λ - Reference surface tension (at scale line), λ

Transcription

1 XVIII IMEKO WORLD CONGRE Mtrology for a ustainabl Dvlopmnt ptmbr, 7, 006, Rio d Janiro, Brazil THE ROLE OF EXPERIMENTAL DEIGN IN THE HYDROMETER FIELD Maria do Céu Frrira, José Mndonça Dias Portugus Institut for Quality, Cntral Laboratory of Mtrology, CAPARICA, Portugal. mcfrrira@mail.ipq.pt Dpartmnt of Mchanical and Industrial Enginring, Faculty of cincs and Tchnology, Univrsidad Nova d Lisboa, Quinta da Torr, CAPARICA, Portugal. jad@fct.unl.pt Abstract: Th main purpos of this papr is to study th budgt of hydromtrs calibration and th rror of Cuckow s mthod according th ignal-to-nois Ratio Taguchi approach in ordr to analys this two diffrnt kinds of mthodology. Th calibration of dnsity hydromtrs is influncd by som paramtrs; in ordr to rsarch th influnc of ths factors in th masuring valu, an xprimntal dsign with dynamic charactristics was applid. With this mthodology, fiv control factors wr considrd, at two lvls ach and a L 3 Orthogonal Array was dfind. It was valuatd th influnc of ach paramtr (factor) and th intraction btwn thm according Cuckow s mthod and Taguchi mthodologis. Th simultanous analysis of th rror and th snsitivity lads to a propr choic of significant control factors and rspctiv bst lvls in ordr to rduc th variability of th masurmnt systm. Kywords: Dnsity, Hydromtrs, Taguchi, ignal-to-nois Ratio, Dynamic Charactristics.. INTRODUCTION Hydromtrs ar dnsity liquids masuring instrumnts usd in svral filds which tracability to I is providd by National Laboratory of Mtrology. A basic tool in nsuring this tracability is masuring instrumnt calibration. Masurmnt of liquid dnsity plays a major rol in th quality assuranc of production procsss. Th dnsity of liquids is masurd in a wid rang of procss industris, such as pharmacutical, food, agrochmical and ptrochmical. Bsid othr masurmnts mthods th us of hydromtrs is on of th most mployd by majority industrial laboratoris. A hydromtr can b calibratd with th rfrnc liquids, whr it frly floats in on of its graduation marks at th sam plan of th lvl surfac of th liquid. Portugus Institut for Quality adopts th hydrostatic wighing mthod for hydromtrs calibration using a singl liquid whos dnsity is not dlibratly altrd throughout th whol procss. In this study, a dnsity hydromtr was calibratd in thr scal graduation marks using n-nonan as rfrnc liquid by Cuckow mthod, which is basd on th Archimds principl war + ϕ λ pt ( ρliq ρal ) [ + α ( tliq trf )] ρal ρ = + ρar ρ al war wliq + ϕ λliq ρ wt whr: ρ liq - Th dnsity of th liquid in which th hydromtr is wighd, ρ - Th dnsity of th air in which th hydromtr is al wighd, w - Th wight of th hydromtr in air of dnsity ρ, ar w - Th apparnt wight of th hydromtr in liquid of liq dnsity ρ al, λ - Rfrnc surfac tnsion (at scal lin), λ pt liq - Liquid surfac tnsion of tst liquid, α - Cubic xpansion cofficint (5E-06), d - whr d is th diamtr of hydromtr stm at ϕ = g mniscus lvl and g th gravity acclration. For th calibration of hydromtrs, all paramtrs must b known, including thir.. CALIBRATION REULT AND UNCERTAINTY BUDGET From quation (), th dnsity of ach rfrnc valu was obtaind by xprimntal mthod at 0 ºC which valus ar shown in Tabl. Tabl : Calibration rsults at 0 ºC Indication,0035,0065,0095 Convntional tru valu,0035,0069,00935 aa ()

2 In ordr to valuat th of calibration in accordanc to th GUM [], th valus of th svral contributions to combind ar summarizd in Tabls, 3 and 4 for th thr convntional tru valu. Tabl : Expandd for th rfrnc valu of,0035 tandard componnt Unit Valu (x i ) uncrtaint y snsitivity coficint C i Contribution to th combind u i (y) Tabl 4: Expandd for th rfrnc valu of,00935 tandard componnt Unit Valu (x i ) snsitivity coficint C i Contribution to th combind u i (y) u (t rf ) ºC 0,50E-03 -,50E-05-6,30E-08 u (t liq ) ºC 0,,50E-03,50E-05 6,30E-08 u (l pt ) (mn/m) 75 4,9E-0 9,534E-06 4,68E-07 u (t rf ) ºC 0,50E-03 -,505E-05-6,6E-08 u (t liq ) ºC 0,,50E-03,505E-05 6,6E-08 u (l pt ) (mn/m) 75 4,9E-0 9,477E-06 4,65E-07 u (l liq ) (mn/m),9,50e-0 -,35E-05-3,3E-07 u (r liq ) 0, ,64E-06,398E+00 6,48E-06 u (r al ) 0,0070 3,5E-06-3,448E-0 -,E-06 u (r ar ) 0,0070 3,5E-06-4,977E-0 -,75E-07 u (r wt ) 8,89E-05 0,000E+00 0,00E+00 u (w liq ) g 4,7500 6,58E-05 3,066E-0,0E-06 u (w ar ) g 50,3,57E-04 -,65E-03-4,7E-07 u (a) /K,5E-05,45E-05,00E-0,45E-06 u (d) cm 0,444,0E-03 9,75E-04,0E-06 u c = 7,6E-06; Dgrs of frdom =,95E+3; Expandd U(y) for a confidnc lvl of 95% =,43E-05 Tabl 3: Expandd for th rfrnc valu of,0069 tandard componnt Unit Valu (x i ) snsitivity coficint C i Contribution to th combind u i (y) u (t rf ) ºC 0,50E-03 -,53E-05-6,8E-08 u (t liq ) ºC 0,,50E-03,53E-05 6,8E-08 u (l liq ) (mn/m),9,50e-0 -,34E-05-3,35E-07 u (r liq ) 0, ,64E-06,407E+00 6,5E-06 u (r al ) 0,0070 3,5E-06-3,554E-0 -,5E-06 u (r ar ) 0,0070 3,5E-06-5,4E-0 -,80E-07 u (r wt ) 8,89E-05 0,000E+00 0,00E+00 u (w liq ) g 43,400 6,6E-05 3,03E-0,94E-06 u (w ar ) g 50,3,6E-04 -,74E-03-3,43E-07 u (a) /K,5E-05,45E-05,008E-0,46E-06 u (d) cm 0,444,0E-03 9,89E-04,0E-06 u c = 7,8E-06; Dgrs of frdom =,4E+3; Expandd U(y) for a confidnc lvl of 95% =,44E-05 Figur summarizs calibration rsults with th valu of. u (l pt ) (mn/m) 75 4,9E-0 9,505E-06 4,66E-07 u (l liq ) (mn/m),9,50e-0 -,333E-05-3,33E-07 u (r liq ) 0, ,64E-06,40E+00 6,50E-06 u (r al ) 0,0070 3,5E-06-3,58E-0 -,3E-06,00935,0069 VCV (g/cm^3) u (r ar ) 0,0070 3,5E-06-5,045E-0 -,77E-07 u (r wt ) 8,89E-05 0,000E+00 0,00E+00 u (w liq ) g 43,0756,0E-04 3,085E-0 3,0E-06 u (w ar ) g 50,3,6E-04 -,687E-03-3,38E-07 u (a) /K,5E-05,45E-05,005E-0,46E-06 u (d) cm 0,444,0E-03 9,8E-04,0E-06 u c = 7,55E-06;,0035 -,50E-04 -,50E-04-5,00E-05 5,00E-05,50E-04,50E-04 Figur : Uncrtainty of calibration Dgrs of frdom =,84E+3; Expandd U(y) for a confidnc lvl of 95% =,5E-05

3 3 ANALYI OF THE DENITY HYDROMETER ERROR UING TAGUCHI N RATIO Aftr th analysis of th paramtrs rlatd with th calibration of th dnsity hydromtr, was carrid out an xprimntal layout with fiv control factors, at two lvls ach, as follows: A Ambint tmpratur in th rang of 7 ºC to 3 ºC; B Atmosphric prssur in th rang of 900 hpa to 900 hpa; C uprficial tnsion of n-nonan in th rang of mn/m to 5 mn/m; D Liquid tmpratur in th rang of 8 ºC to ºC; E- Humidity in th rang of 45% to 80%. Tabl 5 shows th Orthogonal Array (L 3 ) obtaind for this calibration study. Tabl 5: Dsign Array n A B C D E Following th approach in Yano, H. (99), a linar rgrssion modl is fittd for ach row of Tabl 5 using th st of obsrvations obtaind for xprimntal rsults. Thus, for i th row of Tabl 6 th modl ( M ) ε y = m + β M + i i () m - Man of indication valu, M - Tru valu, M - Man of tru valu β - Ratio of covarianc s ε - Calibration rror Th valu of β is slctd in ordr to minimiz th total of th squars of diffrncs btwn th lft sid and th right sid of th quation. Th valu of β that minimizs quation is givn by: [( M - M) y + ( M - M ) y ( M k - M ) yk ˆ β = ] (3) r Whr r is stimatd by: K r 0 ( Mi M ) (4) i= r = and M by: M ( M + M k M ) + k = (5) Th variation causd by linar ffct, dnotd by givn by: is β K β = [ ( M i M )y i ] (6) r i= Th rror variation, including th dviation from linarity V, is: = T β (7) Th rror varianc is th rror variation dividd by its dgrs of frdom. V = kr = T β 0 kr0 For this application, signal-rspons is stimatd in dcibls by: ( β V ) = 0log r ( db) V η (9) Th raw data obtaind for ach xprimntal run, along with its associatd ignal-to-nois Ratio ar summarizd in Tabl 6. (8) is fittd using last squars rgrssion, whr: y - Indication valu,

4 Tabl 6: Raw data for ach xprimnt al rsults n Mdium of convncional tru FC r b T Vrror rror valu (db),0046,00765,007 9,74 5,87E-05 5,4E ,68E-07,58E-06 63,73,0048,0073,0038 9,74 6,6E-05 4,90E ,08E-07 4,96E-06 60,449 3,00398,0070,0008 9,74 6,5E-05 4,90E-05 5,40E-05 7,8E-07 5,0E-06 60,387 4,0043,00736,004 9,74 5,95E-05 5,07E ,70E-07 3,9E-06 6,548 5,00479,00783,0089 9,74 5,78E-05 5,3E-05 0,38E-07,67E-06 65,778 6,00445,00750,0056 9,74 6,00E-05 5,03E ,E-07 3,66E-06 6,007 7,00449,00753,0059 9,74 5,70E-05 5,9E-05 0,57E-07,0E-06 67,699 8,00467,0077,0077 9,74 5,8E-05 5,9E ,0E-07,E-06 64,686 9,008,00484, ,74 5,77E-05 5,E-05 0,7E-07,90E-06 65, 0,0047,0045, ,74 5,76E-05 5,E-05 0,74E-07,9E-06 65,64,009,0043,0078 9,74 5,8E-05 5,6E ,48E-07,44E-06 64,033,0038,0044, ,74 5,66E-05 5,9E-05 0,49E-07,04E-06 67,958 3,0043,00446,0075 9,74 5,78E-05 5,9E-05 0,9E-07,04E-06 64,869 4,0048,0045, ,74 5,7E-05 5,5E-05 0,0E-07,40E-06 66,608 5,006,0049, ,74 6,5E-05 5,0E-05 0,80E-07,96E-06 64,709 6,005,00455, ,74 5,85E-05 5,E ,90E-07,73E-06 63,476 7,00440,00745,005 9,74 6,3E-05 4,93E ,74E-07 4,7E-06 60,707 8,0044,00745,005 9,74 6,3E-05 4,93E ,76E-07 4,73E-06 60,695 9,004,0075,00 9,74 6,0E-05 5,0E ,58E-07 3,90E-06 6,69 0,004,0076,00 9,74 6,0E-05 5,0E ,6E-07 3,94E-06 6,65,0043,00736,004 9,74 6,8E-05 4,89E ,33E-07 5,3E-06 60,69,0043,00737,0043 9,74 6,8E-05 4,89E ,35E-07 5,4E-06 60,5 3,00448,00753,0059 9,74 5,7E-05 5,8E-05 0,70E-07,9E-06 67,350 4,0044,00746,005 9,74 5,75E-05 5,4E-05 0,E-07,56E-06 66,4 5,0087,00497, ,74 5,95E-05 5,5E ,5E-07,46E-06 63,879 6,0087,0049, ,74 5,8E-05 5,7E ,4E-07,7E-06 64,360 7,0054,00458, ,74 5,86E-05 5,E ,07E-07,85E-06 63,84 8,0046,00449, ,74 5,9E-05 5,06E ,85E-07 3,39E-06 6,4 9,0073,00477,0078 9,74 5,85E-05 5,3E ,85E -07,69E-06 63,545 30,0073,00476,0078 9,74 5,9E-05 5,08E ,5E-07 3,6E-06 6,760 3,005,00455, ,74 6,0E-05 4,98E ,00E-07 4,0E-06 6,34 3,0053,00456,0076 9,74 6,0E-05 4,98E ,94E-07 4,6E-06 6,39 3. ANALYI OF VARIANCE Th main purpos of this of this mthodology is to idntify th principal factors that can induc rror incras in ordr to crat conditions for minimiz th variability of th masurmnt systm. Analysis of Varianc (ANOVA), is an important tatistic Tool that can validatd th slction of factors and lvls which minimiz th ratio. A rsult of this condnsd analysis is prsnts in Tabl 7. Th so-calld nt contribution (ρ) is also prsntd for significant ffcts. Tabl 7: ANOVA Factor g.l. M F0 p D,78,78 4,763 0,038 E 3,808 3,808 0,054 0,004 B x D,55,55 8,934 0,006 C x D 7,945 7,945,80 0,00 B x C x D 6,4 6,39 6,858 0,05 Erro 6,566 6,368 Total 6,99 3

5 According ANOVA rsults, th following conclusions ar rachd: i) Thr is vidnc that factors D and E hav an impact on valu and th bst lvls to rduc variation will b lvl for liquid tmpratur (D) and lvl to humidity (E) (Figur. ) ,5 65,0 64, ,0 63,5 63,0 60 P TL TL Figur 4: Joint ffcts plots for Intraction btwn prssur and liquid tmpratur 6,5 6,0 6,5 -,, Figur : plot for liquid tmpratur TL , ,0 6 64,5 6 64,0 63, ,0 6,5 59 T TL TL 6,0 6,5 6,0 Hu Figur 3: plot for humidity ii) Th intraction btwn BxD and CxD hav significant ffcts on ; for th first intraction th bst lvls to rduc variation will b th lvl for prssur (B) and lvl for liquid tmpratur (D). For th intraction CxD, th bst lvls to rduc variation will b th lvl for surfac tnsion (C) and lvl for liquid tmpratur (D). Figur 5: Joint ffcts plots for intraction btwn Intraction btwn surfac tnsion and liquid tmpratur iii) Th intraction btwn BxCxD is also ambiguous. Th bst lvls for prssur (B) and surfac tnsion (C) is th lvl and th lvl optimiz th contribution of liquid tmpratur (D). A possibl xplanation for ths rsults can b rlatd by th ffcts of th factors whn intract which significant influncs ar modify T: TL T: Figur 6: Joint ffcts plots for Intraction btwn prssur, liquid tmpratur and surfac tnsion TL P P

6 Th plot of Figur 7 allows assuming that all rsiduals ar in accordanc with a Normal distribution. Expctd Normal Valu 3,0,5,0,5,0 0,5 0,0-0,5 -,0 -,5 -,0 -,5 Normal Prob. Plot; Raw Rsiduals **(5-0) dsign; M Rsidual=,36793 DV: -3, Rsidual Figur 7: Normality of rsidus As a rsult of this study, th bst lvls of control factors to incras th ratio of this calibration procss ar: TL ; HU ; P TL ; T TL ; P T TL. 3. TAGUCHI MEAUREMENT ERROR According Yano (99), th masurmnt rror is dfind by, V =, and / N = 0 logη = 67, 43 η η = 0 V V 67,479 0 = 3,69E + 06 = =,7E 07 3,69E + 06 =,7E 07 witch valu was obtaind by th confirmatory xprimnt for th bst lvls of th significant factors. Thus, th dviation of masurmnt rror, s, is th squar root of varianc, s =,7E 07,,99,95,75,55,35,5,05,0 4. CONCLUION Th analysis of data from paramtr dsign xprimnts for signal-rspons systms could hlp to bttr undrstand th importanc of incrasing linarity and snsitivity, and rducing variability. In this papr it was applid Taguchi s mthodology to a dynamic procss dnsity hydromtr calibration- whr was charactrisd th bst lvls of control factors which optimiz th calibration procdur. A comparativ analysis btwn th rror and valus obtaind from hydromtr calibration by Cuckow s mthod and Taguchi mthodology, show a gradint of 8x0-04, which valu is in agr with rproducibility conditions. Th Taguchi rror valu includs th masurmnt rror and th sum of variancs in all rang of dnsity hydromtr scal. This study dals with on common typ of signal-rspons; othr typs xist and many rquir a somwhat modifid analysis. REFERENCE [] F.W. Cuckow: Nw Mthod of High Accuracy for of th Calibration of tandard Hydromtrs, J. oc. Chm. Ind., 68, pp , 949. [] BIPM, IEC, IFCC, IO, IUPAC, IUPAP, OIML, Guid to th xprssion of in masurmnt, first dition, IO Gnva, 995. [3] Taguchi, G., (99), Taguchi Mthods: Rsarch and Dvlopmnt, Amrican upplir Institut, Inc. [4] Taguchi, G. (993), Taguchi on Robust tchnology Dvlopmnt: Bringing Quality nginring Upstram, Th Amrican ocity of Mchanical Enginrs, 345 East 47th trt, Nw York, NY [5] Yano, H. (99), Mtrological Control, Quality Rsourcs, Watr trt, Whit Plains,, NY 060 s = 5,E 04 ( g / cm 3 )