DETERMINATION OF THE DISTORTION COEFFICIENT OF A 500 MPA FREE-DEFORMATION PISTON GAUGE USING A CONTROLLED-CLEARANCE ONE UP TO 200 MPA

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1 XX IMEKO World Congrss Mtrology for Grn Growth Stmbr 9 14, 2012, Busan, Rublic of Kora DETERMINATION OF THE DISTORTION COEFFICIENT OF A 500 MPA FREE-DEFORMATION PISTON GAUGE USING A CONTROLLED-CLEARANCE ONE UP TO 200 MPA Yuanchao Yang, Jin Yu National Institut of Mtrology (NIM), Bijing , China yangyc@nim.ac.cn Abstract: Cross-float xrimnts btwn a 500 MPa fr-dformation (FD) iston gaug and a 200 MPa controlld-claranc (CC) on wr carrid out at a sris of systm rssurs and jackt rssurs. Th two charactristics of th CC iston gaug, th zro claranc jackt rssur and th jackt rssur cofficint, wr dtrmind basd on th Hydmann-Wlch modl. Aftr that, th rssur dndnc of th ffctiv ara for th FD iston was calculatd using th cross-float data, from which th rssur distortion cofficint was dtrmind to b MPa -1. Th rsult was about 19% largr than th on obtaind by th Lam quation. Kywords: distortion cofficint; controlldclaranc iston gaug; Hydmann-Wlch modl; ffctiv ara; cross-float xrimnt. 1. INTRODUCTION Th rssur scal at th National Institut of Mtrology (NIM) is u to 500 MPa, which is ralizd by a fr-dformation (FD) iston gaug of 5 MPa/kg mad by DH Instrumnts, USA 1, rfrrd to as FD500. Th uncrtainty of its ffctiv ara du to th distortion bcoms mor significant in this rssur rang, scially abov 100 MPa. To obtain adquat small uncrtainty in th rssur rgion of 500 MPa, th ky oint is to accuratly dtrmin th distortion cofficint of FD500. Gnrally, th distortion cofficint can b valuatd by thory or xrimnt. In thory, th mthods includ siml calculations basd on th Lam quation and finit lmntal analysis (FEA) [1-4]. Th thortical calculations nd good knowldg of th lastic constants and th iston-cylindr dimnsions, th boundary loading conditions and th fluid rortis. With th lastic constants unknown and using litratur valus, th calculation basd on th Lam quation could rsult a 1 Crtain commrcial quimnt, instrumnts or matrials ar indntifid in this ar to fostr undrstanding. Such idntification dos not imly ndorsmnt by th authors nor dos it imly that th quimnt or matrials ar ncssarily th bst for th uros. disagrmnt of 20% [5]. Larg disagrmnt also rvald whn using th FEA mthod [6]. In xrimnt, th distortion cofficint can b obtaind by th similarity mthod [7] or calibrating against a wll charactrizd iston gaug. Th controlldclaranc (CC) iston gaug is widly usd as a rimary rssur standard abov 100 MPa [8, 9], of which th distortion cofficint can b wll charactrizd by an indndnt xrimnt using th Hydmann-Wlch (HW) mthod [10-12]. In this ar, cross-float xrimnts wr rformd btwn th FD500 and a 200 MPa CC iston gaug of 2 MPa/kg, which is also mad by DH Instrumnt 1 and dnotd as CC200. On on hand, th CC200 was charactrizd using th HW modl. Its rssur dndnc of th ffctiv ara and uncrtainty wr valuatd. On th othr hand, th rssur dndnc of th ffctiv ara for FD500 was also obtaind in th rssur rang of 200 MPa. 2. PRINCIPLE AND EXPERIMENTAL SETUP Th FD500 and CC200 wr connctd to th rssur lin with valvs and a digital rssur monitor gaug. Anothr fr-dformation iston gaug of 100 MPa was usd to aly th jackt rssur. Th rssur-transmitting fluid in both lins was Di(2-htylhxyl) sbacat. In th HW modl, th ffctiv ara of th CC iston gaug is xrssd as ACC, = A0, T[1 + ( α )( 0 + αc T T0)](1 + bs)[1 + d( z j)] (1) whr A 0,T0 is th iston ara at zro rssur and th rfrnc tmratur, s is th systm rssur, b is th distortion cofficint of th iston, α and α c ar th thrmal xansion cofficints of th iston and cylindr, T is th tmratur of th iston-cylindr and T 0 th rfrnc tmratur (20 ), j is th jackt rssur. Th jackt rssur cofficint d and th zro claranc jackt rssur z ar two modlling aramtrs. Th jackt rssur cofficint d is th rlativ chang in th ffctiv ara in rsons to th jackt rssur and can b dtrmind as

2 1 ΔA 1 ΔM d = (2) Δj A Δj M whr ΔA and ΔM ar th chang in th ffctiv ara of CC iston gaug and th chang in mass to achiv nw quilibrium undr Δ j. Th zro claranc jackt rssur z mans th jackt rssur at which th claranc btwn th iston and cylindr vanishs, and th iston fall-rat turns 0. Undr laminar-flow conditions, th cubic root of th iston fall-rat v is roortional to th istoncylindr claranc h, which is also roortional to ( z - j ). Consquntly, th rlationshi btwn j and v can b writtn as 1/3 j = kv + (3) z whr k is a constant dnding on s. Th zro claranc jackt rssur z is dtrmind by masuring th iston fall-rat v undr diffrnt j and xtraolating th lot of j vrsus v 1/3 to v 1/3 =0. Th iston fall-rat was masurd by a built-in linar variabl diffrntial transformr (LVDT), which was calibratd using a cathtomtr. Th iston osition was rcordd using a comutr through th rmot communication ort ovr 3 min to 8 min, and th fallrat was calculatd from th linar fit of th osition vrsus tim data. To liminat th tmratur ffct, all th fall-rat data wr corrctd for th tmratur dndnc of th viscosity of th rssur fluid [13]. Th cross-float and fall-rat masurmnts wr carrid out at systm rssurs s from 40 MPa to 200 MPa in sts of 20 MPa. At ach s, th jackt rssur j was initially st as 0, and thn varid from 10% to 50% of s in sts of 4 MPa. 3. RESULTS OF THE EXPERIMENTS 3.1 Charactrization of CC Zro claranc jackt rssur z Fig. 1 shows th rsults of iston fall-rat masurmnts, and j is lottd against v 1/3 at ach systm rssur. According to quation (3), th xrimntal data ar fittd to linar functions (solid lins in Fig. 1), and z is stimatd by th intrct on y-axis of th fittd lin. At s abov 100 MPa, th linar fits show som dviations from th xrimntal data at j =0 and ur limits of j. Th dtrmind z is lottd against s, as shown in Fig. 2, and a quadratic olynomial is fittd to th data giving 4 2 z = s (4) s Th valu of z rdictd by quation (4) is usd in th HW modl, with th uncrtainty stimatd by th combination of th on originatd from th fit of j vrsus v 1/3 and th on from th fit of z vrsus s. Th valus of z at ach systm rssur with thir uncrtaintis ar summarizd in Tabl 1. Fig. 1. Plot of j against v 1/3 at systm rssurs from 40 MPa to 200 MPa. Solid lins rrsnt th linar fits to th data. Fig. 2. Zro claranc jackt rssur z as a quadratic function of systm rssur s Jackt rssur cofficint d To dtrmin th jackt rssur cofficint d, cross-float xrimnts wr rformd btwn CC200 and FD500 at diffrnt jackt rssur j. With j incrasd, th ffctiv ara of CC200 was dcrasd and th mass addd on CC200 was rducd. Comaring to th ffctiv ara of CC200 at zro jackt rssur, th rlativ chang in ffctiv ara at j, Δ A / A, was calculatd basd on th rlativ chang in mass taking account of th corrction for th buoyancy and tmratur ffct. As an xaml, Fig. 3 shows th lot of th rlativ chang in ffctiv ara vrsus j at th systm rssur of 200 MPa. Alying linar fit to th data, d is drivd from th slo of th fittd straight lin. Howvr, Fig. 3 also rvals dviation from th linar fit. As a linar function of j, th bhavior of th rlativ chang in ffctiv ara at vry low j is diffrnt from that at vry high j, imlying a j dndnc of d. Th nonlinarity aars at s abov 120 MPa, whil th data k good linarity at s of 120 MPa and blow. Th nonlinar bhavior in th lot of Δ A / A vrsus j at highr s is cohrnt with th nonlinarity in th lot of j vrsus v 1/3 mntiond abov. Th influnc of nonlinarity on th valu of z and d is considrd as an uncrtainty comonnt, and will b discussd in Sction

3 ( σ ) b= 3 1 / E (6) whr σ and E ar Poisson s ratio and Young s modulus of th matrial, rsctivly. Th iston is mad from tungstn carbid. With σ of and E of 560 GPa rovidd by th manufacturr, b is obtaind as MPa -1. Th smi-rang of b is valuatd as 10% and assuming a uniform distribution. Th standard uncrtainty of b is takn as u(b)= MPa -1. Fig. 3. Rlativ chang in ffctiv ara as a linar function of j at s =200 MPa Prssur dndnc of th ffctiv ara of CC200 Aftr th dtrmination of z, d and b, th rssur dndnc of th ffctiv ara of CC200 is charactrizd according to quation (1) as C = (1 + bs)[1 + d( z j)] (7) Fig. 5 shows th lot of C against s at jackt rssur of 0%, 25% and 50%, rsctivly. Th rror bars rrsnt th standard uncrtainty of C. It can b sn that th distortion cofficint is almost zro whn th jackt rssur is alid as 25% s. Fig. 4. Plot of d against s. Solid lin is th quadratic fit to th data. Th jackt rssur cofficint d obtaind at ach systm rssur is lottd against s, as shown in Fig. 4. Fitting to a quadratic function, d can b xrssd as d = ( s s ) 10 (5) Th uncrtainty of th fittd d is th combination of th on originatd from th fit of Δ A / A against j and th on from th fit of d against s. Th fittd valus of d and thir uncrtaintis ar listd in Tabl 1. Tabl 1. Fittd valus of z, d and thir standard uncrtaintis. s z u( z ) d u(d) MPa MPa MPa 10-6 MPa MPa Piston distortion cofficint b Th iston distortion cofficint b is calculatd from th lastic thory as Fig. 5. Prssur dndnc of th ffctiv ara of CC200 at j =0, 0.25 s, and 0.5 s, rsctivly. Th uncrtainty of C is considrd to hav two comonnts. On comonnt, u 1 (C ), is th root sum squar of u(b) s, u(d) ( z - j ), and u( z )d. Th othr comonnt, u 2 (C ), is drivd from th nonlinarity in th dtrmination of z and d obsrvd at highr systm rssur. Bcaus of th nonlinarity, th z and d valu is dndnt on th slctd rang of th linar fit. If th data at highr j wr usd, z would b ovrstimatd and d undrstimatd. Convrsly, z would b undrstimatd and d ovrstimatd if data at lowr j wr usd. Th influnc of th slctd rang on z and d is corrlatd, and thir ffcts on C ar canclld artly. To calculat u 2 (C ), z and d at s abov 100 MPa ar rdtrmind by linar fitting to th data at highr j, and nw rsult of C is obtaind. Th diffrnc btwn th nw rsult of C and th original valu is takn as u 2 (C ). At j =0, 0.25 s, and 0.5 s, th two uncrtainty comonnts u 1 (C ), u 2 (C ) and th combind uncrtainty u(c ) ar summarizd in Tabl 2.

4 Tabl 2. Uncrtainty of C in th cas of j =0, 0.25 s, and 0.5 s. j =0 j =0.25 s j =0.5 s s u 1 u 2 u u 1 u 2 u u 1 u 2 u MPa Th uncrtainty comonnt u 1 (C ), du to z, d and b, is similar in all th cas of j. Th uncrtainty u 2 (C ) du to th nonlinarity is vry small whn th jackt rssur is 0. Howvr, at highr j, u 2 (C ) bcoms dominant whn s is abov 100 MPa. As a rsult, th combind uncrtainty of C at highr j and highr s is significantly largr than that in th cas of j =0. Sinc th uncrtainty is rlativly small at j =0, th iston gaug CC200 dscribd hr is rcommndd to b usd with zro jackt rssur. Basd on th charactrization of CC200, th crossfloat data usd to dtrmin d wr usd again to calculat th ffctiv ara of FD500 at ach s. Using th rsult at j =0, th ffctiv ara A is lottd against s, as shown in Fig. 7. Th ffctiv ara is modlld by a linar function of s as A (1 ) = A0 + λ (8) s whr A 0 is th zro-rssur ffctiv ara and λ is th distortion cofficint. A linar fit to th data givs th rsults of A 0, λ and thir standard dviations as 2 6 A0 = mm sr ( A0) = (9) λ = MPa s( λ) = MPa whr s r (A 0 ) is th rlativ standard dviation of A 0 and s(λ) is th standard dviation of λ. Taking into account th uncrtainty du to th aramtrs of CC200, which ar discussd in Sction and 3.1.5, th combind uncrtainty of A 0 and λ ar stimatd as 5 ur ( A0 ) = (10) 8-1 u( λ) = MPa Fig. 6. Plot of th uncrtainty of C and its comonnts as a function of systm rssur. In th cas of j =0, th uncrtainty comonnts and th combind uncrtainty of C ar lottd as a function of s, as shown in Fig. 6. For CC200 usd with j =0, th uncrtainty du to th rssur dndnc of th ffctiv ara is lss than MPa -1 at s of 80 MPa and abov. Howvr, whn CC200 is usd with j =0.5 s, this valu would b MPa Piston ara A 0,T0 At j =0, CC200 was calibratd against a 100 MPa FD rssur standard, of which th ffctiv ara was tracabl to th NIM rimary rssur standard of 10 MPa [14]. Dtails of th tracability will b rsntd lswhr. From th calibration, th iston ara A 0,T0 was dtrmind to b mm 2 with a rlativ standard uncrtainty of Charactrization of FD500 Fig. 7. Prssur dndnc of th ffctiv ara of FD500 from 40 MPa to 200 MPa. Th rsult is obtaind at j =0. Fig. 7 shows good linarity, and th dviation from th linar fit is found to b lss than ± Strictly saking, th dtrmind distortion cofficint is only valid in th rssur rang of 200 MPa, which is only 40% of th full scal of FD500. Howvr, w assumd th linarity could b xtndd u to 500 MPa. Th assumtion is rasonabl, sinc th dviation from th linarity for a similar iston gaug has bn rortd to b lss than ± in th full scal of 500 MPa [12]. 4. DISCUSSION Th ffctiv ara and distortion cofficint of FD500 dscribd hr ar tracabl to th NIM rimary standard and CC200, rsctivly. Th distortion cofficint was dtrmind xrimntally using a CC iston gaug basd on th HW mthod. On th thory oint of viw, th distortion cofficint can b calculatd basd on th lastic thory with th information of th matrial rortis. For siml FD

5 iston gaug, a simlifid calculation basd on th Lam quation is xrssd as 2 2 3σ 1 1 Rc + R λ = + + σ 2 2 c (11) 2E 2E c Rc R whr σ, σ c is th Poisson s ratio of iston and cylindr, E, E c is th Young s modulus of iston and cylindr, R is th radius of iston or th innr radius of cylindr, R c is th outr radius of cylindr. In th cas of FD500, th iston and cylindr all ar mad from tungstn carbid, th Poisson s ratio and Young s modulus for iston ar 0.218, 560 GPa, and 0.218, 620 GPa for cylindr rovidd by th manufacturr. R and R c ar masurd to b 0.79 mm and 10 mm, rsctivly. Using ths data and quation (11), th distortion cofficint is calculatd to b MPa -1. Taking th uncrtainty of Poisson s ratio and Young s modulus as 0.01 and 50 GPa, th uncrtainty of th calculatd distortion cofficint is stimatd to b MPa -1. Comaring to th valu dtrmind by xrimnt, which is MPa -1 with an uncrtainty of MPa -1, th two rsults ar badly consistnt. Th diffrnc btwn th two rsults is MPa -1, about 19% of th xrimntal valu, and can caus a rlativ dviation of in ffctiv ara at 500 MPa. Th distortion cofficint calculatd hr is unrliabl, bcaus quation (11) is an aroximation undr idal conditions and th lastic aramtrs ar not masurd dirctly. It may imrov th thortical rsult by alying FEA to th ral structur of th iston-cylindr unit and using mor accurat lastic aramtrs. Using xrimntal mthod to dtrmin th distortion cofficint, th nonlinarity of th CC iston gaug at high systm rssur is a roblm. This roblm may b solvd using a highr ordr HW modl. Howvr, it nds to xtnd th jackt rssur rang to much highr rgion clos to th zro claranc jackt rssur, and that may damag th iston and it s difficult to masur th xtrmly slow fall-rat. 5. CONCLUSIONS Th distortion cofficint of FD500 was studid using CC200 basd on th HW mthod. Piston fallrat masurmnts and cross-float xrimnts wr carrid out at jackt rssurs from 0 to 50% of systm rssur. Nonlinarity was obsrvd at systm rssur abov 100 MPa. Th uncrtainty of th rssur dndnc of th ffctiv ara of CC200 was found to b smallst in th cas of j =0 than in othr cas. Using th cross-float data at j =0, th distortion cofficint of FD500 was dtrmind to b MPa -1 with an uncrtainty of MPa - 1. Th rsult is about 19% largr than th on obtaind from th simly calculation basd on th Lam quation. Th xtraordinary discrancy may aris from th ovrsimlifid calculation and inaccurat lastic aramtrs. In ordr to imrov th consistncy btwn th rsults of xrimnt and thory, futur work will focus on FEA and masurmnts of matrial rortis. REFERENCES [1] G. Buonanno, M. Dll Isola, G. Iagulli and G. Molinar, A Finit Elmnt Mthod to Evaluat th Prssur Distortion Cofficint in Prssur Balancs, High Tm. High rssurs, vol. 31, , [2] G. Molinar, G. Buonanno, G. Giovinco, P. Dlajoud and R. Hains, Effctivnss of Finit Elmnt Calculation Mthods (FEM) on High Prformanc Prssur Balancs in Liquid Mdia u to 200 MPa, Mtrologia, vol. 42,. S207-S211, [3] W. Sabuga, G. Molinar, G. Buonanno, T. Esward, J. C. Lgras and L. Yagmur, Finit Elmnt Mthod Usd for Calculation of th Distortion Cofficint and Associatd Uncrtainty of a PTB 1 GPa Prssur Balanc EUROMET Projct 463, Mtrologia, vol. 43, , [4] S. Dogra, S. Yadav and A. K. Bandyoadhyay, Comutr Simulation of a 1.0 GPa Piston-Cylindr Assmbly Using Finit Elmnt Analysis (FEA), Masurmnt, vol. 43, , [5] G. F. Molinar, An Old Instrumnt in th Nw Tchnological Scnry: Th Piston Gaug in Liquid Mdia u to 1 GPa, Mtrologia, vol. 30, , 1993/94. [6] M. P. Fitzgrald and C. M. Sutton, Prssur Balanc Elastic Distortion: Exrimnt vrsus Thory, Mtrologia, vol. 42,. S239-S241, [7] R. S. Dadson, R. G. P. Grig and A. Hornr, Dvlomnts in th Accurat Masurmnt of High Prssurs, Mtrologia, vol. 1, , [8] A. K. Bandyoadhyay and A. C. Guta, Ralization of a National Practical Prssur Scal for Prssurs u to 500 MPa, Mtrologia, vol. 36, , [9] J. K. L. Man, Establishmnt of an Australian High Prssur Standard, Mtrologia, vol. 42,. S169- S172, [10] P. L. M. Hydmann and B. E. Wlch, Exrimntal Thrmodynamics. London: Intrnational Union of Pur and Alid Chmistry, Ch. 4, , [11] A. K. Bandyoadhyay and D. A. Olson, Charactrization of a Comact 200 MPa Controlld Claranc Piston Gaug as a Primary Prssur Standard using th Hydmann and Wlch Mthod, Mtrologia, vol. 43, , [12] H. Kajikawa, K. Id and T. Kobata, Prcis Dtrmination of th Prssur Distortion Cofficint of Nw Controlld-Claranc Piston-Cylindrs basd on th Hydmann-Wlch Modl, Rv. Sci. Instum., vol. 80, no. 9, , St [13] P. Vrgn, Nw High-rssur Viscosity Masurmnts on Di(2-thylhxyl) Sbacat and Comarisons with Prvious Data, High Tm. High rssurs, vol. 22, , [14] Y. C. Yang and J. Yu, Calculation of Effctiv Ara for th NIM Primary Prssur Standards, PTB- Mittilungn, vol. 121, , 2011.