HYDROMETER CALIBRATION: LINKING SIM NMIs TO THE EURAMET KEY COMPARISON REFERENCE VALUE OF EURAMET.M.D-K4

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1 Smposo de Metrología 00 7 al 9 de Octubre HYROMETER CALIBRATION: LINKING SIM NMIs TO THE EURAMET KEY COMPARISON REFERENCE VALUE OF EURAMET.M.-K4 Becerra L O, Lorefce S, Pennecch F Centro Naconal de Metrología (CENAM), km 4,5 Carretera a los Cués, Mpo. El Marqués, Qro. Méxco Insttuto Nazonale d Rcerca Metrologca () Strada delle Cacce, Torno, Italy lbecerra@cenam.mx, slorefce@nrm.t. INTROUCTION Calbraton of hydrometers s an mportant actvty of metrology due to mportance of densty n qualty of products, (sugar concentraton, percentage of alcohol per volume, etc.,) as well as part of measurements for quantty (flow and volume), among others. ue to the above reasons t s mportant for laboratores to demonstrate ther capablty n densty meters calbraton. A possblty s to partcpate n comparsons. Wthn the EURAMET regon a key comparson on hydrometer calbraton was carred out from October 003 to May 004 under the dentfcaton of EURAMET.M.-K4. Another key comparson between SIM NMIs on hydrometer calbraton was carred out from Aprl 007 to October 008 dentfed as SIM.M.-K4. -Italy and CENAM- Mexco were the plot laboratores for the two key comparsons respectvely. Wth the ntenton of lnkng these two key comparsons, a blateral comparson between CENAM-Mexco and -Italy was carred out from June to October 007; ths blateral comparson was dentfed as SIM.M.-S. In the present paper the methodology s presented to lnk the results of the partcpant NMIs of the SIM.M.-K4 comparson to EURAMET.M.-K4 key comparson reference value (KCRV EU ), trough the blateral comparson SIM.M.-S.. PARTICIPANTS AN TRAVELLING STANARS OF COMPARISONS.. EURAMET.M.-K4 Partcpants IMGC Italy (now ) OMH Hungary PTB Germany BNM-LNE France IPQ Portugal MIKES Fnland BEV Austra UME Turkey GUM Poland SMU Slovaka VNIIM Russa Travellng standards For the comparson two sets of hydrometers were used wthn the followng ranges Table Set for EURAMET.M.-K4 Range Scale dvson Table Set for EURAMET.M.-K4 Range Scale dvson SIM.M.-K4 Partcpants CENAM-Mexco BSJ Jamaca CENAMEP Panama CESMEC Chle IBMETRO Bolva INECOPI Peru Centro Naconal de Metrología

2 Smposo de Metrología 00 7 al 9 de Octubre INEN Ecuador INMETRO Brazl INTI Argentna LACOMET Costa Rca LATU Uruguay NIST Unted States NRC Canada SIC Colomba Travellng standards For the comparson, two smlar sets of hydrometers were used wthn the followng ranges, Table 3 Set for SIM.M.-K4 Range Scale dvson SIM.M.-S Partcpants Italy CENAM Mexco Travellng standards For ths supplementary comparson two hydrometers were used wth the followng characterstcs, Table 4 Travellng standards for the SIM.M.-S Range Scale dvson ANALYSIS OF RESULTS OF THE THREE COMPARISONS From the results of partcpants of EURAMET.M.- K4, a KCRV EU for each nomnal value was calculated. These KCRV EU were calculated ether as the weghted mean or as the medan from results of partcpants, accordng to []. egrees of equvalence oes between the values reported by partcpants and the correspondng KCRV EU,.(EURAMET) were calculated as well as the degrees of equvalence does for each par of partcpants of EURAMET.M.-K4 []. As concerns SIM.M.-K4 a smlar analyss was done [3]. A KCRV SIM for each nomnal value was calculated as the weghted mean of results of partcpants. oes between partcpatng NMIs and the KCRV SIM were calculated as well as does among partcpants of SIM.M.-K4. Fnally, n the analyss of SIM.M.-S [4] lke ths was a blateral comparson there was not need to calculate RV but does between both partcpants were calculated. In order to lnk the results of the SIM NMIs to the KCRV of EURAMET, the followng analyss was done: - oes between results reported by and the KCRV EU n EURAMET.M.-K4, = KCRV () - does between results reported by NMI and CENAM n SIM.M.-K4, d EU EU ( SIM ) = X ( SIM ) CENAM SIM () - does between results reported by CENAM and n SIM.M.-S d = CENAM (3) n order to calculate the oes between the SIM NMI and the KCRV EU we have to calculated as follow, = d( SIM ) + d + = X SIM ) KCRV EU ( (4) oes between results reported by SIM NMI and the KCRV EU X Value reported by SIM NMI (SIM ) In order to calculate (4), t s mportant to assume that results reported by CENAM n both comparson SIM.M.-K4 and SIM.M.-S are strongly correlated. The same s assumed for, results reported by n both comparsons EURAMET.M.-K4 and SIM.M.-S are strongly correlated too. The nomnal values measured n the dfferent comparsons are not the same, and due to ths Centro Naconal de Metrología

3 Smposo de Metrología 00 7 al 9 de Octubre reason t s not possble to calculate (4) for each ndvdual value. The lnkng procedure used s descrbed n [5], t conssts n calculatng a regresson curve for the correspondng oes. Accordng to the specfc purpose of ths work a straght lnes (frst order curves) were calculated for, d, d (SIM ). The straght lne functon s the frst opton for the fttng curves, where oes are consdered proportonal to the nomnal densty. In order to ntroduce all set of values a matrx equaton takes the form, = ρβ ε (5) where s the column vector of degrees of equvalence, ρ s the matrx of nomnal values of densty, ε s the column vector of measurement errors and β s the column vector whch contans the slope m and the ntercept b of the proposed functon, straght lne, = M n n, ρ ρ m ρ = M M, = ρ n b ρ n β, ε ε ε = M ε n ε n The weghted least squares method (WLS) was used to calculate the regresson curves for the correspondng oes,. The soluton of (5) by WLS s [5] β= T T ( ρ ψ ρ) ρ ψ ˆ (6) The weghng matrces ψ, were formed by the varance (and covarance) of the correspondng oes, [5, 6]. Varances were calculated as the square of the uncertantes of oes reported n each comparson, and the covarances were calculated wth pars of the uncertantes of oes and the correspondng correlaton coeffcent. For the correlaton coeffcents t was assumed 0.9 for oes correspondng to the calbraton of same hydrometer and 0.3 for oes correspondng to the calbraton of dfferent hydrometers. These correlaton coeffcents were assumed based on the experence of the authors. The calculated curves take the followng form, where: ( ) mρ + b ε = (7) ρ oes calculated for the correspondng comparson, m slope of the straght lne calculated, dmensonless b ε ntercept of the straght lne calculated, adjustment error for the proposed functon, In order to check the agreement between predcted and observed values the ch-square test χ was used. 3.. Calculated contnuous functons of oes n The calculated curve coeffcents for the domnum from to.995 s, b = 5.809E-06 m = E-05 d n the domnum from 0.80 to.98 s, The calculated curve coeffcents for b = E-05 m =.7067E-05 both approxmatons were checked by χ test. For SIM NMIs the followng coeffcents for curves to d were calculated n the descrbe (SIM ) domnum from 0.60 to.99, except d whch s the domnum from 0.80 for NRC (SIM ) to.99 because NRC dd not partcpate n the lower range n SIM comparson. The domnum of each functon s fxed by the nomnal values measured n the correspondng comparson. Centro Naconal de Metrología 3

4 Smposo de Metrología 00 7 al 9 de Octubre (SIM ) d descrbes the dfference () as a contnuous functon. Table 5. Fttng coeffcents for the contnuous d (SIM ) functons (SIM ) d b,, m d.539e E-05 SIC(SIM ) d 6.07E E-05 LATU (SIM ) d 6.463E E-05 INECOPI (SIM ) d 4.655E E-05 INTI (SIM ) d.463e-05-8.e-06 NIST (SIM ) d.070e E-04 NRC(SIM ) d.397e e-05 CENAMEP(SIM ) The other sx partcpants of SIM.M.-K4, dd not pass the χ test for the proposed functon. 3.. Calculaton of oes of SIM NMIs respect to the KCRV EU The contnuous functons of oes of SIM NMIs respect to KCRV EU, as contnuous functons were calculated by the followng formula [5], = d + d SIM ), (. + (8) ths takes the followng form, ρ = ( m'' + m' + m) ρ + b' ' + b' + b ε (9a) or ( ) ( ) ( ρ ) = m*) ρ + ( b ) ε * ( (9b) where ε s the adjustment error for the resultng functon n, m * = m + m + m and b * = b + b + b. For CENAM s oes wth respect to KCRV EU,, the contnuous functon was calculated CENAM for the followng formula, = d + (0) CENAM or presented as the addton of the coeffcents, = ( m' + m) + b' + b () CENAM ρ ( ) ε The calculated coeffcents for the contnuous are lsted n table 6. functons of Table 6. Fttng coeffcents for the contnuous functons CENAM SIC LATU INECOPI INTI NIST NRC b*, m* -.730E E E E E E E E-05.95E E E E E E-04 CENAMEP.4E E-04 The domnum of s from 0.80 to.98, due to the fact that the lnkng comparson SIM.M.-S s wthn the same are shown n Fg. domnum. The calculated Uncertanty of oes of SIM NMIs respect to the KCRV EU, In order to calculate the uncertanty of numercal smulatons by Monte Carlo method were [6,7], see Fg.. done for each For the numercal smulatons, oes and ther correspondng standard uncertantes were consdered as the mean value and the standard devaton of normal probablty densty dstrbutons of the nput quanttes. For the numercal smulaton trals were done for each The normalzed errors to an approxmated 95% of confdence level were calculated wth the followng expresson, E n U ' ( ' ) = () In Fg. 3 are shown the curves of the normalzed ρ, the accepted values are. errors of ( ) Centro Naconal de Metrología 4

5 Smposo de Metrología 00 7 al 9 de Octubre ', In table 7 are shown the values of ( ) U ', and n E calculated for selected denstes wthn the domnum of the functons. 5. CONCLUSIONS In the present paper were presented the oes of. SIM NMIs to the KCRV EU, From fourteen partcpants of SIM.M.-K4, only eght partcpants were lnked to the KCRV of the EURAMET.M.-K4. The other sx partcpants faled the χ test for the proposed functon. All partcpants that passed the normalzed errors smaller than. χ test had The straght lne functon s the frst opton for the fttng curves, where oes are consdered proportonal to the nomnal densty. Indeed t s possble to calculated other functons that could ft better than straght lne (e.g. by a second order polynomal functon), but the straght lne s one of the smplest assumpton. In the straght lne functon, the systematc effects could be dvded two groups, the constant effects whch could be ncluded n the ntercept (b) and the relatve effects whch could be ncluded n the slope (m), see formula (7). In Fgure 4 are shown the oes between the SIM NMIs consdered for ths work and the KCRV SIM n SIM.M.-K4 [3]. In Fgure 4, t could be observed the systematc effects besdes to the random effects. In Fgure 5 s shown the same nformaton but presented n a dfferent way. In Fgure 5, t could be noted the dsperson of the calculated oes for SIM NMIs n SIM.M.-K4, whch could be reasonable be attrbuted to random effects arsng from measurement systems of the SIM NMIs. These dspersons are not ncluded n the lnkng functons, but do are ncluded n the uncertanty of these functons. The uncertanty of the was domnated by the uncertantes of the oes of the lnkng comparson SIM.M.-S due to the scale dvson of the travellng standards used n such blateral comparson. A smlar analyss could be done for lnkng SIM.M.- K4 to the CCM key comparson CCM.-K4 whch wll be started n 00, actvty watng to be done, once results from CCM.-K4 be avalable. REFERENCES [] Cox M. G., The evaluaton of key comparson data, Metrologa, 00, 39, [] Lorefce S, Malengo A, Vámossy C, Toth H, Bettn H, o Céu Ferrera M, Madec T, Gosset A, Henonen M, Buchner C, Lenard E, Spurny R, Akcadag U, omostroeva N 008 EUROMET Project No. 70 (EUROMET.M.- K4) Comparson of the calbratons of hghresoluton hydrometers for lqud densty determnatons, Metrologa 45, Tech. Suppl., [3] Becerra L.O. 009 Fnal report of comparson of the calbratons of hydrometers for lqud densty determnaton between SIM laboratores: SIM.M.-K4, Metrologa, 46, Tech. Suppl., [4] Becerra L.O., Lorefce S., 009 Report of the blateral comparson of the calbratons of hydrometers for lqud densty determnaton between CENAM Mexco and Italy: SIM.M.-S, Metrologa, 46, Tech. Suppl., [5] Lorefce S, Becerra L O, Pennecch F, Method for lnkng a blateral to a key comparson: applcaton to the SIM.M.-S and the EUROMET.M.-K4. n preparaton. [6] JCGM 00:008 - Evaluaton of measurement data Gude to the expresson of uncertanty n measurement Frst edton September 008. [7] JCGM 0:008 - Evaluaton of measurement data Supplement to the Gude to the expresson of uncertanty n measurement Propagaton of dstrbutons usng a Monte Carlo method Frst Edton 008. Centro Naconal de Metrología 5

6 Smposo de Metrología 00 7 al 9 de Octubre ' (SIM NMI -KCRV EU ) 6.0E E-05.0E E E E E-05 ' (CENAM) ' (SIC) ' (LATU) ' (INECOPI) ' (INTI) ' (NIST) ' (NRC) ' (CENAMEP) -8.0E E Nomnal Values,. Fg.. oes between SIM NMIs and KCRV EU, U 95% (' ) (SIM NMIs - KCRV EU ).40E-04.0E-04.00E E E E-05 ' CENAM ' SIC ' LATU ' INECOPI ' INTI ' NIST ' NRC ' CENAMEP.00E E Nomnal Values, Fg.. Expanded uncertanty (approx. 95%) of oes between SIM NMIs and KCRV EU, Centro Naconal de Metrología 6

7 Smposo de Metrología 00 7 al 9 de Octubre En (SIM NMIs -KCRV EU ) ' (CENAM) ' (SIC) ' (LATU) ' (INECOPI) ' (INTI) ' (NIST) ' (NRC) ' (CENAMEP) Nomnal Values, Fg. 3. Normalzed errors between SIM NMIs and KCRV EU oes (SIM).0E E E E E-04 CENAM (SIM) SIC (SIM) LATU (SIM) INECOPI (SIM) INTI (SIM) NIST (SIM) NRC (SIM) CENAMEP (SIM) -.5E Nomnal Values ( ) Fg. 4. oes between selected SIM NMIs and KCRV SIM as calculated n SIM.M.-K4. Centro Naconal de Metrología 7

8 Smposo de Metrología 00 7 al 9 de Octubre oes (SIM).0E-04 oes ( ) 5.0E E E E E-04 CENAM SIC LATU INECOPI INTI NIST NRC CENAMEP SIM NMIs Fg. 5. oes between selected SIM NMIs and KCRV SIM as calculated n SIM.M.-K4. Centro Naconal de Metrología 8

9 Smposo de Metrología 00 7 al 9 de Octubre Table 7. Calculated values of oes of SIM NMIs regards to the KCRV EU, ther expanded uncertantes assocated and the calculated normalzed errors, for selected values of densty wthn the domnum of the functons. ρ ' CENAM U (95%) En ' SIC U (95%) En ' LATU U (95%) En ' INECOPI U (95%) En E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05 8.E E E E E E E E E E E E E E E E E E E E E E E E E E E E-06 9.E E E E E E E E E E E E E E E E E E E E E E E E E ρ ' INTI U (95%) En ' NIST U (95%) En ' NRC U (95%) En ' CENAMEP U (95%) En E E E E E E E E E E E E E E E E E E E E E E E E E-05 7.E E E E-05.00E E E E E E E E-05.03E E E E E E E E-05.05E E E E E E-05 6.E E-05.07E E E E E E E E-05.09E E E E E E E E-05.E E E E-05 8.E E E E-05.4E E E E E E E E-05.6E E E E E E E E-05.8E E E E E E E E-05.0E E E E E E E E-05.3E E E E E E E E-05.5E E E E E E E E-05.7E E E E E E E E-05.9E E E Centro Naconal de Metrología 9