased acoustc path length, by about mm. Consequently, n the Ormen Lange applcaton the effect of changed transducer port length and changed transducer length partly cancel each other, and by combnng them, one fnds an ncreased acoustc path length, n the range from mm to mm, dependng on the path no., cf. Table 6 [1]. Frst, consder a smplfed analyss of the effect of the ncreased acoustc path length, to llustrate the basc analyss. The average calculated length change for the fve acoustc paths s about, say, 0.04 mm, as a very rough fgure. If ths average length change s not corrected for, the measurement error (msreadng) due to transducer port and transducer length changes becomes, approxmately [1], Δv ΔL mm 2 = %. (10) v L 780mm If ths error s not corrected for, the ultrasonc flow meter thus underestmates the volumetrc flow rate. Consequently, the effect of ths average acoustc path length change on the USM s postve, %.

15 15 The above smplfed analyss effectvely llustrates the basc dea of the analyss, but does not account for changes n the acoustc path lengths of the ndvdual paths. A more thorough analyss has thus been made, where calculated changes n the ndvdual acoustc path lengths are accounted for. Ths analyss - gven n the followng - reveals that the tentatve fgure gven by Eq. (10) may represent a rough but stll reasonable estmate. Table 6 gves the calculated change n acoustc path length based on the calculated change n transducer port length and the calculated change n transducer length, calculated usng FEM n Sectons 5.1 and 5.2, respectvely. Table 6. Combned effect of (a) calculated transducer port length change and (b) calculated transducer length change, due to pressure and temperature effects, from flow calbraton (Westerbork) to feld operaton (Ormen Lange) condtons, for each of the 5 paths of the Q-Sonc 5. Results both for ndvdual paths and for the ntegrated effect are gven. (After [1].) Change n Change n Change n Interrogaton Effect on Path no. Port no. port length transducer acoustc path length acoustc path [mm] length [mm] length [mm] [mm] [%] 1A B A B A B A B A B Integrated effect, usng the assumed USM ntegraton weght factors: Based on these numbers, the effect on the velocty measured on each acoustc path can be calculated smlar to Eqs. (8)-(10), for each of the 5 paths of the Q-Sonc 5. The ntegrated value s then found based on the tentatve Q-Sonc 5 ntegraton weghts dscussed n Secton 3.3. The ntegrated value (typcal value for the effect on the flow meter n total) s found to be about % %. The postve sgn s an effect of ncreased transt tmes, meanng that the USM s underestmatng the volumetrc flow rate f ths acoustc path length effect s not corrected for. 6. P RESSURE AND TEMPERATURE INFLUENCES ON THE REYNOLDS NUMBER CORRECTION The Reynolds number used at the Westerbork flow calbraton s lower than at Ormen Lange operatng condtons, cf. Table 2. At Westerbork, the Reynolds number was n the range , and at Ormen Lange, t s about For constant flow velocty, the Reynolds number s about a factor 2 larger at Ormen Lange than at Westerbork condtons. Ths means that the flow profle s dfferent at Westerbork condtons than at Ormen Lange condtons, for the same flow velocty. In the Q-Sonc 5 software, a Reynolds number correcton s made by Elster-Instromet, wth ntenton to account for ths dfference [12]. The Reynolds number correcton for the Ormen Lange meterng staton s dscussed n the followng. The effect of the change n Reynolds number from Westerbork to Ormen Lange condtons s here studed by a set of measured axally symmetrc flow profles, reported by a seres of

16 16 laboratores [13-17]. In one laboratory, both smooth and rough ppe walls were used. In the other laboratores smooth ppes were used. The Reynolds number range covered by these experments s about For each Reynolds number (flow velocty profle), the devaton from reference s calculated usng the tentatve Q-Sonc 5 ntegraton weghts dscussed n Secton 3.3. The results are shown n Fg. 8a over a wde Reynolds number regon. (a) (b) Fg. 8. Tentatve Q-Sonc 5 USM ntegraton method used on a set of measured symmetrc flow profles taken from the lterature, over (a) a wde Reynolds number range, and (b) the Reynolds number range of nterest here. The Q-Sonc 5 results are shown wth blue dot markers. In (b), the Reynolds number correcton curve proposed by Eq. (11) s shown usng the blue lne. The Westerbork and Ormen Lange Reynolds number ranges are ndcated wth a red rectangular box and a red marker, respectvely. Correspondng Reynolds numbers at Westerbork and Ormen Lange for constant flow velocty of 15 m/s are shown wth red markers. In Fg. 8b, the same results are shown over the Reynolds number regon of nterest here. In addton, a ftted straght lne s added, gven as 6 ln(10 Re) RC = , (11) ln10 where RC s the Reynolds number correcton, and Re s the Reynolds number. Fg. 8b may be nterpreted as follows: At Westerbork, the flow calbraton s ntended to ensure that at 15 m/s, the USM gves 15 m/s as output value, at the Reynolds number At Ormen Lange condtons, however, use of the USM at 15 m/s gves (erroneously) a lower output flow velocty value than 15 m/s, snce the Reynolds number correspondng to 15 m/s s , and the curve n Fg. 8b decreases by ncreasng Reynolds number. At Ormen Lange condtons, thus, the USM underestmates the flow velocty. Ths s a property of the ntegraton method of the USM (.e. the ntegraton wegth factors). The underestmaton made by the USM can be corrected usng a Reynolds number correcton factor for the Q-Sonc 5 flow meter, from Westerbork to Ormen Lange condtons, where q = q USM C W, (12) RE Number OL RE Number RCWesterbork C W OL = (13) RC OrmenLange v = 15 m s

17 17 s the Reynolds number correcton factor from Westerbork to Ormen Lange condtons, evaluated at 15 m/s flow velocty at both locatons (Re = and , respectvely). The Reynolds number correcton from Westerbork to Ormen Lange condtons s thus about %. It can be shown that ths Reynolds number correcton of about % apples to all flow veloctes of relevance here, 1 19 m/s. The actual Reynolds number correcton carred out n the Q-Sonc 5 has not been avalable for the present work. However, Elster-Instromet clams that the Reynolds number correcton they use s 90 % correct [12]. Now, assume that the magntude of the correcton s about 0.04 %, as argued above, and assume that the Reynolds number correcton made n the Q-Sonc 5 s at least 50 % correct. Then the error of the Q-Sonc 5 Reynolds number correcton wll be less than 0.02 %. Consequently, the Reynolds number correcton already carred out n the Q-Sonc 5 s assumed to be suffcent, and no further correcton s ntroduced here. (It would also be dffcult to devce any addtonal Reynolds correcton.) 7. CORRECTION MODEL FOR PRESSURE - TEMPERATURE EFFECTS Correcton models for pressure and temperature effects are proposed on bass of the calculaton results gven n Sectons 3-6. Correcton factors have been desgned to be mplemented at the flow computer level, and not at the USM level (.e. not n the Q-Sonc 5 software). 7.1 Combned measurement error Table 7 gves an overvew of the calculated contrbutons, and ther combned effect on the USM [1]. Table 7. Varous contrbutons to the measurement error, and ther combned effect, caused by pressure and temperature changes, from flow calbraton (Westerbork) to feld operaton (Ormen Lange) condtons, for the Q-Sonc 5 ultrasonc flow meter. (Cf. also Table 8, whch s an extract of Table 7.) (After [1].) Contrbutng factor to measurement error, due to pressure and temperature changes Path no. Contrbuton to error Integrated contrbuton to error (all 5 paths) Combned contrbuton to error Source (further detals) Cross-sectonal area and acoustc path geometry % % % Fg. 5 (nclnaton angles & lateral chord postons), % effect on paths 1-5: % % % Expanson transducer ports, effect on paths 1-5: % % Table % % % % Expanson transducers, effect on paths 1-5: % % Table % % % % Combned ntegrated effect, expanson transducer ports % Table 6 & expanson transducers, all 5 paths: Reynolds number correcton (assumed devaton from 0 % Sect. 6 Elster-Instromet Reynolds number correcton): Combned effect, total (%) %

18 18 The contrbuton to the measurement error from the cross-sectonal area and the acoustc path geometry (the nclnaton angles and the lateral chord postons), ntegrated over all 5 paths, s calculated to be %, cf. Fg. 5. The contrbuton from expanson of the transducer ports, ntegrated over all 5 paths, s calculated to be %. The contrbuton from expanson of the transducers themselves, ntegrated over all 5 paths, s calculated to be %. The contrbuton from Reynolds number correcton (assumed devaton from the Elster-Instromet Reynolds number correcton), s taken to be 0 %, cf. Secton 6. Thus, accordng to the models used, the combned effect accumulates to %. If effects caused by pressure and temperature changes are not corrected for, the Q-Sonc 5 wll underestmate the volumetrc flow rate by the same amount. A graphcal vsualzaton of the same data s gven n Fg. 9. The estmates of the varous contrbutons are obtaned by calculatons, and are of course assocated wth uncertantes. The estmates used here are consdered to be the best possble on bass of the nformaton at hand % 0.25 % Contrbutons to USM measurement error, Westerbork to Ormen Lange condtons, due to P&T effects % Cross-sectonal area and acoustc path geometry Transducer port expanson % Error (%) 0.20 % 0.15 % 0.10 % 0.05 % % Transducer expanson Reynolds number correcton (assumed) TOTAL ERROR, INTEGRATED 0.00 % % % Contrbuton Fg. 9. Varous contrbutons to the measurement error, and ther combned total effect, caused by pressure and temperature changes, from flow calbraton (Westerbork) to feld operaton (Ormen Lange) condtons, for the Elster-Instromet Q-Sonc 5 ultrasonc flow meter. (After [1].) 7.2 Correcton factors % Correcton of the volumetrc flow rate for the effects of pressure and temperature changes from flow calbraton (Westerbork) to feld operaton (Ormen Lange) condtons, as dscussed n Chapters 4-6 and summarzed n Secton 7.1, s proposed to be done by multplcaton of the measured volumetrc flow rate wth two correcton factors, as outlned n the followng: PT nom (1) A "nomnal P&T correcton factor", C W OL, representng the man correcton: - Westerbork (63 barg, 7 o C) Ormen Lange (nomnal P&T, 230 barg, 40 o C),

19 - Fxed correcton factor, based on the FEM calculatons. - Accounts for the effects A - E n Table 3. - Implemented n the flow computer (not n the USM software). (2) An "nstantaneous P&T correcton factor", 19 PT nst C OL, representng an adjustment for smaller P&T changes: - Ormen Lange (nomnal P&T, 230 barg, 40 o C) Ormen Lange (actual P&T), - Westerbork (63 barg, 7 o C) Possble new flow calbraton (new P&T condtons). - Onlne "lvng" correcton factor, based on the (smplfed) analytcal model B. - Accounts for the effects A and B n Table Implemented n the flow computer (not n the USM software). The followng correcton model s thus proposed for the Ormen Lange applcaton, where q = q K C C, (14) USM Flow Westerbork PT nom W OL PT nst OL q : Corrected volumetrc flow rate at Ormen Lange lne condtons [m 3 /s]. q USM : Output volumetrc flow rate of the Q-Sonc 5 at Ormen Lange lne condtons [m 3 /s]. K C Flow Westerbork PT nom W OL : Flow calbraton correcton factor [dmensonless], establshed under flow calbraton at Westerbork. Flow dependent. Not addressed here. : "Nomnal P&T correcton factor" [dmensonless], for pressure and temperature changes from flow calbraton (Westerbork) to feld operaton (Ormen Lange) condtons (nomnal), accountng for: Changes of cross-sectonal area (dameter) (based on FEM) Changes of acoustc path geometry (nclnaton angles and lateral chord postons) (based on FEM) Expanson / contracton of transducer ports (based on FEM) Expanson / contracton of transducers (based on FEM) Changes of the Reynolds number. PT nst C OL : "Instantaneous P&T correcton factor" [dmensonless], for (a) nstantaneous (small) changes of Ormen Lange pressure and temperature condtons,.e. devaton between actual and nomnal Ormen Lange P&T condtons, and (b) n case of a future recalbraton at Westerbork or another flow calbraton lab, pressure and temperature changes from "old" to "new" flow calbraton, both accountng for Changes of cross-sectonal area (dameter) Changes of acoustc path geometry (nclnaton angles and lateral chord postons), both based on the analytcal model B.