1、Reference number ISO/TR 20461:2000(E) ISO 2000 TECHNICAL REPORT ISO/TR 20461 First edition 2000-11-01 Determination of uncertainty for volume measurements made using the gravimetric method Dtermination de lincertitude de mesure pour les mesurages volumtriques effectus au moyen de la mthode gravimtri
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7、s Page Foreword.iv 1 Scope 1 2 Modelling the measurement .1 3 Standard uncertainty of measurement associated with the volume V 20 .4 4 Sensitivity coefficients4 5 Standard uncertainty associated with the volume delivered by a piston-operated volumetric apparatus6 6 Standard uncertainties of measurem
8、ent.7 7 Expanded uncertainty of measurement associated with volume V 20 7 8 Example for determining the uncertainty of the measurement 7 Bibliography10ISO/TR 20461:2000(E) iv ISO 2000 All rights reserved Foreword ISO (the International Organization for Standardization) is a worldwide federation of n
9、ational standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. Inter
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11、 rules given in the ISO/IEC Directives, Part 3. The main task of technical committees is to prepare International Standards. Draft International Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by
12、 at least 75 % of the member bodies casting a vote. In exceptional circumstances, when a technical committee has collected data of a different kind from that which is normally published as an International Standard (“state of the art”, for example), it may decide by a simple majority vote of its par
13、ticipating members to publish a Technical Report. A Technical Report is entirely informative in nature and does not have to be reviewed until the data it provides are considered to be no longer valid or useful. Attention is drawn to the possibility that some of the elements of ISO/TR 20461 may be th
14、e subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. ISO/TR 20461 was prepared by Technical Committee ISO/TC 48, Laboratory glassware and related apparatus, Subcommittee SC 1, Volumetric instruments.TECHNICAL REPORT ISO/TR 20461:2000(E) ISO 200
15、0 All rights reserved 1 Determination of uncertainty for volume measurements made using the gravimetric method 1 Scope This Technical Report gives the detailed evaluation of uncertainty for volume measurements according to the Guide to the Expression of Uncertainty in Measurement (GUM) 1. It uses th
16、e gravimetric method specified in ISO 8655-6 2 as the reference method for calibrating piston-operated volumetric apparatus. It has been arranged in paragraphs to facilitate direct access to different aspects of this kind of evaluation as follows: modelling the measurement by describing the physical
17、 equations necessary to calculate the volume using the gravimetric method of measurement; determination of the standard uncertainty of measurement associated with the volume V 20 by describing the calculation procedure according to the GUM; determination of the sensitivity coefficients with an examp
18、le of the calculation of all sensitivity coefficients by using complete equations, approximations of equations and by giving numerical values for standard conditions; determination of the standard uncertainty associated with the volume delivered by a piston-operated volumetric apparatus giving the c
19、ombination of the standard uncertainty associated with the volume V 20 measured using the gravimetric measuring system and the experimental standard deviation associated with the volume delivered by the apparatus; determination of the standard uncertainties of measurement with a brief insight into t
20、he calculation of uncertainties of measuring devices according to GUM; determination of the expanded uncertainty of measurement associated with volume V 20 ; example of the determination of the uncertainty for volume measurements. 2 Modelling the measurement The equation for the volume V 20 of the d
21、elivered water at 20 Ci sg i v e nb y V 20 = m Z Y (1) with m = m 2 m 1 m E (2) where m is the balance reading of delivered water; m 1 is the balance reading of the weighing vessel before delivery of the measured volume of water;ISO/TR 20461:2000(E) 2 ISO 2000 All rights reserved m 2 is the balance
22、reading of the weighing vessel after delivery of the measured volume of water; m E is the balance reading of the mass loss due to evaporation of liquid during the measurement; Z is the combined factor for buoyancy correction and conversion from mass to volume; Y is the thermal expansion correction f
23、actor of the delivering device. Equation (1) combines the measurement results yielded by the balance (m), air and liquid densities yielded by measurements of air and liquid temperatures, air pressure and relative humidity of air in conjunction with tables or equations for the factor (Z), and paramet
24、ers of the delivering device (Y). Z is given by a bb a a wb w a w 1 11 1 Z (3) where w is the density of water; a is the density of air; b is the density of the standard weight used to calibrate the balance according to OIML (Organisation Internationale de Mtrologie Lgale), b = 8 000 kg/m 3 for stee
25、l weights. The density of water w (in kg/m 3 ) is given by an equation 3 which is a very useful approximation of the equation of Kell 4,5 in the temperature range 5 Ct o4 0C. The relative deviation between this equation and the original equation of Kell (given in reference 5 in terms of the ITS-90 t
26、emperature scale and valid for temperatures between 0 C and 150 C) is less than 10 6 in the temperature range 5 Ct o4 0C. 4 w w 0 i i i t a (4) where t w is the water temperature in degrees Celsius; with the constants (ITS-90 temperature scale): a 0 is equal to 999,853 08 kg/m 3 ; a 1 is equal to 6,
27、326 93 10 2 C 1 kg/m 3 ; a 2 is equal to 8,523 829 10 3 C 2 kg/m 3 ; a 3 is equal to 6,943 248 10 5 C 3 kg/m 3 ; a 4 is equal to 3,821 216 10 7 C 4 kg/m 3 . Any additional corrections for the pressure dependence and gas saturation of the water density are negligible as they are very small.ISO/TR 204
28、61:2000(E) ISO 2000 All rights reserved 3 The density of air a (in kg/m 3 ) is given by 5: 1a 2a 3 a aa 0 kp kt k tt (5) where t a0 is equal to 273,15 C; p a is the pressure, expressed in hectopascals (hPa); is the relative humidity, expressed as a percentage; t a is the air temperature, expressed i
29、n degrees Celsius; with the constants (ITS-90 temperature scale): k 1 is equal to 0,348 44 (kg/m 3 ) C/hPa; k 2 is equal to 0,002 52 kg/m 3 ; k 3 is equal to 0,020 582 (kg/m 3 ) C. The correction for the thermal expansion of the delivering device is given by cdd 2 0 1( ) Yt t (6) where c is the cubi
30、c expansion coefficient in C 1 ; t d is the device temperature in degrees Celsius; t d20 is equal to 20 C. The temperatures t w , t a ,a n dt d a r ea s s u m e dt ob eu n c o r r e l a t e d ,a st h ea c t u a lv a l u e so ft w and t d do not only depend on t a , but also strongly depend on the ha
31、ndling by the user. Considerable effects of evaporation-cooling and hand-warming when using handheld apparatus are to be taken into account. The resulting temperature differences are often larger than the uncertainty in the temperature measurement. Equations (1) to (6) show that one may write: ba dd
32、 2 0 c 20 bwa 1( ) m tt V (7) This model shows that the measured volume V 20 is a function of m, t w , t a , p a , , c , t d , and some constants. 20 w a a c d ( ) ( , , , , , , ; constants) i Fx Fmt t p t V (8)ISO/TR 20461:2000(E) 4 ISO 2000 All rights reserved 3 Standard uncertainty of measurement
33、 associated with the volume V 20 According to the GUM the standard uncertainty of measurement associated with the value V 20 may be written as: 2 22 2 2 20 ( ) () () ii i i ii F uV c ux ux x (9) 22 2 2 22222 20 w a a waa () () () () (). . . FFFF uV um ut ut up mttp (10) where u 2 (x i ) are the stan
34、dard uncertainties referred to the measurement of each quantity which contributes to the final result (described by the model); c i 2 are the sensitivity coefficients giving the weight of each individual standard uncertainty. The sensitivity coefficients may be determined by calculating the partial
35、derivatives as indicated in equation (9), by numerical calculations, or by experiment. As the uncertainties of the constants equation (8) and the uncertainties of equations (4) and (5) for w and a are very small compared to other uncertainties, they may be neglected in the evaluation of uncertainty.
36、 4 Sensitivity coefficients The evaluation of the uncertainty of measurement does not require such exact values and exact solutions of the mathematical model for the measurement, as the determination of the volume V 20 itself. Approximations are tolerable, but they have to be used only for this unce
37、rtainty evaluation. In the following the approximations w a w , b a b , w 1000kg/m 3 ,1 c (t d t d20 ) 1, and b w b are used without special notation. Keep in mind that the first approximations are of the order 10 3 or less, whereas the last approximation is of the order 10 1 . This last approximati
38、on is justified as it is affecting only the air buoyancy correction. The sensitivity coefficients c i in equation (9) are calculated as partial derivatives using equations (11) to (29). The sensitivity coefficient c w related to the balance reading m is calculated as follows: 20 w V F c mm (11) ww F
39、 c m (12) 3 3 w mn l 10 1 kgg F c m (13) The sensitivity coefficient c c related to the cubic expansion coefficient c of the piston-operated volumetric apparatus is calculated as follows: c ba dd 2 0 cbwa () Fm ct t (14)ISO/TR 20461:2000(E) ISO 2000 All rights reserved 5 c dd 2 0 cw () Fm ct t (15)
40、c 1 3 d 3 c kg 10 K ( 20 C) m F cm t (16) It should be emphasized that c is not a well defined value for a compound system. The sensitivity coefficient c t d related to the temperature t d of the piston-operated volumetric apparatus is calculated as follows: d ba c dbwa t Fm c t (17) d c dw t Fm c t
41、 (18) If c =1 0 5 K 1 is used: d 1 8 3 d kg 10 K m t F cm t (19) It should be emphasized that the temperature t d of the piston-operated volumetric apparatus is neither spatially nor temporally constant because of hand-warming at the middle and the top, and evaporation-cooling at the bottom of the a
42、pparatus. The sensitivity coefficient c t w related to the water temperature t w is calculated as follows: w 4 1 cd d 2 0 ba w 2 wb wa 1 1( ) () () i ti i tt Fm ci a t t (20) w 4 1 w w 22 ww ww 1 i ti i Fm m ci a t tt (21) It is possible to use the expression 41 w w w 2,1 10 K t instead of the sum g
43、iven in equation (21) in the temperature range of 19 Cto2 1C with sufficient accuracy. w 1 41 7 3 ww kg 2 , 11 0 K 2 , 11 0 K m t Fm cm t (22) It should be emphasized that t w may also be affected by evaporation-cooling as by hand-warming. The sensitivity coefficient c p a related to the air pressur
44、e p a is calculated as follows: a bw 1 cd d 2 0 2 ab aa 0 wa 1( ) () p k Fm ct t pt t (23)ISO/TR 20461:2000(E) 6 ISO 2000 All rights reserved a 1 2 aa a 0 w p k Fm c pt t (24) If t a =20C is used: a 1 9 3 a kg 1,2 10 K m p F cm p (25) The sensitivity coefficient c related to the relative air humidit
45、y is calculated as follows: bw2 a3 cd d 2 0 2 ba a 0 wa 1( ) () kt k Fm ct t tt (26) 2a 3 2 aa 0 w kt k Fm c tt (27) If t a =20C is used: 1 10 3 kg 11 0 % m F cm (28) The sensitivity coefficient c t a related to the air temperature t a is calculated as follows: a bw 2 a 01 a3 cd d 2 0 22 ab wa aa 0
46、1( ) ()() t kt kp k Fm ct t t tt (29) a 2a 0 3 1a 22 a wa a 0 () () t kt k kp Fm c t tt (30) If =50%,p a = 1013 hPa, and t a =20Ca r eu sed : a 1 9 3 a kg 4,5 10 K m t F cm t (31) 5 Standard uncertainty associated with the volume delivered by a piston-operated volumetric apparatus As mentioned in annex B of ISO 8655-6: 2 there are two sources of uncertainty. One source is the uncertainty of the measurement of the delivered volume by the gravimetric method, the other is the uncertainty of the delivery process itself. By combining both, the standard uncertainty associated with the vo
