# ISO 5168-2005

Reference numberISO 5168:2005(E)©ISO 2005INTERNATIONAL STANDARD ISO5168Second edition2005-06-15Measurement of fluid flow — Procedures for the evaluation of uncertainties Mesure de débit des fluides — Procédures pour le calcul de l incertitudeBA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A the PDF-creation parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below. © ISO 2005 All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISO s member body in the country of the requester. ISO copyright office Case postale 56 • CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail copyright@iso.org Web www.iso.org Published in Switzerland ii © ISO 2005 – All rights reservedBA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A see D.12 K meter factor K mean meter factor jK jth K-factor; lblength of crest lhgauged head l1distance from the upstream tapping to the upstream face L1l1divided by the pipe diameter, dp2l′ distance from the downstream tapping to the downstream face 2L′ 2l′ divided by the pipe diameter, dpm particular item in a set of data m′ number of data sets to be pooled m″ number of verticals 2M′ ( )221L β′ − n number of repeat readings or observations n′ exponent of lh, usually 1,5 for a rectangular weir and 2,5 for a V-notch BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A Q flow, expressed in cubic metres per second, at flowing conditions R specific gas constant RedpReynolds number related to dpby the expression Vdpρ/µ smt,popooled experimental standard deviation of the orifice plate readings spestandard deviation of a larger set of data used with a smaller data set spostandard deviation pooled from several sets of data sr,popooled experimental standard deviation for the radiator readings s(x) experimental standard deviation of a random variable, x, determined from n repeated observations ( )s x experimental standard deviation of the arithmetic mean, x t Student’s statistic T0upstream absolute temperature T0,xtemperature at which measurement x is made Topoperating temperature uc,corr(y) combined uncertainty for those components for multiple meters that are correlated uc,uncorr(y) combined uncertainty for those components for multiple meters that are uncorrelated u*calinstrument calibration uncertainty from all sources, formerly called systematic errors or biases u*crirelative uncertainty in point velocity at a particular depth in vertical i due to the variable responsiveness of the current meter u*drelative standard uncertainty in the coefficient of discharge BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A usmstandard uncertainty of a single value based on past experience u(xi,corr) correlated components of uncertainty in a single meter u(xi,uncorr) uncorrelated components of uncertainty in a single meter u*(xi) standard uncertainty associated with the input estimate, xi*c()uy combined standard uncertainty associated with the output estimate, y u*(xi) relative standard uncertainty associated with the input estimate xi*c()uy combined relative standard uncertainty associated with the output estimate, y U*(y) relative expanded uncertainty associated with the output estimate U(y) expanded uncertainty associated with the output estimate, y UCMCcombined uncertainty of the calibration rig AS-overall-EU type A uncertainty in meter error *AS-overall-KU type A uncertainty in the K-factor V mean velocity in the pipe Vi mean velocity associated with a vertical i xiestimate of the input quantity, Xixmmth observation of random quantity, x x0dimension at temperature T0,xx arithmetic mean or average of n repeated observations, xm, of randomly varying quantity, x y estimate of the measurand, Y ∆xiincrement in xiused for numerical determination of sensitivity coefficient BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A correlated sources require different treatment (see Annex F). Consideration should also be given to the time over which the measurement is to be made, taking into account that flow-rate will vary over any period of time and that the calibration can also change with time. If the functional relationship between the input quantities X1, X2, …, XN, and output quantity Y in a flow measurement process is specified in Equation (1): ( )12, ,.,NYfXX X= (1) then an estimate of Y, denoted by y, is obtained from Equation (1) using input estimates x1, x2, … xN, as shown in Equation (2): ( )12, ,.,Ny fxx x= (2) Provided the input quantities, Xi, are uncorrelated, the total uncertainty of the process can be found by calculating and combining the uncertainty of each of the contributing factors in accordance with Equation (3): () ( )2c1Niiiuy cux==∑(3) Where the extent of interdependence is known to be small, Equation (3) may be applied even though some of the input quantities are correlated; ISO 5167-1:2003 [1]provides an example of this. Each of the individual components of uncertainty, u(xi), is evaluated using one of the following methods: Type A evaluation: calculated from a series of readings using statistical methods, as described in Clause 6; Type B evaluation: calculated using other methods, such as engineering judgement, as described in Clause 7. Uncertainty sources are sometimes classified as “random” or “systematic” and the relationship between these categorizations and Type A and Type B evaluations is given in Annex I. The sensitivity coefficients, ci, provide the links between uncertainty in each input and the resulting uncertainty in the output. The methods of calculating the individual sensitivity coefficients, ci, are described in detail in Clause 8. BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A see D.1: ,11niimmxxn==∑(4) b) Calculate the standard deviation of the sample in accordance with Equation (5); see D.2: ()()()2,111niimimsx x xn==−−∑(5) The standard uncertainty of a single sample is the same as its standard deviation and is given by Equation (6): ( ) ( )iiux sx= (6) c) Calculate the standard deviation of the mean value in accordance with Equation (7); see D.4: ()()iis xsxn= (7) The standard uncertainty of the mean value is then given by Equations (8): ( ) ( )iiux sx= (8) BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A the shapes of the distributions are shown in Annex B. 7.3 Rectangular probability distribution Typical examples of rectangular probability distributions include maximum instrument drift between calibrations, error due to limited resolution of an instrument’s display, manufacturers tolerance limits. The standard uncertainty of a measured value, xi, is calculated from Equation (9): ()3iiaux = (9) where the range of measured values lies between xi− ai andxi+ ai. The derivation of Equation (9) is given by Dietrich [2]. BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A k is the quoted coverage factor; see Annex C. Where a coverage factor has been applied to a quoted expanded uncertainty, care should be exercised to ensure that the appropriate value of k is used to recover the underlying standard uncertainty. However, if the coverage factor is not given and the 95 % confidence level is quoted, then k should be assumed to be 2. 7.5 Triangular probability distribution Some uncertainties are given simply as maximum bounds within which all values of the quantity are assumed to lie. There is often reason to believe that values close to the bounds are less likely than those near the centre of the bounds, in which case the assumption of rectangular distribution could be too pessimistic. In this case, the triangular distribution, as given by Equation (11), may be assumed as a prudent compromise between the assumptions of a normal and a rectangular distribution. ()6iiaux = (11) 7.6 Bimodal probability distribution When the error is always at the extreme value, then a bimodal probability distribution is applicable and the standard uncertainty is given by Equation (12): ( )iiux a= (12) Examples of this type of distribution are rare in flow measurement. 7.7 Assigning a probability distribution When the source of the uncertainty information is well defined, such as a calibration certificate or a manufacturer’s tolerance, the choice of probability distribution will be clear-cut. However, when the information is less well defined, for example when assessing the impact of a difference between the conditions of calibration and use, the choice of a distribution becomes a matter of the professional judgement of the instrument engineer. 7.8 Asymmetric probability distributions The above cases are for symmetrical distributions, however, it is sometimes the case that the upper and lower bounds for an input quantity, Xi, are not symmetrical with respect to the best estimate, xi. In the absence of information on the distribution, GUM recommends the assumption of a rectangular distribution with a full range equal to the range from the upper to the lower bound. The standard uncertainty is then given by Equation (13): ()12iiiabux′+= (13) where (xi− ai) Xi (xi+ b′i). BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A see Annex K. A common example of an asymmetric distribution is seen in the drift of instruments due to mechanical changes, for example, increasing friction in the bearings of a turbine meter or erosion of the edge of an orifice plate. 8 Sensitivity coefficients 8.1 General Before considering methods of combining uncertainties, it is essential to appreciate that it is insufficient to consider only the magnitudes of component uncertainties in input quantities, it is also necessary to consider the effect each input quantity has on the final result. For example, an uncertainty of 50 µm in a diameter or 5 % in a thermal expansion coefficient is meaningless in terms of the flow through an orifice plate without knowledge of how the diameter or thermal expansion impact the measurement of flow-rate. It is, therefore, convenient to introduce the concept of the sensitivity of an output quantity to an input quantity, i.e. the sensitivity coefficient, sometimes referred to as the influence coefficient. The sensitivity coefficient of each input quantity is obtained in one of two ways: analytically; or numerically. 8.2 Analytical solution When the functional relationship is specified as in Equation (1), the sensitivity coefficient is defined as the rate of change of the output quantity, y, with respect to the input quantity, xi, and the value is obtained by partial differentiation in accordance with Equation (15): iiycx∂=∂(15) However, when non-dimensional uncertainties (for example percentage uncertainty) are used, non-dimensional sensitivity coefficients shall also be used in accordance with Equation (16): * iiixycxy∂=∂(16) In certain special cases where, for example, a calibration experiment has made the functional relationship between the input and output simple, the value of cior c*ican be unity. Example 1 in Annex G gives an example for a calibrated nozzle. 8.3 Numerical solution Where no mathematical relationship is available, or the functional relationship is complex, it is easier to obtain the sensitivity coefficients numerically, by calculating the effect of a small change in the input variable, xi, on the output value, y. BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A the treatment of correlated uncertainties is discussed in C.6. Correlation arises where the same instrument is used to make several measurements or where instruments are calibrated against the same reference. BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A by referring to Figure 1, it can be seen that, with an effective k factor of 1, the bandwidth defined by a standard uncertainty will only have a confidence level of about 68 % associated with it. There is, therefore, a 2:1 chance that the true value will lie within the band, or a 1 in 3 chance that it will lie outside the band. Such odds are of little value in engineering terms and the normal requirement is to provide an uncertainty statement with 90 % or 95 % confidence level; in some extreme cases, 99 % or higher might be required. To obtain the desired confidence level, an expanded uncertainty, U, is used in accordance with Equation (21): ( )cUkuy= (21) or, where relative uncertainties are being used, in accordance with Equation (22): ()**cUkuy= (22) Key X1standard deviation X2coverage factorY percent of readings in bandwidth Figure 1 — Coverage factors for different levels of confidence with the normal, or Gaussian, distribution It is recommended that for most applications a coverage factor, k = 2, be utilized to provide a confidence level of approximately 95 %; the choice of coverage factor will depend on the requirement of the application. Values of k for various levels of confidence are given in Table 2. BA178AF3EC677050DBAC9B8DA5349567ADC990EEFF93F3B562D565BBCCE551B04FB30348A9F905D2BEBE2EE710159EC8B560C0BF30F077189D3B68E26A9F3C8A6364EAF5157E4DFF4A2F011E10AD206AF0BC6437B3B1BD91F7746FD35E915923E633A see Annex E. Estimate the standard uncerta