Handbook-USM-fiscal-gas-metering-stations.pdf

Research forIndustrial Development HANDBOOK OF UNCERTAINTY CALCULATIONS Ultrasonic fiscal gas metering stations 2 12 1.8 Relative Expanded Uncertainty 10 1.6 Axial flow velocity [m/s] (95 % confidence level) of mass flow rate [%] 1.4 8 1.2 1 6 0.8 4 0.6 0.4 2 0.2 0 0 0 50000 100000 150000 200000 250000 Mass flow rate [kg/h] Relative Expanded Uncertainty Axial flow velocity December 2001 Handbook of Uncertainty Calculations Ultrasonic Fiscal Gas Metering Stations December 2001 Prepared for The Norwegian Society for Oil and Gas Measurement (NFOGM) The Norwegian Petroleum Directorate (NPD) by Per Lunde and Kjell-Eivind Frøysa Christian Michelsen Research AS The Handbook and the Excel program EMU - USM Fiscal Gas Metering Stations, Version 1.0, are freeware and can be downloaded from the NFOGM web pages Norwegian Petroleum Norwegian Society for Christian Michelsen Research Directorate Oil and Gas Measurement (CMR) (NPD) (NFOGM) Secretariat: Norwegian Society P.O. Box 600, P.O. Box 6031 Postterminalen, of Chartered Engineers, N-4003 Stavanger, Norway N-5892 Bergen, Norway P.O. Box 2312 Solli, Phone: (+47) 51 87 60 00 Phone: (+47) 55 57 40 40 N-0201 Oslo, Norway Fax: (+47) 51 55 15 71 Fax: (+47) 55 57 40 41 Phone: (+47) 22 94 75 48 / 61 Web-page: www.npd.no Web-page: www.cmr.no Fax: (+47) 22 94 75 02 Web-page: www.nfogm.no ISBN 82-566-1009-3 Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 1 TABLE OF CONTENTS PREFACE 6 PART A - USER'S GUIDE 1. INTRODUCTION 8 1.1 Background 8 1.1.1 USM fiscal metering of gas 8 1.1.2 Contributions to the uncertainty of USM fiscal gas metering stations 12 1.1.3 Uncertainty evaluation of USMs and USM gas flow metering stations 16 1.2 About the Handbook 18 1.3 About the program EMU - USM Fiscal Gas Metering Station 21 1.4 Overview of the Handbook 25 1.4.1 Part A - User's Guide 26 1.4.2 Part B - Appendices 27 1.5 Acknowledgements 27 2. USM FISCAL GAS METERING STATION 28 2.1 Description of USM fiscal gas metering station 28 2.1.1 General 28 2.1.2 Gas metering station equipment considered in the Handbook 32 2.2 Functional relationship - USM fiscal gas metering station 34 2.2.1 Basic operational functional relationships 35 2.2.2 Flow calibration issues 37 2.2.3 Alternative functional relationship 38 2.3 Functional relationship - Multipath ultrasonic gas flow meter 40 2.3.1 Basic USM principles 40 2.3.2 Alternative USM functional relationships 42 2.3.3 Choice of USM functional relationship and meter independency aspects 44 2.3.4 Pressure and temperature corrections of USM meter body 46 2.3.5 Transit time averaging and corrections 52 5 Calorific value 79 3.3.4 Density measurement 76 3.2 Temperature measurement 71 3.3 USM repeatability in flow calibration 82 3.5 Signal communication and flow computer calculations 96 3.4 Functional relationship .2.3 Gas compressibility factors Z and Z0 72 3.2.4.6 USM integration method 53 2.Pressure measurement 62 2.Gas densitometer 53 2.4.Handbook of uncertainty calculations .4.4.1 General density equation (frequency relationship regression curve) 56 2. UNCERTAINTY EVALUATION EXAMPLE 102 4.1 Instrumentation and operating conditions 102 .3 USM integration uncertainties (installation conditions) 94 3.5 Functional relationship .3.2 Temperature correction 57 2.4.6 Functional relationship .2.4 Installation correction 59 2.4.4.2 Deviation factor 81 3.5 Corrected density 62 2.2 Gas measurement uncertainties 69 3.2.1 Pressure measurement 69 3.2 USM transit time uncertainties 89 3.3 VOS correction 57 2.3.1 USM gas metering station uncertainty 66 3.USM fiscal gas metering stations NPD.4.Temperature measurement 64 3.4 USM field uncertainty 83 3.4.2.1 Flow calibration laboratory 81 3.1 USM meter body uncertainty 84 3.3. UNCERTAINTY MODEL OF THE METERING STATION 66 3.6 Summary of input uncertainties to the uncertainty model 96 4. NFOGM.3 Flow calibration uncertainty 80 3.4 Miscellaneous USM effects 95 3. CMR (December 2001) 2 2. 2.7 Authors’ comments to the uncertainty evaluation example 144 5.2.Expanded uncertainty of USM in field operation 137 4.4.5 Calorific value 125 4.3.4.2 Detailed level 156 .4.3.2.1 Pressure measurement 104 4.5 Signal communication and flow computer calculations 139 4.6 Summary .4 Summary .4 Energy flow rate 143 4.1 Repeatability (random transit time effects) 128 4.1 General 150 5. CMR (December 2001) 3 4.6.2 Meter body 130 4.4 Integration method (installation effects) 136 4.5 Miscellaneous USM effects 137 4.2 Temperature measurement 110 4.2. PROGRAM "EMU .6.Handbook of uncertainty calculations .4.USM FISCAL GAS METERING STATION" 150 5.4.3 Flow calibration uncertainty 125 4.4.2.4 USM field uncertainty 128 4.3 Mass flow rate 142 4.1 Flow calibration laboratory 126 4.3 Systematic transit time effects 134 4. NFOGM.4.1 Volumetric flow rate.2 Gas measurement uncertainties 104 4.4 Pressure measurement uncertainty 155 5.3 Gas compressibility factors 116 4. line conditions 139 4.2 Gas parameters 152 5.6.Expanded uncertainty of flow calibration 127 4.1 Overall level 155 5.2 Volumetric flow rate.2 Deviation factor 126 4.USM fiscal gas metering stations NPD.3 USM setup parameters 153 5.6.4.Expanded uncertainty of USM fiscal gas metering station 139 4.3 USM repeatability in flow calibration 127 4.6 Summary .4 Density measurement 118 4.3.3. standard reference conditions 140 4. flow calibration 168 5.10.7 Density measurement uncertainty 160 5.1 Overall level 161 5.5.5 Temperature measurement uncertainty 157 5.APPENDICES APPENDIX A .15 Program information 185 6.10.9 Flow calibration uncertainty 164 5.8 Calorific value uncertainty 163 5.1 Terminology for evaluating and expressing uncertainty 202 .7.SOME DEFINITIONS AND ABBREVIATIONS 199 APPENDIX B .12 Graphical presentation of uncertainty calculations 173 5.2 USM systematic deviations re.1 Uncertainty curve plots 173 5.USM fiscal gas metering stations NPD.1 USM field repeatability 166 5.2 Uncertainty bar-charts 174 5.Handbook of uncertainty calculations .7.13 Summary report . USM MANUFACTURER SPECIFICATIONS 188 7.10 USM field uncertainty 165 5.1 Overall level 157 5.12.Expanded uncertainty of USM fiscal gas metering station 180 5.6 Compressibility factor uncertainty 159 5.14 Listing of plot data and transit times 185 5.5.2 Detailed level 161 5.FUNDAMENTALS OF UNCERTAINTY EVALUATION 202 B.2 Detailed level 158 5. NFOGM. CONCLUDING REMARKS 193 PART B . CMR (December 2001) 4 5.12.11 Signal communication and flow computer calculations 172 5. UNCERTAINTY MODEL FOR THE GAS DENSITOMETER 256 REFERENCES 263 .4 Uncertainty model of the USM fiscal gas metering station 249 APPENDIX F .2 Symbols for expressing uncertainty 208 B.2.2 The “decomposition method” 253 F.1 NPD regulations (selection) 216 C.3.3 The “variance method” 255 APPENDIX G .2 NORSOK I-104 requirements (selection) 220 APPENDIX D .3 Identification of USM uncertainty terms 246 E.3.SELECTED REGULATIONS FOR USM FISCAL GAS METERING STATIONS 216 C.3.2 Simplified USM uncertainty model 240 E.2 Uncertainty model for the un-flow-corrected USM 234 E.1 The “covariance method” 252 F.3.4 Documentation of uncertainty evaluation 214 APPENDIX C .PRESSURE AND TEMPERATURE CORRECTION OF THE USM METER BODY 222 APPENDIX E .Handbook of uncertainty calculations .1 Consequences of flow calibration on USM uncertainty contributions 243 E.2.ALTERNATIVE APPROACHES FOR EVALUATION OF PARTIALLY CORRELATED QUANTITIES 252 F.3 Combining the uncertainty models of the gas metering station and the USM 243 E.1 Basic USM uncertainty model 235 E.THEORETICAL BASIS OF UNCERTAINTY MODEL 229 E. NFOGM. CMR (December 2001) 5 B.2 Modified uncertainty model for flow calibrated USM 243 E.1 Basic uncertainty model for the USM fiscal gas metering station 230 E.USM fiscal gas metering stations NPD.3 Procedure for evaluating and expressing uncertainty 211 B. Per Lunde. Tore Løland. the Norwegian Society for Oil and Gas Measurement (NFOGM) and Christian Michelsen Research (CMR). MetroPartner. The GUM was jointly developed by the International Organisation of Standardi- zation (ISO). CMR (December 2001) 6 PREFACE Norwegian regulations relating to fiscal measurement of oil and gas require that the overall measurement uncertainty is documented to be within defined limits. the different methods used have given different results. The reference group has consisted of: Reidar Sakariassen. would satisfy the need for a modern method of uncertainty evaluation in the field of fiscal oil and gas measurement. based upon the principles laid down in the GUM. This report is providing general rules for evaluating and expressing uncertainty in measurement. JME Consultants and Holta-Haaland. so that different measurement systems could be directly and reliably compared. addressing fiscal metering of oil using turbine meters. A reference group consisting of nine persons with a broad and varied competence from oil and gas measurement has evaluated and commented the handbook. Phillips Petroleum Company Norway. and to the members of the reference group for their contribution to this handbook. In 1993 the ISO report “Guide to the expression of uncertainty in measurement” (commonly referred to as the “Guide” or the “GUM”) was published. December 2001 Norwegian Petroleum Directorate (NPD) Norwegian Society for Oil and Gas Measurement (NFOGM) Einar Halvorsen Phillips Petroleum Company Norway Svein Neumann . The Norwegian Metrology and Accreditation Service. Jostein Eide. Erik Malde. A consistent and standardised method of uncertainty evaluation has been required.Fiscal metering stations” was developed by the Norwegian Petroleum Directorate (NPD). Norsk Hydro. Hans Arne Frøystein.Ultrasonic fiscal gas metering stations” is reliable and in conformity with the GUM. The intention of this initiative was that a user-friendly handbook together with an Excel program. intended for a broad scope of measurement areas. However. Norsk Hydro.Handbook of uncertainty calculations . a follow-up project was initiated by the same partners to develop a handbook addressing the uncertainty of fiscal gas metering stations using ultrasonic meters (USM). and fiscal metering of gas using orifice meters. the International Organization of Legal Metrology (OIML) and the International Bureau of Weights and Measurement (BIPM). As a further development with respect to fiscal metering of gas. Statoil. We wish to express our thanks to the project leader at CMR. the International Electrotechnical Commision (IEC). John Eide.USM fiscal gas metering stations NPD. Kongsberg Fimas. Trond Folkestad. Endre Jacobsen. In 1999 a “Handbook of uncertainty calculations . Håkon Moestue. The reference group concludes that the “Handbook of uncertainty calculations . Statoil. with a revision in 1995. NFOGM. Handbook of uncertainty calculations . CMR (December 2001) 7 PART A USER'S GUIDE . NFOGM.USM fiscal gas metering stations NPD. and have already been developed to a level where they are competitive alternatives to more conventional technology as turbine and orifice meters.1. 1. 2000]. no pressure loss. Better than ±1 % uncertainty of mass flow rate (measured value) is being reported. tentatively). the Norwegian Petroleum Directorate (NPD) and Christian Michelsen Research AS (CMR). 2000]. no obstruction of flow. there has earlier been worked out a "Handbook on uncertainty calculations . reducing the need for a multi- plicity of meters to cover a wide flow range.. and the short upstream/downstream requirements with respect to disturbances. and self-checking capabilities (from sound velocity.Fiscal metering stations" [Dahl et al. cf. As a further development with respect to fiscal gas metering stations.1 USM fiscal metering of gas Multipath transit-time ultrasonic gas flow meters (USMs) are today increasingly taken into use for fiscal metering of natural gas. etc. such as flow monitoring (e. USMs have demonstrated their capability to provide metering accuracy within national regulation requirements [NPD. velocity of sound (VOS) profile). flow velocity profile. INTRODUCTION 1. Compact metering stations can be constructed on basis of the large turn-down ratio of USMs (35:1 or larger.g. 2000]. Fig.Handbook of uncertainty calculations . 1998]. signal level. NFOGM.USM fiscal gas metering stations NPD. CMR (December 2001) 8 1. USM technology offers significant operational advantages such as no moving parts. There are also potentials of utilizing additional information such as for gas density and calorific value determination. and bi-directional operation (reducing need for pipework). 2001]. a follow-up project has been initiated between the same partners for developing a handbook on uncertainty calculations for gas metering stations which are based on a flow calibrated multipath ultrasonic gas flow meter [Lunde. Measurement possibilities are offered which have not been available earlier. [Kongsberg. 1999]. [Instromet. 2000]. as required for custody transfer [NPD. pulsating flow.1. and fiscal gas metering stations based on an orifice flow meter. In ap- .). That handbook concentrated on fiscal oil metering stations based on a turbine flow meter. Three manufacturers offer such meters for gas fiscal metering today [Daniel. The first generation of USMs has been on the market for about a decade. 2001].1 Background In a cooperation between the Norwegian Society of Oil and Gas Measurement (NFOGM). [AGA. 1. Handbook of uncertainty calculations . Used by permission of Daniel Industries. users were to a large extent awaiting the AGA-9 document. based e. FMC Kongsberg Metering and Instromet. USMs still represent "new" technology. issued in 1998. 1998a]. (c) In- stromet Q-Sonic [Instromet. 9 [AGA-9. In the USA.1 Three types of multipath ultrasonic flow meters for gas available today: (a) Daniel Senior- Sonic [Daniel. (a) (b) (c) Fig. work has been initiated for ISO standardization of ultrasonic gas flow measurement. on the reports ISO/TR 126765:1997 [ISO. 2001]. which requires competence both in production and use. .USM fiscal gas metering stations NPD. [NORSOK.. The technology is expected to mature over time. etc. Canada. and the need for standardization and improved traceability is significant. Internationally. multi-path ultrasonic meters can offer significant cost bene- fits. which in Europe has been tried out over a period of about 10 years. 1998] provides guidelines for practical use of USMs for fiscal gas metering. The world market for precision metering of gas is significant. In spite of the considerable interest from oil and gas industry users. 1997] and AGA-9. USM technology is increasingly gaining acceptance throughout the industry. 1997. 2001].g. 2000. 1. CMR (December 2001) 9 propriate applications. 2000]. 2000. 2001]. and several hundred meters are today delivered each year on a world basis (including fiscal and "check” meters). and is today in use in gas metering stations onshore and offshore. NFOGM. Compared to conventional turbine and orifice meters. In Norway the use of USMs is already included in the national regulations and standards for fiscal metering of gas [NPD. (b) Kongsberg MPU 1200 [Kongsberg. USM technology is relatively young compared with more traditional flow metering technologies. and potentials and needs exist for further robustness and development. A USM is an advanced "high-technology" electronics instrument. In the USA the AGA report no. As there is a great number of ways to determine the flow of natural gas. in the NPD regulations entering into force in January 2002 [NPD. In practical use of USMs in fiscal gas metering stations. the seller and buyer of a gas product often chooses to measure the gas with two different methods.1). For example. systematic errors may over time accumulate to significant economic values. This will often result in two different values for the measured gas flow. wear. deviation in conditions from flow calibration to field operation are needed. involving different equipment. . the need for developing standardized and accepted uncertainty evaluation methods for such metering stations is significant. Methods and tools to evaluate the consequences for the uncertainty of the metering station due to e.Handbook of uncertainty calculations . flow velocity profiles. bends. in service. An uncertainty analysis shall be prepared for the measurement system within a 95 % confidence level” (cf. demonstrated in flow testing. meter orientation re. This concerns both installation conditions (bends. Moreover. Section C.g. NFOGM. For example. the lack of an accepted method for evaluating the uncertainty of such metering stations has thus represented a "problem". an uncertainty analysis is planned to be included. the uncertainty figure found in flow calibration of a USM does not necessarily reflect the real uncertainty of the meter when placed in field operation. wall corrosion. The many different approaches to calculating the uncertainty is also a source of confusion.USM fiscal gas metering stations NPD. etc. Although it is common knowledge that each value of the measured gas flow has an uncertainty. creating a dispute as to the correctness of each value. flow conditioners. CMR (December 2001) 10 For example. methods and tools to evaluate the accuracy of USMs will be important in coming years. Along with the increasing use of USMs in gas metering stations world wide. pitting.) and .varying practice in this respect has definitely been experienced among members in the group of USM manufacturers. A great challenge is now to be able to be confident of the in-service performance over a significant period of time and changing operational conditions. conditions may be different from the test situation. it often causes contractual disputes. In the ongoing work on ISO standardization of ultrasonic gas flow meters. 2001] it is stated that “it shall be possible to document the total uncertainty of the measurement system. Even within such high accuracy figures as referred to above (±1 % of mass flow rate). and practical problems may occur so that occasionally it may be difficult to ensure that such accuracy figures are actually reached. engineering companies (metering system designers) and operating companies. • With respect to pressure and temperature correction of the meter body dimensions. especially for transit time "dry calibration". some manufacturers have argued that "dry calibration" is sufficient (i. temperature. also with respect to uncertatinty evaluation1.4). 3 A possible approach with reduced dependence on flow calibration in the future would impose higher requirements to the USM technology and to the USM manufacturer. (The NORSOK I-104 standard states that “flow conditioner of a recognized standard shall be installed. However. without flow calibration of individual meters). Appendix A. and whether it could be accepted by national authorities for fiscal metering applications4. Today. • Systematic transit time effects (cf. actual upstream lengths are in practice included in flow calibration. Another perspective in future development of USM technology which also affects evaluation of the measurement uncertainty.4) would not be calibrated “away” to the same extent. and 20 oC) instead of the flow calibration P and T. Table 1. CMR (December 2001) 11 gas conditions (pressure. a meter is in general subject to both "dry calibration" and flow calibration prior to installing the USM in field operation. etc. 1998a]. the reference pressure and temperature would be the “dry calibration” P and T (e. and (3) the contributions of such "dry calibration" uncertainties to the total USM measurement uncertainty. (2) change of the "dry calibration" parameters with operational conditions (pressure. is the role of "dry calibration" and flow calibration2. There are several reasons for that.e. Table 1.). and AGA-9 does also in principle open up for such a practice. Today. 4 A possible reduced dependency on flow calibration in the future would necessitate the establish- ment of a totally new chain of traceability to national and international standards for the USM measurement. provided the USM manufacturer can document sufficient accuracy3. and an increased dependency on "dry calibration". cf. the traceability of the individual meter manufacturer’s "dry calibration" procedures would become much more important and critical than today.) 2 For an explanation of the terms "dry calibration" and flow calibration. which might impose larger and thus more important corrections. gas composition and transducer distance.g. analysis using an accepted 1 For instance. “dry calibration” is not a calibration. the USM technology itself would have to be sufficiently robust with respect to the range of axial and transversal flow profiles met in practice. cf. temperature. such as: • Installation effects would not be calibrated “away” (i. 1 atm.USM fiscal gas metering stations NPD. relative to at "dry calibration" conditions). in such a way that the overall uncertainty requirements are exceeded” [NORSOK. This involves both (1) the measurement uncertainty of the "dry calibration" methods. • Traceability aspects (see below). Such tools should be based on internationally recognized and sound measurement practice in the field. However. and it is definitely a question whether such a practice would be sufficiently traceable. the traceability is achieved through the accreditation of the flow calibration laboratory. and flow conditioner used and included in the flow calibration.Handbook of uncertainty calculations . In any case. NFOGM.e. . unless it is verified that the ultrasonic meter is not influenced by the layout of the piping upstream or downstream. unless the production of the USM technology is extremely reproducible (with respect to transit time contributions). Today USM measurement technology is not at a level where the traceability of the "dry calibration" methods has been proved. With a possible reduced dependence on flow calibration. • Documentation of sufficient accuracy in relation to the national regulations. 6. and are developed for different use. and with the intention to formulate the model in a best possible meter independent way which preferably meets the input uncertainty terms commonly used in the field of USM technology. 6 If the USM is flow calibrated in a flow calibration facility. it represents an adaptation and extension of the GARUSO model to the scenario with flow calibration of the USM. NFOGM. The uncertainty model developed here for fiscal gas metering stations which are based on a flow calibrated USM.1. and an uncertainty model for a USM which is both ”dry calibrated” and flow calibrated. The model is to some extent simplified relative to the physical effects actually taking place in the USM. also with respect to uncertainty evaluation. Among others.1. the USM is assumed to be "dry calibrated" and flow calibrated. If the USM is not flow calibrated (only "dry calibrated"). In the present Handbook. the AGA-9 report [AGA. and then operated in a metering station.2 Contributions to the uncertainty of USM fiscal gas metering stations In the present Handbook an uncertainty model for fiscal gas metering stations based on a flow calibrated USM is proposed and implemented in an Excel uncertainty calculation program. a flow computer.USM fiscal gas metering stations NPD. but has been done mainly to simplify the user interface. compressibility factors calculated from 5 In this context it is important to distinguish between an uncertainty model for a USM which is only ”dry calibrated”. In such contexts an uncertainty model of the GARUSO type is relevant. A possible future sceneario with “dry calibrated" USMs only is not covered by the present Handbook and calculation program. It consists basically of a USM. CMR (December 2001) 12 uncertainty evaluation method would be central. This is by no means a necessity. However. The two approaches definitely require two different uncertainty models5. uses the GARUSO model as a basis for the development. 1998] recommends that the USM shall meet specific minimum measurement performance requirements before the application of any correction factor adjustment. Appendix E. The two uncertainty models are related but different. use of "dry calibration" only is quite another scenario than the combined use of "dry calibration" and flow calibration.2 and 2. 1. and instrumentation such as pressure transmitter.Handbook of uncertainty calculations . These requirements (deviation limits) therefore in practice represent "dry calibration" requirements. . AGA-9 recommends that the manufacturer shall provide sufficient test data confirming that each meter shall meet the minimum performance requirements. 1997.. The uncertainty model GARUSO [Lunde et al. cf. only "dry calibrated". 2000a] represents an uncertainty model for USMs which have not been flow calibrated. temperature element and transmitter. to avoid a too high “user treshold” in the Excel program (with respect to specification of input uncertainties). The type of fiscal gas metering stations considered in the present Handbook is described and motivated in Sections 1. a vibrating element densitometer. 1-1.). say. In this context. In reality. it is worth mentioning that even a relatively “small” reduction of a meter’s systematic error by. Therefore. inter- acting with the flow over a finite volume. All of these factors influence on the USM integration method as well as the measured transit times. and a calorimeter for measurement of the calorific value. as indicated in Table 1. etc. 7 It should be noted that some of the effects listed in Table 1. may over time transfer to sic- nific economic values [NPD. Tables 1. Moreover. 0. infinite Reynolds number.1-0.USM fiscal gas metering stations NPD. Table 1.1 gives contributions to the uncertainty of the instruments used for gas measurement.4 are not sufficiently well understood today. there clearly exist potentials for further improved accuracy of this technology. Table 1. cross-flow. but.3 gives uncertainty contributions due to the signal communication and flow computer calculations.4 give an overview of some effects which may influence on a USM fiscal gas metering station. and to "eliminate" or reduce a number of systematic effects in the USM. NFOGM. as systematic effects. pitting. Improved control and corrections could be achieved if better understanding and a more solid theoretical basis for the USM methodology was available. as required for fiscal measurement of gas.2 gives uncertainty contributions related to flow calibration of the USM. wall corrosion.Handbook of uncertainty calculations . deposits. and simplified (if any) treatment of diffraction effects. Table 1. . eliminate or reduce certain systematic effects related to the meter body dimensions. CMR (December 2001) 13 gas chromatography (GC) analysis. 2001]. uniform or no transversal flow. assumed that the meter and instruments otherwise function according to manufacturer recommendations.4 gives some uncertainty contributions related to the USM in field operation. the measured transit times and the integration method (installation effects). use of flow conditioner. in reality the acoustic beam has a finite beam width. Flow calibration of USMs is used to achieve a traceable comparison of the USM with a traceable reference measurement. several effects may still be influent.e. such as uniform axial flow (i. etc.2 %. interaction of infinitely thin acoustic beams (rays) with the flow. the axial flow profile will change both with Re and with the actual installation conditions (such as bend configurations. The expressions forming the basis for present-day USMs are based on a number of assumptions which are not fulfilled in practice. by improved understanding and correction for such systematic effects.). wear. although the USM technology has definitely proven to be capable of measuring at very high accuracy already. Transversal flow is usually significant and non-uniform (swirl. Re). That is. Uncertainty contributions such as those listed in these tables are accounted for in the uncertainty model for USM gas metering stations described in Chapter 37. and with acoustic diffraction effects (due to the finite transducer aperture). Table 1. despite flow calibration. Flow calibration of the USM may eliminate or reduce a number of the systematic effects.4. ) Calorific value • Calorimeter uncertainty Table 1. Uncertainty contributions related to signal communication and flow computer calculations. condensation. including repeatability of the flow calibration laboratory Table 1. NFOGM. flow rate in by-pass density line. sampling representa- tivity.2. calibration constant. variation in gas composition) Density • Densitometer uncertainty (calibrated) • Repeatability • Temperature correction (line and calibration temperatures) • VOS correction (line and densitometer sound velocities. Source Uncertainty contribution (examples) Flow computer • Signal communication (analog (frequency) or digital signal) • Flow computer calculations . power supply effects.g. Source Uncertainty contribution (examples) USM flow calibration • Calibration laboratory • Deviation factor • USM repeatability in flow calibration. stability. Uncertainty contributions related to flow calibration of the USM. and uncertainty of basic data unerlying the equation of state) • Analysis uncertainty (GC measurement uncertainty. vibration. self induced heat. viscosity. deposits. Measurand Uncertainty contribution (examples) Pressure • Transmitter uncertainty (calibration) • Stability of pressure transmitter • RFI effects on pressure transmitter • Ambient temperature effects on pressure transmitter • Atmospheric pressure • Vibration • Power supply effects • Mounting effects Temperature • Transmitter/element uncertainty (calibrated as a unit) • Stability of temperature transmitter • Stability of temperature element • RFI effects on temperature transmitter • Ambient temperature effects on temperature transmitter • Vibration • Power supply effects • Lead resistance effects Compressibility • Model uncertainty (uncertainty of the equation of state itself.1. Uncertainty contributions related to measurement / calculation of gas perameters in the USM fiscal gas metering station. CMR (December 2001) 14 Table 1. pressure effect.Handbook of uncertainty calculations . periodic time) • Installation (by-pass) correction (line and densitometer P and T) • Miscellaneous effects (e. etc. corrosion.3.USM fiscal gas metering stations NPD. flow calibration) • Meter orientation relative to pipe bends (possible difference rel. are not to be included in the uncertainty evaluation. b) If the USM is flow calibrated together with the upstream pipe section to be used in field operation. drift. wax.g. etc. pitting (influence on flow profiles) • Possible wall deposits (lubricant oil.) • Sound refraction (flow profile effects on transit times) • Possible beam reflection at the meter body wall Eliminated Integration Systematic • Pipe bend configurations upstream of USM (possible method difference rel. liquid.) • Coherent noise (acoustic cross-talk through meter body. CMR (December 2001) 15 Table 1. pipe vibrations. to some tion method extent a) Only uncertainties related to changes of conditions from flow calibration to field operation are in question here. grease. electromagnetic cross-talk.) • Finite clock resolution • Electronics stability (possible random effects) • Possible random effects in signal detection/processing (e. liquid. effects of possible transducer exchange) • Possible systematic effects in signal detection /processing (e. effects. That means. Ultrasonic gas flow meter (USM) in field operation Uncertainty Type of Eliminated Uncertainty contribution (examples) by flow term effect calibration? a) Meter body Systematic • Measurement uncertainty of dimensional quantities Eliminated (at "dry calibration" conditions) • Out-of-roundness Eliminated • P & T effects on dimensional quantities (after possible P & T corrections) Transit times Random • Turbulence (transit time fluctuations due to random (repeatability) velocity and temperature fluctuations) • Incoherent noise (RFI.4. erronous signal period identification). and integra. nated. erronous signal period identification) • Possible cavity delay correction • Possible deposits at transducer front (lubricant oil. Uncertainty contributions to an ultrasonic gas flow meter (USM) in field operation. ambient temperature effects.g. wear. drift. pressure control valves. • Power supply variations Systematic • Cable/electronics/transducer/diffraction time delay. including finite-beam effects (line P and T effects. . grease. c) If the USM is flow calibrated together with the flow conditioner to be used in field operation. Possibly elimi- laneous cal model).USM fiscal gas metering stations NPD. reciprocity. etc. which is relatively common practice. wax. etc. flow calibration) • Initial wall roughness (influence on flow profiles) Eliminated b) • Changed wall roughness over time: corrosion.Handbook of uncertainty calculations . with respect to transit times. flow calibration) (installation • In-flow profile to upstream pipe bend (possible effects) difference rel. NFOGM. Systematic • Inaccuracy of the functional relationship (mathemati. etc. acoustic reverberation in gas. ambient temperat. transducer exchange) • ∆t-correction (line P and T effects. systematic USM uncertainty contributions which are practically eliminated by flow calibration.) (influence on flow profiles) • Possible use of flow conditioner Eliminated c) Miscel. 1995].1.. and is from 1998 marketed as MPU 1200 by FMC Kongsberg Metering. In the following. temperature and density measurements. 1993]. In 1995 an uncertainty model for Fluenta’s multipath ultrasonic gas flow meter FMU 7008 was developed [Lunde et al. The uncertainty analysis is to account for the propagation of all input uncertainties which influence on the uncertainty of the station. cf. Bergen.. etc. some more space has thus been necessary to use for description of the model itself. In the Handbook. on recommendations given in the GERG TM 8 [Lygre et al. V. Today there exists no established and widely accepted uncertainty model for USMs.Handbook of uncertainty calculations . The uncertainty model for USM fiscal gas metering stations presented here has been developed on basis of earlier developments in this field. From 2001 Fluenta AS. a group of 9 participants in the GERG Project on Ultrasonic Gas Flow Meters9 initi- 8 Fluenta’s FMU 700 technology was in 1996 sold to Kongsberg Offshore (now FMC Kongsberg Metering). Distrigaz (Belgium). and uncertainties of additional measurements and models used (e. 9 Participating GERG (Groupe Européen de Recherches Gazières) companies in the project were BG plc (UK). CMR (December 2001) 16 1. ENAGAS (Spain). Norway. Table 1. It has thus been considered necessary to develop such a model in the present work. NFOGM.g. 1988]. Gaz de France DR (France). Z-factor measurement/calculation. pressure. Ruhrgas AG (Germany). SNAM (Italy) and Statoil (Norway).g.3 Uncertainty evaluation of USMs and USM gas flow metering stations For fiscal gas metering stations. N. derived from the basic functional relationship of such stations.USM fiscal gas metering stations NPD. 1995].4). a brief historic review is given. In 1987-88. In general. Nederlandse Gasunie (Holland). These are the uncertainty of the flow meter in question (in the present case the USM. Based e. NAM (Holland). calorific value measurement). . 2001]. is a part of Roxar Flow Measurement AS. than would have been needed if a more established and accepted uncertainty model of USM fiscal gas metering stations was available. a measured value is to be accompanied with a statement of the uncertainty of the measured value. as a part of the scope of work. the uncertainty of the reference meter used by the flow calibration laboratory at which the gas meter was calibrated. a sensitivity study of multipath USMs used for fiscal gas metering was carried out by CMR for Statoil and British Petroleum (BP) [Lygre et al.. an uncertainty analysis is needed to establish the measurement uncertainty of the metering station [NPD. An uncertainty model for Fluenta’s FGM 100 single-path flare gas meter was prepared by CMR in 1993 [Lunde. nor an uncertainty model for a USM used as part of a fiscal gas metering station. temperature. 1997] has been used as a fundament for development of an uncertainty model and a handbook on uncertainty calculations for fiscal gas metering stations which are based on 10 In later years (1998-2000) the GARUSO model has beeen further developed at CMR e..1. 1999]. NFOGM. 1997.USM fiscal gas metering stations NPD. All uncertainty contributions were thus assumed to be uncorrelated.4 were accounted for. and fiscal gas metering stations based on an orifice flow meter. . Uncertainty contributions such as most of those listed in Table 1. [Hallanger et al. containing an uncertainty analysis of an ultrasonic flow meter used in a gas metering station (for mass flow rate measurement). In UK a British Standard BS 7965:2000 [BS.Handbook of uncertainty calculations . density. One major intention was to develop an uncertainty model in conformity with accepted international standards and recommendations on uncertainty evaluation. 2001]. and density obtained from GC analysis).. using sensitivity coefficients apparently set equal to 1 for the USM part (or not described).. For an orifice gas metering station. 1995a]. The connection between the USM functional relationship and the USM uncertainty model was not described. such as the GUM [ISO. compressibility and calorific value) is often the same as in USM fiscal gas metering stations. CMR (December 2001) 17 ated in 1996 the development of a relatively comprehensive uncertainty model for USMs. to enable use of CFD (“computational fluid dynamics”) .. 2000b]. 1999]10. the secondary instrumentation (pressure. combined with secondary instrumentation (pressure and temperature measurements. 2000] was issued in 2000. the theoretical basis for the GARUSO model [Lunde et al. the GARUSO model relates to USMs which are “dry calibrated” but not flow calibrated. [Wild. to study installation effects [Lunde et al. named GARUSO [Lunde et al. As described in a footnote of Section 1. and apparently (for the USM part) to contribute with equal weights to the total uncertainty. 2000a] [Sloet.. 1998]. On basis of an initiative from the Norwegian Society of Oil and Gas Metering (NFOGM) and the Norwegian Petroleum Directorate (NPD). This model has not been found to be sufficiently well documented and accurate to be used as a basis for the present Handbook. In this Handbook. That handbook concentrated on fiscal oil metering stations based on a turbine flow meter.1.calculations of 3-dimensional flow velocity profiles as input to the uncertainty calculations.g. a "Handbook on uncertainty calculations .Fiscal metering stations" was developed in 1999 [Dahl et al. A number of uncertainty contributions were identified and combined in the uncertainty model by a root-mean-square approach. • The Microsoft Excel program EMU . providing a traceable weighting and propagatation of the various uncertainty contributions to the overall uncertainty. Also. Tables 1.Handbook of uncertainty calculations . cf.2 . the present description is adapted to a less detailed level. a different model is proposed here. for a number of the uncertainty contributions. In relation to the uncertainty model for the USM described in the British Standard BS 7965:2000 [BS. the footnote on the GARUSO model in Section 1. That is.USM fiscal gas metering stations NPD. NFOGM. with respect to user input. density measurements and Z-factor evaluation). in comparison with the GARUSO model.. for traceability purposes. The present model is based on an approach where the various contributions are derived from the metering station’s functional relationship. 12 The description of the pressure. and the calculation of their expanded uncertainties. Chapter 3. 11 Cf. 2000]. but in different manners. 1999]. in addition to some others. the same types of basic uncertainty contributions may be described in the two models13.1. are similar to the descriptions given in [Dahl et al. cf. the same types of input uncertainty contributions are accounted for as in the BS 7965:2000 model.1.1.USM fiscal gas metering stations consists of • The Handbook (the present document). with some modifications. Also the calorific value uncertainty is accounted for here. and individual instruments of such stations. The resulting expressions for the uncertainty model are different from the ones proposed in [BS. CMR (December 2001) 18 use of flow calibrated multipath USMs11. . sensitivity coefficients are derived and documented. 13 In the uncertainty model for the USM proposed here.4. temperature. However. The present description is also combined with the uncertainty evaluation of the secondary instrumentation12 (pressure. temperature and density measurements.2 About the Handbook The Handbook of uncertainty calculations . 2000].USM Fiscal Gas Metering Station for performing uncertainty calculations of fiscal gas metering stations based on USM flow meters. correlated as well as uncorrelated effects are described and documented. 1. Essentially. CMR (December 2001) 19 The Handbook and the Excel program address calculation of the uncertainty of fiscal metering stations for natural gas which are based on use of a flow calibrated multipath ultrasonic transit-time flow meter (USM). • Pressure transmitter. constructed and operated according to NPD regulations [NPD. Q.2: • USM (flow calibrated). • Standard volume flow (i. four flow rate figures are normally to be calculated [NORSOK.1. Section 2. However. cf. • Temperature element and transmitter. also the standard volume flow and the energy flow rate have been addressed in the present Handbook. • Compressibility factors calculated from gas chromatography (GC) analysis.2. • Calorific value measurement (calorimeter). cf. The Handbook was originally intended to address two of these four flow rate figures [Lunde.e. The USM fiscal gas metering stations addressed in the present Handbook are assumed to be designed. the 14 For the energy flow rate. 2000]: the actual volume flow and the mass flow rate.e. For fiscal gas metering stations. in which the calorific value measurement has been assumed to be uncorrelated with the standard volume flow measurement.USM Fiscal Gas Metering Station14. Sections 3.1 and 3. • Mass flow rate.5. qm. the volumetric flow rate at standard reference conditions). the volumetric flow rate at line conditions). By propagation of input uncertainties the model calculates among others the metering station’s “expanded uncertainty” and “relative expanded uncertainty”. 2001].g. qv. NFOGM. qe (application specific).. For USM fiscal metering of gas.USM fiscal gas metering stations NPD. . the NPD regulations refer to e. so that the expanded uncertainty can be calculated for all four flow rate figures using the program EMU . The equipment and instruments considered are. and • Energy flow rate. • Flow computer.Handbook of uncertainty calculations . a simplified approach has been used here. • Densitometer (on-line installed vibrating element). 1998a]: • Actual volume flow (i. CMR (December 2001) 20 NORSOK I-104 national standard [NORSOK. They are also considered to be in consistence with the proposed revision of ISO 5168 [ISO/CD 5168. the present Handbook and the computer program EMU . manufacturers of ultrasonic gas flow meters. Appendix C and Section B. 9 [AGA-9.3). 15 In the AGA-9 report [AGA-9. the NPD regulations and the NORSOK I-104 standard state that the expanded uncertainty of the measurement system shall be specified at a 95 % confidence level. Both the NPD regulations and the NORSOK I-104 standard refer to the GUM (Guide to the expression of uncertainty in measurement) [ISO. calculation of compressibility factors.20. With respect to uncertainty evaluation and documentation. 1998] as recognised standards (“accepted norm”). temperature element/transmitter. 5 calibration points are recommended. 1998a] and the AGA Report No. as well as others with interests within the field. and is primarily written for experienced users and operators of fiscal gas metering stations. are fully analytical.1 (typically M = 5 or 6)15. engineering personnel. pressure transmitter. with a possibility to draw a curve between these points.USM fiscal gas metering stations NPD. The Handbook and the accompanying Excel program provides a practical approach to the field of uncertainty calculations of ultrasonic fiscal gas metering stations. 5. 6 calibration test flow rates (“calibration points”) have been recommended. using a coverage factor k = 2 (cf.e. Appendix C). In the NORSOK I-104 industry standard [NORSOK. the uncertainty models for the USM. . 1995a] as the “accepted norm” with respect to uncertainty analysis (cf. with expressions given and documented for the model and the sensitivity coefficients (cf. NFOGM. Consequently. Chapter 3.Handbook of uncertainty calculations . as a compromise between cost and performance [Ref Group. That is. 1998] and in [OIML. 1989]. and calorimeter. 2001]. It has been chosen [Ref Group.USM Fiscal Gas Metering Station are based primarily on the recommended procedures in the GUM. i. F and G). which is based on Ap- pendices E.USM Fiscal Gas Metering Station. An example of such output is given in Fig. 2000] (which is based on the GUM). 1998a]. 2001] to calculate the uncertainty of the metering station only in the “flow calibration points”. This has consequently been adopted here and used in the Handbook and the program EMU . densitometer. The uncertainty model for the USM gas metering station developed here is based on an analytical approach. at the M test flow rate figures described in Section 2. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 21 1.3 About the program EMU - USM Fiscal Gas Metering Station As a part of the present Handbook, an Excel program EMU - USM Fiscal Gas Me- tering Station has been developed for performing uncertainty evaluations of the fiscal gas metering station16. The program is implemented in Microsoft Excel 2000 and is opened as a normal workbook in Excel. The program file is called “EMU - USM Fis- cal Gas Metering Station.xls”. The abbreviation EMU is short for “Evaluation of Metering Uncertainty” [Dahl et al., 1999]. It has been the intention that the Excel program EMU - USM Fiscal Gas Metering Station may be run without needing to read much of the Handbook. However, Chapter 5 which gives an overview of the program, as well as Chapter 4 which - through an uncertainty evaluation example - is intended to provide some guidelines for specifying input parameters and uncertainties to the program, may be useful to read together with running the program for the first time. At each “input cell” in the program a comment is given, with reference to the relevant section(s) of the Hand- book in which some information and help about the required input can be found. As delivered, the program is “loaded” with the input parameters and uncertainties used for the example calculations given in Chapter 4. As the Excel program EMU - USM Fiscal Gas Metering Station is described in more detail in Chapter 5, only a brief overview is given here. It is organized in 24 worksheets, related to: gas parameters, USM setup, pressure, temperature, compressibility factors, density, calorific value, flow calibration, USM field operation, flow computer etc., various worksheets for plotting of output uncertainty data (graphs, bar- charts), a worksheet for graph and bar-chart set-up, a summary report, two additional worksheets listing the plotted data and the calculated USM transit times, and two program information worksheets. As described in Section 1.2, the program calculates the expanded and relative expanded uncertainties of a gas metering station which is based on a flow calibrated USM, for the four measurands in questions, qv, Q, qm and qe. The theoretical basis for the uncertainty calculations is described in Chapters 2 and 3. A calculation example is given in Chapter 4, including discussion of input uncertain- 16 In the earlier Handbook [Dahl et al., 1999], the two Excel programs were named EMU - Fiscal Gas Metering Station (for gas metering stations based on orifice plate), and EMU - Fiscal Oil Metering Station (for oil metering stations based on a turbine meter). Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 22 ties and some guidelines to the use of the program. An overall description of the program is given in Chapter 5. In addition to calculation/plotting/reporting of the expanded uncertainty of the gas metering station (cf. Figs. 5.20 and 5.27-5.31), and the individual instruments of the station, the Excel program can be used to calculate, plot and analyse the relative importance of the various contributions to the uncertainty budget for the actual instruments of the metering station (using bar-charts), such as: • Pressure transmitter (a number of contributions) (cf. Fig. 5.21), • Temperature element / transmitter (a number of contributions) (cf. Fig. 5.22), • Compressibility ratio evaluation (a number of contributions) (cf. Fig. 5.23), • Densitometer (including temperature, VOS and installation corrections) (Fig. 5.24), • USM flow calibration (laboratory uncertainty, meter factor, repeatability) (Fig. 5.25), • USM field operation in the the metering station (deviation from flow calibration, with respect to P & T corrections of meter body dimensions, repeatability, systematic transit time effects and integration/installation effects) (cf. Fig. 5.26), • The gas metering station in total (cf. Fig. 5.27). In the program the uncertainties of the gas density, pressure and temperature measurements can each be specified at two levels (cf. Tables 1.5, 3.1, 3.2 and 3.4; see also Sections 5.4, 5.5 and 5.7): (1) “Overall level”: The user specifies the combined standard uncertainty of the gas density, pressure or temperature estimate, u c ( ρˆ ) , u c ( P̂ ) or u c ( T̂ ) , respectively, directly as input to the program. It is left to the user to calculate and document u c ( ρˆ ) , u c ( P̂ ) or u c ( T̂ ) first. This option is general, and covers any method of obtaining the uncertainty of the gas density, pressure or temperature estimate (measurement or calculation)17. 17 The “overall level” options may be of interest in several cases, such as e.g.: • If the user wants a “simple” and quick evaluation of the influence of u c ( ρˆ ) , u c ( P̂ ) or u c ( T̂ ) on the expanded uncertainty of the gas metering station, • In case of a different installation of the gas densitometer (e.g. in-line), • In case of a different gas densitometer functional relationship than Eq. (2.25), • In case of density measurement using GC analysis and calculations instead of densitometer measurement. • In case the input used at the “detailed level” does not fit sufficiently well to the type of input data / uncertainties which are relevant for the pressure transmitter or temperature element/transmitter at hand. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 23 (2) “Detailed level”: u c ( ρˆ ) , u c ( P̂ ) or u c ( T̂ ) , respectively, is calculated in the program, from more basic input for the gas densitometer, pressure transmitter and temperature element transmitter, provided by the instrument manufacturer and calibration laboratory. Such input is at the level described in Table 1.1. For the input uncertainty of the compressibility factors, the user input is specified at the level described in Table 1.1. See Tables 1.5, 3.3 and Section 5.6. For the input uncertainty of the calorific value measurement, only the “overall level” is implemented in the present version of the program, cf. Tables 1.5, 3.5 and Section 5.818. With respect to USM flow calibration, the user input is specified at the level described in Table 1.2. See Tables 1.5, 3.6 and Sections 4.3, 5.9. Table 1.5. Uncertainty model contributions, and optional levels for specification of input uncertainties to the program EMU - USM Fiscal Gas Metering Station. Uncertainty contribution Overall level Detailed level Pressure measurement uncertainty ü ü Temperature measurement uncertainty ü ü Compressibility factor uncertainties ü Density measurement uncertainty ü ü Calorific value measurement uncertainty ü USM flow calibration uncertainty ü USM field uncertainty ü ü Signal communication and flow computer calculations ü With respect to USM field operation, a similar strategy as above with “overall level” and “detailed level” is used for specification of input uncertainties, see Table 1.5 and Section 5.10. For the “detailed level”, the level for specification of input uncertainties is adapted to data from "dry calibration" / flow calibration / testing of USMs to be provided by the USM manufacturer (cf. Chapter 6). In particular this concerns: • Repeatability. The user specifies either (1) the repeatability of the indicated USM flow rate measurement, or (2) the repeatability of the measured transit times (cf. Tables 1.4, 3.8 as well as Sections 4.4.1 and 5.10.1). Both can be given as flow rate dependent. 18 Improved descriptions to include a calculation of the calorific value uncertainty (with input at the “detailed level”) may be implemented in a possible future revision of the Handbook, cf. Chapter 7. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 24 • Meter body parameters. The user specifies whether correction for pressure and temperature effects is used by the USM manufacturer, and the uncertainties of the pressure and temperature expansion coefficients. Cf. Tables 1.4 and 3.8 as well as Sections 4.4.1 and 5.10.2. • Systematic transit time effects. The user specifies the uncertainty of uncorrected systematic effects on the measured upstream and downstream transit times. Ex- amples of such effects are given in Table 1.4, cf. Table 3.8 and Sections 4.4.3, 5.10.2. • Integration method (installation effects). The user specifies the uncertainty due to installation effects. Examples of such are given in Table 1.4, cf. Table 3.8 and Sections 4.4.4, 5.10.2. It should be noted that for all of these USM field uncertainty contributions, only uncertainties related to changes of installation conditions from flow calibration to field operation are in question here. That means, systematic USM uncertainty contributions which are practically eliminated by flow calibration, are not to be included in the uncertainty model (cf. Table 1.4). Consequently, with respect to USM technology, the program EMU - USM Fiscal Gas Metering Station can be run in two modes: (A) Completely meter independent, and (B) Weakly meter dependent19. Mode (A) corresponds to choosing the “overall level” in the “USM” worksheet (both for the repeatability and the systematic deviation re. flow calibration), as described above. Mode (B) corresponds to choosing the “detailed level”. These options are further described in Section 5.3. See also Section 5.10 and Chapter 6. It is required to have Microsoft Excel 2000 installed on the computer20 in order to run the program EMU - USM Fiscal Gas Metering Station. Some knowledge about 19 By “weakly meter dependent” is here meant that the diameter, number of paths and the number of reflections for each path need to be known. However, actual values for the inclination angles, lateral chord positions and integration weights do not need to be known. Only very approximate values for these quantities are needed (for calculation of certain sensitivity coefficients), as described in Chapter 6 (cf. Table 6.3). Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 25 Microsoft Excel is useful, but by no means necessary. The layout of the program is to a large extent self-explaining and comments are used in order to provide online help in the worksheets, with reference to the corresponding sections in the Handbook. In the NPD regulations it is stated that “it shall be possible to document the total uncertainty of the measurement system. An uncertainty analysis shall be prepared for the measurement system within a 95 % confidence level” [NPD, 2001] (cf. Section C.1). The GUM [ISO, 1995a] put requirements to such documentation, cf. Appendix B.4. The expanded uncertainties calculated by the present program may be used in such documentation of the metering station uncertainty, with reference to the Hand- book. That means, provided the user of the program (on basis of manufacturer information or another source) can document the numbers used for the input uncertainties to the program, the Handbook and the program gives procedures for propagation of these input uncertainties. It is emphasised that for traceability purposes the inputs to the program (quantities and uncertainties) must be documented by the user, cf. Section B.4. The user must also document that the calculation procedures and functional relationships implemented in the program (cf. Chapter 2) are in conformity with the ones actually applied in the fiscal gas metering station21. 1.4 Overview of the Handbook The Handbook has been organized in two parts, "Part A - User's Guide" and "Part B - Appendices". Part A constitutes the main body of the Handbook. The intention has been that the reader should be able to read and use Part A without needing to read Part B. In Part A one has thus limited the amount of mathematical details. However, to keep Part A 20 The program is optimised for “small fonts” and 1152 x 864 screen resolution (set in the Control Panel by entering “Display” and then “ Settings”). For other screen resolutions, it is recommended to adapt the program display to the screen by the Excel zoom functionality, and saving the Excel file using that setting. 21 If the “overall level” options of the program are used, the program should cover a wide range of situations met in practice. However, note that in this case possible correlations between the estimates which are specified at the “overall level” are not accounted for (such as e.g. between the calorific value and the Z-factors, when these are all obtained from GC analysis). In cases where such correlations are important, the influence of the covariance term on the expanded uncertainty of the metering station should be investigated. giving necessary and essential details. which to some extent may also serve as a guideline to use of the program EMU . The model is derived in detail in Appendix E. Chapter 6 summarizes proposed uncertainty data to be specified by USM manufacturers.1 Part A .USM Fiscal Gas Metering Station. including background for the work and a brief overall description of the Hand- book and the accompanying uncertainty calculation program EMU .been included in Chapter 3. In Chapter 7 some concluding remarks are given.Handbook of uncertainty calculations .USM fiscal gas metering stations NPD. It is believed that this approach simplifies the practical use of the Handbook. CMR (December 2001) 26 “self-contained”. including description of and necessary functional relationships for the instruments involved.USM Fiscal Gas Metering Station. to make Chapter 3 “self contained” so that it can be read independently. Part B has been included as a more detailed documentation of the theoretical basis for the uncertainty model. In Chapter 4 the uncertainty model is used in a calculation example. . on basis of the functional relationships given in Chapter 2.User's Guide Part A is organized in Chapters 1-7.4. 1. That is. temperature and density measurements. The type of instrumentation of the USM fiscal gas metering stations addressed in the Handbook is described in Chapter 2. with an overview of the various worksheets involved. the USM itself as well as pressure. NFOGM. all the expressions which are implemented in the uncertainty calculation program EMU . which is described in Chapter 3.USM Fiscal Gas Metering Station have been included and described also in Part A. Also definitions introduced in Appen- dix E have .when relevant . for completeness and traceability purposes. without needing to read the appendices. These descriptions and functional relationships serve as a basis for the development of the uncertainty model of the USM fiscal gas metering station.USM Fiscal Gas Meter- ing Station have been included in Chapter 3. Chapter 1 gives an introduction to the Hand- book. The program itself is described in Chapter 5. The gas compressibility factor calculations and calorific value measurement/calculation are also addressed. Only those expressions which are necessary for documentation of the model and the Excel program EMU . consisting of (in arbitrary order) Reidar Sakariassen (Metropartner).Appendices Part B is organized in Appendices A-G. Appendices E and F. This constitutes the main basis for Chapter 3. Trond Folkestad (Norsk Hydro). John Magne Eide (JME Consultants. In Appendix G details on the uncertainty model of the vibrating element gas densitometer are given. CMR (December 2001) 27 1. cf. 22 To the knowledge of the authors. Highly acknowledged are also the useful discussions in and input from the technical reference group. 1995a] are given in Appendix B. Selected national regulations for USM fiscal gas metering are included in Appendix C. selected national regulations and the theoretical basis for the uncertainty model described in Chapter 3. 1995a]. Håkon Moestue (Norsk Hydro). and different approaches used by USM meter manufacturers are discussed and compared.USM fiscal gas metering stations NPD. Appendix D addresses the pressure and temperature correction of the USM meter body. 1. .4. of Physics) and Eivind Olav Dahl (CMR) are greatly acknowledged. Endre Ja- cobsen (Statoil). Some definitions and abbreviations related to USM technology are given in Appen- dix A. Hans Arne Frøystein (Justervesenet). Tore Løland (Statoil). represented by Svein Neumann (NFOGM) and Einar Halvorsen (NPD). selected definitions and procedures for evaluation of uncertainty as recommended by the GUM [ISO.Handbook of uncertainty calculations . Discussions with and comments from Magne Vestrheim (University of Bergen. In Appendix F three alternative approaches for evaluation of partially correlated quantities are discussed. representing Holta and Håland) and Jostein Eide (Kongsberg Fimas). Dept. NFOGM.5 Acknowledgements The present Handbook has been worked out on an initiative from the Norwegian So- ciety of Oil and Gas Metering (NFOGM) and the Norwegian Petroleum Directorate (NPD). as a basis for Chapter 2. Erik Malde (Phillips Petroleum Company Norway). giving definitions.2 Part B . The theoretical basis of the uncertainty model for the USM fiscal gas metering station is given in Appendix E. the method which is here referred to as the “decomposition method” (with decomposition of partially correlated quantities into correlated and uncorrelated parts) has been introduced by the authors. and it is shown that the “decomposition method”22 approach used in Appendix E is equivalent to the “covarance method” approach recommended by the GUM [ISO. For reference. Lit- erature references are given at the end of Part B. and • gas chromatograph (GC) measurement. For fiscal gas metering stations. the volumetric flow rate at line conditions). four flow rate figures are normally to be calculated [NORSOK.USM Fiscal Gas Metering Station described in Chapter 5. CMR (December 2001) 28 2. For USM fiscal metering of gas.g. the volumetric flow rate at standard reference conditions). the uncertainty evaluation example given in Chapter 4.Handbook of uncertainty calculations . 2. This includes a brief description of metering station methods and equipment (Section 2. the gas densitometer (Section 2.e. USM FISCAL GAS METERING STATION The present chapter gives a description of typical USM fiscal gas metering stations.5 and 2. and • Energy flow rate. • Mass flow rate. 1998a] this includes • sales and allocation measurement of gas. • Standard volume flow (i.. the NPD regulations refer to e. The USM fiscal gas metering stations to be evaluated in the present Handbook are assumed to be designed. qe (application specific).1 Description of USM fiscal gas metering station 2. 2001].e.1 General Fiscal measurement is by [NPD.USM fiscal gas metering stations NPD. the USM instrument (Section 2.3). qm. By [NORSOK. as well as the functional relationships of the metering station (Section 2.1). Q. NFOGM. • measurement of fuel and flare gas.4) and the pressure and temperature instruments (Sections 2.1. and the Excel program EMU . • sampling. 1998a]: • Actual volume flow (i. constructed and operated according to NPD regulations [NPD. 2001] defined as measurement used for sale or calculation of royalty and tax.6). the . qv. serving as a basis for the uncertainty model of such metering stations described in Chapter 3.2). Section C. 1998] as recognised standards (“accepted norm”)23. the NPD regulations [NPD.USM fiscal gas metering stations NPD. Appendix C. The NORSOK I-104 standard states that “flow conditioner of a recognized standard shall be installed. Density shall be measured by at least two densitometers in the metering station. CMR (December 2001) 29 NORSOK I-104 national standard [NORSOK. Section C. 2001] state that (cf. Cf.Handbook of uncertainty calculations . temperature. 9 [AGA-9. whilst the rest of the meter runs operate within their specified operating range.2): ”Pressure and temperature shall be measured in each of the meter runs. (The selection is not necessarily complete. density and composition analysis shall be measured in such way that representative measurements are achieved as input signals for the fiscal calculations”. As a basis for the uncertainty calculations of the present Handbook. a selection of NPD regulations and NORSOK I-104 requirements which apply to USM metering stations for sales and allocation metering of gas. this means that on USM gas sales metering stations a minimum of two parallel meter runs shall be used. this means that density can be measured using densitometers in each meter run. each with at least one USM. which provide possible (but not necessary) ways of fulfilling the stated requirements. In practice. or by two densitometers located at the gas inlet and outlet of the metering station. As an “accepted norm”. the NORSOK I-104 standard [NORSOK. 1998a] and the AGA Report No. For allocation metering stations there is no such requirement of two parallel meter runs. actual upstream lengths and flow conditioner are included in the flow calibration.1): “On sales metering stations the number of parallel meter runs shall be such that the maximum flow of hydrocarbons can be measured with one meter run out of service. thus. have been summarized in Appendix C. in such a way that the overall uncertainty requirements are exceeded” [NORSOK.” It is further stated that “The density shall be measured by the vibrating element technique. 2001] the requirements are to a large extent stated as functional requirements. In the comments to the requirements. 1998a] state that (cf.” In practice. NFOGM. .” In practice. it is referred to “accepted norms” including industry standards.) With respect to instrumentation of gas sales metering stations. Only regulations which influence on the uncertainty model and calculations of USM metering stations are included. unless it is verified that the ultrasonic meter is not influenced by the layout of the piping upstream or downstream. The NPD regulations further state that “pressure. 1998a]. 23 In the revised NPD regulations [NPD. Three such combination methods are indicated in Table 2. Several methods are in use for measurement of Q.1.e. or used only as back-up or redundancy measurements (i.USM fiscal gas metering stations NPD. That is. The fiscal gas metering stations considered here are based on flow rate measurements using one or several flow calibrated multipath ultrasonic gas flow meters (USM). is addressed in Section 2.1. 1. 2001]. .2. and the density at standard reference conditions. NFOGM. not used in the reported measurement). The USM measures basically the average axial gas flow velocity. Table 2. in some stations two USMs are used in series. if such determination can be done within the uncertainty requirements applicable to density measurement. or calculated from GC measurement of the gas composition.Handbook of uncertainty calculations .e. Q can be calculated from qv. the NPD regulations also open up for calculation of density from GC measurements: “density may be determined by continuous gas chromatography. qv. This will provide independent control of the density value and that density is still measured when GC is out of operation” (cf. cf. Section C. which when multiplied with the pipe's cross-sectional area yields the volumetric flow rate at line conditions.1 [Sakarias- sen. ρ 0 . With regard to sales gas metering stations two independent systems shall be installed” (cf.1. the line density. they are both used in the reported measurement).. ρ. a common approach is to measure ρ using a densi- 24 Symbols used in Tables 2. a comparison function against for example one densitometer should be carried out. CMR (December 2001) 30 On the other hand. In the North Sea. With respect to measurement of energy contents.. If only one gas chromatograph is used.3 are explained in Section 2. the NPD regulations state that “Gas composition from continuous flow proportional gas chromatography or from automatic flow proportional sampling shall be used for determination of energy content. Moreover. for such metering stations two GCs in parallel are required. The implications of such relatively complex instrument configurations and varying company practice on the evaluation of the metering station’s measurement uncertainty.124.1).2. cf. Such relatively complex instrument configurations may be further complicated by the fact that the multiple measurements may be used differently by different companies: they may be averaged (i. the volumetric flow rate at standard reference condition.1). Section C. Each of ρ and ρ 0 can either be measured by densitometers. while for allocation stations use of a single GC may be sufficient.1-2. Fig. CMR (December 2001) 31 tometer and to calculate ρ 0 from the gas composition output of a GC (method 2 in Table 2. NFOGM. Measurement of ρ 0 using a densitometer is less common (method 1 in Table 2.2.USM fiscal gas metering stations NPD. Different methods of measuring the volumetric flow rate at standard reference conditions. Primary measurement Method Functional Line conditions Std. Different methods of measuring the energy flow rate. qm.1).Handbook of uncertainty calculations . m calculated 2 qm = qv from GC analysis ZRT Table 2. qe. According to the NPD regulations [NPD. 1997] and thus the mass flow rate. m calculated H S calculated 3 qe = H S qv mP0 using densitometer from GC analysis from GC analysis Z calculated from Z 0 calculated from H S calculated 4 PT0 Z 0 GC analysis GC analysis from GC analysis qe = H S qv P0 TZ Z calculated from Z 0 calculated from H S measured 5 GC analysis GC analysis using calorimeter . Table 2. relationship q m = ρq v ρ measured 1 using densitometer mP Z .1). Primary measurement Method Functional Line conditions Std. Q. Several methods can be used for measurement of the line density ρ [Tambo and Søgaard. the line denstity ρ can either be measured using densitometer(s). m calculated 2 Q= qv mP0 using densitometer from GC analysis PT0 Z 0 Z calculated Z 0 calculated 3 Q= qv from GC analysis P0 TZ from GC analysis Table 2. relationship ρ ρ measured ρ 0 measured Q= qv 1 ρ0 using densitometer using densitometer ρZ 0 RT0 ρ measured Z 0 .2. cf. Primary measurement Method Functional Line conditions no. relationship conditions ρ measured ρ 0 measured H S measured 1 ρ using densitometer using densitometer using calorimeter qe = H S qv ρ0 ρ measured ρ 0 measured H S calculated 2 using densitometer using densitometer from GC analysis ρZ 0 RT0 ρ measured Z 0 . Different methods of measuring the mass flow rate. Table 2. reference Calorific value no. 2001]. or calculated from GC measurement of the gas composition. reference conditions no.3. qm.1. and Z and Z0 are calculated from GC analysis. Method 1 in Table 2. cf. That is. The conversion from qv therefore involves ρ. since the NORSOK standard I-104 state that "the density shall be measured using the vibrating element technique". cf. the energy flow rate. According to the NPD regulations [NPD. would be beyond the scope of the present Handbook [Ref. correspondig to the three methods for measurement of Q shown in Table 2. 2001]. 2001]. NFOGM. This has been done to reduce the complexity of the uncertainty model. Table 2. qm. For measurement of the mass flow rate.1.3 is considered in the present Handbook [Ref. Method 3 of Table 2. Group.3. Section C.USM fiscal gas metering stations NPD. [Ref. or measured directly on basis of gas combustion (using a calorimeter) (cf.1.Handbook of uncertainty calculations . cf.2. Five combination methods for measurement of qe are indicated in Table 2.1.1 and C.2 Gas metering station equipment considered in the Handbook From the more general description of USM gas metering stations given in Section 2.1 may be included in a possible later revision of the Handbook. 26 Note that the NORSOK I-104 industry standard represents an “accepted norm” in the NPD regulations (i. Method 5 in Table 2. gives possible ways of fulfilling the NPD requirements). P and T are measured. 2000]. P and T are meas- 25 Other methods for measurement of Q listed in Table 2. cf.1). Sections 2. by densitometer26. and the implications of these for the uncertainty model of the gas metering station. Chapter 7. i. Group. ρ 0 and Hs. a common method is to measure ρ using a densitometer and to calculate ρ 0 and Hs using the gas composition output from a GC (method 3 in Table 2. the superior (gross) calorific value Hs can either be calculated from the gas composition measured using GC analysis.2 is considered here [Ref. 2001]. qe. That is. 2001]. A more general treatment of all three approaches. 2. CMR (December 2001) 32 Several methods are in use also for measurement of qe.1. With respect to measurement of the energy flow rate. qe can be calculated from Q and the superior (gross) calorific value Hs. the gas metering station equipment considered in the present Handbook is addressed in the following.1.1 is considered here [Lunde. both specified at standard reference conditions. Group. 2001]25. Sec- tion C. With respect to measurement of the volumetric flow rate at standard reference conditions. In the North Sea.e.1.3 [Sakariassen. Group.e. . Q. and that the NPD regulations also open up for use of calculated density from GC analysis.3). 2001]. .3 may be included in a possible later revision of the Handbook. T Not specified. Otherwise not specified. and a calorimeter (Hs) [Ref. temperature element/transmitter. Otherwise not specified Calorific value. densitometer. USM. the type of fiscal gas metering stations considered in the present Handbook consist basically of one or several USMs (qv). Otherwise not specified Only flow calibrated USMs are considered here. NFOGM. would be beyond the scope of this Handbook [Ref. Otherwise not specified. Table 2. Table 2. Z0 and Hs. which means that the Handbook typically addresses sales and allocation metering stations. 2001]27. Group. This has been done to reduce the complexity of the uncertainty model. and a calorimeter is used to measure Hs. Hs Calorimeter (combustion method). Chapter 7. flow calibrated USM. 1998a]. Measurement Instrument Ultrasonic meter (USM) Multipath. ρ On-line installed vibrating element densitometer. 2000]. are not covered by the present Handbook [Ref. Example: Solartron 7812 Gas Density Transducer [Solartron. USM fiscal gas metering station equipment considered in the Handbook. 1999]. Z and Z0 are calculated from GC analysis. Z and Z0 Calculated from GC measurements. and instrumentation such as pressure transmitter (P). cf. Flow computer Not specified.Handbook of uncertainty calculations . Pressure (static). as well as fuel gas metering stations in which a flow calibrated USM is employed. Consequently. Included is also example instrumentation used for uncertainty evaluation of a fiscal gas metering station. A more general treatment of all five methods. Example: Rosemount 3051P Reference Class Pressure Transmitter [Rosemount. 2001].4. P Not specified. compressibility factors calculated from GC analysis (Z and Z0). In general. Example: Pt 100 element (EN 60751 tolerance A) [NORSOK. the flow computer. 2000]. without treating possible correlations between Z. Group. gas chromatograph and calorimeter are not limited to 27 Other methods for measurement of qe listed in Table 2. Sampling and the GC measurement itself are not addressed. Compressibility. and the implications of these.4. a flow computer. Temperature. CMR (December 2001) 33 ured. temperature element and transmitter (T). vibrating element densitometer (ρ). Ultrasonic flare gas meters and fuel gas meters which are not flow calibrated. pressure transmitter. Group.USM fiscal gas metering stations NPD. Density. 2001]. cf. Rosemount 3144 Smart Temperature Transmitter [Rosemount. These are then re-written to an equivalent form which is more convenient for the purpose of accounting (in the uncertainty model) for the relevant uncertainties and systematic effects related to flow calibration of the USM. These example instruments represent typical equipment commonly used today when upgrading existing fiscal metering stations and designing new fiscal metering stations. 1999].4. The example temperature transmitter evaluated is the Rosemount Model 3144 Smart Tem- perature Transmitter [Rosemount.Handbook of uncertainty calculations . Group. 2.2 Functional relationship . one densitometer and one calorimeter. 2001] for the example uncertainty evaluation of a USM fiscal gas metering station described in Chapter 4. a conservative “worst-case” approach may be sound and useful. 2000]. For uncertainty evaluation of such USM gas metering stations. The example pressure transmitter evaluated is the Rosemount Model 3051P Reference Class Pressure Transmitter [Rosemount. Such an approach should give the highest uncertainty of the metering station. The example densitometer evaluated is the Solartron 7812 Gas Density Transducer [Solartron. CMR (December 2001) 34 specific manufacturers or models. NFOGM. The calorific value is assumed to be measured using a calorimeter (since possible correlation between Z. This instrument configuration should cover the relevant applications. 2000]. Table 2. cf. Group. in which only one instrument of each type is considered.USM Fiscal Gas Metering Station should cover a relatively broad range of instruments. For the densitometer only on-line installed vibrating element density transducers are considered [NORSOK. Table 2.4 also gives some equipment chosen by NPD. the Handbook and the program EMU . the functional relationships which are used in practice are given. 2001]. Within these limitations. The temperature element and transmitter are assumed to be calibrated together. one USM. in combination with a Pt 100 4-wire RTD (calibrated together).USM fiscal gas metering stations NPD. First. Z0 and Hs is not accounted for). one pressure transmitter. . NFOGM and CMR [Ref. With respect to measurement of P. and is used here for the purpose of uncertainty evaluation [Ref.USM fiscal gas metering station The functional relationships of the USM fiscal gas metering station which are needed for expressing the uncertainty model of the metering station are discussed in the following. one temperature element/transmitter. T and ρ. 1998a]. That means. 3. thus. . In the present Handbook.g.4) for the axial volumetric flow rate at line conditions.Handbook of uncertainty calculations . (2. are meter factors.1) PT0 Z 0 Q= qv [Sm3/h] .2. of the M meter factors.1. for reasons explained in the following. 1998. 29 By [AGA-9. 2001b]. conditions (Q). p. • Volumetric flow rate at line conditions (qv) to mass flow rate (qm).1. to reduce possible confusion between USM and turbine meter terminologies. The curve K established on basis of these M meter factors (e. CMR (December 2001) 35 2.2) will here be referred to as the “correction factor”. q v ) (2. method 1 of Table 2. j = 1.2]. 1987. and the energy flow rate. respectively28.. as discussed in Section 2. paragraph 5.2.1. By the same IP reference.. . the axial volumetric flow rate at standard reference conditions. Here this terminology will be avoided. ….2.3) qe = H S Q [MJ/h] . Tables 2.2) P0 TZ q m = ρq v [kg/h] . using the methods described in Section 2.2 and method 5 of Table 2.4] a meter factor is defined as “the ratio of the actual volume of liquid passing through the meter to the volume indicated by the meter”.5) is some function. For flow meters the term “calibration factor” is essentially cover- ing what is meant by the “K-factor” [Eide. and • Volumetric flow rate at line conditions (qv) to energy flow rate (qe).8.1- 2. The K-factor thus has the unit of pulses per volume. K M . q m . involving different types of instrumentation. q v . Kj. also [API. can be made using different methods.1) q v = 3600 ⋅ K ⋅ qUSM [m3/h] . the axial mass flow rate. Q. K 2 .USM fiscal gas metering stations NPD.3. (2. A-4] the function K is referred to as “calibration factor”. By [IP. the K-factor (meter output factor) is defined as “the number of signal pulses emitted by the meter while the unit volume is delivered”. the numbers Kj.2. Here. M. measured at different nominal test flow rates (“calibration points”). method 3 of Table 2. Section 2. NFOGM.1 Basic operational functional relationships The measurement of natural gas using a flow calibrated ultrasonic gas flow meter (USM) can be described by the equations (cf. respectively.3 are used. cf.. (2. q e . the correction factor29 K ≡ f ( K 1 . In this terminology (mainly originating from use of turbine meters). The terms “K-factor” and “calibration factor” will be avoided in the present document. paragraph 2. (2. 28 It should be noted that the conversion from • Volumetric flow rate at line conditions (qv) to volumetric flow at standard ref. cf. 1987. f. P0 = 1 atm. j : axial volumetric flow rate measured by the USM under flow calibration. T0. 30 In the present Handbook. for the actual gas composition. i. qUSM . (2. CMR (December 2001) 36 q ref . Pcal and Tcal. P: static gas pressure in the meter run (“line pressure”). M.e. Hs : superior (gross) calorific value per unit volume [MJ/Sm3]. …. and T0 = 0 oC. and at combustion reference temperature 25 oC [ISO 6976. T. …. P0 = 1 atm. at standard reference conditions. j : reference value for the axial volumetric flow rate under flow calibration of the USM (axial volumetric flow rate measured by the flow calibration laboratory). T0 : gas temperature at standard reference conditions: T0 = 15 oC = 288. P0. at flow calibration pressure and temperature. qe : axial energy flow rate. and T0 = 15 oC.USM fiscal gas metering stations NPD. qUSM : axial volumetric flow rate measured by the USM under field operation (at line conditions). temperature (T) and gas compositional conditions (“line conditions”). Z: gas compressibility factor in the meter run (“line compressibility factor”). ρ: gas density in the meter run (“line density”). for the actual gas.15 K. NFOGM. j. j Kj ≡ . Pcal and Tcal. = 1. Z0 : gas compressibility factor at standard reference conditions. j = 1. P0 : static gas pressure at standard reference conditions30: P0 = 1 atm. as the acual values for P0 and T0 are used only for calculation of Q and qe.6) qUSM . q ref . . 1995c]. M. j = 1. at test flow rate no. However. at flow calibration pressure and temperature. their calculated uncertainties at standard reference conditions will be representative also for normal reference conditions. Q and qe and their uncertainties are specified at standard reference conditions. …. M . Tcal: gas temperature at USM flow calibration. T0 (for the actual gas composition). P0. at P.01325 bara. j. m: molar mass [kg/mole]. T: gas temperature in the meter run (“line temperature”). Pcal : static gas pressure at USM flow calibration. at test flow rate no. before the correction factor K is applied.Handbook of uncertainty calculations . Q: axial volumetric flow rate at standard reference conditions (P0 and T0). P and T. j = 1. j and qv : axial volumetric flow rate at actual pressure (P). actual gas and actual flow rate. qm : axial mass flow rate. where FWME = 100 + FWME M åq j =1 USM . ….g. j. 1998]32. NFOGM. respectively). 1998]. In this case. 2. As discussed e. 1998] å (q ) M USM .Handbook of uncertainty calculations . as: • the flow-weighted mean error (FWME) [AGA-9. the weight factor is 0. j qmax DevU . j and qUSM are partially correlated. the “single factor correction” is referred to as “single calibration-factor correction”. • the weighted mean error (WME) [OIML. this terminology will be avoided here. j Kj is the (uncorrected) deviation at test flow rate no. j and qUSM are correlated. M.USM fiscal gas metering stations NPD. calculated e. .g: 31 By [AGA-9. In this case.0. j − q ref . e. (b) Multi-factor correction. M. 1989]. That is. a single (constant) correcion value may be used over the meter’s expected flow range. j = 1. Some suggested methods of establishing the correction factor from the meter factors are31: (a) Single-factor correction. j = 1. in the AGA-9 report [AGA-9. with the exception that at the highest flow rate. 33 According to the “OIML method” [OIML. or • the average meter factor. multiple correcion values are used over the meter’s expected flow range. j qmax is the flow-weighted mean error. (2. the weighted mean error (WME) is calculated as the flow-weighted mean error (FWME) (see the footnote above).g.4 instead of 1. there are in general several ways to calculate the correction factor K from the M meter factors Kj. …. calculated using a more sophisticated error correction scheme over the meter’s range of flow rates. j and qUSM are measurement values obtained using the same meter (at the flow laboratory and in the field. 32 The (constant) correction factor calculated on basis of the FWME is given as [AGA-9. As explained in the footnote accompanying Eq. 1989]33. qUSM .5). j 100 j =1 K= . j = = −1 q ref .2 Flow calibration issues The correction factor is normally calculated by the USM manufacturer or the operator of the gas metering station. CMR (December 2001) 37 Since qUSM . qUSM . some contributions to qUSM . 1998]. Single-factor correction is effective especially when the USM’s flow measurement output is linear (or close to linear) over the meter’s flow rate operational range. j qmax ≈ 1 . j 1 DevU . and qUSM .2. while others are uncorrelated. . From Eqs. K 2 . or • a regression analysis method. In the vicinity of this test point no. is defined as the ratio of the correction factor and the meter factor no.3 Alternative functional relationship Eqs.9) qUSM . for the USM field measurements in the type of gas metering stations addressed here (cf.j are all very close to unity. 2. (2. (2. K M . (2.7) êë Kj úû where the deviation factor at test flow rate no.8) Kj K. j . • a multi-point (higher-order) polynomial algorithm [AGA-9.4) are the expressions used in practice. Section 2. In the uncertainty model one needs to account (among others) for the uncertainty related to use of the correction factor. K 2 . Kdev.1) as it stands is not very well suited as a functional relationship to be used as the basis for deriving an uncertainty model of the USM gas metering station. the uncertainty of the flow calibration laboratory. j.4). j q v = 3600 K dev . (2.USM fiscal gas metering stations NPD. Eq. Kj and Kdev. (2. Eq. .1) becomes q ref ..8) the functional relationship (2. j ≡ . j (2. j be the calibration point closest to the actually measured flow rate. .6)-(2.1. covers both method (a) (single factor correction) and the more general method (b) (multi-factor correction).2. K M . However. the partial correlation between the USM field measurement and the USM flow calibration measurement. the correction factor K may conveniently be written as é f ( K 1 . To model such effects by the functional relationship.1)-(2. . j qUSM . Multi-factor correction is useful when the USM’s flow measurement output is nonlinear over the meter’s flow rate operational range.. q v )ù K ≡ f ( K 1 . j. 1998]. (2. q v ) = K j ê ú ≡ K j ⋅ K dev .Handbook of uncertainty calculations .1)-(2. (2. K. j. K K dev . . The correction method formulated in Eqs. NFOGM. CMR (December 2001) 38 • a piecewise linear interpolation method [AGA-9.1) is re- arranged to some extent in the following.. 1998]. Let test flow rate point no. and the cancelling of systematic effects by performing the flow calibration.2).j. and used below. the term “error” will be avoided in this context. j Kj Now.2). 2. the correction factor K is not necessarily equal to the meter factor Kj at test flow rate no. j = 0. and Kdev.1). 1998]. j by a constant correction factor.2 = 1. CMR (December 2001) 39 It should be noted that Eqs.1 = 1. since error refers to comparison with the (true) value of the flow rate.j is needed as a basis for specifying the necessary uncertainty of Kdev.9) and (2. the above FWME example. So will be the case if e. at q / q max = 0. Section 3. M . That will depend on the method used for calculation of K (cf. 2. j . The close relationship between DevC . While Eq.10) q ref . NFOGM. Eq. from Eqs.j as the “deviation factor”. j = 1 + DevC . ….00689 and Kdev.1 refers to comparison with the reference measurement of the flow calibration laboratory (cf.11) For example.8).1 is called “percent error”. j and Kdev. For this purpose an example of a USM flow calibration correction discussed by [AGA-9. j .1. 0. j = ( qUSM .g. Fig. and hence the term “percentage relative deviation” is preferred here. where K ≠ Kj). …. 34 By [AGA-9. in this example (Fig. j − q ref . e. The corrected relative deviation is defined as KqUSM . 2. It should be noted that in general.j is the background for referring to Kdev.1) are equivalent. (2. j ) q ref . j K DevC . An interpretation of the deviation factor Kdev.10)). if K is calculated so that K = Kj at test flow rate no. Eq. the deviation factor can be written as K dev . Fig. (2. respectively. Shown is also the percentage corrected relative deviation at test point j after multiplication of qUSM . 1998]. one will have DevC . the ordinate of Fig.01263.USM fiscal gas metering stations NPD.99943. Kdev. defined as DevU .10) and (2.1 and 1. open cir- cles) at six test rate points. Fig. j = 1.0.1) the applied correction factor is constant and equal to K = 1. Here. j − q ref . 35 In the present example (cf.9) is better suited for deriving the uncertainty model for the USM gas metering station. 1998] serves to be convenient. (2. (2.0031 [AGA-9. j = = −1. However. 6 (M = 6 here).Handbook of uncertainty calculations . 2.10 = 0.05.g. (2. cf. (2. as motivated above.j = 1. a piecewise linear interpolation method is used for calculation of K.3. j. .j as input to the uncertainty calculations (cf. (2. The figure shows the percentage relative deviation34 for a set of flow calibration data (uncorrected. j = 1. K (in this example calculated on basis of the FWME35) (shown with triangles).1) is the expression used in practice for the field measurement. 2. j. one finds Kdev. 2 and 2.Multipath ultrasonic gas flow meter The functional relationship of the USM gas metering station. respectively. Figs.. Measurement of Gas by Multipath Ultrasonic Meters [AGA-9.2)-(2.j) and FWME-corrected (DevC. CMR (December 2001) 40 Fig.1. j . cf.Handbook of uncertainty calculations .1 Example of uncorreced (DevU. using gas-tight seals (o-rings) to contain the pressure in the pipe. 1998]. Texas.9) and (2. [AGA-9. j .j) flow-calibration data for a 8” diameter USM. (The figure was based on data provided by Soutwest Research Institute.3 Functional relationship .1 Basic USM principles A multipath ultrasonic transit time gas flow meter is a device consisting basically of a cylindrical meter body (“spoolpiece”). (2. the functional relationship of the USM is needed.3. qUSM and qUSM . To account for the systematic effects which are eliminated by flow calibration. American Gas Association.) 2. 1998]. Copied from the AGA Report No. 2000a]. involves the volumetric flow rates measured by the USM in the field and at the flow calibration laboratory. 1997]. an electronics unit with cables and a flow computer [ISO.4). given by Eqs. NFOGM. 2. with the permission of the copyright holder. 1. as well as correlated and uncorrelated effects between qUSM and qUSM . 9.USM fiscal gas metering stations NPD.3 The transducers are usually mounted in transducer ports and in direct contact with the gas stream. 2. San Santonio. 2. ultrasonic transducers typically located along the meter body wall. . [Lunde et al. at an angle with respect to the pipe axis. CMR (December 2001) 41 USMs derive the volumetric gas flow rate by measuring electronically the transit times of high frequency sound pulses. . 2. (c) Instromet Q-Sonic 5-path reflecting-path configuration [Instromet. (a) (b) (c) Fig.3). The gas flow velocities along the N acoustic paths are used to calculate the average axial gas flow velocity in the meter run.2 and 2. Transit times are measured for sound pulses propagating across the pipe. 1. 2. The present Handbook accounts for USMs with reflecting paths as well as USMs with non-reflecting paths. FMC Kongsberg Metering and Instromet. Figs.Handbook of uncertainty calculations .2 illustrates some concepts used. corresponding to Fig.2 Three different types of USM path configurations used today: (a) Daniel SeniorSonic 4-path non-reflecting-path configuration [Daniel. including non-reflecting and reflecting path configurations. Copied from the respective sales brochures with the permission of Daniel Industries. 2. NFOGM.USM fiscal gas metering stations NPD. 2000]. and upstream against the gas flow (cf. the difference between the upstream and downstream propagating transit times is proportional to the average gas flow velocity along the acoustic path. 2000]. For each of the N acoustic paths. Multiple paths are used to improve the measurement accuracy in pipe configurations with complex axial flow profiles and transversal flow. which (in the USM) is multiplied with the pipe cross-sectional area to give the volumetric axial gas flow rate at line conditions.1. downstream with the gas flow. 2000]. (b) Kongsberg MPU 1200 6-path non-reflecting-path configuration [Kongsberg. Different types of path configurations are used in USMs available today. Fig. The four formulations are equivalent.g. (Left: centre path example (yi = 0). This has implications for the meter body uncertainty (cf..1). [AGA-9.200 bar. or pressures of 150 . . 2.2 Alternative USM functional relationships A schematic illustration of a single path in a multipath ultrasonic gas flow meter (USM) is shown in Fig.4. The USM metering principle is described elsewhere (cf. NFOGM.) 36 Note that due to limitations in pressure and temperature of current accredited flow calibration laboratories (e.2).g. 2000a]). 2. but differ with respect to which input quantities that are used for describing the geometry of the USM37. e.3. Receiving Pulse cables & detection Receiving electronics transducer z L pi yi x 2R φi Transmitting transducer Signal Transmitting cables & TOP VIEW FRONT VIEW generator electronics Fig. Four different formulations of the functional relationship of the USM measurement are relevant. 10 oC and 50 bar maximum pressure).USM fiscal gas metering stations NPD. 37 With respect to the meter independency of the uncertainty model for the USM fiscal gas metering station. CMR (December 2001) 42 Normally. C and D. 1998]. USMs used for sales metering of gas are (1) “dry calibrated” in the factory and (2) flow calibrated in an accredited flow calibration laboratory36.3.4. as well as for systematic effects on transit times (cf. before installation for duty in the gas line [AGA-9. Sections 2.3 Schematic illustration of a single path in a multipath ultrasonic transit time gas flow meter with non-reflecting paths (for downstream sound propagation). which is needed for the uncertainty analysis of Chapter 3. Sections 3. except for the functional relationship described in the following. 1998].Handbook of uncertainty calculations .3.3. 2. These are here referred to as Formula- tions A. Sec- tion 3. B. cf. especially for line applications with temperatures of 50-60 oC. the deviations between line conditions and flow calibration conditions may be significant.4 and 3. Right: path at lateral chord position yi.3 and 5. and details of that are not further outlined here. [Lunde et al. Nrefl.i + 1) L2i (t1i − t 2i ) C qUSM = πR 2 å i =1 wi 2 xi t1i t 2i R. N.i + 1) Li (t1i − t 2i ) B qUSM = πR 2 åw i =1 i 2t1i t 2i cos φ i R.5. φi: inclination angle (relative to the pipe axis) of path no.i+1) of the total propagation distance (interrogation length) of the acoustic path inside the pipe. This length is 1/(Nrefl. along with the geometrical quantities involved in the respective formulations. t2i: transit time for downstream sound propagation of path no. N. i = 1. i. N. i = 1. Alternative and equivalent functional relationships for multipath ultrasonic transit time flow meters. i = 1. i = 1. i = 1. i. …. for path no. (b) For reflecting paths (Nrefl. …. yi and xi Here. ….5. yi and φi N ( N refl . C and D are summarized briefly in Table 2. i. ….. i. yi: lateral distance from the pipe center (lateral chord position) for path no.USM fiscal gas metering stations NPD. xi: length of the projection of Li along the pipe axis (x-direction). i. wi : integration weight factor for path no.i = 0): interrogation length of path no. t1i: transit time for upstream sound propagation of path no. Li and xi N ( ( N refl . ….i + 1) R 2 − yi2 (t1i − t2i ) A qUSM = 2πR 2 åw i =1 i t1it2i sin 2φi R. N. .i: number of wall reflections for path no.e.i + 1) xi2 + 4 R 2 − y i2 ) (t1i − t 2i ) D qUSM = πR 2 åw i =1 i 2 xi t1i t 2i R. i = 1. i.Handbook of uncertainty calculations .i > 0): the portion of the distance from the transmitting transducer front to the first reflection point. N. …. NFOGM. …. Formu. the following terminology is used: R: inner radius of the USM meter body. i = 1. i. …. i. Li and φi N ( N refl . the length of the portion of the inter- transducer centre line lying inside a cylinder formed by the meter body’s inner diameter). for path no. B. …. N). Li: (a) For non-reflecting paths (Nrefl. N. which is lying inside a cylinder formed by the meter body’s inner diameter. N. Functional relationship Geometrical quantities involved lation (i = 1. N. i = 1. i = 1. Table 2. i. CMR (December 2001) 43 The expressions for the four formulations A. N ( N refl . N (i. …. C or D.. C or D is not essential or critical. 2000]. the measurement uncertainties and out-of roundness effects on the meter body geometry (systematic effects) would have to be accounted for in the model (such as in the GARUSO model. qUSM .. In this case. with the exception that the factor (Nrefl. and Nrefl. By the flow calibration. it is thus not important which geometry parameters that have actually been measured on the USM meter body at hand. 2000] meters. cf.4). NFOGM. 1997. and Nrefl.i = 1 and 2. B.i = 0. Table 2. the situation would be somewhat different. in the "dry calibration"38. this factor has been introduced here to account both for USMs with reflecting paths and USMs with non-reflecting paths (cf. Table 1. except for the factor (Nrefl. 2001]). . The description of such dimensional changes is independent of USM formulation A. or D differ only by which meter body geometry parameters that are used as input quantities. Formulation C corresponds to the expression given by [ISO. B. 2000a]. With respect to (Nrefl. CMR (December 2001) 44 N: number of acoustic paths. For the uncertainty model of a flow calibrated USM. three single. For the flow calibration measurement of the USM.3 Choice of USM functional relationship and meter independency aspects It should be highly emphasized that for the uncertainty model of the USM gas metering station presented in Chapter 3.i+1). cf. B. 1997] (Eq. measurement uncertainties and out-of roundness effects on the meter body geometry (systematic effects) are eliminated (cf. For example. and can be described similarly using any of these. 2000] and the 6-path Kongsberg MPU 1200 [Kongsberg. similar expression as given in Table 2. For the 5-path Instromet Q-sonic [Instromet.3.USM fiscal gas metering stations NPD. i..e. the actual choice of USM formulation A. no reflecting paths are used. so that only dimensional changes due to possible pressure and temperature differences are left to be accounted for in the model..i+1). [Frøysa et al. The formulations A. and for evaluation of the expanded uncertainty of the metering station using the Excel program EMU . the USM has been subject only to "dry calibration".Handbook of uncertainty calculations . j . 38 For a USM for which flow calibration has not been made. C.5 apply. (23) of that document). Formulation A was used in [Lunde et al. 2.i+1) has been included here to account also for USMs with reflecting paths.and two double reflecting paths are used. respectively.5. in the 4-path Daniel SeniorSonic [Daniel. 1997] and [Lunde et al.USM Fiscal Gas Metering Station described in Chapter 5. [Lunde et al. 6). this meter independent uncertainty model.2. NFOGM. and (2) the USM field operation of the USM.e. In this operational mode. (3. Li. In this operational mode the uncertainty model is completely meter independent (does not involve formulations A. Eq. where the relative standard uncertainties Erept and EUSM.20)-(3. one of the formulations A.∆. are calculated from more basic input (cf. after this process. the uncertainty of such USMs may be evaluated in terms of formulation A by calculating the combined standard uncertainties of R. This compli- cation may be overcome e. Eqs.USM fiscal gas metering stations NPD. so that the resulting uncertainty model. B.∆ or both. Eq. becomes essentially independent of USM formulation. and how they shall appear in the model (correlation aspects. φi}. However. cf. C or D that were used for the formal derivation of the model. EUSM. . xi} or {R. B. Erept. EUSM.19)-(3. Eqs. where Erept.i. or D at all).10): (a) An “overall level”.6). would be the result irrespective of which USM formulation A. yi and φi are used as input quantites. (3. (3. such as one of the sets {R. CMR (December 2001) 45 The main reason for introducing the four formulations. the choice of formulation is arbitrary.3. φi}. {R. xi}. This process is described in Appendix E. and using these combined standard uncertainties of R.g. (2) With respect to USM related input uncertainties. Li. etc. the user has the option to run the program EMU . 39 Sensitivity coefficients involving the USM functional relationship formulation appear in the uncertainty terms related to (1) repeatability of the USM in field operation. C and D would then be more important. B.). Li. yi and φi from the standard uncertainties of the sets {R. Therefore. yi and φi as input to the uncertainty calculations.47)). Li. Eq. C.20). to sort out which uncertainty terms that are to be accounted for and not. {R. 2000]). sensitivity coefficients have to be calculated in the program. (3. a specific USM formulation is needed in the process of deriving a mathematically and formally valid uncertainty model.∆ (cf. respectively.19)) are given directly. The choice of formulation A.Handbook of uncertainty calculations . yi. xi}. (3. However. For USMs for which other geometry quantities than R. . for calculation of these coefficients. C and D that is actually used39. and for making a choice of one of them for the following description. (b) A more “detailed level”.USM Fiscal Gas Metering Station at two levels (cf. 3. as follows: As an example. C and D has to be chosen.3. B.4 and 5. uncertainty terms are associated and assembled in groups. the uncertainty model does not depend on which of the formulations A. B. is threefold: (1) Firstly. xi} or {R. Sections 1. as done in Section 2. yi. In fact. assume formulation A is used. (2. (2. 2.∆ is independent of which of the formulations A.i + 1) R 2 − y i2 (t1i − t 2i ) qUSM = 2πR 2 å wi . Section 5.20)-(2. Note that other models for β (cf. 2.43)-(3.27). given by Eqs. cf. For Ebody. B. i. the sensitivity coefficients are completely independent of formulation. N ( N refl .44) and the text accompanying Eq. or a pressure difference of 36 bar (for a 12” USM example with wall thickness 8. Formulation A is used.22) below.∆. For these three reasons. the lateral chord positions and the inclination angles of the paths. one of the formulations A. (3.g.∆.USM fiscal gas metering stations NPD. C and D that is used. it can be shown that Ebody.20)-(2. B. the sensitivity coefficients are completely independent of which formulation A. sensitivity coefficients appear in connection with the uncertainty terms Etime. In the case studied here.12) i =1 t1i t 2i sin 2φ i involving the geometrical quantities R. with flow calibration of the USM. cf.Handbook of uncertainty calculations . cf. one has to choose one of the four formulations A. only uncertainties related to pressure and temperature expansison/contraction of the meter body are to be accounted for in Ebody.1. relative to the P and T conditions at flow calibration40. With respect to EUSM. yi and φi.21)). Eqs. N.3.∆ and Ebody. For Etime. C and D that is actually used. as the USM functional relationship.4 Pressure and temperature corrections of USM meter body Pressure and temperature effects on the meter body dimensions (expansion/contraction..3. B.3.∆.28). 1998. i.4). the meter body inner radius.1 %.e. i = 1. In the present document.6) would give other figures for the pressure expansion/contraction. since Eqs. (2.15).. . β. (2. Eq. B.g. In this case.4) act as systematic effects and appear directly as errors in the flow rate measurement result.. Section 2. qv. (3. if not corrected for.∆. 40 These results can be calculated e. since the P and T effects on the geometrical quantities are correlated (cf.20). C or D is needed. (2.∆ accounts for uncertainties related to meter body quantities. for description of uncertainties related to dimensional changes of the meter body. (3. Eq. (3. Table 1.e.4.12)-(2. Table 2. .∆ (cf.19) used for the radial pressure expansion coefficient. (3. Such errors in qv become significant. for a temperature difference of 24 oC. If such temperature and pressure effects With respect to Erept. about 0. Fig. cf.4 mm). and Eq.1]. for the same reason as given in connection with Erept above. CMR (December 2001) 46 (3) Thirdly. Ebody. The results are consistent with the analysis and results given in the AGA-9 report [AGA-9. the sensitivity coefficients are given by Eqs.25)-(3. using Eqs. NFOGM.22) are not generally valid. with USM meter body material data as given in Table 4. e. C or D. caused by possible pressure and temperature differences from flow calibration to field operation.. relative to the reference conditions). i in the USM. and inclination angles. some meter manufacturers have implemented correction factors for such dimensional changes. Section 3. showing lateral change with T and P (sche- matically). the NPD regulations [NPD. . NFOGM. and that may be included in possible future upgrades of the Handbook. Note that today such corrections address the dimensional changes of the meter body. Over time a systematic error of this magnitude may accumulate to significant economic values. 2. Fig. This concerns the average inner diameter of the meter body. For example. Pressure and temperature correction is also addressed by the AGA-9 report41. the effects of transducer expansion/contraction due to pressure and temperature effects can be accounted for by including such effects in the standard uncertainties of the coefficients of linear thermal and pressure expansion. so correcting for the error is often recommended.1). expansion due to pressure increase and contraction due to a temperatur decrease. but not necessarily the effect of dimensional changes of the transduc- ers42. that would result in an error of 0. Cf. CMR (December 2001) 47 work in opposite “directions” (e.1.Handbook of uncertainty calculations . An alternative approach would be to account for such effects in the functional relationship of the USM. if they work in the same “direction”. 42 In the program EMU . respectively. However.02 % of the total volume” (cf.4. the meter body radius (R). they will can- cel each other.USM fiscal gas metering stations. the axial distance between transducer pairs. To correct for the small dimensional changes (expansion/contraction) of the meter body caused by the operational pressure and temperature in the field (cf. u(α) and u(β).g. Section C. 1998].2 %. the AGA-9 report recommends measurement and documentation of relevant dimensions of the meter at atmospheric conditions and a temperature 20 oC. as a part of the “dry calibration” of the USM [AGA-9. are (cf. 2001] state that “Correction shall be made for documented measurement errors. Correction shall be carried out if the deviation is larger than 0. the length of each acoustic path between transducer faces. Fig.USM fiscal gas metering stations NPD. 2.4). At a temperature T and pressure P. and the inclination angles (φi).4 Geometry for acoustic path no. Appendix D) 41 For each individual USM. Some recommendations for measurements of these quantities in the factory are given in the AGA-9 report. the lateral chord positions (yi). e. Pdry = 1 atm. 1989]. cf. 44 The radial pressure and temperature correction factors for the USM meter body. For convenience. yi0 and φi0.17) respectively. respectively. 1995]. R0. ∆Pdry ≡ P-Pdry .21)-(2. The correction factors for the inner radius of the meter body due to dimensional changes caused by temperature and pressure changes relative to “dry calibration” conditions.4. …. where Pdry and Tdry are the pressure and temperature at “dry calibration” conditions.14) æ tan( φ i0 ) ö φ i ≈ tan −1 çç ÷÷ . Cpsm and Ctsm. N (2. and corresponding expressions for β* are given in Appendix D. This is so because the relevant reference temperature and pressure with respect to meter body expansion / contraction are not the “dry calibration” temperature and pressure. 1998]) K T ≡ 1 + α∆Tdry . KP and KT are here referred to as the radial pressure and temperature correction factors for the USM meter body. i = 1. Appendix D) 43 Note that the actual values of the temperature and pressure at “dry calibration” (Pdry and Tdry ) are never used in the uncertainty model of the gas metering station.e. [IP. NFOGM. e.6 gives different models in use for β. Table 2. and Tdry = 20 oC 43. e. (2. 1981.15) è 1 − ( 1 − β β )( K P − 1 ) ø * where subscript “0” is used to denote the relevant geometrical quantity at "dry calibration" conditions. e. For thin-walled cylindrical and isotropi- cally elastic meter bodys. α is the coefficient of linear thermal expansion of the meter body material.g. β and β* depend on the type of support provided for the meter body installation (i.22).e. . (2. Note that all models in use represent simplifications. the model used for the meter body pressure expansion / contraction).Handbook of uncertainty calculations .g. but the temperature and pressure at flow calibration of the USM. i.USM fiscal gas metering stations NPD. ∆Tdry ≡ T-Tdry . [API. KP and KT. [AGA-9.16) K P ≡ 1 + β∆Pdry . Eqs. are given as (cf. β and β* are the radial and axial linear pressure expansion coefficients for the meter body. i.13) yi ≈ K T K P y i0 (2.1. CMR (December 2001) 48 R ≈ K T K P R0 (2. should not be confused with the corresponding volumetric pressure and temperature correction factors of the meter body. Section 3. (2. cf.USM Fiscal Gas Metering Station. respectively44.g. they are not used at all in the program EMU . β and β* are releated as (cf. giving C psp ≈ 1 + ( 2 R0 wY )( 1.Handbook of uncertainty calculations . 2001]. 592] β= 0 (ends free) wY • Thin wall.235 for the for the cylindrica cylindrical l tan k(ends tank model mod el (ends capped ) capped). Table 2. [IP. 1989] state that their expression is based upon a number of approximations that may not hold in practical cases. For steel (σ = 0. w < R0 /10 45 In [API. for linear pressure expansion of the inner radius of the USM meter body (isotropic material assumed). a tank prover. under uniform internal pressure. β assumptions [AGA-9. p.00025.3( R0 + w ) + 0. 1989] the volumetric pressure correction factor Cpsp for a cylindrical container (such as a conventional pipe prover. [API. This is relatively close to the volumetric pressure correction factor Cpsp for a cylindrical tank (pipe with ends capped) which is given as C psp = ( 1 + β∆P ) 2 ( 1 + β * ∆P ) ≈ 1 + ( 2 β + β * )∆P .25 − σ )∆P . 2001. For values of ( C psp − 1 ) that do not exceed 0. the uncertainty may be larger.3). 1981]. 2001. R • Cylindrical pipe section model [Roark.95 R0 wY )∆P . 1995]. NFOGM.5 ⋅ radial expansion for ends- R free model β = 0 .3 forthe for thecylindrica cylindricall pipe pipe sec section tion model mod el(ends (endsfree). For higher values of ( C psp − 1 ) (such as for high pressure differences). 593] • Thin wall. 593] β ≈ ( R0 wY )( 1 − σ 2 ) and β * ≈ ( R0 wY )( 0. p.3 (steel)) wY Instromet No P or T correction used.5 − σ ) . 2001] clamped. is given as C psp ≈ 1 + ( 2 R0 wY )∆P . w < R0 /10 R0 ( β = 0. or a test measure).4 R0 2 2 • Cylindrical tank model Daniel Industries β= (pipe with ends capped) ( R0 + w ) 2 − R0 2 Y [Daniel. the value of ( C psp − 1 ) may sometimes be subject to an uncertainty of ±20 %.USM fiscal gas metering stations NPD. clamped ) (2. no axial displacement) Pressure expansion analysis based • Radial expansison assumed to be on: = 0. w < R0 /10 1 1. 1996. Reference / Models for the coefficient of USM meter body USM manufacturer linear radial pressure expansion. î 2 −σ where the values inserted for β * β apply to steel (σ = 0.3) ( β ≈ 0. CMR (December 2001) 49 ì ï− σ = −0. [IP. 2001.6. 1998].3) this gives C psp ≈ 1 + ( 1.5 0 wY • Thin wall.85 0 for w << R0) wY FMC Kongsberg Metering R0 æ σ ö • Cylindrical tank model [Kongsberg.85 for σ = 0. • Infinitely long pipe model (ends [Autek. where β * is the coefficient of linear pressure expansion for the axial displacement of the USM meter body. For a thin-walled cylindrical tank one has [Roark. p. 2001] • Thick wall R • Steel material (σ = 0. Models used by USM manufaturers etc.18) β ï 1 − 2σ ï = 0. β= ç1 − ÷ (pipe with ends capped) 45 wY è 2ø [Roark. . free) β* ï ≈ í0 for theinite for inf − length cylindrica ininite-length l pipe cylindrical pipe mod el (ends model (ends clamped). various models for β may be implemented. p. cf. The ends-clamped model may be relevant for installation of the USM in a long straight pipeline.e. as can easily be seen from the table. i = 1. for thin-walled steel pipes. …. φ i 0 = ±45o. and as a possible “worst case” approach (since it gives about 15 % larger pressure expansion than the cylindrical tank model). i.USM fiscal gas metering stations NPD. 2001]. Chapter 7. Appendix D) qUSM ≈ qUSM . CMR (December 2001) 50 Here the following terminology has been used. The infinitely-long-pipe model (ends clamped) used by Instromet [Autek. The “cylindrical tank model” (pipe with ends capped) [Roark.Handbook of uncertainty calculations . (2. Y = Young’s modulus (modulus of elasticity) of the meter body material. p. σ = Poisson’s ratio of the meter body material (dimensionless). at "dry calibration" conditions [m]. N. It should also be noted that for USMs where all inclination angles are equal to ±45o. R0 = Average inner radius of the meter body. (2.20) 46 The assumption that the radial displacement is at maximum half the of the value of the ends-free model was a tentative estimate [Autek.17) can be written as (cf. and does not account for flanges or very long pipe installations. bends.12)-(2. Relative to the ends-free model. namely R0 β= . the expression corresponding to the “cylindrical pipe section model (ends free)”. etc. 2001] assumes no axial displacement and that the radial displacement is at maximum half the value of the ends-free model46. NFOGM.0 ⋅ C tsm ⋅ C psm . Both models allow for displacement of the pipe in axial direction.e. 2001. it has for the present Handbook been chosen to use the expression for β which has been applied by [AGA-9. pp. 1998]. 592] applies to a finite-length pipe section with free ends and does not account for flanges. but in different directions [Roark. 592-3]. 593] may to some extent apply to installation in a pipe section between bends. 47 In a possible future revision of the Handbook. (2. Eqs. .19) wY i. w = Average wall thickness of the meter body [m]. The “cylindrical pipe section model” [Roark. the ends-capped model and the ends-clamped model used by Instromet differ by about 15 % and 50 %. For simplicity (to limit the number of “cases”)47. 2001. 2001. (2. Li.. 1981]. [API. (2. for which it is a valid expression since all inclination angles in the meter are 45o.3 %).20)-(2. s and m refer to “temperature”. Appendix D). 2001].22). 50 By one manufacturer [Autek. (2. for which they represents an approximation (cf. FMC Kongsberg Metering are using a correction approach of the type given by Eqs. [API. (2. p. Hence. 1989]. C psm ≈ 1 + 4 β∆Pdry .20)-(2. p. cf.17) and Eqs.20)-(2. 2001]. if the same β-model is used). “pressure”. for such meters Eqs.12)-(2. B. 49 Eqs. It turns out that for moderate pressure deviations ∆Pdry (a few tens of bars).USM fiscal gas metering stations NPD. Eq. the P and T corrections of the geometrical quantities of the meter body can be separated from the basic USM functional relationship and put outside of the summing over paths.0 is given by Eqs.22) may be neglected.21)-(2.22) are the volumetric thermal and pressure correction factors of the USM meter body48.13)-(2. e.22) are used by Daniel Industries for the SeniorSonic USM [Daniel. (2. “steel” and “meter”. …. It should be noted that Eqs. On basis of these expressions (which in fact predict larger dimensional changes than Eqs. [Dahl et al. 2001]. CMR (December 2001) 51 where C tsm = K T3 = ( 1 + α∆Tdry )3 ≈ 1 + 3α∆Tdry . where subsripts t.21) C psm = K P3 = ( 1 + β∆Pdry )3 ≈ 1 + 3 β∆Pdry . the errors made by using Eqs. 2001] that the errors due to pressure and temperature expansion / contraction are negligible except under extreme situations (such as when the errors approach the uncertainty of the best flow calibration laboratories. . about 0. The relationship (2.20)-(2. a notation is used according to “common” flow metering terminology. and qUSM . C and D. it is argued [Autek. (D.17) [Kongsberg.15). which in the notation used here can be written as C tsm ≈ 1 + 4α∆Tdry . [IP. yi0. 50. C-29]. (2.22) is strictly not valid expressions for meters involving inclination angles different from 45o. alternative expressions for the temperature and pressure correction factors of the USM meter body (alternatives to Eqs. NFOGM. yi.22)) have been presented. (2. 1998. Li0.21)- (2. 1995]. (2. xi0 and φi0 inserted instead of the quantities R. N. xi and φi. (2.12)-(2. fully consistent with the less general discussion on this topic given in [AGA- 9.22) are equivalent49. (2. That means.Handbook of uncertainty calculations . with the "dry calibration" quantities R0. i = 1. (2.20) has been shown in Appendix D for all formulations A.28) gives the relative error by using this approximation.g. and these equa- 48 For the correction factor of the meter body. 1999]. respectively. possible averagings and corrections of the measured transit times are not adressed here.20)-(2.13)-(2. and a description of these would be meter dependent. pressure deviations of e. the measured upstream and downstream transit times t 1i and t 2 i contain possible time delays due to the electronics.15) are used to describe P and T effects on the meter body in the uncertainty model of Chapter 3. it is left to the USM manufacturer to specify and document the following input uncertainties: (1) With respect to repeatability (random transit time effects): One specifies either Erept (the relative standard uncertainty of the repeatability of the measured flow rate. Fig.15) and (2. 1997. (2.3. Appendix D). (2.20)-(2. cables.006 % and 6⋅10-4 = 0. employs ±45o inclination angles. However. Today. For example. 1996. for the 12” USM data given in Table 4. and possible cavities in front of the transducers. (2. the only USM manufacturer using Eqs.12) are time averaged transit times. by using Eqs. 40o to 60o. for which the two formulations (2.Handbook of uncertainty calculations . the transit times t 1i and t 2 i for upstream and downstream sound propagation appearing in Eq.g. and since both formulations are in use.USM fiscal gas metering stations NPD. 2001].13)-(2. ∆Pdry = 10 and 100 bar yield relative errors in flow rate of about 6⋅10-5 = 0. 2.3 [Lunde et al. the error introduced by using Eqs. To achieve sufficient accuracy of the USM. or u( t̂ 1random i ) (the standard uncertainty of the tran- .22) have been shown here to be practically equivalent (Appendix D).20)-(2. 2. (2.22) (cf. CMR (December 2001) 52 tions may be used for inclination angles in the range of relevance for current USMs. For this reason. cf..20)-(2. In order to avoid meter dependent functional relationships in the uncertainty model of the gas metering station. such time averagings and time corrections have been implemented in different ways by the different USM manufacturers.5 Transit time averaging and corrections In practice.06 %. these additional time delays may have to be corrected for in the USM. NFOGM. respectively. Daniel Industries [Daniel.22). the more generally valid equations (2. and especially for inclination angles approaching 60o. 2000a]. However. In the uncertainty model described in Chapter 3. transducers and diffraction effects (including finite beam effects). at a particular flow rate). for larger pressure deviations.22) increases.3 and angles equal to 60o. Moreover. …. diffraction. .352. at a particular flow rate. transducer cavities.∆ . In order to avoid meter dependent functional relationships in the uncertainty model of the gas metering station. that would be meter dependent.4.3. also Chapter 5): 51 If specific time averaging algorithms were accounted for in the repeatability of the transit times. CMR (December 2001) 53 sit time t 1i .3).351. 52 If transit time corrections used in USMs were accounted for (such as correction for one or several time delays due to transducers. one specifies the standard uncertainties related to systematic effects on the transit times t 1i and t 2 i . Section 2. 2.e. This approach is meter independent. and nonzero transit time difference (∆t) at zero flow). (2) With respect to systematic transit time effects: u( t̂ 1systematic i ) and u( t̂ 2systematic i ) . Section 3.6 USM integration method Different USM manufacturers use different integration methods.3).2). cf. N. 2. the relative standard uncertainty of the USM integration method due to change of installation conditions from flow calibration to field operation (cf.3 and Chapter 6 (Table 6. cf. the specific USM integration method is not adressed here53.4 Functional relationship . different integration weights. Section 3. i = 1.4. that might be meter dependent.3. after possible time correction) (cf. elecronics. N. Section 2. A description of these would thus be meter dependent.USM Fiscal Gas Metering Station be specified at two levels (cf. Both approaches are meter independent.Handbook of uncertainty calculations . it is left to the USM manufacturer to specify and document E I . i = 1. respectively (cf. i. Section 3.Gas densitometer As described in Section 1. 53 Very approximate values for the integration weights wi.USM fiscal gas metering stations NPD.4. the uncertainty of the gas densitometer can in the program EMU . cf. In the uncertainty model described in Chapter 3.3. …. wi.3. NFOGM.2). Section 5. are used only to calculate certain sensitivity coefficients in the “weakly meter dependent” mode of the program. By this method. 1999]. E ρ . • In case of density measurement using GC analysis and calculation instead of densitometer measurement(s). from more basic input uncertainties for the vibrating element gas densitometer. [ISO/CD 15970.28).Handbook of uncertainty calculations . 1997]. The output signal is a frequency or a periodic time (τ).3. which is primarily dependent upon density. vibrating in the cylinder’s Hoop vibrational mode. the density transducer is 54 The “overall level” option may be of interest in several cases. This option is completely general. such as e.2. (2) “Detailed level”: E ρ is calculated in the program. • In case of a different installation of the gas densitometer (e. 2. They consist of a measuring unit and an amplifier unit. temperature and gas composition [Tambo and Søgaard. It is left to the user to calculate and document E ρ first.555. only on-line installation of the densitometer is considered. and secondarily upon other parameters. Gas densitometers considered in the “Detailed level” are based on the vibrating cylinder principle. 1998a. e. NFOGM. such as pressure. CMR (December 2001) 54 (1) “Overall level”: The user specifies the relative combined standard uncertainty of the density measurement.g.: • If the user wants a “simple” and quick evaluation of the influence of u c ( ρˆ ) on the expanded uncertainty of the gas metering station. . Here. • In case of a different gas densitometer functional relationship than Eq. From the densitometer the sample flow can either be returned to the pipe (to the sample probe or another low- pressure point) or sent to the atmosphere (by the flare system).g. Any change in the natural frequency will represent a density change in the gas that surrounds the vibrating cylinder. The following discussion concerns the “Detailed level”. cf. cf. (2. using a by-pass gas sample line. provided by the instrument manufacturer and calibration laboratory. and covers any method of obtaining the uncertainty of the gas density estimate (measurement or calculation)54. Fig. 55 The NORSOK regulations for fiscal measurement of gas [NORSOK. To reduce the temperature differences between the densitometer and the line. in-line). §5. gas is extracted (sam- pled) from the pipe and introduced into the densitometer. In this case a functional relationship of the gas densitometer is needed.g. The vibrating cylinder is situated in the measuring unit and is activated at its natural frequency by the amplifier unit. directly as input to the program.USM fiscal gas metering stations NPD.7] state that “the density shall be measured by the vibrating element technique”. 2. (b) Principle sketch of possible on-line installation of a gas densitometer on a gas line (figure taken from [ISO/CD 15970. 1999]).5 (a) The Solartron 7812 gas density transducer [Solartron. 1999] (example). pressure and temperature [ISO/CD 15970. (a) (b) Fig. The functional relationship involves a set of calibration constants. NFOGM. velocity of sound (VOS) correction. Handbook of uncertainty calculations . and installation correction (see below). 1999].USM fiscal gas metering stations NPD. For USM metering stations where the flow meter causes no natural pressure drop in the pipe. CMR (December 2001) 55 installed in a pocket in the main line. so that the pressure difference between the sample inlet hole and the sample return hole can create sufficient flow through the sample line / densitometer to be continuously representative with respect to gas. and the whole density transducer installation including the sampling line is thermally insulated from the ambient. the sampling device (probe) may be designed to form a pressure drop. . as well as temperature correction. [Solartron.sound velocity of the gas surrounding the vibrating element [m/s].c . note that the form of the regression curve is actually not used in the densitometer uncertainty model. The relation between the density and the periodic time is obtained through calibration of the densitometer at a given calibration temperature (normally 20 °C). 1999. in density transducer [kg/m3]. Section 3. argon or methane.USM fiscal gas metering stations NPD. on a known pure reference gas (normally nitrogen. a commonly used densitometer in North Sea fiscal gas metering stations. The present uncertainty model is thus independent of the type of regression curve used. 1999]. is a function of density and varies typically in the range 200 - 900 µs [Tambo and Søgaard. P ) [Tambo and Søgaard. NFOGM. 1997].indicated (uncorrected) density. it should be emphasized that the functonal relationship described in the following is relatively general.regression curve constants (given in the calibration certificate). 1997]. and at several points along the densitometer’s measuring range.periodic time (inverse of the resonance frequency. τ. K1. K1 and K2 are not needed as input to the uncertainty model. The form of the regression curve can vary from manufacturer to manufacturer. CMR (December 2001) 56 In the following.2.4. 1999]. reference will be made to the Solartron 7812 Gas Density Trans- ducer [Solartron. K2 . However. . ρ u = f ( τ .23) where ρu . (2. and that K0. §6. the output is the periodic time of the resonance frequency of the cylinder’s Hoop vibrational mode.Handbook of uncertainty calculations .1 General density equation (frequency relationship regression curve) For gas density transducers based on the vibrating cylinder principle.23) is one example of such a curve. (2. due to their acknowledged properties). 2. This is also the densitometer used for example calculations in Chap- ter 4. The periodic time. cf.T . K0. One common regression curve is [ISO/CD 15970. output from the densitometer) [µs].4] ρ u = K 0 + K 1τ + K 2τ 2 . However. c . τ . and should apply to any on-line installed vibrating element gas density transducer. The calibration results are then fitted with a regression curve.4. and Eq. ρ d = ρT ⋅ ê è c 2ú (2. §6.23)-(2.23) should be made for best accuracy. 1999.24) where ρT . (2. and it may be desirable to introduce VOS corrections to maintain the accuracy of the transducer [Solartron. 1999.Handbook of uncertainty calculations .2 Temperature correction When the densitometer operates at temperatures other than the calibration temperature. In the Solartron 7812 gas density transducer. §6. a small calibration offset may be experienced. a correction to the density calculated using Eq. §6. 1999]. 2. 1999]. one of the correction methods are suggested”. The equation for temperature correction uses coefficient data given on the calibration certificate. when it is necessary. Appendix E] é æK ö ù 2 ê1 + çç d ÷÷ ú ê τc ø ú .USM fiscal gas metering stations NPD. 1999]. 57 It is stated in [Solartron. 1999. [Solartron. a 4-wire Pt 100 temperature element is incorporated. Eqs. The basic relationship for VOS correction is [ISO/CD 15970. This offset is pre- dictable.4.6. the need to apply VOS correction is less likely.calibration temperature [Κ]. 1999. τ. 1999] for the calibration constants K18 and K19 is used. of the vibrating cylinder is influenced by the gas compressibility (or.3 VOS correction The periodic time. However. the gas composition). Tc .25) ê1 + æç K d ö ú ÷÷ ê çè τcd ø úû ë 56 Here.1] that “the 7812 Gas Density Transducer is less sensitive to VOS influence than previous models of this instrument and. §E. NFOGM. Td . K18 .constants from the calibration certificate56. CMR (December 2001) 57 2. . K19 . and is given as [ISO/CD 15970. for installation and check purposes [Solartron. the notation of [Solartron. in consequence. [Solartron.5] ρ T = ρ u [1 + K 18 (Td − Tc )] + K 19 (Td − Tc ) .24) do not account for such effects. in density transducer [kg/m3]. when the vibrating element gas densitometer is used on gases other than the calibration gases (normally nitrogen or argon).6 and Appendix E]57. (2.4.temperature corrected density.gas temperature in density transducer [Κ]. in other words. (2. Consequently. and thus on the VOS in the gas. For this method. For measurement of gas which has a reasonably well defined composition. 1999. K18 and K19. For metering stations involving a USM. Ap- pendix E]. at calibration temperature and pressure conditions [m/s]. This method has been recommended for measurement of different gases at varying operating conditions. There are several well established methods of VOS correction. and four common methods are: 1.25) is used. 2001a]). 1999.temperature and VOS corrected density. cc . 4. K2. cd .10⋅104 µm for 7812. 1999]. The “User Gas Equation” method [Solartron. . So- lartron can supply a “User Gas Calibration Certificate” [Solartron. The “Pressure/Density method” [ISO/CD 15970. [Solartron. This method is here referred to as the “USM method”. in density transducer [kg/m3]. This equation is shown on nitrogen or argon calibration certificates. This correction method is recommended by Solartron for applications where pressure data is not available. This specifies modified values of K0. and applies a correction based on two coefficients that define the VOS characteristic. but where gas composition and temperature do change.transducer constant [µm] (characteristic length for the Hoop mode resonance pattern of the vibrating element [Eide. in order to include the effects of VOS for the given gas composition. K1.USM fiscal gas metering stations NPD. a different (and approximate) expression for the VOS correction than Eq.Handbook of uncertainty calculations . The User Gas Equation is an approximate correction for a typical mixture of the calibration gas (normally nitrogen or argon) and methane. 1. equal to 2. NFOGM.VOS for the calibration gas. 3.VOS for the measured gas. Appendix E] calculates the VOS (cd) based on the specific gravity and the line temperature. 1999.35⋅104 µm for 7810 and 2.62⋅104 µm for 7811 sensors [Solartron. 1999]. Kd . and may be useful for measurement of different gases at varying operating conditions. the VOS measured by the USM (averaged over the paths) is often used for cd. Ap- pendix E] calculates the VOS (cd) based on the line pressure and density and applies the required correction. in the density transducer [m/s]. 2. CMR (December 2001) 58 where ρ d . (2. and precautions to avoid pressure loss. in dry gas at 110-150 bar and 10 oC) [Sakariassen. Installation errors result from the sample gas in the density transducer not being at the same temperature or pressure as the gas in the line. [Geach.g.7] state that (1) “The density shall be corrected to the conditions at the fiscal measurement point”. 60 A tempreature change of 1 oC can correspond to much more than 0. Temperature is a critical installation consideration as a 1 oC temperature error represents a 0. VOS correction methods based directly on Eq. CMR (December 2001) 59 In the following. In some cases the change can be as large as 0. Despite thermal insulation of the by- pass density line. [Tambo and Søgaard. and hence its density is different. downstream of the USM. NFOGM. Other VOS correction algorithms may be included in later possible revi- sions of the Handbook. (2. since the temperature also changes the compressibility. §5.3 % in density change. 59 The NORSOK I-104 industry standard for fiscal measurement of gas [NORSOK. Z. external loop pipework should be in direct contact with the main line.9 % (e.25) was chosen by [Ref. 1994]. Temperature measurement is available in the density transducer since a Pt 100 element is integrated in the 7812 densitometer [Solartron. In this connection it is worth remembering that the densitometer will always give the density for the gas in the density transducer. With respect to temperature deviation between the density transducer and the main flow due to ambient temperature effects. [Geach. the gas conditions at the density transducer may be different from the line conditions (at the USM).3 % density error [Matthews.4 Installation correction The vibrating element density transducer is assumed to be installed in a by-pass line (on-line installation). There may thus be need for an installation correction of the density59.2. but also possibly with respect to pressure (cf. This includes the “USM method” and the “pressure/density method”58.g. To aid density equalization. 1994]). Group. density transducers should be installed in a thermal pocket in the main line”. 2001]. (2.3. 2.4. . and (2) “if density is of by-pass type.25) are considered. temperature compensation shall be applied”. 1997] or more [Sakariassen. 1994] state that “The pipework should be fully insulated between these two points to reduce temperature changes and. 1998a. The temperature 58 The VOS correction algorithm given by Eq. especially with respect to temperature (due to ambient temperature influence). where possible. Unfortunately. e. this can be difficult to achieve. 1999].USM fiscal gas metering stations NPD.Handbook of uncertainty calculations . 2001]60. 2001] for use in the present Handbook. are li- able to cause unacceptable pressure drops”. 1997]. the following instrumentation is considered: 61 In practice. but fast enough to represent the changes in gas composition [Tambo and Søgaard. NFOGM. 1999] æT öæ P öæ Z d ö ρ = ρd ç d ÷çç ÷÷ç ÷ . 1999]. From the real gas law.4. With respect to possible pressure deviation. gas compressibility factor for the gas in the density transducer.7. installed in the external loop.Handbook of uncertainty calculations . gas pressure in the pipe. and the temperature element and transmitter in the densitometer are calibrated together (at the same time as the densitometer). P . if installed between the flow element measuring point and the density transducer. 1994] that “careful consideration should be given to any flow control valves.26) èT øè Pd øè Z ø where T . . etc.. correction for deviation in gas conditions at the densitometer (in the by-pass line) and at the USM (line conditions) is made according to [ISO. [ISO/CD 15970. 1999]. The flow through the densitometer must be kept low enough to ensure that the pressure change from the main line is negligible. These devices. at USM location (line conditions) For the uncertainty analysis of the densitometer described in Sections 3. Pd . gas compressibility factor for the gas in the pipe. 1994]. and resorts to overcome the problem of satisfying pressure and temperature equilibrium are discussed by [Geach. Zd . gas temperature in the pipe. 1994]. 1994] state that “such instrumentation should only be used as a last resort where it is not possible to ensure good pressure equalization with the meter stream”. 4. [Geach. in the flow computer. Procedures for pressure shift tests are discussed by [ISO/CD 15970. at the USM location (line conditions) [K]. pressure measurement is not available in the density transducer [Geach. pressure in the density transducer [bara]. to minimize the uncertainty of the densitometer’s temperature reading. (2. at the USM location (line conditions) [bara]. Normally.4 and 5. the densitometer’s temperature transmitter is usually located in the densitometer. Z .2.USM fiscal gas metering stations NPD. filters (including transducer in-built filters). it is emphasized by [Geach. CMR (December 2001) 60 transmitter for this Pt 100 element may be located close to the densitometer61 or further away.2. the operator of the metering station typically assumes that the densitometer pressure is equal to the line pressure. P̂ ≈ P̂d . and the pressure deviation is relatively small63. The line pressure P is the pressure in the metering volume.27) èT øè 1 + ∆Pd P øè Z ø 62 Note that by one gas USM manufacturer. pressure measurement is not available in the density transducer. which for a pressure of 100 bar corresponds to 20 mbar. or outside the cone (outside the meter body).02 % of the line pressure [Eide.4. In practice. Differences in pressure will have more influence on low pressure systems than high-pressure systems. for measurement of the line temperature T). To account for this situation. Td). and in the density transducer (for measurement of the temperature at the densitometer. only one pressure transmitter is here assumed to be installed: in the meter run (close to the USM..2. from pressure shift tests. NFOGM. cf. CMR (December 2001) 61 For a densitometer of the by-pass type. Consequently. so that the gas compressibility factors in the line and in the densitometer (Z and Zd) can differ significantly. etc.4 and 4. with negligible loss of accuracy. In this description. then.USM fiscal gas metering stations NPD.e. for measurement of the line pressure P). if the density sampling probe is located in the cone. i.e. Correction for deviation in gas compressibility factors is thus normally made. there will be an uncertainty associated with that assumption. ∆Pd may be estimated empirically. However. (2. Sections 3. 63 Tests with densitometers have indicated a pressure difference between the densitometer and the line of up to 0. i. USMs are available today for which the meter body is coned towards the ends. .26) for installation correction is here replaced by æT öæ 1 öæ Z d ö ρ = ρd ç d ÷çç ÷÷ç ÷ . where ∆Pd is the relatively small and unknown pressure difference between the line and the densitometer pressures (usually negative).2.Handbook of uncertainty calculations . ∆Pd represents the uncertainty of assuming that Pd = P. the additional pressure difference between the line pressure and the pressure at the density sampling point has to be included in ∆Pd. Pd is not measured. let Pd = P + ∆Pd. Consequently. the temperatures in the densitometer and in the line can vary by several oC . In practice. the inner diameter decreases slightly over some centimeters from the ends towards the metering volume (introduced for flow profile enchancement purposes). (2. How- ever.62 Two temperature transmitters are assumed to be installed: in the meter run (close to the USM. the expression Eq. That is. or just taken as a “worst case” value. the gas composition is the same at the USM as in the densitometer. 2001a]. 64 Note that alternative (but practically equivalent) formulations of the VOS correction may possibly be used in different densitometers.g. also Chapter 5): (1) “Overall level”: The user gives u c ( P̂ ) directly as input to the program. (2. the VOS correction and the installation correction) are accounted for in a single expression. such as e.Handbook of uncertainty calculations .28) is a relatively general functional relationship for on-line installed vibrating element gas densitometers.4 and 4.2. the indicated (uncorrected) density ρu has been used as the input quantity related to the densitometer reading instead of the periodic time τ. (2. [ISO/CD 15970.28).Pressure measurement As described in Section 1.3.USM Fiscal Gas Metering Station be specified at two levels (cf.4. which apply to the Solar- tron 7812 Gas Density Transducer [Solartron.4.28) æ ö è T øè 1 + ∆P P øè Z ø ê1 + ç K d ÷ ú d ê çè τcd ÷ø ú ë û in which all three corrections (the temperature correction. cf. Note that in Eq.: • If the user wants a “simple” and quick evaluation of the influence of u c ( P̂ ) on the expanded uncertainty of the gas metering station. the functional relationship of the corrected density measurement becomes é æ K ö2 ù ê1 + çç d ÷÷ ú ê τc úæ T öæ öæ Z d ö ρ = {ρu [1 + K18 (Td − Tc )] + K19 (Td − Tc )}ê è c ø 2 úç d ÷çç 1 ÷÷ç ÷ .5 Functional relationship . and not u( τˆ ) . 1999]. and covers any method of obtaining the uncertainty of the pressure measurement65. It is left to the user to calculate and document u c ( P̂ ) first. NFOGM.3. cf. . This option is completely general. Sections 3. as well as other densitometers of this type64. 65 The “overall level” option may be of interest in several cases. 1999]. the uncertainty of the pressure transmitter can in the program EMU . e.4.24)-(2. CMR (December 2001) 62 2. (2. 1999] (used in the example calculations in Chapter 4).g. 2. Eq.5 Corrected density By combining Eqs.USM fiscal gas metering stations NPD.2. (2.27). That has been done since u( ρˆ u ) is the uncertainty specified by the manufacturer [Solar- tron. as mentioned in Section 2. 6. cf. NPD and CMR [Ref. Used by permission.4 and Fig. 2. 2001] to be used in the present Handbook for the uncertainty evaluation example of Chapter 4 is the Rosemount 3051P Reference Class Pressure Transmitter [Rose- mount. Group.USM fiscal gas metering stations NPD. Publication No. The example pressure transmitter chosen by NFOGM. as the transmitter is calibrated and given a specific “accuracy” in the calibration data sheet.g. from more basic input uncertainties for the pressure transmitter. NFOGM.6 The Rosemount 3051P Reference Class Pressure Transmitter (example). i. The Rosemount 3051P is a widely used pressure transmitter when upgrading existing North Sea fiscal gas metering stations and when designing new metering stations. Table 2. However.USM Fiscal Gas Metering Station. 1999]. the pressure relative to the atmospheric pressure [barg]. Measurement principles of gauge pressure sensors and transmitters are described e. . It has been found convenient to base the user input to the program on the type of data which are typically specified for common pressure transmitters used in North Sea fiscal gas metering stations. 2. The following discussion concerns the “Detailed level”. no functional relationship is actually • In case the input used at the “detailed level” does not fit sufficiently well to the type of input data / uncertainties which are relevant for the pressure transmitter at hand. 2000]  2000 Rosemount Inc. 2000]. 00822-0100-1025 [Rose- mount. provided by the instrument manufacturer and calibration laboratory. in [ISO/CD 15970.Handbook of uncertainty calculations . Published in the Rosemount Comprehensive Product Catalog. This transmitter is also chosen for the layout of the presssure transmitter user input to the program EMU . CMR (December 2001) 63 (2) “Detailed level”: u c ( P̂ ) is calculated in the program. Fig. The pressure transmitter output is normally the overpressure (gauge pressure).e. the temperature transmitter and the Pt 100 element shall be calibrated as one system (cf.5]. cf.2. 1998a. 1999]).11). As for the pressure measurement.USM Fiscal Gas Metering Station be specified at two levels (cf. also [Dahl et al.: • If the user wants a “simple” and quick evaluation of the influence of u c ( T̂ ) on the expanded uncertainty of the gas metering station. A 3-wire tem- 66 Cf. The temperature loop considered here consists of a Pt 100 or 4-wire RTD element and a smart temperature transmitter. §5.Temperature measurement As described in Section 1.3. CMR (December 2001) 64 used here for calculation of the uncertainty of the pressure measurements (cf. • In case the input used at the “detailed level” does not fit sufficiently well to the type of input data / uncertainties which are relevant for the temperature element / transmitter at hand. and the uncertainty due to the functional relationship is included in the calibrated “accuracy” of the transmitter66.2.2). 67 The “overall level” option may be of interest in several cases.3..Handbook of uncertainty calculations .2. also Chapter 5): (1) “Overall level”: The user gives u c ( T̂ ) directly as input to the program. This option is completely general. (3. It is left to the user to calculate and document u c ( T̂ ) first. The functional relationship is only internal to the pressure transmitter. 2. or as one unit [NORSOK. the footnote accompanying Eq. By [NORSOK. provided by the instrument manufacturer and calibration laboratory The following discussion concerns the “Detailed level”.g. §5. the uncertainty of the temperature transmitter can in the program EMU .USM fiscal gas metering stations NPD.1 and Appendix C. Section C. and covers any method of obtaining the uncertainty of the gas temperature measurement67. from more basic input uncertainties for the temperature element / transmitter.3. 1998a. . NFOGM.5].6 Functional relationship . The Pt 100 temperature element is required as a minimum to be in accordance with EN 60751 tolerance A. installed either as two separate devices. (2) “Detailed level”: u c ( T̂ ) is calculated in the program. Section 2. it has been found convenient to base the user input to the program on the type of data which are typically specified for common temperature transmitters used in North Sea fiscal gas metering stations. such as e. 7. (3. 1999]. where the Pt 100 element is screwed directly into the transmitter. Published in the Rosemount Comprehensive Product Catalog. 00822-0100-1025 [Rosemount. no functional relationship is actually used here for calculation of the uncertainty of the temperature measurements (cf. the footnote accompanying Eq.. NPD and CMR [Ref.Handbook of uncertainty calculations . 2. i. 68 Cf. The functional relationship is only internal to the temperature element/transmitter. also [Dahl et al. the “digital accuracy” is used. 1999]). The measurement principle and functional relationship of RTDs is described e. However. The Rosemount 3144 transmitter is widely used in the North Sea when upgrading existing fiscal gas metering stations and when designing new metering stations. 2001] to be used in the present Handbook for the example uncertainty evaluation of Chap- ter 4 is the Rosemount 3144 Smart Temperature Transmitter [Rosemount.4 and Fig. This transmitter is also chosen for the layout of the temperature transmitter user input to the program EMU . The temperature transmitter chosen by NFOGM. Fig. 2000]  2000 Rosemount Inc.g. CMR (December 2001) 65 perature element may be used if the temperature element and transmitter are installed as one unit.USM fiscal gas metering stations NPD.e. 2. cf. and the uncertainty due to the functional relationship is included in the calibrated “accuracy” of the element/transmitter68. as the element/transmitter is calibrated and given a specific “accuracy” in the calibration data sheet.12). . 2000]. Used by permission. Group. in [ISO/CD 15970.USM Fiscal Gas Metering Station. NFOGM.7 The Rosemount 3144 Temperature Transmitter (example). Publication No. The signal is transferred from the temperature transmitter using a HART protocol. Table 2. 4) and the signal communication and flow computer calculatations (Section 3. Section 3. On the other hand. For convenience. q m and q e (Section 3. Z/Z0 and Hs. the axial mass flow rate.2). for completeness and traceability purposes.been included also in the present chapter. . 3. the detailed mathematical derivation of the uncertainty model is given in Appendix E. and the axial energy flow rate. (2. the axial volumetric flow rate at standard reference conditions.Handbook of uncertainty calculations . ρ.USM fiscal gas metering stations NPD. the USM field uncertainty (Section 3.USM Fiscal Gas Metering Station described in Chapter 5. NFOGM. Appendix E has been included as a documentation of the theoretical basis for the uncertainty model. the basic functional relationships are given by Eqs. . q e . CMR (December 2001) 66 3. The intention has been that the reader should be able to use the present Handbook without needing to read Appendix E. Q.when relevant . The main expressions of the model are summarized in the present chapter. Q. Hence. q m .1). to make Chapter 3 self-consistent so that it can be read independently. qV . definitions introduced in Appendix E have .1 USM gas metering station uncertainty For USM measurement of natural gas. the flow calibration uncertainties (Section 3.the expressions which are necessary for a full documentation of the calculations made in the Excel program have been included here. The model is developed in accordance with the terminology and procedures described in Appendix B. As outlined in Section 1.USM Fis- cal Gas Metering Station is given in Section 3.4. That means. qV .2)-(2.4).6. for the four measurands in question.5).3). respectively. and the Excel program EMU . as a basis for the uncertainty calculations of Chapter 4. UNCERTAINTY MODEL OF THE METERING STATION The present chapter summarizes the uncertainty model of the USM fiscal gas metering station. The chapter is organized as follows: The expressions for the uncertainty of the USM fiscal gas metering station are given first. The uncertainty model is based on the description and functional relationships for the metering station given in Chapter 2. T.9) and (2. a summary of the input uncertainties to be specified for the program EMU . for the axial volumetric flow rate at line conditions. Detailed expressions are then given for the gas measurement uncertainties (related to P. E qm and E qe are the relative combined standard uncertainties of qV . (3. Section B. where k is the coverage factor69 (cf. defined as u c ( q̂ v ) u c ( Q̂ ) u c ( q̂ m ) u c ( q̂ e ) E qv ≡ .6) EQ2 = E P2 + ET2 + E Z2 / Z 0 + E q2v (3. the relative expanded uncertainties are given as U ( q̂ v ) u c ( q̂ v ) =k = kE qv .7) E q2m = E ρ2 + E q2v (3. . NFOGM.USM fiscal gas metering stations NPD. (3. It can be shown (Appendix E) that these may be expressed as E q2v = E cal 2 + EUSM 2 + Ecomm 2 + E 2flocom (3.Handbook of uncertainty calculations . Q. q m and q e . EQ ≡ . Here.1) q̂ v q̂ v U ( Q̂ ) u c ( Q̂ ) =k = kEQ .9) 69 In the present Handbook k = 2 is used [NPD.3). Section B. E qe ≡ (3. E qv .8) E q2e = E H2 S + EQ2 (3. EQ .3) q̂ m q̂ m U ( q̂ e ) u c ( q̂ e ) =k = kE qe . (3. E qm ≡ .2) Q̂ Q̂ U ( q̂ m ) u c ( q̂ m ) =k = kE qm . (3.4) q̂ e q̂ e respectively. CMR (December 2001) 67 For these four measurands. corresponding to a 95 % confidence level (approximately) and a normal probability distribution (cf.5) q̂ v Q̂ q̂ m q̂ e respectively. 2001].3). cf. ET ≡ relative combined standard uncertainty of the line temperature estimate.3. j is assumed to be uncorrelated with the other quantities appearing in Eq. B. C or D. (3. E Z / Z 0 ≡ relative combined standard uncertainty of the estimate of the compressibility ratio between line and standard reference conditions. the following assumption have been made: • The deviation factor estimate K̂ dev . (3. ρ̂ . Note also that to obtain Eqs. CMR (December 2001) 68 respectively.g.9). in flow calibration and field operation (assembled in on term). The various terms involved in Eqs. EUSM ≡ relative combined standard uncertainty of the estimate q̂v . . due to signal communication between USM field electronics and flow computer (e. Note that Eqs. related to field operation of the USM. P̂ . due to flow computer calculations.6)-(3. E comm ≡ relative standard uncertainty of the estimate q̂v . EHS ≡ relative standard uncertainty of the superior (gross) calorific value estimate.6)-(3. Ĥ S .9). use of analog frequency or digital signal output). (2.9) as they stand are completely meter independent. and thus independent of the USM functional relationship (Formulations A.9) are: E cal ≡ relative combined standard uncertainty of the estimate q̂v .Handbook of uncertainty calculations . Eρ ≡ relative combined standard uncertainty of the density estimate. Ẑ Ẑ 0 . in flow calibration and field operation (assembled in on term). (3. Sec- tion 2. NFOGM. EP ≡ relative combined standard uncertainty of the line pressure estimate.USM fiscal gas metering stations NPD. T̂ . related to flow calibration of the USM. E flocom ≡ relative standard uncertainty of the estimate q̂v .6)-(3.2). 1.2. Sections 1. The relative combined standard uncertainties of these gas parameters are defined as u c ( P̂ ) u c ( T̂ ) u c ( Ẑ Ẑ 0 ) EP ≡ .2 and 3. Ĥ S . ET ≡ . where u c ( P̂ ) ≡ combined standard uncertainty of the line pressure estimate. . E Z / Z0 ≡ .2.Handbook of uncertainty calculations . Z.2 Gas measurement uncertainties Gas parameter uncertainties are involved in Eq.5. ρ̂ . and Z0). Ẑ Ẑ 0 . u( Ĥ S ) ≡ standard uncertainty of the superior (gross) calorific value estimate. As an example.9) for E qe (involving Hs). (3.5 and 5. Sections 2. T̂ .6)-(3. (3. (3.9) in Section 4. NFOGM. CMR (December 2001) 69 • Possible correlation between Ĥ S and Q̂ has been neglected. cf. (3. the expanded uncertainty of a USM fiscal gas metering station is evaluated according to Eqs. u c ( T̂ ) ≡ combined standard uncertainty of the line temperature estimate. P̂ T̂ Ẑ Ẑ 0 u ( ρˆ ) u ( Hˆ S ) Eρ ≡ c . u c ( ρˆ ) ≡ combined standard uncertainty of the line density estimate.USM fiscal gas metering stations NPD.7) for EQ (involving P. u c ( P̂ ) .1 Pressure measurement The combined standard uncertainty of the static gas pressure measurement. u c ( Ẑ Ẑ 0 ) ≡ combined standard uncertainty of the estimate of the compressibility ratio between line and standard reference conditions. EHS ≡ (3. Eq.3. cf. 3. T.USM Fiscal Gas Metering Station at two levels: “Overall level” and “Detailed level”.8) for E qm (involving ρ) and Eq. 2.10) ρˆ Hˆ S respectively.6. can be given as input to the program EMU . 3. P̂ .4. [Ref Group. At the “Detailed level”. 1999] (pp. only the “Detailed level” is discussed in the following. An alternative and more correct approach would have been to start from the functional relationship of the pressure measurement. The uncertainty model for the pressure transmitter is quite general. and in accordance with common company practice [Dahl et al. and derive the uncertainty model according to the recommendations of the GUM [ISO. (3. relative to 1 atm. u( P̂temp ) ≡ standard uncertainty of the effect of ambient gas temperature on the pressure transmitter. 2000] u( P̂transmitter ) ≡ standard uncertainty of the pressure transmitter. 1999]. 2001]. . CMR (December 2001) 70 As the “Overall level” is straightforward. the Rosemount 3051P Pressure Transmitter.g.11). 84-89). for the static gas pressure measurement.. such as mounting effects. u( P̂stability ) ≡ standard uncertainty of the stability of the pressure transmitter.5). u( P̂atm ) ≡ standard uncertainty of the atmospheric pressure. terminal-based linearity. 70 Here. u( P̂vibration ) ≡ standard uncertainty due to vibration effects on the pressure measurement. u( P̂RFI ) ≡ standard uncertainty due to radio-frequency interference (RFI) effects on the pressure transmitter. u( P̂misc ) ≡ standard uncertainty due to other (miscellaneous) effects on the pressure transmitter. including hysteresis. and similar transmitters (cf. NFOGM. etc. u c ( P̂ ) is assumed to be given as70 u c2 ( P̂ ) = u 2 ( P̂transmitter ) + u 2 ( P̂stability ) + u 2 ( P̂RFI ) + u 2 ( P̂temp ) (3. as a simplified approach. 1995a].11) + u 2 ( P̂atm ) + u 2 ( P̂vibration ) + u 2 ( P̂power ) + u 2 ( P̂misc ) where [Rosemount. ≡ 1. with respect to drift in readings over time. Section 2. u( P̂power ) ≡ standard uncertainty due to power supply effects on the pressure transmitter.USM fiscal gas metering stations NPD.. and applies to e. for change of ambient temperature relative to the temperature at calibration. repeatability and the standard uncertainty of the pressure calibration laboratory. The description is similar to that given in [Dahl et al.Handbook of uncertainty calculations . due to local meteorological effects. the sensitivity coefficients have been assumed to be equal to 1 throughout Eq.01325 bar. 2 Temperature measurement The combined standard uncertainty of the temperature measurement.2. and similar transmitters (cf. 71 In accordance with common company practice [Dahl et al. At the “Detailed level”. u( P̂stability ) . also Table 3.Handbook of uncertainty calculations . can be given as input to the program EMU .5. and document their traceability. CMR (December 2001) 71 u( P̂ ) needs to be traceable to national and international standards. NFOGM.. The description is similar to that given in [Dahl et al.transm ) ≡ standard uncertainty of the stability of the temperature transmitter. with respect to drift in the readings over time.USM fiscal gas metering stations NPD. 1995a].6). the uncertainty of the Rosemount 3051P pressure transmitter is evaluated in Section 4.. As the “Overall level” is straightforward. the Rosemount 3144 Temperature Transmitter used with a Pt 100 element. Note that this is a simplified approach. 79-83).transm ) ≡ standard uncertainty of the temperature element and temperature transmitter.12) + u 2 ( T̂vinbration ) + u 2 ( T̂ power ) + u 2 ( T̂cable ) + u 2 ( T̂misc ) where [Rosemount.g. An alternative and more correct approach would have been to start from the full functional relationship of the temperature measurement. u c ( T̂ ) is assumed to be given as71 u c2 ( T̂ ) = u 2 ( T̂elem . calibrated as a unit. and derive the uncertainty model according to the recommendations of the GUM [ISO.1.elem ) (3. u c ( T̂ ) .transm ) + u 2 ( T̂stab . and applies to e.6 and 5. u( P̂RFI ) . cf. u( T̂RFI ) ≡ standard uncertainty due to radio-frequency interference (RFI) effects on the temperature transmitter. 2001]. Sections 1.transm ) + u 2 ( T̂RFI ) + u 2 ( T̂temp ) + u 2 ( T̂stab . 3. The uncertainty model for the temperature element/transmitter is quite general.2.USM Fiscal Gas Metering Station at two levels: “Overall level” and “Detailed level”. 1999] (pp.1. (Cf. 2. u( P̂vibration ) and u( P̂power ) . u( T̂stab . (3.12). Section 2. 1999]. 2000] u( T̂elem . u( P̂temp ) . . the sensitivity coefficients have been assumed to be equal to 1 throughout Eq.) As an example.3. It is left to the calibration laboratory and the manufacturer to specify u( P̂transmitter ) . only the “Detailed level” is discussed in the following. [Ref Group. u( T̂RFI ) . the gas compressibility factor at line and standard reference conditions. (Cf. empirical equations of state can be used to calculate Z and Z0.elem ) ≡ standard uncertainty of the stability of the Pt 100 4-wire RTD temperature element. Input to these equations are e.2. u( T̂vibration ) . 1997]. 1994] and the GERG / ISO method [ISO 12213-3.) As an example. and document their traceability. u( T̂ power ) ≡ standard uncertainty due to power supply effects on the temperature transmitter.transm ) . u( T̂stab . such as the AGA- 8 (92) equation [AGA-8.elem ) . NFOGM.2. for change of gas temperature relative to the temperature at calibration.e. pressure. Z and Z0 are given manually. CMR (December 2001) 72 u( T̂temp ) ≡ standard uncertainty of the effect of temperature on the temperature transmitter.g. Instability may relate e. the uncertainty of the Rosemount 3144 temperature transmitter used with a Pt 100 element is evaluated in Section 4. u( T̂cable ) ≡ standard uncertainty of lead resistance effects on the temperature transmitter. u( T̂stab .Handbook of uncertainty calculations . 3. i. u( T̂ power ) and u( T̂cable ) .2. and mechanical stress during operation. u c ( T̂ ) needs to be traceable to national and international standards. u( T̂temp ) . . It is left to the calibration laboratory and the manufacturer to specify u( T̂elem .1. Various equations of state are available for such calculations.3 Gas compressibility factors Z and Z0 For natural gas. u( T̂vibration ) ≡ standard uncertainty due to vibration effects on the temperature transmitter. u( T̂misc ) ≡ standard uncertainty of other (miscellaneous) effects on the temperature transmitter. also Table 3. u( T̂stab .USM fiscal gas metering stations NPD.g. temperature and gas composition. as well as instability and hysteresis effects due to oxidation and moisture inside the encapsulation.transm ) . Table 2. respectively. In the program EMU .2.USM Fiscal Gas Metering Station. to drift during operation. cf. the model uncertainties of the Z-factor estimates Ẑ and Ẑ 0 have been assumed to be uncorrelated. some comments should be given. NFOGM. Ẑ and Ẑ 0 are here treated as being uncorrelated. and are taken to be mutually correlated. The argumentation is then as follows: In many cases Ẑ and Ẑ 0 relate to highly different pressures. the uncertainty of the model (equation of state) used for calculation of Ẑ and Ẑ 0 ).2).mod + E Z2 0. and negative at another pressure). However.USM fiscal gas metering stations NPD. and the uncertainty of the “basic data” underlying the equation of state). If Ẑ is calculated from the AGA-8 (92) equation [AGA-8. E Z . especially in cases where the USM is operated close to standard reference conditions (close to 1 atm. That means.Handbook of uncertainty calculations . A similar argumentation applies if the line temperature is significantly different from 15 oC. (3.13). and the analysis uncertainty (due to the inaccurate determination of the gas composition). two kinds of uncertainties are accounted for here (cf. (3. (2. E Z 0 .ana − E Z 0. CMR (December 2001) 73 For each of the estimates Ẑ and Ẑ 0 . and the uncertainty of the “basic data” underlying the equation of state).13) where E Z .: the error may be positive at one pressure. 1997]): the model uncertainty (i. (3. in a P-T range so narrow that the error of the equation may possibly be expected to be more systematic. 1997]. whereas the analysis uncertainties act as systematic effects. Since the equation of state is empirical.mod ≡ relative standard uncertainty of the estimate Ẑ due to model uncertainty (the uncertainty of the equation of state itself.mod + ( E Z .e. it may not be correct to assume that the error of the equation is systematic over the complete pressure range (e. this is clearly a reasonable and valid approach. That means. and variation in gas composition). [Tambo and Søgaard. ana ≡ relative standard uncertainty of the estimate Ẑ due to analysis uncertainty (measurement uncertainty of the gas chromatograph used to determine the line gas composition. and 15 oC). If Ẑ and Ẑ 0 are estimated using the same equation of state (such as e. this choice may be questionable. and Ẑ 0 is calculated from ISO 6976 [ISO.13). As a conservative approach thus. the AGA-8 (92) equation or the ISO 12213-3 method).ana ) 2 . That means.g. 72 In the derivation of Eq.g. E Z2 / Z 0 = E Z2 .mod ≡ relative standard uncertainty of the estimate Ẑ 0 due to model uncertainty (the uncertainty of the equation of state itself. Ẑ (at high pressure) and Ẑ 0 (at 1 atm.) are not necessarily correlated. 1994] or the ISO / GERG method [ISO 12213-3. In such cases E Z / Z0 may possibly be overestimated by using Eq. . 1995c] (as is often made). For conversion of qv to Q according to Eq.g. e. the model uncertainties are assumed to be mutually uncorrelated72. 3.0 % in the range P = 0-1400 bar and T = 143-473 K (-130 to +200 oC).mod and E Z 0.3 % in the range P = 0-172 bar and T = 213-393 K (-60 to +120 oC). ana ≡ relative standard uncertainty of the estimate Ẑ 0 due to analysis uncertainty (measurement uncertainty of the gas chromatograph used to determine the gas composition. Handbook of uncertainty calculations . 1995c]. • 0. In Options (2) or (3). E Z .mod is automatically determined from uncertainties specified in ISO 6976 [ISO.5 % in the range P = 0-700 bar and T = 143-473 K (-130 to +200 oC).mod is automatically determined from uncertainties specified in [AGA-8. 1994].3. . the relative model uncertainty of the Z-factor calculations have been specified to.0 ⋅ 10 −4 = 0. • E Z .3% 2 = 1. 1994].mod and E Z 0 . NFOGM.1: • 0. CMR (December 2001) 74 E Z 0.USM Fiscal Gas Metering Station. in the range 0-1400 bar and -130 to +200 oC. Option (2): E Z .2. or Option (4): E Z 0 . 1994]. Fig.15 % .mod are calculated from uncertainties specified for the AGA-8 (92) equation of state [AGA-8. For input of the relative standard uncertainties E Z . so only Options (2)-(4) are discussed in the following.1 Model uncertainties For the AGA-8 (1992) equation of state [AGA-8. 1994]. Option (3): E Z 0 . mod = 0.5 ⋅ 10 −3 = 0.mod is automatically determined from uncertainties specified in [AGA-8.5 % . That means: • E Z .mod or E Z 0 .mod in the program EMU .5 ⋅ 10 −3 = 0.mod = 0.1% 2 = 5.1 % in the range P = 0-120 bar and T = 265-335 K (-8 to +62 oC). in the range 0-172 bar and -60 to +120 oC. • E Z . mod = 1. • 1. • E Z .0% 2 = 5 ⋅ 10 −3 = 0.mod = 0.3).5% 2 = 2. the user of the program has four choices: Option (1): E Z . a 95 % level of confidence and a normal probability distribution (k = 2.05 % .USM fiscal gas metering stations NPD. in the range 0-120 bar and -8 to +62 oC. Option (1) is straigthforward. cf.25 % . 3. • 0.mod are given directly as input to the program. cf. by assuming that the expanded uncertainties specified in the AGA-8 report refer to a Type A evaluation of uncertainty. in the range 0-700 bar -130 to +200 oC. Section B. and variation in gas composition). this yields E Z 0. mod = 0. 8.015 2 % 3 ≈ 0.2 Analysis uncertainties With respect to the uncertainty contributions EZ. American Gas Association. and the Z-factor is calculated for each of these gas compositions. CMR (December 2001) 75 Fig. The suggested method for establishing these input uncertainty contributions involves numerical analysis (Monte Carlo type of simulations). The contributions to E Z 0 .2.ana and EZ0. The spread of the Z-factors (for example the standard deviation) calculated in this way.3. Ideally. ISO 6976 can be used for calculation of Ẑ 0 (Op- tion 4). variation limits can be established for each of the gas components as input to the Monte Carlo simulations. Compressibility Factors of Natural Gas and Other Related Hydrocarbon Gases [AGA-8.015 % (equation of state) [ISO.measurement and the natural variations of the gas composition (at least when an online GC is not used). 1995c]. a number of gas compositions within such variation limits for each gas component can then be established (where of course the sum of the gas components must add up to 100 %). both of these are to be entered manually by the user of the program.05 % (basic data) and 0. 3. 1994]. At standard reference conditions.ana . Next. a large number of gas compositions generated randomly .05 2 + 0. cf. This method is used in Section 4. will give information about the analysis uncertainty. 2001]. Handbook of uncertainty calculations . Based on knowledge of the metering station.3). NFOGM. with the permission of the copyright holder. Such limits must take into account the uncertainty of the GC .USM fiscal gas metering stations NPD.030 %. 3. By assuming a 100 % level of confidence and a rectangular probability distribution (k = 3 .2.1 Targeted uncertainty for natural gas compressibility factors using the detail characterization method.0522 % 3 ≈ 0. Section B.3. Copied from the AGA Report No. [Sakariassen.mod are then 0. 4)73. only the “Detailed level” is discussed in the following.∆Pd u 2 ( ∆P̂d ) + s ρ2 . Annex 2 and 3]. In practice. 1997. .P u c2 ( P̂ ) + u 2 ( ρˆ temp ) + u 2 ( ρˆ misc ) where u( ρˆ u ) ≡ standard uncertainty of the indicated (uncorrected) density estimate. It represents an extension of the uncertainty model for gas densitometers presented by [Tambo and Søgaard.T u c2 ( T̂ ) + s ρ2 .Handbook of uncertainty calculations .τ u 2 ( τˆ ) + s ρ2 . including the calibration laboratory uncertainty. and hysteresis. e. Other (less precise) methods may also be used.3.Tc u 2 ( T̂c ) + s ρ2 .4 and 5. the uncertainty model includes sensitivity coefficients derived from the function relationship. NFOGM. ρ̂ u . 2001]. relates mainly to the more detailed approach which has been used here with respect to the temperature.2. u( ρˆ rept ) ≡ standard uncertainty of the repeatability of the indicated (uncorrected) density estimate. E ρ .g. such as e. ρ̂ u . however. (2. Sections 1. Eq. The uncertainty model for the gas densitometer is quite general. Section 2.4 Density measurement The relative combined standard uncertainty of the gas density measurement. and should apply to any on-line installed vibrating-element densitometer.14) + s ρ2 .cc u 2 ( ĉc ) + s ρ2 .USM Fiscal Gas Metering Station at two levels: “Overall level” and “Detailed level”. 2. 3. Appendix G) u c2 ( ρˆ ) = s ρ2u u 2 ( ρˆ u ) + u 2 ( ρˆ rept ) + s ρ2 .USM fiscal gas metering stations NPD. the reading error during calibration. instead of taking them to be equal to 1.g based on the decision of the uncertainty of the molecular weight [Sakariassen. 73 The extension of the present densitometer uncertainty model in relation to the model presented in [Tambo and Søgaard. At the “Detailed level”.7. should be used in this calculation.Td u 2 ( T̂d ) + s ρ2 . the relative combined standard uncertainty E ρ is given as (cf.cd u 2 ( ĉ d ) (3.28). the Solartron 7812 gas density transducer (cf.K d u 2 ( K̂ d ) + s ρ2 . can be given as input to the program EMU . CMR (December 2001) 76 within the limits of each gas component. 1997]. As the “Overall level” is straightforward. a smaller number (less than 10) will often provide useful information about the analysis uncertainty. cf. VOS and installation corrections. Here. (3. ρ̂ u . (2. u( ĉc ) ≡ standard uncertainty of the calibration gas VOS estimate. u( T̂c ) ≡ standard uncertainty of the densitometer calibration temperature estimate.14) by a “lumped” term. . u( ρˆ temp ) ≡ standard uncertainty of the temperature correction factor for the density estimate. u( ∆P̂d ) ≡ standard uncertainty of assuming that P̂d = P̂ . (2. K̂ d . Eq.stability (drift. shift between calibrations75). . [Ref Group. accounting for miscellaneous uncertainty contribu- tions74.possible deposits on the vibrating element. 74 In accordance with common company practice [Dahl et al. 76 For guidelines with respect to uncertainty evaluation of reading error during measurement. due to possible deviation of gas pressure from densitometer to line conditions. Note that this is a simplified approach. An alternative and more correct approach would have been to start from the full functional relationship of the uncorrected density measurement ρ u . u( ĉ d ) ≡ standard uncertainty of the densitometer gas VOS estimate. ρ̂ (represents the model uncertainty of the temperature correction model used. τ̂ . T̂d . 1995a].. 1997. ĉc . . u( ρˆ misc ) .e. 75 For guidelines with respect to uncertainty evaluation of shift between calibrations. Annex 2]. ĉ d . Eq. u( ρˆ misc ) ≡ standard uncertainty of the indicated (uncorrected) density estimate. i. u( τˆ ) ≡ standard uncertainty of the periodic time estimate. cf. 2001]. with a weight (sensitivity coefficient) equal to one. NFOGM.possible liquid condensation on the vibrating element. . 1999].24)). Annex 2]. T̂c .23).Handbook of uncertainty calculations . and derive the influences of such miscellaneous uncertainty contributions on the total uncertainty according to the recommendations of the GUM [ISO. [Tambo and Søgaard.possible corrosion of the vibrating element. [Tambo and Søgaard.reading error during measurement (for digital display instruments)76.USM fiscal gas metering stations NPD. with derived sensitivity coefficients. CMR (December 2001) 77 u( T̂d ) ≡ standard uncertainty of the gas temperature estimate in the densitometer. u( K̂ d ) ≡ standard uncertainty of the VOS correction densitometer constant estimate. . such as due to: . 1997. cf. various “miscellaneous uncertainty contributions” listed in the text have been accounted for in the uncertainty model (Eq. neglecting possible pressure dependency in the regression curve. (3.15e) ë K̂ d + ( τˆĉc ) K̂ d2 + ( τˆĉ d ) 2 û K̂ d d 2 é 2 K̂ d2 2 K̂ d2 ù ρˆ s ρ .15c) é s ρ .Td = ê1 + [ T̂d ρˆ u K̂ 18 + K̂ 19 ] ù ρˆ ë [ ] ú .K =ê 2 − ú .15f) ë K̂ d + ( τˆĉc ) 2 K̂ d + ( τˆĉ d ) û τˆ 2 2 K̂ d2 ρˆ 2 K̂ d2 ρˆ s ρ .model uncertainty of the VOS correction model.23). (3. .cd = 2 .flow in the bypass density line. 5.Handbook of uncertainty calculations .τ = − ê 2 − 2 ú (3.22). . and so are also the estimates P̂ and ∆P̂d .15h) P̂ + ∆P̂d P̂ + ∆P̂d P̂ .Tc = − ê [ T̂c ρˆ u K̂ 18 + K̂ 19 ] ù ρˆ [ ] ë ρˆ u 1 + K̂ 18 ( T̂d − T̂c ) + K̂ 19 ( T̂d − T̂c ú ) û T̂c (3.15a) ρˆ s ρ .T = − . s ρ . s ρ . ρˆ u 1 + K̂ 18 ( T̂d − T̂c ) + K̂ 19 ( T̂d − T̂c ) û T̂d (3.15g) K̂ d + ( τˆĉc ) ĉc 2 K̂ d + ( τˆĉ d ) ĉ d 2 ρˆ ∆P̂d ρˆ s ρ .25).∆Pd = − . Eq. . cf.14) are defined as (cf. (3. (3.15b) T̂ é s ρ . Appendix G) s ρu = [ ρˆ 1 + K̂ 18 ( T̂d − T̂c ) ] [ ] ρˆ u 1 + K̂ 18 ( T̂d − T̂c ) + K̂ 19 ( T̂d − T̂c ) .cc =− 2 . . (2. Eq. (3. NFOGM. the estimates T̂ .self-induced heat.USM fiscal gas metering stations NPD.possible gas viscosity effects.P = . CMR (December 2001) 78 . T̂d and T̂c are assumed to be uncorrelated (since random effects contribute significantly to the uncertainty of the temperature measurement.15d) é 2 K̂ d2 2 K̂ d2 ù ρˆ s ρ . Table 4. (3. (2. . .mechanical (structural) vibrations on the gas line. The sensitivity coefficients appearing in Eq. In this model.8 and Fig.variations in power supply. or (2) measured directly by combustion (using a calorimeter). u( τˆ ) and u( K̂ d ) . (3. the uncertainty of cd is to be calculated from the expressions used to calculate cd and the input uncertainties to these. (3. Note that the uncertainty of the Z-factor correction part of the installation correction described in Section 2. u( ρˆ rept ) .4. u c ( P̂ ) is given by Eq.USM Fiscal Gas Metering Station to specify u( ĉ c ) . cd: the “USM method” and the “pressure/density method”.1. u c ( T̂ ) and u c ( T̂d ) are in the program set to be equal and are given by Eq. there are basically two contributions to the uncertainty of cd: (1) the uncertainty of the USM measurement of the line VOS.3) at least two methods in use today to obtain the VOS at the density transducer. the calorific value is usually either (1) calculated from the gas composition measured using GC analysis. It is left to the calibration laboratory and the manufacturer to specify u( ρˆ ρ u ) . u( ρˆ rept ) is referred to as “the standard uncertainty of type A component”.5 Calorific value As described in Section 2. u( Ẑ d Ẑ ) . Annex 2 and 3].USM fiscal gas metering stations NPD. In this approach. 3. . - u( ĉ d ) is to be calculated and given by the user of the program. the Solartron 7812 Gas Density Transducer is evaluated in Section 4. As an example of density uncertainty evaluation. the uncertainty model is independent of the particular method used to estimate cd in the metering station. Section C.4. u( T̂c ) .4. and document their traceability. cf. 2001].2. and has thus been neglected here. Table 2. to be obtained by determining the density (at stable conditions) at least 10 times and deriving the standard deviation of the mean. and (2) the deviation of the line VOS from the VOS at the densitometer. CMR (December 2001) 79 respectively. Table 3. 1997. For the “pressure/density method”. For the “USM method”.11).Handbook of uncertainty calculations . u c ( ρˆ ) needs to be traceable to national and international standards.3 ([NPD.1).12). In [Tambo and Søgaard. It is left to the user of the program EMU . there are (as described in Section 2. u( ρˆ temp ) . u( ∆P̂d ) and u( ρˆ misc ) .2.4. is negligible (Appendix G). Evaluation of u( ĉ d ) according to these (or other) methods is not a part of the present Handbook.4. With respect to u( ĉ d ) . u( ĉ d ) . NFOGM. cf. j + E K2 dev .USM fiscal gas metering stations NPD. and Table 2. (3.j . Table 1. i. is uncorrellated to the uncertainty of the volumetric flow rate at standard reference conditions.9). the uncertainties of the actual calorimeter and measurement method will contribute.1). (2. j.4). at the “overall level”). (3.3 (method 5). j.5 and Section 5.8. NFOGM. cf. E rept . ISO 6976 is used [ISO. j . Appendix E) 2 E cal ≡ E q2ref . E K dev . j . cf. j + E rept 2 . including reproducability). Q̂ . and uncertainties in the calorific value can be due to uncertainties in the gas composition and model uncertainties of the ISO 6976 procedure.16) where E qref .g. it has been assumed that the uncertainty of the calorific value estimate. Eq. K̂ dev . including repeatability of the flow laboratory reference measurement).e. as a simplification. CMR (December 2001) 80 In the first case. cf. Further. M (due to random transit time effects on the N acoustic paths of the USM. the calorific value is thus implicitly assumed to be measured using a method which is uncorrelated with gas chromatography (such as e. j ≡ relative standard uncertainty of the deviation factor estimate. q̂ ref . j ≡ relative standard uncertainty of the reference measurement. E H S . 2000]. 2001]. cf. (3. Table 2. In the second case. Such an analysis would be outside the scope of work for the Handbook [Lunde.3 Flow calibration uncertainty The relative combined standard uncertainty of the flow calibration which appears in Eq. . relative standard deviation) of the USM flow calibration measurement (volumetric flow rate). The user of the program is to specify the relative standard uncertainty E H S directly as input to the program (i. M (representing the uncertainty of the flow calibration laboratory. 3. at test flow rate no. j = 1.Handbook of uncertainty calculations . As the conversion from line conditions to standard reference conditions for the volumetric flow is assumed here to be carried out using a gas chromatograph (calculation of Z and Z0. ….6) is given as (cf. In the present Handbook no detailed analysis is carried out for the relative standard uncertainty of the calorific value. [Ref Group. a footnote accompanying Eq.e. …. Ĥ s . 1995c]. a calorimeter). at test flow rate no. j ≡ repeatability (relative standard uncertainty. j = 1. cf.j.9.3. NFOGM. E qref .e. and after correction of the USM measurement data using K (cf. Sections 4. As an example. at test flow rate no.3. M (representing the uncertainty of the flow calibration laboratory. j ) u( K̂ dev . where u( q̂ ref . E rept .17) q̂ ref . j ) . at test flow rate no. u( K̂ dev . j. will serve as an input uncertainty to the program EMU . j ) ≡ standard uncertainty of the deviation factor estimate. j ≡ . E cal is evaluated in Section 4. j.Handbook of uncertainty calculations . It is calculated by the program EMU . CMR (December 2001) 81 The relative standard uncertainties in question here are defined as rept u( q̂ ref . j ) ≡ standard uncertainty of the reference measurement. i. u( K̂ dev . K. It is left to the flow calibration laboratory to specify E qref . K̂ dev . M (due to random transit time effects on the N acoustic paths of the USM. E qref . standard deviation) of the rept u( q̂USM USM flow calibration measurement (volumetric flow rate). the uncertainty of the resulting deviation curve is to be determined.1). j respectively.USM Fiscal Gas Metering Station. j K̂ dev .3. …. j and document its traceability to national and international standards. j ) ≡ repeatability (standard uncertainty.3.2 and Fig. q̂ ref . 3. j . j ≡ .2 Deviation factor With respect to the deviation factor Kdev. Section 2.4. (3. j = 1. including repeatability of the flow laboratory reference measurement). 2.3.USM Fiscal Gas Metering Station on basis of the .1 Flow calibration laboratory The relative standard uncertainty of the flow calibration laboratory. j = 1. j ≡ . the task here is the following: For a given correction factor. j . j is discussed in Section 4. and Table 3. E K dev . This uncertainty is the standard uncertainty of the deviation factor. and is to be given by the program user. j .1. As an example. including reproducability).USM fiscal gas metering stations NPD.1 and 5.6. …. . j q̂USM . j ) u( q̂USM .j ) E qref . 3. (3. Tables 4. j is evaluated in Section 4. j = 1. E K dev . Eq.4). j respectively. (2.11). which ranges from 1 to DevC . and is to be given by the program user. j ) 1 DevC .6. at the M test flow rates. 3. accounts for random transit time effects on the N acoustic paths of the USM in flow calibration (standard deviation of the spread).3. j u( K̂ dev . CMR (December 2001) 82 deviation data DevC . respectively (cf. Table 1.9. …. The repeatability in field operation. j = 1. Sections 4. will serve as an input uncertainty to the program EMU . As an example. j ) is determined by the “span” of the deviation factor K̂ dev . cf.6. cf. by different symbols.18) 3 K̂ dev . cf. It is left to the USM manufacturer to specify and document the deviation data DevC . E K dev .9. E rept . M. Sections 4. j (k = 3 . Table 1. cf. cf. since in general they may account for different effects and therefore may be different in magnitude. the repeatability in flow calibration and in field operation have both been accounted for. j ) = . cf. j = = . E rept .2. E rept . E rept . u( K̂ dev ..3 USM repeatability in flow calibration The relative combined standard uncertainty of the USM repeatability in flow calibration. j .4. Section B. ….2 and 5.USM fiscal gas metering stations NPD. etc. By assuming a Type A uncertainty.Handbook of uncertainty calculations .USM Fiscal Gas Metering Station.2. at the M test flow rates for which flow calibration has been made (“calibration points”). which may also include effects not present in flow calibration. j . j . j . j . at test flow rate no.3). j and E rept .4.3.12 and 6. The two uncorrelated repeatability terms are not assembled into one term. M (cf. . j = 1. Eq. j u( K̂ dev .10)). as explained in the following: The repeatability in flow calibration. a 100 % confidence level and a rectangular probability distribution within the range ± DevC . NFOGM. j. cf. accounts for random transit time effects on the N acoustic paths of the USM in field operation (standard deviation of the spread). Note that in the uncertainty model of the USM gas metering station. such as incoherent noise from pressure reduction valves (PRV noise).3. above and Section 3. the standard uncertainty and the relative standard uncertainty of the deviation factor are here calculated as DevC . …. and the repeatability of the flow laboratory itself. (2. j 3 K̂ dev . M. and Table 3. j .3 and 5. and Table 3.3. ∆ is further given as (cf.19) (and elsewhere) denotes that only deviations relative to the conditions at the flow calibration are to be accounted for in the expressions involving this subscript. (3. EUSM .∆ + E I . 3.∆ + E misc 2 . caused by possible deviation in pressure and/or tem- .3. NFOGM. E misc ≡ relative standard uncertainty of miscellaneous effects on the USM field measurement q̂v . etc.e. at the flow rate in question (due to random transit time effects on the N acoustic paths). inaccuracy of the USM functional relationship (the under- laying mathematical model). j .4. which are not eliminated by flow calibration. related to field operation of the USM (due to change of conditions from flow calibration to field operation). Table 6. φ i .6). ŷ i . are not to be included in these expressions. (3. and which are not covered by other uncertainty terms accounted for here (e. is given as (cf.). uncertainty of the meter body inner radius. relative standard deviation) of the USM measurement in field operation. appearing in Eq.4 USM field uncertainty The relative combined standard uncertainty of the USM in field operation.3.∆ ≡ E body . uncertainty contributions which are practically eliminated at flow calibration. and the inclination angles of the N acoustic paths.Handbook of uncertainty calculations .∆ . That means. E rept . cf. j is discussed in Section 4. The subscript “∆” used in Eq. R̂ . The term EUSM .19) where E rept ≡ repeatability (relative standard uncertainty. q̂v .∆ ≡ relative combined standard uncertainty of the estimate q̂v . As an example. Appendix E) .g. due to possible uncorrected change of the USM meter body dimensions from flow calibration to field operation.∆ + E time . …. That is. CMR (December 2001) 83 It is left to the USM manufacturer to specify and document E rept .∆ ≡ relative combined standard uncertainty of the estimate q̂v .USM fiscal gas metering stations NPD. the lateral chord positions of the N acoustic paths. 2 2 2 2 EUSM (3. N.20) where Ebody . i. (3. i = 1. Appendix E) 2 EUSM ≡ E rept 2 + EUSM 2 . ∆ + Echord . CMR (December 2001) 84 perature between flow calibration and field operation. Only those systematic meter body effects which are not corrected for in the USM or not eliminated by flow calibration are to be accounted for in Ebody .20). (3. caused e. Only those systematic transit time effects which are not corrected for in the USM or not eliminated by flow calibration are to be accounted for in Etime . Section 2. and the USM integration method uncertainty (accounting for installation condition effects) (Section 3. N. etc). As an example.∆ .21) where E rad . B.20) as they stand here are completely meter independent. (3.3.∆ . is given as (cf.∆ . caused by possible deviation in pressure and/or temperature between flow calibration and field operation.1 USM meter body uncertainty The relative combined standard uncertainty related to meter body dimensional changes which appears in Eq.4.∆ ≡ relative standard uncertainty of the USM integration method due to possible change of installation conditions from flow calibration to field operation. Note that Eqs. EUSM is evaluated in Section 4. The following subsections address in more detail the contributions to the USM meter body uncertainty (Section 3.3).6. due to uncertainty of the lateral chord positions of the N acoustic paths. transducer ageing. 3.4.4. by possible deviation in conditions from flow calibration to field operation (P. the USM transit time uncertainties (Section 3. …. T.4. C or D. i = . ŷ i . t̂ 1i and t̂ 2i .g.4. (3. Appendix E) Ebody . E chord .∆ ≡ relative combined standard uncertainty of the estimate q̂v . Etime . cf.2).Handbook of uncertainty calculations . E I .∆ + E angle .19)-(3.2).∆ ≡ relative combined standard uncertainty of the estimate q̂v . R̂ . and thus independent of the choice of USM functional relationship (Formulations A. due to uncertainty of the meter body inner radius.1). i = 1.∆ ≡ E rad . due to possible uncorrected systematic effects on the transit times of the N acoustic paths.USM fiscal gas metering stations NPD. NFOGM. transducer deposits.∆ ≡ relative combined standard uncertainty of the estimate q̂v . N.22) N E chord . s *yi and sφ*i are the relative (non-dimensional) sensitivity coefficients for the sensitivity of the estimate q̂USM to the input estimates R̂ .∆ ≡ relative combined standard uncertainty of the estimate q̂v .Handbook of uncertainty calculations . (3.∆ ≡ å sign( φˆ i )sφ*i Eφi . N. ŷ i and φ̂ i . s R* . due to possible deviation in pressure and/or temperature between flow calibration and field operation. 2000a] 1 N æ 1 ö ç2 + ÷. φ i .26) Q̂ 1 − (ŷ R̂ ) yi 2 i . given as [Lunde et al. E yi .∆ . Here.∆ ≡ s*R E R . caused by possible deviation in pressure and/or temperature between flow calibration and field operation. caused by possible deviation in pressure and/or temperature between flow calibration and field operation. (3. due to possible deviation in pressure and/or temperature between flow calibration and field operation. ŷ i . s *R = å Q̂ (3. Appendix E) E rad .. (3.∆ ≡ å sign( ŷ i )s *yi E yi . E angle . These relative combined standard uncertainties are defined as (cf. …. respectively. due to uncertainty of the inclination angles of the N acoustic paths. ŷ i . NFOGM. due to possible deviation in pressure and/or temperature between flow calibration and field operation.23) i =1 N E angle . R̂ .∆ .25) Q̂ i =1 è i ç ( 2 ÷ 1 − ŷ i R̂ ø ) s = − sign( ŷ i * ) Q̂i (ŷ R̂ ) i 2 . i.∆ ≡ relative combined standard uncertainty of the inclination angle of acoustic path no. where E R .USM fiscal gas metering stations NPD.24) i =1 respectively. 1997. Eφi .∆ ≡ relative combined standard uncertainty of the meter body inner radius. CMR (December 2001) 85 1. i = 1. i. ….∆ .∆ ≡ relative combined standard uncertainty of the lateral chord position of acoustic path no. (3. i.27) Q̂ tan 2φˆ i respectively. the ratio Q̂i Q̂ is for convenience set equal to (calculated as) the weight factor of path no. That means. (3.USM fiscal gas metering stations NPD.29) E yi2 . and out-of-roundness of R̂0 . whereas for more realistic (non-uniform) profiles it represents an approximation. are eliminated at flow calibration (cf. Consequently. 2000a] E R2 . 1997. The subscript “∆” used in Eqs.. The main contributions to E rad . i = 1.30) B sin 2φˆi 0 Eφi .24) are given as [Lunde et al. the relative uncertainty terms involved at the right-hand side of Eqs.i + 1 ) R̂ 2 − ŷ i2 ( t̂ 1i − t̂ 2i ) Q̂ ≡ å Q̂i . (3.28) i =1 P0T̂Ẑ t̂ t̂ sin 2φˆ 1i 2 i i have been used.∆ and E angle . N.∆ = E KP 2 + E KT 2 .∆ = E KP .∆ . (3. (3. where for convenience in notation. (3.22)-(3.22)-(3. Q̂i ≡ 7200πR̂ 2 wi . without much loss of generality. Table 1. E chord . the definititions N P̂T0 Ẑ 0 ( N refl .∆ = E KP 2 + E KT 2 . wi.Handbook of uncertainty calculations . …. and are not to be included in these expressions. uncertainty contributions such as measurement uncertainties of R̂0 . ŷ i0 and φ̂ i0 .4). (3.USM Fiscal Gas Metering Station. where the definitions .24) denotes that only deviations relative to the conditions at the flow calibration (with respect to pressure and temperature) are to be accounted for in these expressions.∆ are thus due to possible change of pressure and temperature from flow calibration conditions to field operation (line) conditions.31) 2φˆi 0 respectively. (3. NFOGM. Note that in the program EMU . CMR (December 2001) 86 Q̂i 2 φˆ i sφ*i = − . For the uniform flow velocity profile this is an exact identity. E KT ≡ (3. u c ( ∆P̂cal ) ≡ combined standard uncertainty of the estimate of the difference in gas pressure between line and flow calibration conditions. Section B. CMR (December 2001) 87 u c ( K̂ P ) u c ( K̂ T ) E KP ≡ .34) where u( βˆ ) ≡ standard uncertainty of the estimate of the coefficient of linear pressure expansion for the meter body material. the radial pressure correction factor for the USM meter body is given from Eq. ∆P̂cal .g. yi.19)). (3.33) where Pcal is the gas pressure at flow calibration conditions. By assuming a Type B uncertainty.USM fiscal gas metering stations NPD. a 100 % confidence level and a rectangular probability distribution within the range ± ∆P̂cal (k = 3 .32) K̂ P K̂ T have been used and u c ( K̂ P ) ≡ combined standard uncertainty of the estimate of the radial pressure correction factor for the USM meter body. Meter body pressure effects For deviation in gas pressure between line and flow calibration conditions. ∆Pcal = P − Pcal . (3. This includes propagation of uncertainties of the input quantities to the β model used (e. (3.6). u c ( ∆P̂cal ) is determined by the “span” of the pressure difference. φi. and the uncertainty of the β model itself (cf. These uncertainty contributions are evaluated in the following. In situations where no pressure correction of the dimensional quantities of the meter body (R. Li and xi) is used by the the USM manufacturer.Handbook of uncertainty calculations . two options need to be discussed: (1) Meter body pressure correction not used. cf. β̂ . (2. (2. Table 2. K̂ P .17) as K P = 1 + β∆Pcal . For calculation of u c ( ∆P̂cal ) . the standard uncertainty of the pressure difference is thus calculated as . u c ( K̂ T ) ≡ combined standard uncertainty of the estimate of the radial tem- erature correction factor for the USM meter body. From Eq. Eq.33) one has u c2 ( K̂ P ) = ( ∆P̂cal ) 2 u 2 ( βˆ ) + βˆ 2 u c2 ( ∆P̂cal ) . NFOGM. K̂ T .3). equal to ∆P̂cal . given as u c2 ( ∆P̂cal ) = u c2 ( P̂ ) + u c2 ( P̂cal ) ≈ 2u c2 ( P̂ ) . NFOGM.11). equal to ∆T̂cal . (3. ∆T̂cal . By assuming a Type B uncertainty.USM fiscal gas metering stations NPD. In situations where no temperature correction of the dimensional quantities of the meter body (R.16) as K T = 1 + α∆Tcal . φi. In situations where pressure correction of the dimensional quantities of the meter body (R.Handbook of uncertainty calculations . yi. u c ( ∆P̂cal ) is determined by the measurement uncertainty of the pressure difference estimate ∆P̂cal itself. φi.38) where u( αˆ ) ≡ standard uncertainty of the estimate of the coefficient of linear temperature expansion for the meter body material. α̂ . Meter body temperature effects For deviation in gas temperature between line and flow calibration conditions. Li and xi) is used by the USM manufacturer. (3. a 100 % confidence level and a rectangular probability distribution . (3.37) where Tcal is the gas temperature at flow calibration conditions. For calculation of u c ( ∆T̂cal ) . (3.36) where the pressure measurements in the field and at the flow calibration have been assumed to be uncorrelated. (2. u c ( ∆T̂cal ) is determined by the “span” of the temperature difference. the radial temperature correction factor of the USM meter body is given from Eq.35) 3 (2) Meter body pressure correction is used. ∆Tcal = T − Tcal . u c ( P̂ ) is given by Eq. and their combined standard uncertainties approximately equal. u c ( ∆T̂cal ) ≡ combined standard uncertainty of the estimate of the difference in gas temperature between line and flow calibration conditions. (3. CMR (December 2001) 88 ∆P̂cal u c ( ∆P̂cal ) = .37) yields u c2 ( K̂ T ) = ( ∆T̂cal )2 u 2 ( αˆ ) + αˆ 2 u c2 ( ∆T̂cal ) . two options need to be discussed: (1) Meter body temperature correction not used. Li and xi) is used by the the USM manufacturer. yi. Eq. (3. (3. given as u c2 ( ∆T̂cal ) = u c2 ( T̂ ) + u c2 ( T̂cal ) ≈ 2u c2 ( T̂ ) .3).Handbook of uncertainty calculations . As an example. (3.20).12).4.U ) 2 . (3. The USM manufacturer must specify whether temperature and pressure correction are made or not. Li and xi) is used by the USM manufacturer. cf. yi.39) 3 (2) Meter body temperature correction is used. In situations where temperature correction of the dimensional quantities of the meter body (R. and to systematic transit time effects (representing those parts of the systematic change of transit times from flow calibration to line conditions which are not corrected for in the USM). E rept . and has the choice between the two Options (1) and (2) for calculation of u c ( ∆T̂cal ) and u c ( ∆P̂cal ) . 3.19) and (3.40) where the temperature measurements in the field and at the flow calibration have been assumed to be uncorrelated. φi. the user specifies the relative standard uncertainties u( αˆ ) αˆ and u( βˆ ) βˆ .USM fiscal gas metering stations NPD. the meter body uncertainty is evaluated in Section 4. Etime .∆ .4. In the Excel program EMU . (3.2 USM transit time uncertainties Uncertainties related to transit times measured by the USM are involved in Eqs. u c ( T̂ ) is given by Eq. the standard uncertainty of the temperature difference is calculated as ∆T̂cal u c ( ∆T̂cal ) = . u c ( ∆T̂cal ) is determined by the measurement uncertainty of the temperature difference estimate ∆T̂cal itself. Section B.USM Fiscal Gas Metering Station. and their combined standard uncertainties approximately equal. given as (cf.2.41) i =1 . CMR (December 2001) 89 within the range ± ∆T̂cal (k = 3 . Appendix E) N 2 E rept ≡ 2å ( s *t 1i Et 1i . NFOGM. (3. These are the relative combined standard uncertainties related to random transit time effects (representing the USM repeatability in field operation). is given by Eq. Handbook of uncertainty calculations .41). t̂ 1i .C . 2000a] (cf.42) i =1 respectively. i. (3.2.C ≡ relative standard uncertainty of uncorrected systematic transit time effects on downstream propagation of acoustic path no.USM fiscal gas metering stations NPD. NFOGM. t̂ 2i . given as [Lunde et al. possible random effects in signal detection/processing (e. Et∆2 i . (3. CMR (December 2001) 90 ( ) N Etime . i. which is important especially for the magnitude of the term Etime . Here. s t*1i and st*2i are the relative (non-dimensional) sensitivity coefficients for the sensitivity of the estimate Q̂ to the input estimates t̂ 1i and t̂ 2i . due to possible deviation in pressure and/or temperature between flow calibration and field operation.44) Q̂ t̂ 1i − t̂ 2 i respectively. respectively. noise (coherent and incoherent).C + s t*2i Et∆2i . Here. such as turbulence. Et∆1i .∆ given by Eq.∆ ≡ å st*1i Et∆1i ..C ≡ relative standard uncertainty of uncorrected systematic transit time effects on upstream propagation of acoustic path no. electronics stability (possible random effects). (3.4. representing the USM repeatability in field operation. (3. The coefficients s t*1i and st*2i have opposite sign and are almost equal in magnitude. and power supply variations. (3. where Et 1i .42). 3.g. Appendix E) Q̂i t̂ 2 i st*1i = .1 USM repeatability in field operation (random transit time effects) The relative combined standard uncertainty related to random transit time effects. but not exactly equal. erronous signal period identification). finite clock resolution.43) Q̂ t̂ 1i − t̂ 2i Q̂i t̂ 1i st*2i = − .U ≡ relative standard uncertainty of those contributions to the transit time estimates t̂ 1i and t̂ 2 i which are uncorrelated with respect to upstream and downstream propagation. due to possible deviation in pressure and/or temperature between flow calibration and field operation. the definition . 1997. g. randomly oc- curing erronous signal period identification. • Electronics stability (possible random effects). and (2) noise and turbulence influence. NFOGM. at a given flow rate. • Incoherent noise (from pressure reduction valves (PRV). . electromagnetic noise (RFI). Table 1. Such effects are typically uncorrelated.g. electromagnetic cross-talk. where u( t̂ 1random i ) ≡ standard uncertainty (i. • Finite clock resolution.). the repeatability in flow calibration and in field operation have both been accounted for.U ≡ i (3. and will contribute to u( t̂ 1random i ) by a root- sum-square calculation (if that needs to be calculated).3. by different symbols. Two options are discussed: (1) Specification of E rept directly. j and E rept . CMR (December 2001) 91 u( t̂ 1random ) Et 1i . user input at different levels may be convenient and useful. • Possible random effects in signal detection/processing (e.4): • Turbulence effects (from temperature and velocity fluctuations. E rept . (1) insufficiently robust signal detection solution. u( t̂ 1random i ) may be taken as the standard uncertainty of the spread of transit times. especially at high flow velocities). Note that in the uncertainty model of the USM gas metering station. respectively. This is motivated as described in Section 3. giving alternating measured transit times over the time averaging period.Handbook of uncertainty calculations . Appendix E). by one or several half periods). possible other acoustic interference. at a specific flow rate. pipe vibrations. etc. such as (cf. standard deviation) due to in-field random effects on the transit time t̂ 1i (and t̂ 2i ).3. acoustic reflections in meter body.). In this case the relative standard uncertainty of the repeatability of the in-field USM flow rate reading ( E rept ) is given directly.45) t̂ 1i has been used (cf. Such random effects could occur in case of e.e. • Coherent noise (from acoustic cross-talk (through meter body). and after possible time averaging (representing the repeatability of the transit times). • Power supply variations. etc.USM fiscal gas metering stations NPD. For user input / calculation of E rept . In practice. g. the repeatability information available from USM manufacturers is related directly to E rept . however. E rept may have a minimum at intermediate flow rates. from information which may preferably be specified by the USM manu- facturer77. In this context. the standard deviation) is information which should be readily available in USM flow computers today. (3. Table 6. which are relatively constant with respect to flow rate. in case of option (2). Today. Eq. i. due to the increasing relative sensitivity coefficient s1*i . (3.45).43). with an increase both at the low and high flow rates. In the lower end of the flow rate range.e. for different flow rates. turbulence effects may be small. Eqs.1). This will give an uncertainty contribution to the USM measurement ( E rept ) which increases by increasing flow rate. i. CMR (December 2001) 92 from information specified by the USM manufacturer. 77 The repeatability of the transit times (e. turbulence effects. and where E rept in practice often is given to be a constant (cf. (2) Specification of u( t̂ 1random i ) . flow noise and possible PRV noise become more dominant so that u( t̂ 1random i ) may increase by increasing flow rate. and calculation of E rept . In spite of such “complications”.Handbook of uncertainty calculations .USM fiscal gas metering stations NPD. so that u( t̂ 1random i ) is normally dominated by the background noise and/or time detection uncertainties. thus.41).e. both Options (1) and (2) in the program EMU - USM Fiscal Gas Metering Station may be useful. if u( t̂ 1random i ) is the input specified. thus.1). In this case the repeatability of the transit times of the in-field USM flow rate reading ( u( t̂ 1random i ) ) is given. this will give a uncertainty contribution to the USM measurement from the USM repeatability which is constant over the flow velocity range.45) are not used in this case. This result is simplified. At lower flow rates. and if E rept is given to be constant over the flow velocity range (as a relevant example. it should be noted that in case of option (1). In practice. as a user’s choice. At higher flow rates. On the other hand. on basis of USM manufacturer information provided today. this will give an uncertainty contribution to the USM measurement ( E rept ) which increases at low velocities. and may be incorrect. cf. if E rept is the input uncertainty specified. (3. cf. Table 6.43) and (3. E rept is then calculated using Eqs. (3. . the situation is more complex. NFOGM.41) and (3. . Consequently.4.4. is given by Eq. due to possible deviation in conditions between flow calibration and field operation. In both cases. However. As an example. reciprocity. due to possible deviation in conditions between flow calibration and field operation. where u( t̂ 1systematic i ) ≡ standard uncertainty of uncorrected systematic effects on the upstream transit time. drift. Chapter 6)78.10. in the program. Handbook of uncertainty calculations . such as signal-to- noise ratio (coherent and incoherent noise). may be available from USM manufacturers (cf. the USM manufacturer might provide information on the variation of E rept with flow rate. NFOGM. u( t̂ 2systematic i ) ≡ standard uncertainty of uncorrected systematic effects on the downstream transit time. information about u( t̂ 1random i ) . the variation of E rept with flow rate would be described by the present uncertainty model. Et∆2 i .13).C ≡ i . 1997].4): • Cable/electronics/transducer/diffraction time delay. ambient temperature effects. and its variation with flow rate. it is a hope for the future that also information on u( t̂ 1random i ) . Such systematic effects may be due to (cf. t̂ 1i . Table 1. 5. 78 In fact. effects of possible transducer exchange).C ≡ i . u( t̂ 1random i ) might actually be calculated from more basic input. E rept or u( t̂ 1random i ) serve as optional input uncertainties. 3. etc. the USM repeatability in field operation is evaluated in Section 4. Sections 4. t̂ 2i . [Lunde et al. • Possible ∆t-correction (line pressure and temperature effects.1.4. (3. drift.2 Systematic transit time effects The relative combined standard uncertainty related to those parts of the systematic change of transit times from flow calibration to line conditions which are not corrected for in the USM.1 and 5. as possible additional information to USM manufacturer repeatability. including finite-beam effects (due to line pressure and temperature effects.46) t̂ 1i t̂ 2 i have been used. CMR (December 2001) 93 However. ambient temperature effects. this is not considered here. In this expression the definitions u( t̂ 1systematic ) u( t̂ 2systematic ) Et∆1i .2.USM fiscal gas metering stations NPD. Alternatively.1 (Figs. cf. (3. clock resolution. It is left to the USM manufacturer to specify and document E rept or u( t̂ 1random i ) .12 and 5..42). effects of possible transducer exchange). • Sound refraction (flow profile effects (“ray bending”). cf. etc. 5. Such systematic effects could occur in case of (1) insufficiently robust signal detection solution. the uncertainty due to systematic transit time effects is discussed in Section 4. by one or several half periods). • Possible transducer deposits (lubricant oil. 3. due to e. Note that in Eqs. NFOGM. (3.g. Sections 4.2. for one or several transducers). As an example. (3.).10. and by changed installation conditions) [Frøysa et al. and is here defined as ∆ u( q̂USM .g. the relative standard uncertainty E I . or (2) significantly changed pulse form. u( t̂ 1systematic i ) and u( t̂ 2systematic i ) serve as input uncertainties.47) q̂USM where . wax.42) and (3.g.20). giving systematically shifted measured transit times over the time averaging period.USM fiscal gas metering stations NPD. systematic erronous signal period identification.3.∆ accounts for installation effects on the uncertainty of the USM fiscal gas metering station. and will contribute to each of u( t̂ 1systematic i ) and u( t̂ 2systematic i ) by a root-sum-square calculation. sound velocity effects). and it is therefore left to the USM manufacturer to specify and document u( t̂ 1systematic i ) and u( t̂ 2systematic i ) .17).2 (Fig. especially at high flow velocities.4. • Possible cavity time delay correction (e.I ) E I .∆ ≡ .4 and 5. (c) transducer exchange (different transducer properties relative to original transducers). the symbol “∆” denotes that only deviations relative to the conditions at the flow calibration are to be accounted for in these expressions.USM Fiscal Gas Metering Station.Handbook of uncertainty calculations . 2001]. The individual effects described above are typically uncorrelated.. (a) noise and turbulence influence. In the program EMU .4.3 USM integration uncertainties (installation conditions) In Eq. grease. (3. These may be meter specific. CMR (December 2001) 94 • Possible systematic effects in signal detection/processing (e.46).4. (b) transducer change or failure (changed transducer properties. It is left to the USM manufacturer to specify and document E I .19). and • Change of transversal flow velocity profiles (from flow calibration to field operation).possible different in-flow profile to the upstream pipe bend. Such installation effects on the USM integration uncertainty may be due to: • Change of axial flow velocity profile (from flow calibration to field operation). (3. cf.∆ . . Such contributions could be e.).I ) by a root-sum-square calculation.USM Fiscal Gas Metering Station. The individual effects described above may typically be uncorre- ∆ lated.g. liquid. CMR (December 2001) 95 ∆ u( q̂USM . 3. inaccuracy of the USM functional relationship (cf.4. Chapters 4 and 5.4.4.4 Miscellaneous USM effects In Eq. both due to e. Note that the subscript “∆” denotes that only changes of installation conditions from ∆ flow calibration to field operation are to be accounted for in u( q̂USM . .4): . . Table 1. etc.).I ) and E I . pitting.19). wear.∆ serves as an input uncertainty. lubricants. in the pipe and meter body.∆ .4). Table 1. the relative uncertainty term E misc accounts for possible miscellaneous uncertainty contributions to the USM measurement which have not been accounted for by the other uncertainty terms involved in the uncertainty model of the gas metering station. and will then contribute to u( q̂USM .possible different pipe bend configuration upstream of the USM. As an example. Cf. or other uncertainty contributions. (3.g. .USM fiscal gas metering stations NPD.I ) ≡ standard uncertainty of the USM integration method due to change of installation conditions from flow calibration to field operation.Handbook of uncertainty calculations . In the program EMU . E I . the definition of E misc accompanying Eq. the uncertainty of the integration method is discussed in Section 4. .possible wall deposits / contamination in the pipe and meter body (grease. NFOGM.possible changed wall roughness over time (corrosion.possible change of meter orientation relative to pipe bends. etc. (cf. • Compressibility factor uncertainty (Table 3.2 (Fig.2). Table 1. but in the program EMU . Cf.5 Signal communication and flow computer calculations In Eq.3 (cf. the flow computer calculation of frequency in case of analog frequency output). Chapter 5): • Pressure measurement uncertainty (Table 3.7). in case that is found to be necessary. (3.USM Fiscal Gas Metering Station the user has the possibility to specify such uncertainty contributions.5). • Signal communication and flow computer calculations (Table 3.6). For some of the quantities. NFOGM. • Density measurement uncertainty (Table 3. the input uncertainties to be given as input to the program EMU - USM Fiscal Gas Metering Station are specified. the relative uncertainty term E comm accounts for the uncertainties due to the signal communication between the USM field electronics and the flow computer. 3. Examples of input uncertainties given for the “detailed level” are described in Chapter 4. The various uncertainties are defined in Sections 3. Section 5.3). cf. • Temperature measurement uncertainty (Table 3.6).g. (3. cf. in the uncertainty model of the gas metering station (e. .USM Fiscal Gas Metering Station the user has the possibility to specify a value accounting for such uncertainty contributions. input uncertainties can be specified at two levels. and in Chapters 3 and 5. (1) “overall level” and (2) “detailed level”.1-3. 3.4). the definition of these terms accompanying Eq. but in the program EMU . cf. • Flow calibration uncertainties (Table 3.5. These two uncertainty contributions are not adressed further here. • USM field uncertainty (Table 3.6 Summary of input uncertainties to the uncertainty model In Tables 3. • Calorific value measurement uncertainty (Table 3. and is normally relatively small.1-3.Handbook of uncertainty calculations . E flocom accounts for the uncertainty of the flow computer calculations. 5.USM fiscal gas metering stations NPD.1). in case that is found to be useful.5). Section 5. 5.8).8. CMR (December 2001) 96 Such miscellaneous uncertainty contributions are not adressed further here.2.10.19). as discussed in Section 1. and are here organized in eight groups (following the structure of the worksheets in the program.6).11 (Fig.17). 2 level temperature transmitter.“ ----- u( P̂vibration ) Standard uncertainty due to vibration effects on the Manufacturer ----.4 u( P̂stability ) Standard uncertainty of the stability of the pressure Manufacturer ----.2.2. P Input Input To be specified Ref. u( P̂misc ) Standard uncertainty due to other (miscellaneous) ef. CMR (December 2001) 97 Table 3.2. u( T̂ RFI ) Standard uncertainty due to RFI effects on the tem.“ ----- perature transmitter. 4. and 5. u( T̂cable ) Standard uncertainty of lead resistance effects on the Manufacturer / ----. for calculation of the expanded uncertainty of the gas temperature measurement. Program user 3. Manufacturer / ----. u( P̂atm ) Standard uncertainty of the atmospheric pressure. Program user .2. Manufacturer / ----. Program user ----.4 (2) Detailed u( P̂transmitter ) Standard uncertainty of the pressure transmitter.“ ----- wire RTD temperature element u( T̂vibration ) Standard uncertainty due to vibration effects on the Manufacturer ----. for calculation of the expanded uncertainty of the static gas pressure measurement. u( P̂temp ) Standard uncertainty of the effect of ambient tempera.“ ----- temperature transmitter. NFOGM.transm ) Standard uncertainty of the stability of the temperature Manufacturer ----.2. to Description level uncertainty and documented Handbook by: Section (1) Overall u c ( T̂ ) Combined standard uncertainty of the temperature es. to Description level uncertainty and documented Handbook by: Section (1) Overall u c ( P̂ ) Combined standard uncertainty of the pressure esti. Manufacturer ----.5 (2) Detailed u( T̂elem . 5.“ ----- fects on the pressure transmitter. Manufacturer ----.“ ----- temperature transmitter. Calibration lab.2. T Input Input To be specified Ref.2 and level timate. Gas temperature measurement.USM Fiscal Gas Metering Station.“ ----- ture on the pressure transmitter.elem ) Standard uncertainty of the stability of the Pt 100 4. u( P̂RFI ) Standard uncertainty due to RFI effects on the pressure Manufacturer ----.1.“ ----- temperature transmitter.1 and level mate.“ ----- transmitter (drift). Program user Table 3.USM Fiscal Gas Metering Station. Manufacturer / 3.“ ----- transmitter (drift). Program user u( T̂misc ) Standard uncertainty due to other (miscellaneous) ef.2.2.“ ----- fects on the pressure transmitter. Manufacturer ----.“ ----- pressure transmitter. 5.transm ) Standard uncertainty of the temperature element and Manufacturer / 3. Gas pressure measurement. Handbook of uncertainty calculations .1 level Calibration lab. 4. Program user 3.“ ----- transmitter. T̂ . u( P̂power ) Standard uncertainty due to power supply effects on the Manufacturer ----.5 u( T̂stab . and 5.USM fiscal gas metering stations NPD. u( T̂temp ) Standard uncertainty of the effect of ambient tempera. P̂ .“ ----- ture on the temperature transmitter. calibrated as a unit.1. Program user ----. Input uncertainties to the uncertainty calculation program EMU . u( T̂stab .“ ----- pressure transmitter. u( T̂ power ) Standard uncertainty due to power supply effects on the Manufacturer ----. Input uncertainties to the uncertainty calculation program EMU . Handbook of uncertainty calculations .“ ----- (uncertainty of the model used for calculation of Ẑ 0 ) E Z . ----.USM Fiscal Gas Metering Station. Calculated by pro. Z/Z0. ĉ c Program user u( ĉ d ) Standard uncertainty of the densitometer gas VOS es. ρ̂ . T̂d u c ( T̂d ) = u c ( T̂ ) u( P̂ ) Standard uncertainty of the line pressure estimate.2.3. Manufacturer / ----.4. Manufacturer ----. 5. 4. for calculation of the expanded uncertainty of the gas density measurement. NFOGM.7 (2) Detailed u( ρˆ u ) Standard uncertainty of the indicated (uncorrected) Manufacturer 3.“ ----- tainty for Ẑ 0 (inaccurate determ. Gas compressibility factors.anal Relative standard uncertainty due to analysis uncer.4 and level mate.“ ----- tainty for Ẑ (inaccurate determ.“ ----- mate.“ ----- tainty of the temperature correction u( ρˆ misc ) Standard uncertainty of the indicated density esitmate.6 E Z 0 .“ ----- mate. to Description level uncertainty and documented Handbook by: Section (1) Overall Not available level (2) Detailed E Z . Z and Z0 Input Input To be specified Ref. ρ̂ u u( T̂c ) Standard uncertainty of the calibration temperature Manufacturer ----.mod Relative standard uncertainty due to model uncertainty Program user ----.2. P̂ Calculated by pro.anal Relative standard uncertainty due to analysis uncer. T̂c u c ( T̂ ) Combined standard uncertainty of the line temperature Calculated by pro.2. Gas density measurement.USM Fiscal Gas Metering Station. T̂ gram u c ( T̂d ) Combined standard uncertainty of the densitometer Calculated: ----.“ ----- timate.“ ----- temperature estimate. τ̂ Manufacturer ----. Program user 3.4 level density esitmate.4. to Description level uncertainty and documented Handbook by: Section (1) Overall u c ( ρˆ ) Combined standard uncertainty of the density esti.“ ----- u( K̂ d ) Standard uncertainty of the VOS correction transducer Manufacturer ----. Manufacturer / ----. CMR (December 2001) 98 Table 3. ρ Input Input To be specified Ref.“ ----- gram u( ∆P̂d ) Standard uncertainty of the pressure difference esti. of gas composition) Program user E Z 0 .2. Manufacturer / ----.“ ----- (uncorrected) density esitmate.3.“ ----- estimate. 4. ρ̂ u and 5. K̂ d u( ρˆ temp ) Standard uncertainty representing the model uncer. for calculation of the expanded uncertainty of the gas compressibility factor ratio.“ ----- constant estimate. Manufacturer / ----. of gas composition) Program user Table 3. ----. ĉ d Program user u( τˆ ) Standard uncertainty of the periodic time estimate. Input uncertainties to the uncertainty calculation program EMU . due to other (miscellaneous) effects .“ ----- ρ̂ u .3 level (uncertainty of the model used for calculation of Ẑ ) and 5.USM fiscal gas metering stations NPD.2.7 u( ρˆ rept ) Standard uncertainty of the repetatility of the indicated Manufacturer ----. Program user ----. ----. Input uncertainties to the uncertainty calculation program EMU .“ ----- estimate. ∆P̂d (from densitometer to line conditions) gram u( ĉ c ) Standard uncertainty of the calibration gas VOS esti.mod Relative standard uncertainty due to model uncertainty Program user 3. manufacturer and 5. K̂ dev .3. j.3.5. to Description level uncertainty and documented Handbook by: Section (1) Overall Not available level (2) Detailed E qref .3. M (rep. laboratory and 5. 4. M. 5. CMR (December 2001) 99 Table 3. 4. j = 1.2. ….6.2.USM Fiscal Gas Metering Station. j = 1. ….3.5 and signal communication with flow computer (flow com. in case of analog frequency output) E flocom Relative standard uncertainty of the estimate q̂ v due to USM ----.7. Program user 3. j Relative standard uncertainty of the deviation factor Calculated by 3. 4. manufacturer 5. Input uncertainties to the uncertainty calculation program EMU .“ ----- flow computer calculations manufacturer .9 Erept . NFOGM. level timate.3. for calculation of the expanded uncertainty of the USM flow calibration. USM flow calibration Input Input To be specified Ref. 4. 4. q̂ ref . Signal communication and flow computer calculations Input Input To be specified Ref. E K dev .1 level urement. j = 1. at test flow rate no. j. j. at test flow rate no.5 .11 puter calculation of frequency.1.9 …. to Description level uncertainty and documented Handbook by: Section Overall EHS Relative standard uncertainty of the calorific value es.3 USM at flow calibration. for calculation of the expanded uncertainty of the calorific value measurement. M.3.USM Fiscal Gas Metering Station. Flow calibration 3. at test flow rate no. USM flow calibration Input Input To be specified Ref.USM fiscal gas metering stations NPD.2. j Relative standard uncertainty of the repeatability of the USM 3. j .USM Fiscal Gas Metering Station.5.9 resenting the uncertainty of the flow calibration laboratory). Handbook of uncertainty calculations . to Description level uncertainty and documented Handbook by: Section E comm Relative standard uncertainty of the estimate q̂ v due to USM 3.3. Input uncertainties to the uncertainty calculation program EMU .8 Table 3. program and 5.2 estimate. Ĥ S . Table 3. j . Input uncertainties to the uncertainty calculation program EMU . j Relative standard uncertainty of the reference meas. with respect to signal communication and flow computer calculations. 4. • Possible cavity time delay correction (e. etc. • Electronics stability (possible random effects).g. at the actual flow rate. erronous signal period identification). 4. It represents the repeatability of the transit times. (2) Detailed u( αˆ ) αˆ Rel. pipe vibrations.10. 4.2 level temperature expansion for the meter body material.2.4 and level q̂ v at actual flow rate.) program u c ( ∆P̂cal ) Combined standard uncertainty of the pressure differ.2 USM. at actual flow rate. Input uncertainties to the uncertainty calculation program EMU .“ ----- ence estimate . to Description level uncertainty and documented Handbook by: Section Specified in E rept Relative standard uncertainty of the repeatability of the USM 3.5 (1) Overall E USM . CMR (December 2001) 100 Table 3. etc. ∆P̂ref (flow calibration to line condit. manufacturer and 5.4. • Incoherent noise (from pressure reduction valves (PRV).1 both cases. flow calibration conditions). • Possible systematic effects in signal detection / processing (e. due to change of conditions from flow calibration to field operation. Calculated by ----. Such systematic effects may be due to e. electromagnetic noise (RFI). USM in field operation. after possible manufacturer and 5. • Finite clock resolution.USM Fiscal Gas Metering Station.10.: • Cable/electronics/transducer/diffraction time delay (line P & T effects. Program user 4.USM fiscal gas metering stations NPD. related to field operation of the 5. • Power supply variations.1 (1) and (2) or u( t̂ random ) Standard uncertainty (i. for calculation of the expanded uncertainty of the USM in field operation (uncorrected deviations re. β̂ USM manufacturer u c ( ∆T̂cal ) Combined standard uncertainty of the temperature dif. especially at high flow velocities). reciprocity. due to change of condi.8.4. E misc Relative standard uncertainty of the estimate q̂ v due to USM manufacturer/ 3.g. electromagnetic cross-talk.4.1 1i field random effects on the transit times. ambient temperature effects.4.4.4 and miscellaneous effects on the USM measurement.4.1 time averaging. drift.10. Such random effects may be due to e.2 u( βˆ ) βˆ Rel. Handbook of uncertainty calculations .3 i on the upstream transit times. ambient temperature effects.) program u( t̂ 1systematic ) Standard uncertainty of uncorrected systematic effects USM 3.10. effects of possible transducer exchange).) • Coherent noise (from acoustic cross-talk (through meter body). sound .: • Turbulence effects (from temperature and velocity fluctuations. standard uncertainty of the coefficient of linear Program user / ----. erronous signal period identification).2. possible other acoustic interference. α̂ USM manufacturer and 5.g. USM field operation Input Input To be specified Ref.2. acoustic reflections in meter body. 4.4.1.). • Possible ∆t-correction (line P & T effects.∆ Relative combined standard uncertainty of the estimate USM manufacturer 3.2 tions from flow calibration to field operation.4.g. drift.e. standard uncertainty of the coefficient of linear Program user / 3.4.g. effects of possible transducer exchange). NFOGM. Calculated by ----.“ ----- radial pressure expansion for the meter body. manufacturer and 5. standard deviation) due to in. ∆T̂ref (flow calibration to line cond.10.“ ----- ference estimate. USM 3. 4. • Possible random effects in signal detection/ processing (e. pipe bend.g.4 method. and • Change of transversal flow velocity profiles (from flow calibration to field operation). CMR (December 2001) 101 velocity effects). etc. u( t̂ 2systematic ) Standard uncertainty of uncorrected systematic effects USM ----.10. > possible wall deposits. E I . especially at high flow velocities. wear. due to change of con. due to change of installation conditions from manufacturer and 5.). • Possible deposits at transducer faces (lubricant oil. due to e. NFOGM. Such installation effects on the USM integration uncertainty may be due to e. grease. > possible different meter orientation rel. pitting).4.Handbook of uncertainty calculations .: > possible different upstream pipe bend configuration. to pipe bends > possible changed wall roughness over time (corrosion.g.2 flow calibration to field operation.∆ Relative standard uncertainty of the USM integration USM 3.USM fiscal gas metering stations NPD. contamination (grease. liquid. 4. waxm.4.: • Change of axial flow velocity profile (from flow calibration to field operation). > possible different in-flow profile to upstr. . • Sound refraction (flow profile effects on transit times.3.). etc.“ ----- i on the downstream transit times. and by changed installation conditions). manufacturer ditions from flow calibration to field operation (see above). . . Operating conditions. where for simplicity the data used in the AGA-9 report have been used also here. Data for the densitometer (Solartron 7812 example) are given in Table 4. to some extent the present chapter also serves as a guideline to using the program. 79 The uncertainty models for the pressure transmitter and the temperature element/transmitter are similar to the ones used in [Dahl et al. 4. and (b) the uncertainty model itself. temperature and density instruments specified in the table are the same as those used for uncertainty evaluation of an orifice fiscal gas metering station by [Dahl et al. 2001].USM fiscal gas metering stations NPD.4. here the input uncertainties to the densitometer uncertainty model are somewhat different.Handbook of uncertainty calculations . the uncertainty model for the densitometer is somewhat different here. NFOGM. and propagated in the uncertainty model. Meter body data are given in Table 4.1. For instance. cf. Section 2. CMR (December 2001) 102 4. with analytical expressions for the sensitivity coefficients. With respect to the USM. 1999]. gas chromatograph and calorimeter. Chapters 3 and 5. as an example. 1999]79.1.USM Fiscal Gas Metering Station. However. as a basis for Chapter 5. Input uncertainties are evaluated for the different instruments of the metering station. the densitometer uncertainty model developed and used here is fully analytical. both with respect to (a) the functional relationship used (different formulations used for the VOS and installation corrections). as specified by NFOGM. whereas in [Dahl et al. cf. 1999] the sensitivity coefficients were calculated using a more numerical approach. used for the present uncertainty evaluation example are given in Table 4.. The pressure..2. Hence. flow computer.1 Instrumentation and operating conditions The USM fiscal gas metering station evaluated in the present Handbook consists of the equipment listed in Table 4. etc. with some modifications. is used to evaluate the relative expanded uncertainty of a gas metering station at given operating conditions.3. Also.. UNCERTAINTY EVALUATION EXAMPLE In the present chapter the uncertainty model of the USM gas metering station described in Chapter 3. NPD and CMR for this example [Ref Group. The input uncertainties discussed here are the same as those used as input to the program EMU . no specific equip- ments are considered. 5. Z and Z0 Not specified (calculated from GC measurements).443 kg/m3 Calibration temperature. n-C5: 0.) Conditions Quantity Value Operating Gas composition North Sea example80 Line pressure.1. T Pt 100 element: according to EN 60751 tolerance A [NORSOK.718 %. CO2: 0. absolute) 100 bara Line temperature.943 %. i-C5: 0. P Rosemount 3051P Reference Class Smart Pressure Transmitter [Rosemount.9973 Superior (gross) calorific value. (Corre- sponds to Fig. Table 2. 2000].1. Here. Temperature. The evaluated USM fiscal gas metering station instrumentation (cf. absolute) 50 bara Temperature.008 %. cd 415.756 %.067 %. ρ Solartron Model 7812 Gas Density Transducer [Solartron.013 %. A representative value may be about 10 % of the temperature difference between densitometer and ambient (air) conditions. Hs 41.62 kg/m3 Ambient (air) temperature. Flow computer Not specified.8460 Sound velocity. P (static. c 417. CMR (December 2001) 103 Table 4. cond. .686 MJ/Sm3 80 Example of dry gas composition. taken from a North Sea pipeline: C1: 83. NFOGM. Rosemount 3144 Smart Temperature Transmitter [Rosemount. ρ 81.2. Td 48 oC 81 Sound velocity. Measurement Instrument Ultrasonic meter (USM) Not specified. Table 4. Z 0. 81 Temperature deviation between line and densitometer conditions may be as large as 7-8 oC [Sa- kariassen. Gas parameters of the USM fiscal gas metering station being evaluated (example). Gas compressibility. 2 oC deviation is used as a moderate example. n-C4: 0. Ambient (air) temperature at calibration 20 oC Standard ref. 1999]. Density.24 m/s Indicated (uncorrected) gas density in densitometer. ρu 82. 2001].15 K) Gas compressibility. Z0 0. C2: 13. Tcal 10 oC Pressure transmitter Ambient (air) temperature at calibration 20 oC Temperature transm. Tair 0 oC Densitometer Temperature. 2000].Handbook of uncertainty calculations . T 50 oC (= 323. N2: 0. i-C4: 0. Pressure (static). Hs Not specified (calorimeter measurement).4). Calorific value.USM fiscal gas metering stations NPD. cc 350 m/s Flow calibration Pressure. Tc 20 oC Calibration sound velocity.475 %. 1998a].040 %. Pcal (static.0 m/s Gas density. C3: 0.98 %. Compressibility. . (Data used in Section 4.2. 1998] Average wall thickness.11). 2R0 ("dry calibration" value) 308 mm [AGA-9. 1998] Table 4.3.440⋅10-4 [Solartron. 2001a] K18 Constant -1.4. as specified in the data sheet [Rosemount. 1999] K19 Constant 8. 82 Note that the expanded uncertainties given in the transmitter data sheet [Rosemount.2 and Fig.Handbook of uncertainty calculations . 4. 5.USM fiscal gas metering stations NPD.. (Data used in Section 4. 5.582. temperature and density measurements are evaluated. CMR (December 2001) 104 Table 4. (3.4. 1998] Young’s modulus (or modulus of elasiticity). 2000] are specified at a 99 % confidence level (k = 3). 1998] Material data Temperature expansion coeff. α 14⋅10 K -6 -1 [AGA-9. Y 2⋅10 MPa 5 [AGA-9. is given by Eq. Gas densitometer data used in the uncertainty evaluation (Solartron 7812 example).) Meter body Quantity Value Reference Dimensions Inner diameter. The contributions to the combined standard uncertainty of the pressure measurement are described in the following. NFOGM.360⋅10-5 [Solartron. u c ( P̂ ) . etc. 1999] 4. as well as the Z-factor and calorific value estimates.1 Pressure measurement The combined standard uncertainty of the pressure measurement.2. w 8.2 Gas measurement uncertainties In the present subsection. 1999] Kd Constant (characteristic length) 21000 µm [Solartron.4 mm [AGA-9. Performance specifications for the Rosemount Model 3051P Reference Class Pres- sure Transmitter are given in Table 4. This expression is evaluated in the following.9. Meter body data for the USM being evaluated (AGA-9 example).2. 2000].) Symbol Quantity Value Reference τ Periodic time 650 µs [Eide.4 and Fig. the expanded uncertainties of the gas pressure. 5. For fiscal gas metering: 0.1 % of span from 20 to 3 [Rosemount. not static pressure measurement) Static pressure effect Negligible (influence only on 3 [Dahl et al.125 % of URL for 5 years 3 [Rosemount. Ambient temperature effects (air). see text below). Performance specifications of the Rosemount Model 3051P Reference Class Pressure Trans- mitter [Rosemount.as a . the transmitter uncertainty is to include the uncertainty of the temperature calibration laboratory (which shall be traceable to international standards). (NA) URL (upper range limit) 138 barG . U ( P̂power ) Negligible (less than ±0. Calibration certificate.USM fiscal gas metering stations NPD.006% URL + 0. (NA) Time between calibrations 12 months . ±(0. 2000] 1000 MHz and for field strength up to 30 V/m. 1999] of URL for 1 year (used here). 2000] Vibration effects.1 % . [Rosemount. Power supply effects.03% span) 3 [Rosemount. and up to 69 bar line pressure. Example Span (calibrated) 70 bar . The confidence level and the probability distribution of the reported expanded uncertainty shall be specified. 3 [Rosemount. 1999] differential pressure measurement. if the calibration laboratory states that the transmitter uncertainty (including the calibration laboratory uncertainty) is within the “reference accuracy” given in the manufacturer data sheet [Rosemount. Coverage Reference tainty factor. used as input to the uncertainty calculations given in Table 4. 2000] Transmitter uncertainty. 2000] % of calibrated span per volt). Mounting position effect Negligible (influence only on 3 [Dahl et al. Alternatively. Pressure transmitter uncertainty. 1999] differential pressure measurement.005 3 [Rosemount. Calibration certifi- 20 oC cate. not static pressure measurement) 1. 2000] Stability. U ( P̂vibration ) nance frequencies. CMR (December 2001) 105 Table 4.Handbook of uncertainty calculations . 2000] for 28 oC temperature changes. one may . U ( P̂transmitter ) : If the expanded uncertainty specified in the calibration certificate is used for the uncertainty evaluation. k Calibration ambient temperature (air) .6. [Rosemount. 2000]. 2000] U ( P̂temp ) per 28°C Negligible (except at reso. U ( P̂transmitter ) ±0. U ( P̂RFI ) ±0. RFI effects. U ( P̂stability ) ±0.. NFOGM..05 % of span 3 [Rosemount. 2000]. Quantity or Source Value or Expanded uncer. 035 bar 3 = 0. 83 The manufacturer's uncertainty specification is used here.05 % of span at a 99 % confidence level (cf. the uncertainty is limited to line pressures below 69 barg. This contribution is zero at the time of calibration.e. It is assumed here that this figure refers to the calibrated span. Stability .009- 0.e. 1999]. .1 % of URL for 1 year [Rosemount. The “reference accuracy” of the 3051P pressure transmitter accounts for hysteresis. The alternative stability uncertainty is given to be 0. i. 2001a]. 1999]. and is specified as a maximum value at a given time.pressure transmitter. hysteresis. the manufacturer has therefore provided a more applicable uncertainty specification when it comes to stability of the 3051P pressure transmitter regarding use in fiscal metering stations (the uncertainty specified in the data sheet is still valid). terminal-based linearity and repeatability. repeatability. 2000] as ±0.125 % of URL for 5 years for 28 °C temperature changes and up to 69 barg line pressure (Table 4. 0..use the latter uncertainty value in the calculations. (Note that this uncertainty specification does not include limitations with respect to temperature changes and line pressure.5). This includes linearity. the transmitter uncertainty may be further reduced. In a dialog with the manufacturer.e. the calibrated span is here taken to be 50 .Handbook of uncertainty calculations .) This alternative uncertainty specification is therefore used for the 3051P pressure transmitter in the present evaluation example.5). NFOGM. However. and reference instruments uncertainty. i. Table 4. giving u( P̂transmitter ) = U ( P̂transmitter ) 3 = [70 ⋅ 0.022 bar.120 bar. u( P̂stability ) : The stability of the pressure transmitter represents a drift (increasing/decreasing offset) in the readings with time. 2.012 bar 83. 70 bar (Table 4.018-0.3).USM fiscal gas metering stations NPD.5). this uncertainty becomes artificially low when considering normal calibration intervals at fiscal metering stations of two or three months [Dahl et al. i. By calibration of the pressure transmitter in an accredited calibration laboratory. An example of a calibration certificate specification for the expanded uncertainty U ( P̂transmitter ) may be in the range 0. This approach is used here. with k = 3 (Section B. reading uncertainty.0005]bar 3 = 0. and is given in the manufacturer data sheet as ±0. The stability of the 3051P pressure transmitter is given in the manufacturer data sheet [Rosemount.011 bar for the standard uncertainty. CMR (December 2001) 106 conservative approach . at a 95 % confidence level (k = 2) [Eide. Furthermore. As an example. and the RFI electric field at the actual metering station should be measured in order to document the actual electric field at the pressure transmitter.069 bar. That is. How- ever. 2000] as ±0. but is assumed here to be 95 %.138 bar / 2 ≈ 0. at a normal probability distribution (k = 2. In practice. RFI effects . . 3. effects (RFI) is given in the manufacturer data sheet [Rosemount. The temperature change referred to is the change in ambient temperature relative to the ambient temperature at calibration (to be specified in the calibration certificate). 4.5. cf. the uncertainty due to RFI effects may be reduced. cf. Table 4.006 % URL + 0.pressure transmitter.03 % span) per 28 °C temperature change. the RFI electric field must be documented in order to evaluate if. cf. u( P̂temp ) : The ambient temperature effect on the Rosmount 3051P pressure transmitter is given in the manufacturer data sheet [Rosemount.pressure transmitter.Handbook of uncertainty calculations . Section B. I. It is noted that the specified RFI uncertainty is atually twice as large as the uncertainty of the transmitter itself. NFOGM. the uncertainty specified by [Rosemount. 84 In a worst case scenario.07 bar 3 = 0. and to what extent. However. The confidence level is not specified. and for field strength up to 30 V/m.5.USM fiscal gas metering stations NPD. Consequently. 2000] as ±(0. u( P̂RFI ) : Radio-frequency interference. u( P̂stability ) = U ( P̂stability ) 2 = [138 ⋅ 0. if the transmitter is calibrated every 12 months. this uncertainty contribution may be difficult to evaluate. a linear time dependency has been assumed here84. the uncertainty due to RFI effects must be included in the uncertainty evaluation as given in the data sheet.e. Ambient temperature effects .3).1 % of span for frequencies from 20 to 1000 MHz. Table 4. the uncertainty due to stability may be used directly without using the time division specified. as long as the RFI electric field at the pressure transmitter is not documented by measurement. u( P̂RFI ) = U ( P̂RFI ) 3 = [70 ⋅ 0. for simplicity.001 ⋅ ( 12 12 )]bar 2 ≈ 0. Consequently.023 bar .001]bar 3 = 0. CMR (December 2001) 107 The time dependency of the stability uncertainty is not necessarily linear. 1999] due to stability effects is divided by 12 and multiplied with 12. 2000]. the average atmospheric pressure is about 1008 and 1012 mbar for the winter and summer seasons. The variation of the atmospheric pressure around the value 1 atm.Handbook of uncertainty calculations .pressure transmitter. 1999] and a calibration laboratory [Fimas. ≡ 1013.2). one obtains u( P̂temp ) = U ( P̂temp ) 3 [ = (138 ⋅ 0.USM fiscal gas metering stations NPD.3). The uncertainty of the absolute static pressure u c ( P̂ ) is thus to include the uncertainty of the atmospheric pressure.USM Fiscal Gas Metering Station. "measurement effect due to vibrations is negligible except at resonance frequencies.09 bar 3 = 0. CMR (December 2001) 108 Consequently. Assuming a 99 % confidence level. and a normal probability distribution for the variation range of the atmospheric pressure (k = 3.03 ) ⋅ 10 −2 ⋅( 20 28 ) bar 3] = [0. the vibration level at fiscal metering stations is considered to be very low (and according to recognised standards). the upper and lower parts of this range (beyond about 940 and 1040 mbar) are very rare (not observed every year) [Lothe. and a calibration temperature equal to 20 oC (Table 4. i.1060 mbar. temperature change of 20 oC. vibration effect is less than 0. For convenience.25 mbar is taken as the average value. cf. Hence. Atmospheric pressure. NFOGM. 2001]. Based on communication with the manufacturer [Rosemount. 1 atm. a max. When at resonance frequencies.5). In the North Sea.006 + 70 ⋅ 0. On a world-wide basis. 6.007 bar . the uncertainty due to vibration effects is neglected for the 3051P transmitter: u( P̂vibration ) = 0.1 % of URL per g when tested from 15 to 2000 Hz in any axis relative to pipe-mounted process conditions" (Table 4.5). 1999]. as a conservative approach. 1994]. u( P̂vibration ) : According to the manufacturer data sheet [Rosemount. due to day-by-day atmospheric pressure variations. the uncertainty due to vibration effects may be neglected.03 bar .25 mbar is here taken to be 90 mbar. one obtains u( P̂atm ) = U ( P̂atm ) 3 = 90 m bar 3 = 0. In the program EMU . .0209 bar 3 ≈ 0. u( P̂atm ) : The Rosemount 3051P pressure transmitter measures the excess pressure.however.e. . the observed atmospheric pressure range includes the range 920 . Section B. Vibration effects .0150 ]bar 3 ≈ 0.0059 + 0. for a possible “worst case” example of ambient North Sea temperature taken as 0 oC (Table 4. ≡ 1013. 5. relative to the atmospheric pressure. respectively (averaged over the years 1955- 1991) [Lothe. respecively.11). The calculated expanded and relative expanded uncertainties (specified at a 95 % confidence level and a normal probability distribution. the uncertainty due to power supply effects is neglected for the 3051P transmitter: u( P̂power ) = 0. 2000] as ±0.5). NFOGM.31 kPa = 0. Power supply effects .pressure transmitter: The mounting position effect [Rosemount. The zero error can be calibrated out at line pressure.15 %. The zero error is given in the data sheet [Rosemount.6 for evaluation of the expanded uncertainty of the pressure measurement according to Eq.005 % of the calibrated span per volt (Table 4.pressure transmitter. CMR (December 2001) 109 7.. in the program.pressure transmitter: The static pressure effect [Rose- mount. The mounting position error is specified in the data sheet [Rosemount. and not on static pressure measurements. which can be calibrated out. Static pressure effect . 9.04 % of URL per 69 barg.16 bar and 0. which was not always the case for the older transmitters [Dahl et al. 1999].. and (b) the span error [Rosemount.. 1999]86. 2000] will only influence on a differential pressure transmitter.USM fiscal gas metering stations NPD. 2000] as less than ±0. and not on static pressure measurements.3) are 0.0031 bar). . 86 Mounting position effects are due to the construction of the 3051P differential pressure transmitter with oil filled chambers [Dahl et al. 2000].25 inH2O (0.10 % of reading per 69 barG. Mounting position effects . The span error is given in the data sheet [Rosemount.. (3.Handbook of uncertainty calculations . These may influence the measurement if the transmitter is not properly mounted. 8. According to the supplier [Rosemount. as considered here [Dahl et al. No span effect”. 1999]85. 1999] this uncertainty is specified to indicate that the uncertainty due to power supply effects is negligible for the 3051P transmitter. These values are further used in Section 4. 2000] as ±0. A sample uncertainty budget is given in Table 4. Hence.6. Section B. u( P̂power ) : The power supply effect is quantified in the manufacturer data sheet [Rosemount. 2000] will only influence on a differential pressure transmitter. with k = 2. The figures used for the input uncertainties are those given in the discussion above. 85 The static pressure effect influencing on 3051P differential pressure transmitters consists of (a) the zero error. 1999]. 2000] as “zero shifts up to ±1. as considered here [Dahl et al. cf. 007 bar 1 4. & fact. (Corresponds to Figs. (3. u c ( T̂ ) . is given by Eq.853⋅10-5 bar2 effects. as specified in the data sheet [Rosemount.) Input uncertainty Combined uncertainty Source Expand. transmitter Atmospheric pressure 0. (3. Standard Sens.16 bar Operating pressure P̂ 100 bara Relative expanded uncertainty (95 % confidence level) U ( P̂ ) P̂ 0. 2000] are specified at a 99 % confidence level (k = 3).2.361⋅10-4 bar2 Stability.070 bar 99 % (norm) 3 0.4 and 5.0799 bar Expanded uncertainty (95 % confidence level. uncertainty coeff.035 bar 99 % (norm) 3 0.761⋅10-3 bar2 RFI effects 0. 5.12). Sample uncertainty budget for the measurement of the absolute static gas pressure using the Rose- mount Model 3051P Pressure Transmitter [Rosemount. The discussion is similar to that given in [Dahl et al..21. level Cov.443⋅10-4 bar2 Ambient temperature 0. transmitter 0. 1999]. NFOGM. This expression is evaluated in the following.069 bar 1 4.787. Variance uncert.138 bar 95 % (norm) 2 0.. etc. k Transmitter uncertainty 0. 87 Note that the expanded uncertainties given in the transmitter data sheet [Rosemount. CMR (December 2001) 110 Table 4.023 bar 1 5.090 bar 99 % (norm) 3 0. 2000].030 bar 1 9.6.39⋅10-3 bar2 Combined standard uncertainty u c ( P̂ ) 0.USM fiscal gas metering stations NPD.16 % 4.021 bar 99 % (norm) 3 0. Distribut. 2000]. calculated according to Eq. The contributions to the combined standard uncertainty of the temperature measurement are described in the following. Performance specifications for the Rosemount Model 3144 Smart Temperature Transmitter and the Pt 100 4-wire RTD element are given in Table 4.012 bar 1 1.0⋅10-4 bar2 Vibration Negligible 0 1 0 Power supply Negligible 0 1 0 Sum of variances u c2 ( P̂ ) 6.11). Conf. .Handbook of uncertainty calculations .2 Temperature measurement The combined standard uncertainty of the temperature measurement. k = 2) U ( P̂ ) 0. 1998a]. Ambient temperature effects . 2000] transmitter. U ( T̂cable ) pendent on lead resistance). U ( T̂vibration ) specifications with no effect on performance).02 % of 3 span.temperature transmitter. Coverage Reference tainty factor. 2000] U ( T̂stab . Negligible (no effect.transm ) Stability . Example o [Rosemount. NFOGM. the transmitter/element uncertainty (calibrated as a unit) will include the uncertainty of the temperature calibration laboratory (to be traceable to international standards). 3 per 1 oC. . 2000] Vibration effects.transmitter. 3 [Rosemount. When first recording the characteristics of the temperature element and then loading this characteristic into the transmitter prior to the final calibration. 2000] % of span per volt).Handbook of uncertainty calculations . with unshielded 3 [Rosemount. ±0.050 oC U ( T̂stab . used as input to the uncertainty calculations given in Table 4.0015 oC 3 [Rosemount. Worst case.8. inde. U ( T̂RFI ) cable: equivalent to the transmitter “accuracy”.10 C 3 calibrated as a unit). U ( T̂elem . 1999]. U ( T̂elem . U ( T̂ power ) Negligible (less than ±0. If the expanded uncertainty specified in the calibration certificate is used for the uncertainty evaluation. Power supply effects. the uncertainty due to the element can be mini- mised [Fimas. Performance specifications of the Rosemount Model 3144 Temperature Transmitter [Rose- mount. [BIPM.7. U ( T̂elem . 1998] Lead resistance effects. Calibration certifi- 20 °C cate (NA) Time between calibrations 12 months . 3 [Rosemount.elem ) Negligible (tested to given 3 [Rosemount. D/A effect: 0.1 % of reading or 0.001 % of span. Quantity or Source Value or Expanded uncer.transm ) “D/A accuracy”: ±0. 2000] RFI effects . 1997] 0. U ( T̂temp ) per 1 oC.005 3 [Rosemount.transm ) whichever is greater. Transmitter/element uncertainty (calibrated as a unit). CMR (December 2001) 111 Table 4. k Calibration ambient temperature (air) . Transmitter/element uncertainty NA Calibration certifi- NA cate (NA) (calibrated as a unit). for 24 months. 1. .1 oC.USM fiscal gas metering stations NPD.temperature element. 2000] and the Pt 100 4-wire RTD element. Stability . 2000] Transmitter/element uncertainty (not “Digital accuracy”: 0.transm ) : The temperature element and the temperature transmitter are assumed to be calibrated as a unit [NORSOK. The confidence level of the reported expanded uncertainty is to be specified. “Digital accuracy”: 0. USM fiscal gas metering stations NPD. one may . The output signal is accessed using a HART protocol. and is specified as a maximum value at a given time. 2001a]. Table 1]. whichever is greater for 24 months. Table 4. u( T̂elem .use the latter uncertainty value in the calculations.10 o C 3 = 0.10 °C at a 99 % confidence level (k = 3. However.e. corresponding to 0. and including the calibration laboratory uncertainty) is within the “accuracy” given in the manufacturer data sheet [Rose- mount. Stability .7). As an example. Table 4. the stability of the 3144 temperature transmitter is given in the manufacturer data sheet [Rosemount. 88 The manufacturer's uncertainty specification is used here. 2000.1 °C. As this is greater than 0. 2000] as 0.033 o C 88.1 % of reading (measured value). the calibration certificate specification for the element/transmitter’s expanded uncertainty U ( T̂elem . at a 95 % confidence level (k = 2) [Eide.7. only the “digital accuracy” is used here (cf.temperature transmitter.transm ) : The stability of the temperature transmitter represents a drift in the readings with time. The expanded uncertainty is then given as 0.transm ) = U ( T̂elem . The value “0. 2. cf. This contribution is zero at the time of calibration. cf.as a conservative approach . a linear time dependency is assumed here89. this uncertainty value is used. the element/transmitter uncertainty may be significantly reduced.323 o C . Section B. . u( T̂stab .3).transm ) 3 = 0. 89 In a worst case scenario.1 °C. for temperature element and transmitter combined. The “accuracy” of the 3144 temperature transmitter used together with a Pt 100 4-wire RTD element is tabulated in the data sheet [Rosemount. for simplicity. i. or 0. 2000]. For use in combination with RTD elements. the uncertainty due to stability may be used directly without using the time division specified. That is. This approach is used here. NFOGM.03 oC.transm ) may be 0.015 oC for the standard uncertainty. CMR (December 2001) 112 Alternatively. By calibration of the the element and transmitter in an accredited calibration laboratory. if the calibration laboratory states that the transmitter/element uncertainty (calibrated as a unit.1 % of reading for 24 months” corresponds to [( 273 + 50 ) ⋅ 0.001] o C ≈ 0. The time dependency is not necessarily linear.Handbook of uncertainty calculations . For fiscal metering stations all cables are shielded. i. the uncertainty given in the data sheet due to stability effects is divided by 24 and multiplied with 12. NFOGM. It is time consuming to predict or measure the actual RFI effects at the metering station.temperature transmitter. the RFI effects should be less than the worst case specified in the data sheet. the expanded uncertainty is specified to be 0.7. for a possible “worst case” ambient North Sea temperature taken as 0 oC (Table 4.USM fiscal gas metering stations NPD. a .2). 2000]. Consequently. 3.054 o C .10 o C 3 = 0. That is. and a calibration temperature equal to 20 oC (Table 4.0015 °C per 1 °C change in ambient temperature relative to the calibration ambient temperature (the “digital accuracy”). Nevertheless. and difficult to evaluate correctly the influence on the temperature measurement.001 ⋅ ( 12 24 )] o C 3 = 0. 2000]. u( T̂stab . u( T̂temp ) : The Rose- mount 3144 temperature transmitters are individually characterised for the ambient temperature range -40 °C to 85 °C . 4.temperature transmitter. cf. Ambient temperature effects . This uncertainty is tabulated in the data sheet as a function of changes in the ambient temperature (in operation) from the ambient temperature when the transmitter was calibrated.e.7. CMR (December 2001) 113 Consequently. u( T̂RFI ) = U ( T̂RFI ) 3 = 0.transm ) = U ( T̂stab . i. effects (RFI) may cause a worst case uncertainty equivalent to the transmitter’s nominal uncertainty. if the transmitter is calibrated every 12 months. Some uncertainty still arises due to the change in ambient temperature. Table 4. It is therefore recommended to use the worst case uncertainty specified in the data sheet for the uncertainty due to RFI effects.7 The ambient temperature uncertainty for Rosemount 3144 temperature transmitters used together with Pt-100 4-wire RTDs is given in the data sheet as 0.Handbook of uncertainty calculations .e. when used with an unshielded cable [Rosemount. and automatically compensate for change in ambient temperature [Rosemount. u( T̂RFI ) : Radio-frequency interference. Table 4. That is. RFI effects (and also effects due to bad instrument earth) may cause additional uncertainty to the temperature measurement that is hard to quantify.transm ) 3 = [( 273 + 50 ) ⋅ 0. cf.033 o C .1615 o C 3 ≈ 0. RFI effects .10 oC. For the “digital acuracy” of the 3144 transmitter. u( T̂vibration ) : According to the manufacturer data sheet [Rosemount.01 o C . Stability . 1999].elem ) = U ( T̂stab . 1999] and a calibration laboratory [Fi- mas.0015 ⋅ 20 o C 3 = 0. 7. The confidence level of this expanded uncertainty is not given. u( T̂ power ) : The power supply effect is quantified in the manufacturer data sheet [Rosemount.3).21 mm peak displacement for 10-60 Hz. cf. 2000]. the uncertainty due to vibration effects may be neglected. .USM fiscal gas metering stations NPD. u( T̂stab .03 o C 3 = 0. u( T̂stab .050 °C. however. which was not always the case for the older transmitters [Dahl et al. 1999] this uncertainty is specified to indicate that the uncertainty due to power supply effects is negligible for the 3144 transmitter. Table 4.elem ) 2 = 0.. BIPM [BIPM. one obtains u( T̂temp ) = U ( T̂temp ) 3 = 0. Oxidation. and considering that the vibration level at fiscal metering stations shall be very low (and according to recognised standards). Hence. 1999]. Vibration effects .050 oC / 2 = 0.USM Fiscal Gas Metering Station.temperature transmitter.temperature element. 1995]. 2000] as being less than ±0. 1997] has performed several tests of the stability of temperature elements which shows that this uncertainty is typically of the order of 0.7. in communication with the manufacturer [Rosemount. 3g acceleration for 60-2000 Hz". and a 95 % confidence level and a normal probability distribution is assumed here (k = 2. "transmitters are tested to the following specifications with no effect on performance: 0. cf.elem ) : The Pt-100 4-wire RTD element will cause uncertainty to the temperature measurement due to drift during operation. Moreover. That is. CMR (December 2001) 114 max.025 oC. [BIPM. the uncertainty due to vibration effects is neglected for the Rosemount 3144 temperature transmitter. 6.Handbook of uncertainty calculations . 1997]. Power supply effects .temperature transmitter. According to the supplier [Rosemount. Section B. moisture inside the encapsulation and mechanical stress during operation may cause instability and hysteresis effects [EN 60751. in the program EMU .005 % of span per volt. u( T̂vibration ) = 0 . 5. temperature change of 20 o C. NFOGM. 22.USM fiscal gas metering stations NPD.900⋅10-3 oC2 RFI effects 0. in the program EMU . the uncertainty due to power supply effects is neglected for the Rosemount 3144 temperature transmitter.USM Fiscal Gas Metering Station. transmitter 0.1615 oC 99 % (norm) 3 0. calculated according to Eq. Sensor lead resistance effects .10 oC 99 % (norm) 3 0. 4-wire RTDs are normally used in fiscal metering stations.12). element 0.025 oC 1 6. (Corresponds to Figs. Standard Sens. The figures used for the input uncertainties are those given in the discussion above. 5.USM Fiscal Gas Metering Station. in the program EMU . k = 2) U ( T̂ ) 0.15 oC Operating temperature T̂ 50 oC (≈323 K) Relative expanded uncertainty (95 % confidence level) U ( T̂ ) T̂ 0.10 oC 99 % (norm) 3 0. (3.250⋅10-4 oC2 Vibration Negligible 0 1 0 Power supply Negligible 0 1 0 Lead resistance Negligible 0 1 0 Sum of variances 2 u ( T̂ ) c 5.) Input uncertainty Combined uncertainty Source Expand. u( T̂ power ) = 0 .010 oC 1 1. Hence. k Transmitter/element 0. Distribut.047 % . (3.. Conf. transmitter Stability. u( T̂cable ) : According to the manufacturer data sheet for the 3144 transmitter [Rosemount. level Cov.12). uncertainty coeff. Table 4.6 and 5. CMR (December 2001) 115 Hence.054 oC 1 2.033 oC 1 1. 2000] with a Pt 100 4-wire RTD element. & fact.111⋅10-3 oC2 Ambient temperature 0.848⋅10-3 oC2 Combined standard uncertainty u c ( T̂ ) 0. 8.033 oC 1 1.050 oC 95 % (norm) 2 0.8 Sample uncertainty budget for the temperature measurement using the Rosemount Model 3144 Temperature Transmitter [Rosemount. NFOGM. 1998].8 for evaluation of the expanded uncertainty of the temperature measurement according to Eq.0765 oC Expanded uncertainty (95 % confidence level. the uncertainty due to lead resistance effects is neglected for the 3144 transmitter: u( T̂cable ) = 0 .Handbook of uncertainty calculations .temperature transmitter.0⋅10-4 oC2 effects.03 oC 99 % (norm) 3 0. the error due to lead resistance effects is "none" (independent of lead resistance) for 4-wire RTDs.111⋅10-3 oC2 uncertainty Stability. A sample uncertainty budget is given in Table 4. Variance uncert. the variation limits may be smaller.been assumed that the C1-component varies with ±0.0577 %. mod = 0.2.3. Fig. 2001] as natural variations over a time scale of months. by using a Monte Carlo type of simulation where the gas composition is varied within its uncertainty limits.4 % and the C3-component with ±0.2. In the present example the AGA-8 (92) equation of state [AGA-8.Handbook of uncertainty calculations .05 2 + 0.15 oC and 0.048 %. In the case of online measurements (using GC analysis). is in general more complicated to estimate.1 % (of the total gas content).015 2 % 3 = ≈ 0. Section B.3. The relative standard uncertainty of the estimate Ẑ due to analysis uncertainty (inaccurate determination of the line gas composition). it has .in a simplified approach . cf. Both uncertainty contributions (a) and (b) will depend on the specific gas quality in question. (3.g.1 % 3 ≈ 0. the C2-component with ±0. It is here assumed that the corresponding uncertainty figures correspond to rectangular probability distributions and 100 % confidence levels (k = 3 .3 Gas compressibility factors The relative combined standard uncertainty of the ratio of the gas compressibility factors.USM fiscal gas metering stations NPD.2. it can be determined e. Examples have shown that this uncertainty can be all from negligible to around 1 %.1. 1994] is used for calculation of Ẑ .6. 1995c] is used for Ẑ 0 . The corresponding relative standard uncertainty of the estimate Ẑ 0 due to model uncertainty becomes E Z 0 . monthly instead of being measured online. To give a typical value to be representative in all cases is not possible. NFOGM.030 %. cf. For the specific example discussed here.13). CMR (December 2001) 116 The calculated expanded and relative expanded uncertainties (specified at 95 % confidence level and a normal probability distribution. and ISO 6976 [ISO.2. 4. cf. especially for the C2 and C3 components. is given by Eq. The variations in the other gas components are smaller. cf.g. and also on the pressure and temperature in question.3) are 9. Section 3. and can be of relevance here if the gas composition data are fed manually to the flow computer e.5 %. E Z / Z 0 . These values are further used in Section 4. As described in Section 3.3). The relative standard uncertainty of the estimate Ẑ due to model uncertainty is then E Z . Such variation ranges can be observed in practice [Sakariassen. and have been neglected here for simplicity.1. 3. 10 gas composi- .0522 % 3 ≈ 0. Section B. respecively. and is evaluated in the following. with k = 2.mod = 0. The uncertainty will depend on (a) the uncertainty of the GC measurement. and (b) the actual variations in the gas composition. 05 % 1 2.ana (k = 1. The value EZ.878⋅10-6 Relative combined standard uncertainty EZ / Z 0 0.6. Distribut. standard sens.170 % Relative expanded uncertainty (95 % confidence level) k ⋅ EZ / Z 0 0. due to GC measurement uncertainty. NFOGM. the influence of such uncertainty on the analysis uncertainty of Z is quite small at low pressures when Z is very close to unity.01 % 91.1 % 95 % (norm) 2 0. Table 4. This value is further used in Section 4. and the Z-factor has been calculated for each of them.g. or natural variations in gas composition). CMR (December 2001) 117 tions within the limits selected for C1. The figures used for the input uncertainties are those given in the discussion above. Therefore. The calculated relative expanded uncertainty (specified at 95 % confidence level and a normal probability distribution. Z 0.7. Model uncertainty. (3. Relative Rel. k uncertainty coeff. 91 It can be shown (by a Monte Carlo type of statistical analysis. the resulting calculated standard deviation using this method is less than 0. .339 %.76⋅10-8 Analysis uncertainty. such as for Ẑ 0 . level Cov. Z0/Z. variance uncert. 5.50⋅10-7 Model uncertainty. This number has been used here as the relative standard uncertainty EZ.13). (Corresponds to Fig.339 % 90 It should be noted that the selected variation limits are larger than what can often be found in practice. Z0 0 67 % (norm) 1 0 Sum of relative variances E Z2 / Z 0 2.16 % 1 2.16 % 67 % (norm) 1 0.56⋅10-6 Analysis uncertainty. with k = 2. Section B. Z0 0. Z 0..16 % is therefore far from being a best case value.9 Sample uncertainty budget for the ratio of compressibility factors. At standard reference conditions.ana = 0 is used here.ana has been neglected in the current example. The resulting calculated standard deviation for the Z factor at line condition is about 0.USM fiscal gas metering stations NPD.Handbook of uncertainty calculations .13).16 % 90.3) is 0. cf. at line conditions and at standard reference conditions. so that EZ0. calculated according to Eq.) Input uncertainty Combined uncertainty Source Relative Conf. C2 and C3 have been used.ana = 0. Ẑ ≈ 1 .052 % 95 % (norm) 2 0. (3. Section B. Relative expanded & fact. cf.9 for evaluation of the expanded uncertainty of the ratio of the gas compressibility factors according to Eq. A sample uncertainty budget is given in Table 4.3).026 % 1 6. EZ0. as used above) that in spite of possible significant uncertainty in the gas compostion (such as e. especially when an online GC is used. [Solartron. 1999]. U ( ρˆ misc ) Negligible (see text) [Matthews.14). 1999. U ( ρˆ misc ) Not specified (see text) [Solartron. §3. Coverage Reference tainty factor.4 Density measurement The combined standard uncertainty of the density measurement.USM fiscal gas metering stations NPD. U ( ρˆ misc ) Not specified (see text) [Solartron.1] < ±0. §3. §1.3] Deposits.2] Temperature correction model. (3. U ( ρˆ misc ) Negligible (see text) [Solartron. 1999.1. U ( ρˆ rept ) Within ±0.3.8] Gas viscosity. etc.1] Densitometer “accuracy”. §3. 1999. §1. §1. [Solartron. is given by Eq.2. U ( ρˆ misc ) Negligible (see text) [Matthews. §1. u c ( ρˆ ) . This expression is evaluated in the following. U ( ρˆ misc ) Negligible (see text) [Solartron. gas) Repeatability. CMR (December 2001) 118 4. U ( ρˆ misc ) Not specified (see text) [Solartron.1] Self induced heat effects. for the example considered here.1] Full scale density range . [Solartron.400 kg/m3 §A.v. U ( ρˆ u ) < ±0. U ( T̂c ) 0. 1999. §A.3. (nitrogen) [Solartron. U ( ρˆ misc ) Negligible (see text) [Solartron.10 as specified in the data sheet [Solartron.001 kg/m3/oC §A. k Calibration temperature (air) .v. used as input to the uncertainty calculations given in Table 4. 1999. Quantity or Source Value or Expanded uncer. 1999. U ( ρˆ misc ) Negligible (see text) [Solartron. U ( ρˆ misc ) Not specified (see text) [Solartron.2. §A.1 % of m. 1994] The contributions to the combined standard uncertainty of the density measurement are described in the following.01 % of full scale [Solartron.element.1] Stability .8] Corrosion. NFOGM. §1. 1999]. 1994] Vibration effects.10 Performance specifications of the Solartron 7812 Gas Density Transducer [Solartron. U ( ρˆ temp ) < 0. 1999.1] Sample flow effects. 1999. Calibration temperature. §1. 1999.Handbook of uncertainty calculations .3. 1999. 1999] Pressure effect. (nat. 1999. 1999. 1 . Performance specifications for the Solartron 7812 gas density transducer are given in Table 4.8] Condensation. As the confidence level of the expanded uncertainties . §3.1] VOS correction model. §1.1 oC (at 20 oC) §7.3.8] Power supply effects.3.11. 1999.3.3.15 % of m. 20 °C §A.3.3. Table 4.2] density [Solartron. U ( ρˆ misc ) Not specified (see text) [Solartron. 1999. 15 % of reading in natural gas. 1999.USM fiscal gas metering stations NPD.027 % for the relative standard uncertainty of the densitometer “accuracy”. This includes linearity. Line temperature.053 %. Repeatability. 5.1 oC at 20 oC. cf. u( ρˆ rept ) : In the manufacturer instrument manual [Solartron. u( ρˆ u ) = U ( ρˆ u ) 2 [ ] = 0. and reference instruments uncertainty. 1999]. That is. By calibration of the the densitometer in an accredited calibration laboratory. . repeatability. Pressure difference. 7. Line pressure. 6. and less than ±0. The relative standard uncertainty is u( ρˆ u ) ρˆ u = 0. at a 95 % confidence level (k = 2) [Eide. u c ( T̂d ) : The expanded uncertainty of the densitometer temperature is taken from Table 4. In practice the density sampling system is designed so that the pressure deviation 92 The manufacturer's uncertainty specification is used here. That is. densitometer to line. 2001a]. u( ρˆ rept ) = U ( ρˆ rept ) 2 = 0. u( T̂c ) : The uncertainty of the calibration temperature is specified in the manufacturer instrument manual [Solartron. Example of a calibration certificate specification for the densitometer "accuracy" U ( ρˆ u ) may be e. reading uncertainty. u( ∆P̂d ) : The densitometer pressure P̂d is not measured.8. Calibration temperature. hysteresis.2] as 0.02 kg / m 3 . That is.6. 4. NFOGM. Such values correspond to 0. u c ( T̂ ) : The expanded uncertainty of the line temperature is taken from Table 4. §A.443 kg / m 3 2 = 0. (2.15 ⋅ 10 −2 ⋅ 82.075 %. u( ρˆ u ) : The densitometer “accuracy” is specified in the manufacturer instrument manual [Solartron.2. and assumed to be equal to the line pressure P̂ . §1. the repeatability is specified to be within ±0.8.0618 kg/m3 92. This includes uncertainties of Eq. 3. 2.01 % of full scale density. Densitometer temperature.1 o C 2 = 0. 1. the densitometer "accuracy" may be significantly reduced. u c ( P̂ ) : The expanded uncertainty of the line pressure is taken from Table 4. with a normal probability distribution (k = 2. this is here assumed to be 95 %. Densitometer “accuracy”.1 % of reading in nitrogen.014-0.3. CMR (December 2001) 119 is not specified in [Solartron.g.2]. for the density range 25-250 kg/m3. §7. The density range of the 7812 densitometer is given to be 1 to 400 kg/m3. 1999. 0. 1999.23).01⋅ 10 −2 ⋅ 400 2 = 0.05 oC.1] as being less than ±0.3). Section B.15 % 2 = 0.Handbook of uncertainty calculations . u( T̂c ) = U ( T̂c ) 2 = 0.027-0. cf.577 m/s. cf. 10. ∆P̂d = 20 mbar is used as a representative example. 8. one obtains u( τˆ ) = U ( τˆ ) 3 = 0.0577 µs. one obtains u( ∆P̂d ) = U ( ∆P̂d ) 3 = 0.02 % of the line pressure.577 m/s. ∆P̂d can be positive or negative. As explained in Section 3. cf. Assuming a Type A uncertainty. tentatively (10 MHz oscillator) [Eide. u( ĉ d ) : As described in Section 2.3). Section B. 2001]. u( τˆ ) : The uncertainty of the periodic time τ involved in the VOS correction depends on the time resolution of the flow computer. at a 100 % confidence level and a rectangular probability distribution within the range ±0. Section B. at a 100 % confidence level and a rectangular probability distribution within the range ±1 m/s (k = 3 .Handbook of uncertainty calculations . one does not rely on the particular method used to estimate cd in the metering station. at a 100 % confidence level and a rectangular probability distribution within the range ±1 m/s (k = 3 . Periodic time. densitometer gas. is relatively small. evaluation of u( ĉ d ) according to these (or other) methods is not a part of the present Handbook.4. tentatively. Assuming a Type B uncertainty. cd: the “USM method” and the “pressure/density method”.1 µs (k = 3 . depending on the actual installation [Sakariassen. there are at least two methods in use today to obtain the VOS at the density transducer. calibration gas.3). which is here set to 0.3. NFOGM. 2001a]. Hence. ∆P̂d .2). CMR (December 2001) 120 between the densitometer and line.2. 2001a].1 µs.02 bar 3 = 0. For the present case ( P̂ = 100 bara. at a 100 % confidence level and a rectangular probability distribution within the range ±20 mbar (k = 3 .USM fiscal gas metering stations NPD. (2. VOS. VOS. one obtains u( ĉ c ) = U ( ĉc ) 3 = 1 3 ≈ 0. cf. . Assuming a Type B uncertainty. one obtains u( ĉ d ) = U ( ĉ d ) 3 = 1 3 ≈ 0.3). Section B.4. u( ĉc ) : The uncertainty of the sound velocity estimate of the calibration gas is tentatively taken to be 1 m/s. The uncertainty of the VOS estimate in the density transducer is here taken to be 1 m/s.0115 bar. cf.1 µs 3 = 0. Assuming a Type B uncertainty. Table 4. when using Eq.3). P̂ [Eide. Section B. 9. Tests with densitometers have indicated a pressure deviation ∆P̂d of up to 0.25) for VOS correction. one obtains u( K̂ d ) = U ( K̂ d ) 3 = 2100 µm 3 = 1212 µm. (2.Handbook of uncertainty calculations .1] as being less than 0. Eq.25). NFOGM. For K̂ d = 21000 µm (cf. is specified in the manufacturer instrument manual [Solartron. 93 For densitometers with vibrating element made from other materials than Ni-span C.25). 1994]. and the temperature correction model given by Eq. Section B. 1994]. 1994].001⋅ 48 ] 2 = 0.4). Temperature correction model. at a 100 % confidence level and a rectangular probability distribution within the range ±2100 µm (k = 3 . and will have an uncertainty. That is. Table 4. is not specified in the manufacturer instrument manual [Solartron.24). Eq. That is. u( ρˆ temp ) = U ( ρˆ temp ) 2 = [0.024 kg / m 3 . In the present calculation example the uncertainty of the VOS correction model is neglected for the Solartron 7812 densitometer. . VOS correction constant. that gives U ( K̂ d ) = 2100 µm. However. u( K̂ d ) : A figure for the uncertainty of the dimensional constant K̂ d used in the VOS correction has not been available for the present study. and will have an uncertainty.24). §A. as a reasonable example [Eide. For high-accuracy densitometers like the Solartron 7812. 1999. cf. CMR (December 2001) 121 11. the VOS correction model itself is not perfect (among others due to use of a calibration gas.001 kg/m3/oC. The uncertainty of the temperature correction model itself. this effect is largely eliminated using the VOS correction model given by Eq. Assuming a Type B uncertainty. (2. Both of these affect the resonance frequency [Matthews. The uncertainty of the VOS correction model itself. the VOS correction model contribution to u( ρˆ misc ) is set to zero. 1999].3). this effect is largely eliminated using Ni-span C stainless steel93. VOS correction model.USM fiscal gas metering stations NPD. 2001a]. For high-accuracy densitometers like the Solartron 7812. u( ρˆ misc ) : For gas densitometers the fluids are very compressible (low VOS). (2. 12. (2. u( ρˆ temp ) : Temperature changes affect both the modulus of elasticity of the vibrating element. the temperature effect may be considerably larger [Matthews. However. 1999]. A tentative uncertainty figure of 10 % is used here. the temperature correction model itself is not perfect. [Matthews. and its dimensions. with another VOS than the line gas in question). 13. and VOS correction is important [Solartron. Consequently. That is. 1999]. the pressure effect contribution to u( ρˆ misc ) is set to zero. will maintain its properties for many years”. but. also [Geach. Pressure effect. §1. 1994]. in the present calculation example this uncertainty contribution is assumed to be negligible for the Solartron 7812 densitometer. NFOGM. the sensing element can be removed and cleaned”94.element. and care should be taken to ensure that the process gas is suitable for use with materials of construction. 17. should deposition take place.” Cf. This cylinder is manu- factured from one of the most stable metals. .3. the uncertainty contribution to u( ρˆ misc ) which is related to deposits is set to zero. “another problem with the gas transducers can be the presence of black dust like particles on the walls of the sensing element. Deposits. 1994]. 15. 1994]. CMR (December 2001) 122 14. In the present calculation example this uncertainty contribution is neglected for the Solartron 7812 densitometer. u( ρˆ misc ) : The uncertainty of the pressure effect is not specified in the manufacturer instrument manual [Solartron. The possibility of deposition is reduced by the use of filters. According to [Campbell and Pinto. Scratches or denting during the cleaning procedure reduces the element to scrap. In the present calculation example this uncertainty contribution is neglected for the Solartron 7812 densitometer. 94 The risk of damaging the element in case of dismantling and clening offshore by unexperienced personnel may be large [Campbell and Pinto.3] “The long term stability of this density sensor is mainly governed by the stability of the vibrating cylinder sensing element.Handbook of uncertainty calculations . u( ρˆ misc ) : The instrument manual states that [Solartron. 16. Stability . and being unstressed.8] “Condensation of water or liquid vapours on the sensing element will cause effects similar to deposition of solids except that the effects will dissappear if re-evaporation takes place.USM fiscal gas metering stations NPD. so all forces are balanced”. u( ρˆ misc ) : The instrument manual states that [Solartron. u( ρˆ misc ) : In the instrument manual it is stated that [Solartron. 1999. 1999. These particles can often cause pitting on the sensing element which renders the cylinder as scrap”. §3. 1994]. §1. “for vibrating cylinders there is no pressure effect on the reson- cance frequency because the fluid surrounds the vibrating element. That is.3. 1999. According to [Mat- thews. Condensation and liquid contamination. That is.3] “Deposition on the cylinder will degrade the long term stability. the long time stability effects contribution to u( ρˆ misc ) is set to zero. In the present calculation example this uncertainty contribution is neglected for the Solartron 7812 densitometer. which has the effect of stopping the transducer vibrating.” In the present calculation example this uncertainty contribution is neglected for the Solartron 7812 densitometer. u( ρˆ misc ) : In the instrument manual it is stated that [Solartron. CMR (December 2001) 123 According to [Campbell and Pinto. That is. NFOGM.8] the 7812 is insensitive to variations in power supply. §3. Power supply effects. the uncertainty contribution to u( ρˆ misc ) which is related to gas viscosity is set to zero. 1999.Handbook of uncertainty calculations . §3.3.3] “Corrosion will degrade the long term stability. That is.USM fiscal gas metering stations NPD. 1999. The presence of this liquid usually indicates a problem with the lub oil seals of the export compressors”. “transducers which are returned for calibration have been found on many occasions to contain large quantities of lu- bricating type oil. Corrosion. Use of anti-vibration gasket will reduce the effects of vibration by at least a factor of 3. at levels up to 10g and 2200 Hz”.5g. u( ρˆ misc ) : In the instrument manual it is stated that [So- lartron. That is. In the present calculation example this uncertainty contribution is neglected for the Solartron 7812 densitometer. Vibration effects. For gas densitometers the effect of viscosity is so small that it is virtually impossible to measure at anything but very low densities [Matthews. Gas viscosity. §1. . 18. the uncertainty contribution to u( ρˆ misc ) which is related to vibration effects is set to zero. and care should be taken to ensure that the process gas is suitable for use with materials of construction. 1994]. 19. In the present calculation example this uncertainty contribution is neglected for the Solartron 7812 densitometer. 20. but levels in excess of this may affect the accuracy of the readings. That is. the uncertainty contribution to u( ρˆ misc ) which is related to corrosion is set to zero. the uncertainty contribution to u( ρˆ misc ) which is related to power supply effects is set to zero. 1994]. In the present calculation example this uncertainty contribution is neglected for the Solartron 7812 densitometer. 21. the uncertainty contribution to u( ρˆ misc ) which is related to condensation is set to zero. u( ρˆ misc ) : Viscosity has the effect of damping all vibrating- element transducers which causes a small over-reading in density. That is.8] “The 7812 can tolerate vibration up to 0. 1999. u( ρˆ misc ) : In the instrument manual it is stated that [Solar- tron. 1 m/s 100 % (rect) 3 0. (3.73⋅10-4 (kg/m3)2 Densitometer tempera.14).11 kg/m3 95 % (norm) 2 0.24⋅10-9 (kg/m3)2 Model constant. 0. (Corresponds to Figs.577 m/s 0. calculated according to Eq.23.16 bar 95 % (norm) 2 0. U ( τˆ ) 0. 1999.76⋅10-4 (kg/m3)2 model. uncert. 0.1 µs 100 % (rect) 3 0.0115 bar -0.8] “since the power consumption is extremely small. Variance Distribut.1 oC 95 % (norm) 2 0. 5. 20 mbar 100 % (rect) 3 0.Handbook of uncertainty calculations .25⋅10-4 (kg/m3)2 Temperature correction 0.07805 kg/m3 Expanded uncertainty (95 % confidence level. 1 m/s 100 % (rect) 3 0.7⋅10-10 (kg/m3)2 Pressure difference.15 oC 95 % (norm) 2 0.00394 5. U ( K̂ d ) 2100 µm 100 % (rect) 3 1212 µm 1.9 and 5.002365 1.048 kg/m3 95 % (norm) 2 0.092⋅10-3 (kg/m3)2 Combined standard uncertainty u c ( ρˆ ) 0. In the present calculation example this uncertainty contribution is neglected for the Solartron 7812 densitometer. Table 4.08 bar -0. 0. calibration gas. That is.77⋅10-4 (kg/m3)2 ture.816037 8. U ( ρˆ u ) Repeatability.05 oC -0.87⋅10-6 (kg/m3)2 U ( ĉ d ) Periodic time. level Cov. u( ρˆ misc ) : In the instrument manual it is stated that [Solartron..252576 3.0765 oC 0. NFOGM.0577 µs -0.18⋅10-6 (kg/m3)2 U ( ĉ c ) VOS. U ( ρˆ rept ) 0.253875 3.19 % .11.0765 oC 0.000163 1.745⋅10-3 (kg/m3)2 racy”. U ( ∆P̂d ) VOS. uncertainty coeff. Self induced heat effects. k = 2) U ( ρˆ ) 0.577 m/s -0.000611 1. Standard Sens. U ( P̂ ) 0.0618 kg/m3 0. uncertainty Neglected 0 kg/m3 1 0 (kg/m3)2 contributions. §3. Sample uncertainty budget for the measurement of the gas density using the Solartron 7812 Gas Density Transducer [Solartron.) Input uncertainty Combined uncertainty Source Expand.15 oC 95 % (norm) 2 0.024 kg/m3 1 5. den. U ( ρˆ misc ) Sum of variances u c2 ( ρˆ ) 6.88⋅10-10 (kg/m3)2 U ( T̂c ) Line temperature. & fact. Conf. 1999]. the self induced heat may be neglected”. U ( ρˆ temp ) Miscell. U ( T̂ ) 0.USM fiscal gas metering stations NPD.989734 3. the uncertainty contribution to u( ρˆ misc ) which is related to heating is set to zero.62 kg/m3 Relative expanded uncertainty (95 % confidence level) U ( ρˆ ) ρˆ 0.88⋅10-5 (kg/m3)2 sitometer-line.02 kg/m3 1 4.89⋅10-5 5. CMR (December 2001) 124 22.04 kg/m3 95 % (norm) 2 0. U ( T̂d ) Line pressure.0⋅10-4 (kg/m3)2 Calibration temperature.000274 1. k Densitometer “accu.1561 kg/m3 Operating density ρ 81. densitometer gas. 3. Section B. with k = 2. This expression is evaluated in the following. and another coverage factor k would alter the densitometer uncertainty. 2001]. NFOGM.15 %. The figures used for the input uncertainties are those given in the discussion above. CMR (December 2001) 125 23.3 Flow calibration uncertainty The relative combined standard uncertainty of the USM flow calibration.3) is 0. “All reso- nant element sensors will be affected by flow rate in some way. This figure is not based on any uncertainty analysis of the calorific value estimate. Section C. it accounts for the contributions from the flow calibration laboratory. 4. Section B.Handbook of uncertainty calculations .19 %. at a level corresponding to NPD regulation requirements [NPD. This value is further used in Section 4.14). cf. the uncertainty contribution to u( ρˆ misc ) which is related to flow in the density sample line is set to zero.5 Calorific value The relative expanded uncertainty of the superior (gross) calorific value.6. u( ρˆ misc ) : According to [Matthews. 2001] (cf. However this effect is very small and providing the manufacturers recommendations are followed then the effects can be ignored”. the output will generally give a positive over-reading of density and the readings will become more unstable.1). Sample line flow effects. the deviation factor. and the USM repeatability in flow calibration.6. That is.16).11 for evaluation of the expanded uncertainty of the pressure measurement according to Eq. k ⋅ E H S . 4. 1994].USM fiscal gas metering stations NPD. at a 95 % confidence level and a normal probability distribution (k = 2. this is an assumption. (3. As Solartron has not specified k. . 1999] data sheet corresponds to a 95 % confidence level (k = 2). As flow rate increases.3). is here taken to be 0. cf. Note that this value is calculated under the assumption that the input uncertainties taken from the [Solartron.2. The relative expanded uncertainty value calculated above is further used in Section 4. E cal . As described in Section 3. The calculated relative expanded uncertainty (specified at a 95 % confidence level and a normal probilility distribution. is given by Eq. In the present calculation example this uncertainty contribution is neglected for the Solartron 7812 densitometer. A sample uncertainty budget is given in Table 4. but is taken as a tentative example [Sakariassen. (3. NFOGM. …. at a 95 % confidence level and a normal probability distribution (k = 2. (2. DevC . j DevC .99937 . at a 100 % confidence level and a rectangular probability distribution within the range ± DevC .10 qmax 1.0. cf.00009 0.00057 0. 2.3 % of the measured value95.15 %. j and E K for this example.70 qmax 0. such as Bernoulli/Westerbork and Advantica [Sakariassen.99991 .USM fiscal gas metering stations NPD.1 Flow calibration laboratory The expanded uncertainty of current high-precision gas flow calibration laboratories is taken to be typically 0.3. 5. j Table 4.005 % 4 0.3. j ) q̂ ref . Section B. As an example evaluation of the uncertainty of the deviation factor estimate K̂ dev .033 % Table 4.25 qmax 0. 95 Some flow calibration laboratories operate with flow rate dependent uncertainty figures. cf. dev . .8).00689 0.00005 0. Input data for these calculations are given in [AGA-9. (Corresp.01263 0.720 % 2 0. j EK rate no. according to Eqs. j = 1.0. the AGA-9 report results given in Fig. j test flow rates discussed in the AGA-9 report. M.12. j = [U ( q̂ ref . respectively.3. j .99995 .3).99943 .2. M. Appendix D]. 2001].00063 0.12 gives K̂ dev . calculated for the M = 6 dev .3). j (k = 3 .40 qmax 0. Section B.3 % 2 = 0. 2.00689 0.0. j = 1. one obtains E ref . 4. as an example.1 are used here. 1998.) dev . The example serves to illustrate the procedure used for evaluation of E K .1. Assuming that this figure applies to all the M test flow rates. j . j 1 qmin 1.2 Deviation factor For a given application.18).Handbook of uncertainty calculations .036 % 6 qmax 0. j ) is taken to be equal to DevC .395 % 3 0. …. the deviation data DevC . CMR (December 2001) 126 4. are to be specified by the USM manufacturer. and a Type B uncertainty is assumed. j .003 % 5 0. j ] 2 = 0. (2. to parts of Fig. the expanded uncertainty U ( K̂ dev . dev . j .10) and (3. As described in Section 3.11. j Test flow Test flow rate K̂ dev .01263 0.0. Evaluation of E K for the AGA-9 example of Fig. cf. cf. and is expected to be significantly less than 0. .10qmax and 0. NFOGM.9. uncertainty evaluation is made for all test flow rates. Section B.2 % corresponds to the standard deviation of the flow rate reading.g. The repeatability in flow calibration is also to include the repeatability of the flow laboratory reference measurement. the 0.13 and 4. In practice. 2000]. j = 1.2 % of measured value. The figures used for the input uncertainties are those given in the discussion above. M.3. 2000].1 %.3. the repeatability may be expected to be flow rate dependent.70qmax.3) are 0.6.3)96. (3. [Kongsberg.36 %.Expanded uncertainty of flow calibration Sample uncertainty budgets are given in Tables 4. Assuming that this repeatability figure corresponds to a 95 % confidence level at a normal probability distribution (k = 2. e.2 %. [Daniel. As this value may be difficult to quantify.70qmax. 4. for evaluation of the expanded uncertainty of the flow calibration according to Eq. cf. 96 If the repeatability figure of 0. . 97 In the program EMU . The calculated relative expanded uncertainties (specified at a 95 % confidence level and a normal probability distribution.16). with k = 2. Section 5.4 Summary .USM fiscal gas metering stations NPD. Confidence level and probability distribution are unfortunately not available from USM manufacturer data sheets. CMR (December 2001) 127 4. for test flow rates 0. Note that in each table only a single test flow rate is evaluated97. Section B. one obtains E rept . j = [U ( q̂ rept USM .10qmax and 0. and to all the M test flow rates. cf.12.87 % and 0.13 and 4.14 apply to test flow rates 0.3 USM repeatability in flow calibration The repeatability of current USMs is typically given to be 0. a coverage factor k = 1 is to be used. cf.14. respectively.2 % figure is here taken to represent both repeatabilities. Table 6.2 % 2 = 0. j ] 2 = 0. cf.however any dependency on flow rate is not specified by manufacturers.Handbook of uncertainty calculations . …. These values are further used in Section 4. respectively. Chapter 6. cf. Tables 4.1. Table 4.USM Fiscal Gas Metering Station. j ) q̂USM . Relative expanded & fact. Relative expanded & fact. Distribut.15 % 1 2. is given by Eq. Variance uncert.41) and (3.Handbook of uncertainty calculations .15 % 1 2.1) Deviation factor 0.14.19).12) USM repeatability in flow 0.072 % 100 % (rect) 3 0.4. standard sens. k = 2) k ⋅ E cal 0. 4.13.00⋅10-6 calibration (Section 4. and effects of changes in conditions from flow calibration to field operation (systematic effects related to meter body dimensions. CMR (December 2001) 128 Table 4. calculated according to Eq.036 % 1 1.1 % 1 1. Flow calibration 0.11.1) Deviation factor 0. 0. Sample uncertainty budget for USM flow calibration (example).1 Repeatability (random transit time effects) The relative combined standard uncertainty representing the repeatability of the USM in field operation.3 % 95 % (norm) 2 0.30⋅10-7 (Table 4. standard sens. and is evaluated in the following. (3. EUSM .38⋅10-6 Relative combined standard uncertainty E cal 0. level Cov. is given by Eqs. (3.10qmax (cf.USM fiscal gas metering stations NPD.4. Input uncertainty Combined uncertainty Source Relative Conf. calculated according to Eq. it accounts for the contributions from the USM repeatability in field operation. Table 4.16).4 USM field uncertainty The relative combined standard uncertainty of the USM in field operation.3. Sample uncertainty budget for USM flow calibration (example). 0.85⋅10-6 Relative combined standard uncertainty E cal 0. 5.25⋅10-6 laboratory (Section 4.. Relative Rel. Variance uncert.25⋅10-6 laboratory (Section 4.45). and is due to random . Relative Rel.) Input uncertainty Combined uncertainty Source Relative Conf.2 % 95 % (norm) 2 0. Distribut.16).3.69 % 100 % (rect) 3 0.37 % 4. (3. Flow calibration 0.12) USM repeatability in flow 0.00⋅10-6 calibration (Section 4.3 % 95 % (norm) 2 0.183 % Relative expanded uncertainty (95 % confidence level. (3. As described in Section 3. for a single test flow rate.70qmax (cf.3. level Cov. k uncertainty coeff.1 % 1 1.. for a single test flow rate. k uncertainty coeff.87 % Table 4.60⋅10-6 (Table 4.3.434 % Relative expanded uncertainty (95 % confidence level.3) Sum of relative variances 2 E cal 18.2 % 95 % (norm) 2 0. NFOGM.395 % 1 15. Table 4. E rept . transit times and integration method).12).3) Sum of relative variances 2 E cal 3. (Corresponds to Fig. k = 2) k ⋅ E cal 0.12). Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 129 transit time effects. In this description, one has the option of specifying either E rept or u( t̂ 1random i ) , cf. Section 3.4.2 and Table 3.8. Both types of input are addressed here. Alternative (1), Specification of flow rate repeatability: In the first and simplest approach, the repeatability of the flow measured rate, E rept , is specified direcly, using the USM manufacturer value, typically 0.2 % of measured value, cf. e.g. [Daniel, 2000], [Kongsberg, 2000], Table 6.1. In practice, the repeatability of the flow rate may be expected to be flow rate dependent, cf. Section 3.4.2, - however a possible dependency on flow rate is not specified in these USM manufacturer data sheets. Confidence level and probability distribution for such repeatability figures are unfortunately not available in USM manufacturer data sheets, cf. Chapter 6. Assuming that this repeatability figure corresponds to a 95 % confidence level at a normal probability distribution (k = 2, cf. Section B.3), and to all the M test flow rates, one obtains E rept = 0.2 % 2 = 0.1 %, at all flow rates, cf. Table 4.15 98. Table 4.15. Sample uncertainty budget for the USM repeatability in field operation (example), for the simplified example of constant E rept over the flow rate range. (Corresponds to Fig. 5.12.) Test Test Test Rel. exp. Conf. level Cov. Relative Rel. exp. rate flow flow uncertainty & fact., standard uncertainty no. velocity rate kE rept Distribut. k uncertainty, (95 % c.l.) (repeatability) E rept 2 E rept 1 0.4 m/s qmin 0.2 % 95 % (norm) 2 0.1 % 0.2 % 2 1.0 m/s 0.10 qmax “ “ “ “ “ 3 2.5 m/s 0.25 qmax “ “ “ “ “ 4 4.0 m/s 0.40 qmax “ “ “ “ “ 5 7.0 m/s 0.70 qmax “ “ “ “ “ 6 10.0 m/s qmax “ “ “ “ “ Alternative (2), Specification of transit time repeatability: In the second approach, E rept is calculated from the specified typical repeatability of the transit times, represented by u( t̂ 1random i ) , the standard deviation of the measured transit times (after possible time averaging), at the flow rate in question. Normally, data for u( t̂ 1random i ) are today not available from USM manufaturerer data sheets. However, such data should be readily available in USM flow computers, and may hopefully be available also to customers in the future. 98 Note that the simplification used in this example calculation is not a limitation in the program EMU - USM Fiscal Gas Metering Station. In the program, E rept can be specified individually at each test flow rate, cf. Section 5.10.1. If only a single value for E rept is available (which may be a typical situation), this value is used at all flow rates. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 130 To illustrate this option, a simplifed example is used here where the standard deviation u( t̂ 1random i ) is taken to be 2.5 ns, at all paths and flow rates. Note that in practice, the repeatability of the measured transit times may be expected to be flow rate dependent (increasing at high flow rates, with increasing turbulence). The repeatability may also be influenced by the lateral chord position. Consequently, this example value represents a simplification, as discussed in Section 3.4.2 99. For this example then, Table 4.16 gives E rept at the same test flow rates as used in Table 4.12, calculated according to Eqs. (3.41), (3.43) and (3.45). Table 4.16. Sample uncertainty budget for the USM repeatability in field operation (example), calculated according to Eqs. (3.41), (3.43) and (3.45), for the simplified example of constant U ( t̂ 1random i ) = 5 ns over the flow rate range. (Corresponds to Fig. 5.13.) Test Test Test Expanded Conf. level Cov. Standard Rel. comb. Rel. exp. rate flow flow uncertainty, & fact., uncertainty, standard uncertainty no. velocity rate U ( t̂ 1random ) Distribut. k u( t̂ 1random ) uncertainty, (95 % c.l.), i i (repeatability) E rept 2 E rept 1 0.4 m/s qmin 5 ns 95 % (norm) 2 2.5 ns 0.158 % 0.316 % 2 1.0 m/s 0.10 qmax “ “ “ “ 0.063 % 0.126 % 3 2.5 m/s 0.25 qmax “ “ “ “ 0.025 % 0.050 % 4 4.0 m/s 0.40 qmax “ “ “ “ 0.016 % 0.031 % 5 7.0 m/s 0.70 qmax “ “ “ “ 0.009 % 0.018 % 6 10.0 m/s qmax “ “ “ “ 0.006 % 0.012 % In Sections 4.4.6 and 4.6, Alternative (1) above is used (Table 4.15), i.e. with specification of the manufacturer value E rept = 0.2 %. 4.4.2 Meter body The relative combined standard uncertainty of the meter body, Ebody ,∆ , is given by Eq. (3.21), and is evaluated in the following. Only uncorrected changes of meter body dimensions from flow calibration to field operation are to be accounted for in Ebody ,∆ . The user input uncertainties to Ebody ,∆ are the standard uncertainties of the temperature and pressure expansion coefficients, u( αˆ ) and u( βˆ ) , respectively, cf. Section 3.4.1. With respect to pressure and temperature correction of the meter body dimen- 99 Note that the simplification used in this example calculation is not a limitation in the program EMU - USM Fiscal Gas Metering Station. In the program, u( t̂ 1random i ) can be specified individually at each test flow rate, cf. Section 5.10. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 131 sions from flow calibration to field operation, two options have been implemented in the program EMU - USM Fiscal Gas Metering Station: (1) P and T corrections of the meter body are not used by the USM manufacturer, and (2) P and T corrections are used. Both cases are described in the calculation examples addressed here. Alternative (1), P & T correction of meter body not used: When pressure and temperature corrections are not used, the dimensional change (expansion or contraction) of the meter body itself becomes the uncertainty, cf. Section 3.4.1. The contributions to the combined standard uncertainty of the meter body are described in the following. 1. Pressure expansion coefficient, u( βˆ ) : Eq. (2.19) is based on a number of approximations that may not hold in practical cases. First, there is the question of which model for β that is most correct for a given installation (pipe support), cf. Section 2.3.4 and Table 2.6. Secondly, there is the question of the validity of the linear model used for KP, cf. Eq. (2.17). Here a tentative figure of 20 % for U ( βˆ ) is taken, as an example. β is there given from Eq. (2.19) and Table 4.3 as βˆ = 0.154 ( 8.4 ⋅ 10 −3 ⋅ 2 ⋅ 10 6 ) bar-1 = 9.166⋅10-6 bar-1. Assuming a Type B uncertainty, at a 100 % confidence level and a rectangular probability distribution within the range ±20 % (k = 3 , cf. Section B.3), one obtains u( βˆ ) = U ( βˆ ) 3 = 0.2 ⋅ 9.166 ⋅ 10 −6 bar −1 3 = 1.05848⋅10-6 bar-1. 2. Pressure difference, u c ( ∆P̂cal ) : In the present example, the pressure difference between flow calibration and line conditions is ∆P̂cal = 50 bar, cf. Table 4.2. Without meter body pressure correction, and assuming 100 % confidence level and a rectangular probability distribution within the range ±50 bar (k = 3 , cf. Section B.3), the standard uncertainty due to the pressure difference is given from Eq. (3.35) as u c ( ∆P̂cal ) = ∆P̂cal 3 = 50 bar 3 ≈ 28.8675 bar. 3. Temperature expansion coefficient, u( αˆ ) : For the uncertainty of the temperature expansion coefficient (including possible inaccuracy of the linear temperature expansion model) a tentative figure of 20 % is taken here. Assuming a Type B uncertainty, at a 100 % confidence level and a rectangular probability distribution within the range ±20 % (k = 3 , cf. Section B.3), one obtains u( αˆ ) = U ( αˆ ) 3 = 0.2 ⋅ 14 ⋅ 10 −6 K −1 3 = 1.61658⋅10-6 K-1. 4. Temperature difference, u c ( ∆T̂cal ) : In the present example, the temperature difference between flow calibration and line conditions is ∆T̂cal = 40 oC, cf. Ta- Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 132 ble 4.2. Without meter body temperature correction, and assuming a 100 % confidence level and a rectangular probability distribution within the range ±40 oC (k = 3 , cf. Section B.3), the standard uncertainty due to the temperature difference is given from Eq. (3.39) as u c ( ∆T̂cal ) = ∆T̂cal 3 = 40 o C 3 ≈ 23.0940 o C. From these basic input uncertainties, one finds, from Eqs. (3.34) and (3.38) respectively, u c ( K̂ P ) = ( ∆P̂cal ) 2 u 2 ( βˆ ) + βˆ 2 u c2 ( ∆P̂cal ) (4.1) = 50 2 ⋅ 1.058 2 + 9.166 2 ⋅ 28.8675 2 ⋅ 10 −6 = 2.6983 ⋅ 10 − 4 u c ( K̂ T ) = ( ∆T̂cal )2 u 2 ( αˆ ) + αˆ 2 u c2 ( ∆T̂cal ) (4.2) = 40 2 ⋅ 1.616 2 + 14 2 ⋅ 23.0940 2 ⋅ 10 −6 = 3.2971 ⋅ 10 −4 Since K̂ P = 1 + 9.166 ⋅ 10 −6 ⋅ 50 = 1.00004583 and K̂ T = 1 + 14 ⋅ 10 −6 ⋅ 40 = 1.00056 (cf. Eqs. (3.33) and (3.37), one finds E KP = u c ( K̂ P ) K̂ P = 2.6983 ⋅ 10 −4 1.00004583 = 2.6971⋅10-4 and E KT = u c ( K̂ T ) K̂ T = 3.2971 ⋅ 10 −4 1.00056 = 3.2952⋅10-4 (cf. Eqs. (3.32). Table 4.17. Sample uncertainty budget for USM meter body, calculated according to Eqs. (3.21)-(3.40) and Table 4.2 for meter body data given in Table 4.3, and no temperature and pressure correction of the meter body dimensions. (Corresponds to Fig. 5.15.) Input Relative Conf. level Cov. Relative uncertainty expanded & fact., standard uncert. Distribut. k uncertainty Coefficient of linear 20 % 100 % (rect) 3 11.5 % temperature expansion, α Coefficient of linear 20 % 100 % (rect) 3 11.5 % pressure expansion, β Rel. comb. standard uncertainty, temperature correction, E KT 3.30⋅10-2 % Rel. comb. standard uncertainty, pressure correction, E KP 2.70⋅10-2 % Rel. comb. Rel. Rel. comb. Contributions to meter body uncertainty standard sens. standard uncertainty coeff. uncertainty Meter body inner radius, Erad ,∆ 4.26⋅10-2 % 3.6 0.1533 % Lateral chord positions, Echord ,∆ 4.26⋅10-2 % - 0.6 - 0.0256 % Inclination angles, Eangle ,∆ - 2.1⋅10-18 % Relative combined standard uncertainty (sum) Ebody ,∆ 0.1278 % Relative expanded uncertainty (95 % confidence level, k = 2) k ⋅ Ebody ,∆ 0.26 % Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 133 Eqs. (3.29)-(3.30) yield E R ,∆ = E yi ,∆ = E KP 2 + E KT 2 = 2.69712 + 3.2952 2 ⋅ 10 −4 = 4.2582⋅10-4. Insertion into Eqs. (3.31) and (3.22)-(3.24) then leads to E rad ,∆ = 0.1533 %, E chord , ∆ = -0.0256 % and E angle ,∆ = -2.1⋅10-18 %, giving Ebody ,∆ = 0.1278 % from Eq. (3.21). Table 4.17 summarizes these calculations. Alternative (2), P & T correction of meter body used: In the present description, pressure and temperature corrections are assumed to be done according to Eqs. (2.13)-(2.19), cf. Section 2.3.4. The contributions to the combined standard uncertainty of the meter body are described in the following. 1. Pressure expansion coefficient, u( βˆ ) : The uncertainty of the pressure expansion coefficient is the same as for Alternative (1) above: u( βˆ ) = 1.05848⋅10-6 bar-1. 2. Pressure difference, u c ( ∆P̂cal ) : When meter body pressure correction is used, the standard uncertainty due to the pressure difference is given from Eq. (3.36) and Table 4.6 as u c ( ∆P̂cal ) = 2u c ( P̂ ) = 2 ⋅ 0.078 = 0.110 bar. 3. Temperature expansion coefficient, u( αˆ ) : The uncertainty of the temperature expansion coefficient is the same as for Alternative (1) above: u( αˆ ) = 1.61658⋅10-6 K-1. 4. Temperature difference, u c ( ∆T̂cal ) : When meter body temperature correction is used, the standard uncertainty due to the temperature difference is given from Eq. (3.40) and Table 4.8 as u c ( ∆T̂cal ) = 2u c ( T̂ ) = 2 ⋅ 0.076 = 0.107 oC. From these basic input uncertainties, one finds, from Eqs. (3.34) and (3.38) respectively, u c ( K̂ P ) = ( ∆P̂cal ) 2 u 2 ( βˆ ) + βˆ 2 u c2 ( ∆P̂cal ) (4.3) = 50 2 ⋅ 1.05848 2 + 9.166 2 ⋅ 0.110 2 ⋅ 10 −6 = 5.2934 ⋅ 10 −5 u c ( K̂ T ) = ( ∆T̂cal ) 2 u 2 ( αˆ ) + αˆ 2 u c2 ( ∆T̂cal ) (4.4) = 40 2 ⋅ 1.61658 2 + 14 2 ⋅ 0.107 2 ⋅ 10 −6 = 6.4681 ⋅ 10 −5 ∆ = E KP 2 + E KT 2 = 5. Only systematic deviations in transit times from flow calibration to field operation are to be accounted for in Etime . k uncertainty Coefficient of linear 20 % 100 % (rect) 3 11.∆ 8.3 Systematic transit time effects The relative combined standard uncertainty Etime .4. k = 2) k ⋅ Ebody . deposits at transducer fronts.0.18. standard uncertainty. one finds E KP = u c ( K̂ P ) | K̂ P | = 5. uncertainty Meter body inner radius.∆ (due to e. (3. Erad . Alternative (1) above is used (Table 4. drift/ageing.37)).2⋅10-19 %. E KP 5.0301 % Lateral chord positions. (3.19)). Rel. Contributions to meter body uncertainty standard sens.0004583 and K̂ T = 1 + 14 ⋅ 10 −6 ⋅ 40 = 1.6 .17). etc. pressure correction. calculated according to Eqs.4644 2 ⋅ 10 −5 = 8. i. 4. Table 4. standard uncert.0251 % Relative expanded uncertainty (95 % confidence level. Eqs. Rel.2909 2 + 6. Insertion into Eqs.00056 = 6.00056 (cf.42)-(3.2 for meter body data given in Table 4. (3.6 and 4.27) then leads to E rad .32)).0.2⋅10-19 % Relative combined standard uncertainty (sum) Ebody .46).21)-(3. Relative uncertainty expanded & fact.∆ = E yi .2909⋅10-5 and E KT = u c ( K̂ T ) | K̂ T | = 6.∆ = 0.05 % In Section 4.0301 %.5 % pressure expansion. and with temperature and pressure correction of the meter body dimensions (according to Eqs. NFOGM.e.13)-(2. Table 4.∆ = -4. giving Ebody . level Cov.166 ⋅ 10 −6 ⋅ 50 = 1. E KT 6. P and T effects. temperature correction.33) and (3. standard uncertainty.025 % from Eq. Input Relative Conf. comb.29)-(3.0050 % Inclination angles.4. (2.46⋅10-3 % Rel. Sample uncertainty budget for USM meter body.∆ is given by Eqs.g. Eqs.∆ = 0.∆ .18 summarizes these calculations. α Coefficient of linear 20 % 100 % (rect) 3 11. Echord .35⋅10-3 % 3. E chord ..0050 % and E angle .5 % temperature expansion.31) and (3.USM fiscal gas metering stations NPD. standard uncertainty coeff. (3. comb.6. comb. CMR (December 2001) 134 Since K̂ P = 1 + 9.0004583 = 5.30) yield E R .21).29⋅10-3 % Rel.3536⋅10-5.4681 ⋅ 10 −5 1.22)-(3. comb. with no P and T correction of the meter body dimensions used by the USM manufacturer.40) and Table 4. (3. ∆ = -0.∆ 0.4644⋅10 (cf.∆ 0.Handbook of uncertainty calculations .). Since such timing errors may in practice not be cor- .6 0. Eangle .44) and (3.4. β Rel.3.2934 ⋅ 10 −5 1. -5 Eqs. (3.∆ 8.35⋅10-3 % . Distribut. (3. u( t̂ 1systematic i ) : u( t̂ 1systematic i ) accounts for change in the upstream transit time correction relative to the value at flow calibration conditions..4. 100 Note that if "dry calibration" is not made at the same transducer distances as used in the USM meter body. cf. 2000a]. It is hoped that. one obtains u( t̂ 1systematic i ) = U ( t̂ 1systematic i ) 3= 600 ns 3 = 346. This includes electronics/cable/transducer/diffraction time delay. However.3). diffraction effects may also need to be taken into account when evaluating u (tˆ1systematic i ) [Lunde et al. Measurement data from [Lunde et al.2). Upstream transit time.g.USM fiscal gas metering stations NPD.2. such effects are neglected in the present example. as improved knowledge on such effects is being developed. Assuming a Type B uncertainty. sound refraction effects. and change of such delays with P. 2000] indicate that for a temperature change of 35 oC a shift in transducer delay of about 1-2 µs may not be unrealistic e. It is assumed that the transducers have been "dry calibrated" at the same transducer distances as used in the USM meter body100. for transducers with epoxy front101. possible transducer cavity delay. at a 100 % confidence level and a rectangular probability distribution within the range ±600 ns (k = 3 . Table 4. 2000b]. Here. the pressure and temperature differences from flow calibration to line conditions are 50 bar and 40 oC. T and time (drift).Handbook of uncertainty calculations . cf. CMR (December 2001) 135 rected for in current USMs.. 1999. 1. .41 ns. possible deposits at transducer front. NFOGM. 1999. Examples of possible contributions are listed in Section 3. In this example. Consequenty. and the knowledge on systematic transit time effects is insufficient today. 2000a..6 µs = 600 ns is used as an example (somewhat arbitrarily).8. 101 The shift will depend among others on the method used for transit time detection [Lunde et al. including documentation of the methods by which they are obtained. such uncertainty data will be available in USM manufacturers data sheets in the future. they are to be accounted for as uncertainties. Data for u( t̂ 1systematic i ) and u( t̂ 2systematic i ) are today unfortunately not available from USM manufaturerer data sheets.2. Section B. it is assumed that all other effects than shift in transducer delay can be neglected. only a tentative calculation example is taken here. for the example of flow calibration at 10 oC and 50 bara (cf. The contributions to the combined standard uncertainty of the meter body are described in the following. In the present example. Table 3. a tentative value U ( t̂ 1systematic i ) = 0. etc. calculated at the same test flow rates as used in Table 4. Section B. calculated from Eqs.42)-(3.10 qmax 0.239 3 2.1197 0. NFOGM.∆ is determined by the uncertainty of the transit times themselves.4 Integration method (installation effects) The relative combined standard uncertainty accounting for installation effects (the integration method). (3.062 5 7. due to the sensitivity coefficients s t1i and st 2i which increase at low flow rates due to the reduced transit time difference.0 m/s qmax 0. Ex- amples of possible contributions are listed in Section 3.41 ns i U (tˆ2systematic ) 590 ns 100 % (rect) 3 340. exp. Assuming a Type B uncertainty.19 gives Etime .46). is given by Eq. Only changes of installation conditions from flow calibration to field operation are to be accounted for in E I . a tentative value U ( t̂ 2systematic i ) = 590 ns is used as an example (somewhat arbitrarily). corresponding to an expanded uncertainty of the ∆t-correction of 10 ns.) Input Given Conf. .. (Corresponds to Fig.3.).∆ is determined by the uncertainty of the ∆t-correction.64 ns. Note that Etime .0 m/s 0.44) and (3.122 4.8. k U (tˆ1systematic ) 600 ns 100 % (rect) 3 346.12.105 6 10. (95 % c.4 m/s qmin 0. Rel. drift. 5.40 qmax 0.44).43)-(3. E time . ∆ 2 E time.4206 0. At higher flow rates Etime .∆ .16. Standard uncertainty expanded & fact.3).). At low flow rates Etime .USM fiscal gas metering stations NPD.25 qmax 0. cf. Uncertainty budget for the systematic contributions to the transit time uncertainties of the 12” USM.4. Table 4.0523 0. Distribut.∆ for this example.0007 0. u( t̂ 2systematic i ) : The difference between u( t̂ 1systematic i ) and u( t̂ 2systematic i ) accounts for the uncertainty of the ∆t-correction relative to the value at flow calibration conditions (P & T effects. one obtains u( t̂ 2systematic i ) = U ( t̂ 2systematic i ) 3 = 590 ns 3 = 340.0 m/s 0.4. CMR (December 2001) 136 2.0308 0.5 m/s 0.∆ increases at low flow rates. Here. E I . Downstream transit time. (3.l. (3. level Cov.64 ns i Test Test Test flow Rel. rate no.Handbook of uncertainty calculations .841 2 1. see also Table 3. cf.70 qmax 0.19. at a 100 % confidence level and a rectangular probability distribution within the range ±590 ns (k = 3 . uncertainty uncert. Table 4.0609 0. comb.∆ . Eqs.001 4 4.0 m/s 0. flow rate standard uncertainty velocity uncertainty.47). ∆ 1 0. 20).Expanded uncertainty of USM in field operation The relative expanded uncertainty of the USM in field operation EUSM is given by Eqs. one has E I . Consequenty. Table 3. Note that in each table only a single test flow .USM fiscal gas metering stations NPD. NFOGM.6 Summary .8).3). cf. inaccuracy of the USM functional relationship. cf. etc.∆ appears to be one of the more difficult uncertainty contributions to specify.∆ . However.∆ should be determined from a (preferably) extensive empirical data set where the type of meter in question has been subjected to testing with respect to various installation conditions (bend configurations.g. CMR (December 2001) 137 In general. A great deal of installation effects testing has been carried out over the last decade.5 Miscellaneous USM effects Uncertainty contributions due to miscellaneous USM effects are discussed briefly in Section 3. cf. for a given type of meter. These are meant to account for USM uncertainties not covered by the other uncertainty terms used in the uncertainty model. where the involved relative uncertainty terms E rept .∆ = 0. Sample uncertainty budgets are given in Tables 4.3 % is used for the effect of changed installation conditions from flow calibration to field operation.6.2. 4. users and research institutions) today posess a considerable amount of information and knowledge on this subject. only a tentative uncertainty figure can be taken here. USM installation effects is today still a subject of intense testing and research. In the present calculation example such miscellaneous USM uncertainties are neglected. for evaluation of EUSM according to these equations. bend. together with theoretical / numerical investigations of such effects. 4.10. Section B. such as e. E I . meter orientation rel. Section 5. and E I .20 and 4.4.4. As an example. This value is further used in Sections 4.∆ . especially when it comes to traceability.6 and 4.USM Fiscal Gas Metering Station. miscellaneous effects can be accounted for in the program EMU .19)-(3. Etime . (3.3 % 2 = 0. However. data for E I .15 %. flow conditioners.∆ and E misc are all evaluated in the sections above.21. a figure of 0.4.Handbook of uncertainty calculations . Assuming a 95 % confidence level and a normal probability distribution (k = 2.∆ are today not available from USM manufaturerer data sheets.4.4. Ebody .. USM manufacturers (together with flow calibration laboratories. E I . Sample uncertainty budget for the USM in field operation (example). Table 4.20).26 % 95 % (norm) 2 0.) Input uncertainty Combined uncertainty Source Relative Conf. k = 2) k ⋅ EUSM 0. Variance uncert. USM repeatability in field 0. Sample uncertainty budget for the USM in field operation (example).105 % 95 % (norm) 2 0.1 % 1 1.70qmax.26 % 95 % (norm) 2 0.4) Miscellaneous effects Neglected .64⋅10-6 (Table 4.) Input uncertainty Combined uncertainty Source Relative Conf. standard sens. (Corresponds to Figs. (Corresponds to Figs. k = 2) k ⋅ EUSM 0.64⋅10-6 (Table 4.3 % 95 % (norm) 2 0.5) Sum of relative variances 2 EUSM 6.128 % 1 1.00⋅10-6 operation (Table 4.32⋅10-6 Relative combined standard uncertainty EUSM 0.25⋅10-6 effects (Section 4. Table 4.USM Fiscal Gas Metering Station. Variance uncert.. . 0.74⋅10-7 effects (Table 4.10qmax.227 % Relative expanded uncertainty (95 % confidence level.128 % 1 1.00⋅10-6 operation (Table 4.1 % 1 1. respectively. Relative expand.43⋅10-6 effects (Table 4. 5.26. Relative expand.251 % Relative expanded uncertainty (95 % confidence level. . USM repeatability in field 0.3 % 95 % (norm) 2 0.052 % 1 2.5) Sum of relative variances 2 EUSM 5. for a single test flow rate.20. for a single test flow rate.21.17) Systematic transit time 0. .20). & fact.15) Meter body uncertainty 0. . Distribut. level Cov.12.19) Installation (integration) 0.20 and 4.18 and 5.USM fiscal gas metering stations NPD. Distribut.19)-(3.12 % 1 1.19) Installation (integration) 0. cf.10.4.Handbook of uncertainty calculations .2 % 95 % (norm) 2 0.4. k uncertainty coeff.10qmax and 0.50 % Table 4. 0.. Relative Rel. cf.239 % 95 % (norm) 2 0. Relative Rel.4.25⋅10-6 effects (Section 4. Tables 4. (3.17) Systematic transit time 0. calculated according to Eqs.18 and 5.2 % 95 % (norm) 2 0.45 % 102 In the program EMU . Section 5. k uncertainty coeff.164⋅10-6 Relative combined standard uncertainty EUSM 0. . 5.4) Miscellaneous effects Neglected . & fact. CMR (December 2001) 138 rate is evaluated102. calculated according to Eqs. 1 0 (Section 4.15) Meter body uncertainty 0. NFOGM.4. (3. standard sens. level Cov.21 apply to test flow rates 0.15 % 1 2. uncertainty evaluation is made for all test flow rates.26.19)-(3.70qmax. 1 0 (Section 4.15 % 1 2. Section B. both effects can be accounted for in the program EMU . respectively. the calculated relative expanded uncertainties (specified at a 95 % confidence level and a normal probability distribution. respectively. Results are given for each of the four measurands qv. with k = 2. Section B. In the present calculation example such uncertainties are neglected. 4. respectively. qm and qe. for test flow rates 0.USM Fiscal Gas Metering Station. Q. E comm and E flocom are evaluated in the sections above.5 Signal communication and flow computer calculations Effects of uncertainties due to signal communication and flow computer calculations are described briefly in Section 3. cf. (3.1) and (3. CMR (December 2001) 139 The calculated relative expanded uncertainties (specified at a 95 % confidence level and a normal probability distribution.50 % and 0.10qmax and 0.11. Section 5. line conditions The relative expanded uncertainty of the volumetric flow rate at line conditions E qv is given by Eqs. However. cf.00 % and 0.Handbook of uncertainty calculations .23.10qmax and 0.USM fiscal gas metering stations NPD.6). for test flow rates 0. EUSM . NFOGM. Sample uncertainty budgets for E qv are given in Tables 4.3) are 1.45 %. cf.70qmax.70qmax. the relative expanded uncertainty of the USM fiscal gas metering station example is calculated.70qmax.22 and 4. 4. 4.58 %.3) are 0.1 Volumetric flow rate. on basis of the calculations given above. These values are further used in Section 4.6 Summary . for test flow rates 0.Expanded uncertainty of USM fiscal gas metering station In the following.10qmax and 0. .5.6.6. For the present example. where the involved relative uncertainty terms E cal . with k = 2. 501 % Relative expanded uncertainty (95 % confidence level.434 % 1 1. Variance uncert.2) and (3. Relative expand.2 Volumetric flow rate. level Cov. .1) and (3.50 95 % (norm) 2 0. for a single test flow rate. and flow Neglected .USM fiscal gas metering stations NPD.70qmax. 5. and flow Neglected . for a single test flow rate.6). (3.20) Signal commun.227 % 1 5. . qv (example). standard reference conditions The relative expanded uncertainty of the volumetric flow rate at standard reference conditions EQ is given by Eqs. Flow calibration 0. NFOGM.24 and 4. respectively. (3.21) Signal commun.6).58 % 4.10qmax.14) USM field operation 0. Relative expand. Sample uncertainty budgets for E Q are given in Tables 4.25 % 1 6. & fact. level Cov. (3.) Input uncertainty Combined uncertainty Source Relative Conf.5) Sum of relative variances E q2v 2. k uncertainty coeff. . 1 0 computer (Section 4. Relative Rel. 0.23. for the volumetric flow rate at line conditions.35⋅10-6 (Table 4. 0. k = 2) k ⋅ E qv 1. Input uncertainty Combined uncertainty Source Relative Conf. . (Corresponds to Fig. calculated according to Eqs.6. Flow calibration 0. Relative Rel. CMR (December 2001) 140 Table 4. Distribut. E cal .51⋅10-5 Relative combined standard uncertainty E qv 0. ET . & fact.45 % 95 % (norm) 2 0. Variance uncert.22. calculated according to Eqs. 1 0 computer (Section 4. standard sens. where the involved relative uncertainty terms E P .5) Sum of relative variances E q2v 8.514⋅10-6 Relative combined standard uncertainty E qv 0.32⋅10-6 (Table 4. Sample uncertainty budget for the USM fiscal gas metering station. For the present example.16⋅10-6 (Table 4.00 % Table 4...88⋅10-5 (Table 4. the calculated relative expanded uncertainties (specified at a 95 % confidence level and a normal . Sample uncertainty budget for the USM fiscal gas metering station.292 % Relative expanded uncertainty (95 % confidence level.28. for test flow rates 0.1) and (3. for the volumetric flow rate at line conditions.70qmax. Distribut. k = 2) k ⋅ E qv 0. qv (example). E Z / Z 0 .87 % 95 % (norm) 2 0. EUSM . E comm and E flocom have been evaluated in the sections above.25.13) USM field operation 0.Handbook of uncertainty calculations .7).37 % 95 % (norm) 2 0.187 % 1 3. standard sens. k uncertainty coeff.10qmax and 0. 5) Sum of relative variances E Q2 2.227 % 1 5.878⋅10-6 (Table 4.9) Flow calibration 0.4⋅10-7 (Table 4.7).87 % 95 % (norm) 2 0. respectively. & fact. 0. CMR (December 2001) 141 probability distribution. level Cov. Sample uncertainty budget for the USM fiscal gas metering station. P 0.07 % and 0. NFOGM. Distribut. k = 2) k ⋅ EQ 0.70qmax. Z/Z0 0. (3. level Cov.348 % Relative expanded uncertainty (95 % confidence level. & fact.25 % 1 6.07 % Table 4.14) USM field operation 0.52⋅10-8 T (Table 4.170 % 1 2. for a single test flow rate.29.16 % 95 % (norm) 2 0.37 % 95 % (norm) 2 0. Q (example). 0. Variance uncert. cf. Q (example).6) Temperature measurement.434 % 1 1.183 % 1 3.047 % 95 % (norm) 2 0. Relative expand. for test flow rates 0.208⋅10-5 Relative combined standard uncertainty EQ 0.USM fiscal gas metering stations NPD.2) and (3.70 % . standard sens. 0. P 0. for a single test flow rate. Relative Rel.8) Compressibility.35⋅10-6 (Table 4.08 % 1 6. Table 4.25. for the volumetric flow rate at standard reference conditions.) Input uncertainty Combined uncertainty Source Relative Conf.88⋅10-5 (Table 4.2) and (3. and flow Neglected . for the volumetric flow rate at standard reference conditions.869⋅10-5 Relative combined standard uncertainty EQ 0. k uncertainty coeff. . k = 2) k ⋅ EQ 1. Z/Z0 0.8) Compressibility.45 % 95 % (norm) 2 0.70qmax. .32⋅10-6 (Table 4.16 % 95 % (norm) 2 0..339 % 95 % (norm) 2 0. Pressure measurement. calculated according to Eqs.Handbook of uncertainty calculations . Sample uncertainty budget for the USM fiscal gas metering station. and flow Neglected . 5. calculated according to Eqs.70 %.20) Signal commun..10qmax.10qmax and 0.24.024 % 1 5.52⋅10-8 T (Table 4. 1 0 computer (Section 4.536 % Relative expanded uncertainty (95 % confidence level. (3.3) are 1. Section B.878⋅10-6 (Table 4. . k uncertainty coeff.9) Flow calibration 0.170 % 1 2.08 % 1 6. (Corresponds to Fig. Relative Rel. Distribut. Input uncertainty Combined uncertainty Source Relative Conf. standard sens. 1 0 computer (Section 4. Variance uncert. 0. with k = 2.339 % 95 % (norm) 2 0.16⋅10-6 (Table 4.047 % 95 % (norm) 2 0. Pressure measurement.024 % 1 5.7).50 95 % (norm) 2 0. .5) Sum of relative variances E Q2 1.6) Temperature measurement.13) USM field operation 0.4⋅10-7 (Table 4. Relative expand.21) Signal commun. k uncertainty coeff. qm (example).. and flow Neglected .61 % Sample uncertainty budgets for E m are given in Tables 4.27. 0. Relative expand. Relative Rel. ρ 0.025⋅10-7 (Table 4. . for a single test flow rate.32⋅10-6 (Table 4. calculated according to Eqs.70qmax. & fact.. CMR (December 2001) 142 4. 1 0 computer (Section 4.88⋅10-5 (Table 4.Handbook of uncertainty calculations .134) USM field operation 0. respectively. for the mass flow rate. Input uncertainty Combined uncertainty Source Relative Conf. . and flow Neglected .306 % Relative expanded uncertainty (95 % confidence level. EUSM .3) and (3. k uncertainty coeff. k = 2) k ⋅ Em 1.02 % Table 4.3) and (3.USM fiscal gas metering stations NPD.6.19 % 95 % (norm) 2 0.25 % 1 6. E cal .8). 5.27 and 5.11) Flow calibration 0.095 % 1 9.30. Variance uncert.8). where the involved relative uncertainty terms E ρ . standard sens.3 Mass flow rate The relative expanded uncertainty of the mass flow rate E m is given by Eqs.87 % 95 % (norm) 2 0.5) Sum of relative variances E m2 2. For the present example.27. Sample uncertainty budget for the USM fiscal gas metering station. level Cov.37 % 95 % (norm) 2 0.70qmax.19 % 95 % (norm) 2 0. Density measurement. .3) and (3.413⋅10-6 Relative combined standard uncertainty Em 0. & fact. Density measurement. Distribut. Table 4.8).602⋅10-5 Relative combined standard uncertainty Em 0. for test flow rates 0.095 % 1 9.10qmax.11) Flow calibration 0.025⋅10-7 (Table 4.10qmax and 0. Relative expand.5) Sum of relative variances E m2 9. for the mass flow rate.227 % 1 5. Variance uncert.183 % 1 3. level Cov. E comm and E flocom are evaluated in the sections above. ρ 0. calculated according to Eqs.20) Signal commun.26. Sample uncertainty budget for the USM fiscal gas metering station.26 and 4. (3.) Input uncertainty Combined uncertainty Source Relative Conf. (Corre- sponds to Figs.35⋅10-6 (Table 4. (3. . standard sens.434 % 1 1. Distribut. Relative Rel. k = 2) k ⋅ Em 0.21) Signal commun.45 % 95 % (norm) 2 0.50 95 % (norm) 2 0.16⋅10-6 (Table 4. (3. 0.13) USM field operation 0. qm (example). the calculated relative expanded uncertainties (specified at a 95 % confidence level and a normal . NFOGM. 1 0 computer (Section 4.510 % Relative expanded uncertainty (95 % confidence level. for a single test flow rate. 10qmax and 0.13) USM field operation 0. CMR (December 2001) 143 probability distribution.) Input uncertainty Combined uncertainty Source Relative Conf. EHs . cf. Sample uncertainty budget for the USM fiscal gas metering station. . standard sens. E comm and E flocom are evaluated in the sections above.075 % 1 5.10qmax and 0. Relative Rel. (3. for test flow rates 0.70qmax. 1 0 computer (Section 4. for the energy flow rate. 4.32⋅10-6 (Table 4. level Cov. Section B. with k = 2.926⋅10-5 Relative combined standard uncertainty E qe 0.88⋅10-5 (Table 4. Hs 0.4 Energy flow rate The relative expanded uncertainty of the energy flow rate E qe is given by Eqs.29.6. where the involved relative uncertainty terms EP .63 %.6) Temperature measurement.63⋅10-7 (Section 4.9) Calorific value. calculated according to Eqs. for test flow rates 0.3) are 1. E cal .70qmax. for test flow rates 0.4) and (3. and flow Neglected .15 % 95 % (norm) 2 0. .10qmax and 0. (Cor- responds to Fig.339 % 95 % (norm) 2 0..3) are 1. respectively. & fact. EUSM .52⋅10-8 T (Table 4.50 95 % (norm) 2 0. P 0. Sample uncertainty budgets for E qe are given in Tables 4.5) Sum of relative variances E q2e 2. 5.170 % 1 2.28. 0.878⋅10-6 (Table 4.71 %. with k = 2.Handbook of uncertainty calculations . qe (example). for a single test flow rate.047 % 95 % (norm) 2 0.20) Signal commun.03 % and 0.87 % 95 % (norm) 2 0. respectively. E Z / Z 0 .70qmax.5) Flow calibration 0.10qmax. Variance uncert.8) Compressibility.16 % 95 % (norm) 2 0. Relative expand. NFOGM.2. Distribut. k = 2) k ⋅ E qe 1.08 % and 0. For the present example.25 % 1 6. 0.31.541 % Relative expanded uncertainty (95 % confidence level.USM fiscal gas metering stations NPD.9).08 % 1 6. Section B. cf.4) and (3. the calculated relative expanded uncertainties (specified at a 95 % confidence level and a normal probability distribution. Pressure measurement.28 and 4.434 % 1 1.9). respectively.024 % 1 5. (3. Table 4. k uncertainty coeff. Z/Z0 0.08 % .4⋅10-7 (Table 4. ET . 0. (3.16 % 95 % (norm) 2 0. standard sens. T and USM flow calibration. where each group corresponds to a worksheet in the program EMU . for a single test flow rate. 0. Pressure measurement. Relative Rel. Input uncertainty Combined uncertainty Source Relative Conf.Handbook of uncertainty calculations ..356 % Relative expanded uncertainty (95 % confidence level.047 % 95 % (norm) 2 0. k = 2) k ⋅ E qe 0.5) Flow calibration 0. qe (example). and the calibration laboratory. calculated according to Eqs. more tentative input uncertainty figures have been used here.8) Compressibility.339 % 95 % (norm) 2 0.21) Signal commun. most of the necessary input uncertainties are normally available from instrument data sheets and calibration certificates.6) Temperature measurement.9). For Z/Z0 and ρ. Z/Z0 0.16⋅10-6 (Table 4. and also used above. only repeatability data have been available. . the input uncertainties are conveniently organized in eight groups.9) Calorific value. USM manufacturer data sheets and flow calibration results.14) USM field operation 0. cf. Where manufacturer specifications have not been available. .29.71 % 4.USM Fiscal Gas Metering Station. level Cov.63⋅10-7 (Section 4. & fact.227 % 1 5. Chapter 6). .2. Chapter 5. for the energy flow rate.5) and in Section 3.other significant input uncertainties (at the "detailed level") are not specified in the manufacturers' data sheets (cf.35⋅10-6 (Table 4.15 % 95 % (norm) 2 0.45 % 95 % (norm) 2 0.265⋅10-5 Relative combined standard uncertainty E qe 0. parts of the necessary specifications of input uncertainties have been available in data sheets or other documents. NFOGM. With respect to P.170 % 1 2. . and flow Neglected . Relative expand. CMR (December 2001) 144 Table 4.183 % 1 3.70qmax.37 % 95 % (norm) 2 0.878⋅10-6 (Table 4.075 % 1 5.6. As described in Section 1. k uncertainty coeff. Sample uncertainty budget for the USM fiscal gas metering station.7 Authors’ comments to the uncertainty evaluation example In the following. Variance uncert.5) Sum of relative variances E q2e 1.08 % 1 6. P 0. with respect to specification of input uncertainties.USM fiscal gas metering stations NPD. Hs 0. With respect to USM field operation.4) and (3.3 (Table 1. 1 0 computer (Section 4. Distribut. some overall comments to the above uncertainty evaluation example are given.4⋅10-7 (Table 4.024 % 1 5.52⋅10-8 T (Table 4. 5. or are calculated from other results (pressure and tem- . At line condition. Table 4. T. 1997]).2. Table 3.ana ).2. all input uncertainties used in the calculation example have been available from the manufacturer data sheet [Rosemount.22. 5. Table 3. cf. 2000]. the transmitter uncertainty and ambient temperature effects on the transmitter are relatively smaller. 2000]. the analysis uncertainty ( E Z . Some of the input uncertainties used in the present example have been available from the manufacturer data sheet [Solartron. 1999]. cf. Z/Z0. 5.6.Handbook of uncertainty calculations . Table 4.3 gives an overview of the input uncertainties to be specified for the compressibility ratio at the "detailed level". • Pressure measurement. Section 4.mod ) has been taken from [AGA-8. Section 4. The stability of the pressure transmitter clearly dominates the uncertainty of the pressure measurement.2. P. Table 3. cf. NFOGM. • Density measurement.ana ) is larger than the model uncertainty in this example. Section 4. Section 3.23.mod ). • Compressibility factor ratio calculation. CMR (December 2001) 145 In the following. 1995c] has been used for the model uncertainty ( E Z 0 . Except for the atmospheric pressure uncertainty.1 gives an overview of the input uncertainties to be specified for the pressure measurement at the "detailed level".4 gives an overview of the input uncertainties to be specified for the density measurement at the "detailed level". 5. in relation to the present example.3.7. 1994]. while the analysis uncertainty is shown to be negligible at standard reference conditions ( E Z 0 . and the stability of the temperature element. • Temperature measurement.6 and Figs. ISO 6976 [ISO.4. The analysis uncertainties are in general more complicated to estimate. Important are also the element and transmitter un- certainy.21.2 gives an overview of the input uncertainties to be specified for the temperature measurement at the "detailed level". 5. cf. ρ. 5. cf. The model uncertainty at line conditions ( E Z . all input uncertainties used in the calculation example have been available from the manufacturer data sheet [Rosemount.2. Ex- cept for the stability of the Pt 100 element (which is taken from [BIPM.2. Table 3. cf.8 and Figs. RFI effects.USM fiscal gas metering stations NPD. The influence of the atmospheric pressure uncertainty. At standard reference conditions. The present example is based on a simplified Monte Carlo type of simulation with natural variations of the gas composition (at a level which has been observed in practice). This means that the model uncertainty is smaller at standard reference conditions than at line conditions. The stability of the temperature transmitter dominates the uncertainty of the temperature measurement. each of these eight groups is commented in some more detail.1. RFI effects on the temperature transmitter.9) and Figs. The influence of ambient temperature effects is relatively smaller.2.3 (Table 4. cf.Handbook of uncertainty calculations . NFOGM.e.1).10. appear to be negligible. At higher flow velocities. 5. Section 2. a number of relevant input uncertainties related to VOS and installation corrections have not been available in data sheets. cf. Section 2. u( ρˆ temp ) . cf. such as for the VOS at calibration conditions (cc). and in the low-velocity range. the VOS correction constant (Kd).10. .USM fiscal gas metering stations NPD. the periodic time. In- put figures for these are discussed in Section 4. Table 4. u( K̂ d ) . all three input uncertainties are significant.6 gives an overview of the three input uncertainties to be specified for the USM flow calibration. and the uncertainty of the temperature correction model. corresponding to the NPD regulation requirements [NPD.5 and Fig. the VOS at densitometer conditions (cd). Table 4.13 and Figs.1). the periodic time (τ). and the pressure deviation between densitometer and line conditions (∆Pd).in general an uncertainty evaluation of the expanded uncertainty of the calorific value would be needed. the actual deviation curve. Section 4. u( ρˆ rept ) .3. Section 4.25. the deviation factor clearly dominates the uncertainty of the USM flow calibration.4. the densitometer temperature. cf. The influences of the pressure difference between densitometer and line conditions. Table 3. • USM flow calibration. Table 3. 5. i. Only the "overall level" is available at present. cf. In the present example. 5. However. the uncertainty of the line temperature. That is.4 and Table 4.2. are also relatively small. 5. • Calorific value measurement. cf.2 (Fig.14. u( T̂c ) . Note that this result will depend largely on the actual correction factor K used.5 gives the input uncertainty to be specified for the calorific value measurement. cf.2. u( τˆ ) .24. u( P̂ ) . u c ( T̂ ) . . 2. That has not been made here. 5.9. and the calibration temperature. u( ∆P̂d ) . Section 4. Table 4. In the present example the "densitometer accuracy" u( ρˆ u ) totally dominates the uncertainty of the pressure measurement. The required input uncertainties are available from flow calibration laboratories and USM manufacturers. the uncertainty of the flow calibration laboratory and the USM repeatability dominate.2.11.2. Hs. 2001]. CMR (December 2001) 146 perature uncertainties). due to the scope of work for the Handbook (cf. u( T̂d ) .11 and Figs. u( ĉc ) and u( ĉ d ) . Less important are the repeatability. Only a tentative example value has been used. The uncertaintes of the line pressure measurement. and the uncertaintes of the VOS in the calibration and densitometer gases. the uncertainty of the VOS correction constant. such information is not available from data sheets today.g. Table 2. to demonstrate the sensitivity to these uncertainty contributions. These are grouped into four groups: USM repeatability in field operation. NFOGM. The four groups are commented in some more detail in the following.4. As correction methods are available. Meter body uncertainty: Pressure and temperature correction of the meter body dimensions are not used by all meter manufacturers today.12.USM fiscal gas metering stations NPD. 5.Handbook of uncertainty calculations . In the present example all four groups contribute significantly to the uncertainty of the USM in field operation.21 and Figs. and due to inaccuracy of the model(s) used for β.4. uncertainty of systematic transit time effects. α and β. this may lead to significant measurement errors (in excess of the NPD requirements [NPD. Table 4. Section 4. the repeatability of the measured transit times could be pecified (standard deviations). the uncertainties of the linear temperature and pressure expansion coefficients. Evaluation of the meter body uncertainty involves specification of the two relative uncertainty terms u( αˆ ) | αˆ | and u( βˆ ) | βˆ | . cf. In general the latter two groups are the most difficult to specify (only the USM repeatability is available from current USM manufacturer data sheets). Chapter 6. Section 4.26.15. Note that for relatively large ∆Pcal . cf. CMR (December 2001) 147 • USM field operation. due to lack of data for u( αˆ ) | αˆ | . such correction might preferably be used on a routinely basis. cf. and Figs.3. cf. These may often be difficult to specify. i. Section 3. the repeatability is taken here to be independent of flow rate (which is probably a simplification).18.1. 5. On lack of other data. that would be preferred. Table 3. In case of pressure and temperature deviation from flow calibration conditions.2.4.8 gives an overview of the input uncertainties to be specified for the USM in field operation (deviation relative to flow calibration conditions). cf.6.18 and Fig. cf. 5. 2001]). 4. cf. cf. Section 2.3. and tentative uncertainty figures have been used in the present calculation example. Table 4. respectively.15. As these uncertainties have not been directly available. e. Section 4. 5.e. 5. cf. If a flow rate dependent repeatability figure were available.4. meter body uncertainty.26.6. and the integration method uncertainty (installation effects). However.4. cf.4. although it should be readily available from USM flow computers. Tables 4.20.2. Alternatively. Section 2. only tentative uncertainty figures have been used in the present example. Table 2. Repeatability: The USM repeatability in field operation has been taken as a typical figure from USM manufacturer data sheets (flow rate repeatability). 2001]104.84 % at 0. Section 4. and 0. NFOGM. as in the present example).27). In the present example these uncertainties have been neglected.USM Fiscal Gas Metering Station.4.4. For the present calculation example a tentative uncertainty figure has been used.12 %) may at first glance appear to be small.Handbook of uncertainty calculations .12 % to the expanded uncertainty of the flow rate reading at 10 m/s. As an uncertainty figure this high-velocity value (0. Systematic transit time effects: Information on uncorrected systematic transit time effects and associated uncertainties are not available from USM manufacturers today. Fig. Section 4. It is difficult to specify representative uncertainty figures for these effects at present.3. Chapter 6.7 gives an overview of the input uncertainties to be specified related to signal communication and flow computer calculations. but yet this knowledge is far from sufficient. Installation (integration) effects: Data on the uncertainty due specifically to installation effects are not available from USM manufacturer data sheets. but far from unrealistic. 2000b]).4 m/s the uncertainty is determined by the difference between these expanded uncertainties (10 ns). Section 4. The uncertainty figure example used here (for 40 oC and 50 bar deviation from flow calibration conditions) are thus very tentative.4.4. cf. cf. cf.19 and Fig. but when remembering that it represents an uncorrected systematic timing error. the resulting error in flow rate reading may over time still represent a significant economic value [NPD. [Lunde et al.g. . 104 The example used in Table 4.17.USM fiscal gas metering stations NPD.19 is by no means a “worst case”..16. Such uncertainty data might preferably be based on a large number of tests and simulations related to varying installation conditions. cf. 5. 2000a. whereas at 0.5 (but can be accounted for in the program EMU . e. cf. cf. • Signal communication and flow computer calculations. The example shows that expanded uncertainties of systematic transit time effects of 600 and 590 ns (upstream and downstream. Table 4. CMR (December 2001) 148 and ∆Tcal (several tens of bars and oC. however (cf.19 and Figs. 5. respectively) contribute by about 0.16)103. 5. Chapter 6.3. 5. 5. More work is definitely required in this area. cf. the actual input uncertainty figures for α and β become important. 5.4 m/s (Table 4. Section 3. 1999. despite the considerable effort on investigating such effects made over the last decade. 103 The uncertainty at 10 m/s is determined by the magnitude of these expanded uncertainties (about 600 ns).26. Table 3.26. Some knowledge is available. Data on such uncertainties have not been available from USM manufacturer data sheets. and Figs. and the compressibility factors. The pressure and temperature measurement uncertainties are less important. CMR (December 2001) 149 • Gas metering station. For the uncertainty evaluation example described in Chap- ter 4. the USM repeatability.31.29. Tables 4. . NFOGM. the systematic deviations relative to flow calibration. It should be emphasized. that this is an example. cf. the dominating contributions to the metering station's expanded uncertainty are due to the deviation factor (at low flow velocities). and that especially with respect to the USM field operation uncertainties.27-5. a number of input uncertainties have only been given tentative example values. however. especially the latter. and Figs. 5. the flow calibration laboratory.22-4. the density measurement.USM fiscal gas metering stations NPD.Handbook of uncertainty calculations . qm. qe.USM FISCAL GAS METERING STATION" The present chapter describes the Excel program EMU . and is based on the uncertainty model for such stations described in Chapter 3. In the following.e. These “input worksheets” are mainly formed as uncertainty budgets. the various worksheets used in EMU . With respect to specification of input parameters and uncertainties. Using the program.Handbook of uncertainty calculations . 5. This evaluation example follows closely the structure of the program. and may thus to some extent serve also as a guideline to using the program. colour codes are used in the program EMU . and • Energy flow rate. which are continuously updated as the user enters new input data. • Standard volume flow (i. using k = 2): • Actual volume flow (i. Other worksheets provide display of the uncertainty calculation results. uncertainty evaluation can be made for the expanded uncertainty of four measurands (at a 95 % confidence level. and are continuously updated in the same way..USM Fiscal Gas Metering Station. The program is implemented in Microsoft Excel 2000 and is based on worksheets where the input data to the calculations are entered by the user.1 General Overall descriptions of the program EMU . • Mass flow rate.1. Exam- ple values are those used in the uncertainty evaluation example given in Chapter 4. NFOGM. with respect to the specification of input uncertainties.2 and 1. according to the following scheme: . in the order they appear in the program. In the following. the volumetric flow rate at standard reference conditions).USM Fiscal Gas Metering Station are presented and described.USM fiscal gas metering stations NPD.USM Fiscal Gas Metering Station which has been implemented for performing uncertainty calculations of USM fiscal gas metering stations.e. CMR (December 2001) 150 5. Q. PROGRAM "EMU . the volumetric flow rate at line conditions). qv.3. some supplementary information is given. The program applies to metering stations equipped as described in Section 2.USM Fiscal Gas Metering Station are given in Sections 1. Handbook of uncertainty calculations . a “cut and paste special” with “picture” functionality may be sufficient for most worksheets. (The latter procedure has been used here. Only parts of the worksheet is copied. 5. In the following subsections the worksheets of the program are shown and explained. NFOGM. The resulting quality is not excellent. if the Word (doc) file is to be converted to a pdf-file. or number read from another worksheet (editing prohibited). An output report worksheet is available. obtained from manufacturers.1 . With respect to uncertainty calculations using the present Handbook and the program EMU .USM Fiscal Gas Metering Station. the ”normal” instrument uncertainties (found in instrument data sheets. Output data are presented in separate worksheets. summarizing the main uncertainty calculation results. The worksheets are designed so that printouts of these can be used directly as a part of the uncertainty evaluation documentation.USM fiscal gas metering stations NPD.4).) are normally to be used. Unfortunately. • Blue font105: Outputs from the program. with reference to the present Handbook (cf. may represent a challenge in 105 These colours refer to the Excel program. use of the “bitmap” feature results in poor-quality pictures. using Corel Photo Paint 7). They may also conveniently be copied into a text document106. In this case use of the “paste special” with “bitmap” feature may solve the problem. The expanded uncertainties calculated by the program may be used in the documentation of the metering station uncertainty. uncertainties.5. etc. Appendix B. by using Microsoft Word 2000.) . graphically (curves and bar-charts). However. pressure or temperature measurement. etc. The user must also document that the calculation procedures and functional relationships implemented in the program (described in Chapter 2) are in conformity with the ones actually applied in the fiscal gas metering station. and by listing. it must be emphasised that the inputs to the program (quantities. but still useful. However. In this case it is recommended to first convert the Excel worksheet in question into an 8-bit gif-file (e. for a number of the figures in Chapter 5. However. 106 For instance. loss of an acoustic path in a USM. confidence levels and probability distributions) must be documented by the user of the program. for some of the worksheets the full worksheet is (for some reason) not being pasted using the “paste special” with “picture” feature. erronous density. calibration laboratories.g. and the necessary input parameters are addressed with an indication of where in Chapters 3 and 4 the input values are discussed. Possible malfunction of an instrument (e.33 the colours have not always been preserved correctly when pasting from the Excel program into the present document (“cut and paste special” with “picture” functionality). and then import the gif-file as a picture into the Word document.). for documentation and reporting purposes. and specific procedures in connection with that.g. CMR (December 2001) 151 • Black font: Value that must be edited by the user. in Figs. Handbook of uncertainty calculations . etc. e. Z0 o 0. etc. the present Handbook and the program may be used to calculate the uncertainty of the metering station also in case of instrument malfunction.MetStat2. ρ 81.62 kg/m 3 Calibration velocity of sound (VOS). Hs 41.xls”. NFOGM.1. the user enters data for EMU . cd Line temperature.USM fiscal gas metering stations NPD. c 417 m/s Calibration temperature. After the desired input parameters and uncertainties have been entered.9973 Ambient (air) temperature at calibration 20 C Gross calorific value.g. the Excel file document may be saved e.MetStat1. a convenient way to use the program may be the following. if e. Tair 0 o C FLOW CALIBRATION CONDITIONS TEMPERATURE TRANSMITTER CONDITIONS Flow calibration pressure.443 kg/m 3 Velocity of sound (VOS).) • The operating gas conditions of the fiscal gas metering station (in the meter run). Td o Line pressure (static).USM Fiscal Gas Metering Station . Tcal o 10 C STANDARD REFERENCE CONDITIONS PRESSURE TRANSMITTER CONDITIONS Compressibility.g. Old evaluations may then conveniently be revisited. CMR (December 2001) 152 this respect. 5. In a practical work situation in the evaluation of a metering station. the instrument manufacturer is able to specify an (increased) uncertainty figure for a malfunctioned instrument. However. used as basis for new evaluations. Pcal o 50 bara Ambient (air) temperature at calibration 20 C Flow calibration temperature.USM Fiscal Gas Metering Station .2 Gas parameters In the worksheet denoted “Gas parameters” shown in Fig. . “EMU . • The gas conditions in the densitometer.2. Tc 20 o C Gas density. The “Gas parameters” worksheet in the program EMU . The file size is about 1. 5. using a modified file name. (Corresponds to Table 4. Z 0.xls”. “EMU .g.4 MB.686 MJ/Sm3 Fig.24 m/s Compressibility at line conditions. T 50 C 415. 5. cc 350 m/s Ambient (air) temperature. METER RUN DENSITOMETER CONDITIONS Temperature at density transducer.1.USM Fiscal Gas Metering Station.846 Indicated (uncorrected) gas density at density transducer. ρυ 82. P 100 bara 48 C o Velocity of sound.USM Fiscal Gas Metering Station Gas parameters OPERATING CONDITIONS. 1 are the same data as specified in Table 4. However. In this mode the “USM setup” worksheet does not need to be specified. NFOGM. and in calculation of the combined gas metering station uncertainty. since then one may conveniently switch between the “overall level” and the “detailed level” in the “USM” worksheet. ….USM fiscal gas metering stations NPD. CMR (December 2001) 153 • The gas conditions at flow calibration of the USM. Only very approximate values for these 107 However. The program uses these data in the calculation of the individual uncertainties of the primary measurements. 5.2 for the calculation example given in Chapter 4. . Mode (B) corresponds to using the “detailed level” in the “USM” worksheet (for the repeatability and/or the systematic deviation re. i = 1.Handbook of uncertainty calculations . cf.USM Fiscal Gas Metering Station can be run in two modes: (A) Completely meter independent. • Some gas conditions at standard reference conditions. Section 5. • The ambient temperature at calibration of the pressure and temperature transmitters.10. Mode (A) corresponds to choosing the “overall level” in the “USM” worksheet (both for the repeatability and the systematic deviation re. flow calibration). In this case some information on the USM is needed. and (B) Weakly meter dependent. cf. lateral chord positions ( y i0 ) and integration weights ( wi ) do not need to be known. since this information is not used in the calculations107.i ) need to be known. since Mode (B) involves the calculation of certain sensitivity coefficients related to the USM. N. N refl . 108 The number of acoustic paths in the USM can be set to any number in the range 1-10. actual values for the inclination angles ( φ i 0 ).10. These depend on φ i 0 y i0 . 5.i . The data used in the input worksheet shown in Fig. Section 5. the program EMU . wi and R0 .3 USM setup parameters With respect to USM technology. By “weakly meter independent” is here meant that the number of paths108 (N) and the number of reflections for each path ( N refl . it is useful to give input to the “USM setup” worksheet in any case. flow calibration). for each path .USM Fiscal Gas Metering Station Title of user default values: USM setup ULTRASONIC FLOW METER CONFIGURATION SPECIFICATIONS METER BODY MATERIAL DATA Inner diameter (spoolpiece). the user can set up his own path configuration(s) and store the data for later use. Y 2.138196601 Configuration name text box (user defined): 2 -45 0 -0. lateral chord positions and integration weights). angle [deg] reflections position [y/R] weight 1 45 0 -0.2.40E-05 K-1 Program default configuration (Gauss .4 mm Integration method: USM configuration menu: Temperature expansion coefficient. and give the desired title of the path setup (for example “USM no.3.138196601 User defined configuration 2 User defined configuration 3 User defined configuration 4 User defined configuration 5 Fig.3. for each path.00E+05 MPa Enter chosen configuration Save current settings as User defined configuration 1: Fill in white boxes below (grey boxes are locked): Save user defined configuration Acoustic Inclination Number of Lateral chord Integration path no.361803399 3 45 0 0. at dry calibration 308 mm Number of acoustic paths 4 Average wall thickness 8.Handbook of uncertainty calculations . 2”). integration weights. (Data partly taken from Table 4. number of reflections. as also described in Chapter 6. The “USM setup” worksheet in the program EMU . integration method. That is. At a later time one may then obtain . number of reflections. 2”). wall thickness. number of reflections. 5.361803399 User defined configuration 1 4 -45 0 0.Jacobi) Young's modulus (modulus of elasticity). 2. …. and Young’s modulus).filling in the white boxes in the table to the left of the worksheet (inclination angles.309016994 0. Then move to the right hand side of the worksheet for storing of the chosen configuration: choose among 1. alpha 1. The input parameters are: number of paths.2. This is done by . lateral chord positions. Table 6. this concerns the path configuration data. NFOGM. That means. One may choose either “program default configuration” (which is the Gauss-Jacobi integration method). lateral chord positions and integration weights. EMU . cf.USM Fiscal Gas Metering Station.) With respect to the integration method.309016994 0. 5.809016994 0. temperature expansion coefficient. CMR (December 2001) 154 quantities are needed (for the calculation of the sensitivity coefficients). inclination angles. or choose among one or several “user default values” (up to 5). The worksheet for setup of the USM parameters is shown in Fig. inclination angles. The worksheet and the program covers both reflecting-path and non-reflecting-path USMs. 5 (for example “User default values no. and meter body material data (usually steel) (diameter.USM fiscal gas metering stations NPD.809016994 0. and the accompanying confidence level / probability distribution. also Table 3. P 100 bar Relative Expanded Uncertainty (95% confidence level. With respect to meter body data.USM Fiscal Gas Metering Station. EMU . NFOGM. k uc(P) 0. 5.0800 bar Expanded Uncertainty (95% confidence level. and press the “Enter chosen configuration” button. the user enters only the relative expanded uncertainty of the pressure measurement.g. This option is used e.0064 bar² Combined Standard Uncertainty.3. These are described separately below. see Fig. uc(P) 0. for the “overall level” and the “detailed level”.1. Table 4.USM fiscal gas metering stations NPD. CMR (December 2001) 155 this stored configuration by going to the “integration method” box. these are usually steel data.0064 bar² 2 Pressure Measurement Sum of variances. 5. Cf.1 Overall level When the “overall level” is chosen for specification of input uncertainties to calculation of the pressure measurement uncertainty. 5.USM Fiscal Gas Metering Station Pressure measurement in meter run Ove ra ll input le ve l Select level of input: De ta ile d input le ve l OVERALL INPUT LEVEL Given Confidence Level Type of Standard Sensitivity Input variable Uncertainty (probability distr. shown for the “overall level” option. 5. respectively.3.4. when the user does not want to go into the “detailed level” of the pressure measurement. cf. k=2). k EP 0.4.3 and 5.1600 % Fig. 5.1600 bar Operating Static Pressure. choose the “User default values no.4 Pressure measurement uncertainty The worksheet “P” for evaluation of the expanded uncertainty of the pressure measurement in the meter run is shown in Figs. 2”. k=2).16 bar 95 % (normal) B 0. uc(P) 0. The “P” worksheet in the program EMU .08 bar 1 0.Handbook of uncertainty calculations .3.) uncertainty Uncertainty Coefficient Variance Pressure measurement 0. or if the “detailed level” setup does not fit sufficiently . 5.1599 bar Operating Static Pressure.0005444 bar² Ambient temperature effect ( 0.1.1599 % Fig. NFOGM.2 Detailed level When the “detailed level” is chosen for specification of input uncertainties to calculation of the pressure measurement uncertainty. shown for the “detailed level” option. the user enters the uncertainty figures of the pressure transmitter in question. k=2). Table 3.) .004761 bar² RFI effects 0.09 bar 99 % (normal) A 0.1 %Span 99 % (normal) A 0.USM Fiscal Gas Metering Station. CMR (December 2001) 156 well to the pressure transmitter at hand. Cf.6 and Fig.) uncertainty Uncertainty Coefficient Variance Transmitter 0.0116667 Bar 1 0. The user must himself document the input value used for the relative expanded uncertainty of the pressure measurement (the “given uncertainty”). k=2).2.86E-05 bar² Atmospheric pressure 0.1 and Section 3.0063902 bar² Combined Standard Uncertainty. 5. uc(P) 0. k uc(P) 0.0009 bar² bar 95 % (normal) B 0 Bar 1 0 bar² 2 Pressure Measurement Sum of variances.4. 5.0069714 Bar 1 4.0799 bar Expanded Uncertainty (95% confidence level.03 %Span ) 99 % (normal) B 0.05 %Span 99 % (normal) A 0. P 100 bar Relative Expanded Uncertainty (95% confidence level. (Corresponds to Table 4.USM fiscal gas metering stations NPD. in addition to the accompanying confidence levels / probability distributions. and its confidence level and probability distribution.Handbook of uncertainty calculations .0233333 Bar 1 0.069 Bar 1 0. EMU . The “P” worksheet in the program EMU .03 Bar 1 0.21.0001361 bar² Stability 0. uc(P) 0.006 %URL + 0.1 %URL / 1 year 95 % (normal) B 0.USM Fiscal Gas Metering Station Pressure measurement in meter run Ove ra ll input le ve l Select level of input: De ta ile d input le ve l DETAILED INPUT LEVEL Type of instrument: Maximum calibrated static pressure 120 barg Minimum calibrated static pressure 50 barg Calibrated span 70 bar Upper Range Limit (URL) 138 barg o Ambient temperature deviation 20 C Time between calibrations 12 months Given Confidence Level Type of Standard Sensitivity Input variable Uncertainty (probability distr. k EP 0.4. Cf.1. This option is used e.6. where the user can specify miscellaneous uncertainty contributions to the pressure measurement not covered by the other input cells in the worksheet. The “ambient temperature deviation” is calculated by the program from data given in the “Gas parameters” worksheet.1 for details.5 and 5. and the accompanying confidence level / probability distribution.2. on basis of a manufacturer data sheet.5. the user specifies only the relative expanded uncertainty of the temperature measurement. respectively. found in instrument data sheets. a “blank cell” has been defined.1 Overall level When the “overall level” is chosen for specification of input uncertainties to calculation of the temperature measurement uncertainty.g.Handbook of uncertainty calculations . cf. a calibration certificate. These are the same as used in Table 4. The “URL” is entered by the user. or other manufacturer information. Table 4. The user must himself document the input uncertainty values used for the pressure measurement (the “given uncertainty”). or if the “detailed level” setup does not fit sufficiently well to the temperature element and transmitter at hand.4 the uncertainty data specified for the Rosemount 3051P Reference Class Smart Pressure Transmitter [Rosemount. 5.USM fiscal gas metering stations NPD. By selecting the “maximum” and “minimum calibrated static pressure”. 5.5. 5. the program automatically calculates the “calibrated span”.g.5 Temperature measurement uncertainty The worksheet “T” for evaluation of the expanded uncertainty of the temperature measurement in the meter run is shown in Figs. for reporting purposes. Table 3. the user must specify a few other data.6. A blank field denoted “type of instrument” can be filled in to document the actual instrument being evaluated. The user must himself . when the user does not want to go into the “detailed” level of the temperature measurement. see Fig. Also the “time between calibrations” has to be specified. In addition to the “usual” pressure transmitter input uncertainties given in the worksheet. NFOGM. for the “overall level” and the “detailed level”.2. CMR (December 2001) 157 In Fig. see Section 4. e. These are described separately below.2. In addition to the input uncertainty values. 2000] have been used. 5. 5.2 and Section 3. T 50 °C Relative Expanded Uncertainty (95% confidence level. shown for the “overall level” option. The user must himself document the input uncertainty values for the temperature measurement. . In Fig.6 the uncertainty figures given for the Rosemount 3144 Smart Temperature Transmitter [Rosemount. the user specifies the uncertainty data of the temperature element and transmitter in question.005625 (°C)² Temperature Measurement Sum of variances. cf. The “T” worksheet in the program EMU . uc(T)2 0. Table 3. NFOGM. for reporting purposes. see Section 4. on basis of a manufacturer data sheet. have been specified.075 °C 1 0. Table 4.) uncertainty Uncertainty Coefficient Variance Temperature measurement 0.2 Detailed level When the “detailed level” is chosen for specification of input uncertainties to calculation of the temperature measurement uncertainty.005625 (°C)² Combined Standard Uncertainty.0464 % Fig. 5.5.15 °C 95 % (normal) B 0. EMU . and its confidence level and probability distribution.2. k uc(T) 0. 2000] used in combination with a Pt 100 temperature element. 5. e. CMR (December 2001) 158 document the input value used for the relative expanded uncertainty of the temperature measurement (the “given uncertainty”). k ET 0.8. 5.g.2. k=2).1. k=2).5. uc(T) 0.USM Fiscal Gas Metering Station Temperature measurement in meter run Ove ra ll input le ve l Select level of input: De ta ile d input le ve l OVERALL INPUT LEVEL Given Confidence Level Type of Standard Sensitivity Input variable Uncertainty (probability distr.USM Fiscal Gas Metering Station.1500 °C Operating temperature.USM fiscal gas metering stations NPD.0750 °C Expanded Uncertainty (95% confidence level. together with the accompanying confidence levels / probability distributions. A blank field denoted “type of instrument” can be filled in to document the instrument evaluated. a calibration certificate. These are the same as used in Table 4.2 for details. or other manufacturer information.Handbook of uncertainty calculations . Cf. 0333333 °C 1 0.0015 °C/°C 99 % (normal) B 0. 5.05 °C 95 % (normal) B 0.0538583 °C 1 0. the user must specify the “time between calibrations”. CMR (December 2001) 159 In addition to the input uncertainty data. where the user can specify miscellaneous uncertainty contributions to the temperature measurement not covered by the other input cells in the worksheet. and its confidence level and probability distribution.7.USM fiscal gas metering stations NPD. In addition to the “usual” temperature transmitter input uncertainties given in the worksheet. uc(T)2 0.temperature element 0.1529 °C Operating temperature.0011111 (°C)² Ambient temperature effect 0. k uc(T) 0. For each of Z and Z0. uc(T) 0.01 °C 1 0.USM Fiscal Gas Metering Station.0029007 (°C)² RFI effects 0.1 °C 99 % (normal) A 0. EMU . shown for the “detailed level” option. 5. two types of input un- .Handbook of uncertainty calculations .025 °C 1 0.0333333 °C 1 0. k=2). a “blank cell” has been defined.6 Compressibility factor uncertainty The worksheet “Z” for evaluation of the expanded uncertainty of the compressibility factor ratio Z/Z0 is shown in Fig. k=2).0011111 (°C)² Stability Max ( 0.1 °C 0.) 5.1 %MV/24months) 99 % (normal) B 0.0473 % Fig. The “ambient temperature deviation” is calculated by the program from data given in the “Gas parameters” worksheet.0058479 (°C)² Combined Standard Uncertainty.) uncertainty Uncertainty Coefficient Variance Temperature element and transmitter 0.8 and Fig.22.6. The user must himself document the input value used for the “miscellaneous uncertainty” of the temperature measurement.000625 (°C)² °C 95 % (normal) B 0 °C 1 0 (°C)² Temperature Measurement Sum of variances.1 °C 99 % (normal) A 0. T 50 °C Relative Expanded Uncertainty (95% confidence level.USM Fiscal Gas Metering Station Temperature measurement in meter run Ove ra ll input le ve l Select level of input: De ta ile d input le ve l DETAILED INPUT LEVEL Type of instrument: o Ambient temperature deviation 20 C Time between calibrations 12 months Given Confidence Level Type of Standard Sensitivity Input variable Uncertainty (probability distr.0765 °C Expanded Uncertainty (95% confidence level. NFOGM. k ET 0. (Corresponds to Table 4.0001 (°C)² Stability . 5. The “T” worksheet in the program EMU . 00000025 Gas compressibility factor at standard conditions (Z0) 0. together with its confidence level and probability distribution.00000 % 2 Gas compressibility calculation Sum of relative variances.00000256 Standard conditions 0 % Standard A 0. k=2). Cf. k EZ0/Z 0. .7.16000 % 1 0. NFOGM. 5. These are described separately below. there is implemented an option of filling in the AGA-8 (92) model uncertainties for Z and Z0 [AGA-8.) Uncertainty Uncertainty Coefficient Variance 1) MODEL UNCERTAINTY Z at line conditions: Fill in AGA-8 uncertainty (option) Z at std.3 and Sections 3. Cf. The “Z” worksheet in the program EMU . the input model uncertainty figures have to be filled in manually.2.8776E-06 Relative Combined Standard Uncertainty. the model uncertainty and the analysis uncertainty.16 % Standard A 0. on basis of pressure and temperature information given in the “Gas parameters” worksheet. 5. Table 3.7. model uncertainty: AGA-8 uncertainty used for the model uncertainty of Z (line conditions) ISO 6976 uncertainty used for the model uncertainty of Z0 (standard reference conditions) Fig. EZ0/Z 0. (EZ0/Z) 2.001696 Relative Expanded Uncertainty (95% confidence level.2.7 Density measurement uncertainty The worksheet “Density” for evaluation of the expanded uncertainty of the density measurement in the meter run is shown in Figs. EMU .) 5. respectively. conditions: Fill in ISO 6976 uncertainty (option) Gas compressibility factor at line conditions (Z) 0.2. (Corresponds to Table 4.USM Fiscal Gas Metering Station Gas compressibility uncertainty DETAILED INPUT LEVEL Relative Relative Given Relative Confidence Level Type of Standard Sensitivity Relative Input variables Uncertainty (probability distr.3 and 4. conditions: Fill in AGA-8 uncertainty (option) Z at std. These are the model uncertainties used in Fig.USM Fiscal Gas Metering Station.05000 % 1 0. for the “overall level” and the “detailed level”. 1994] or the ISO 6976 model uncertainty for Z0 [ISO.3.02600 % 1 6. The user must himself document the uncertainty values used as input to the worksheet.9.76E-08 ANALYSIS UNCERTAINTY Line conditions 0.3393 % 1) Re. 1995c].Handbook of uncertainty calculations .3 for details. CMR (December 2001) 160 certainties are to be specified.USM fiscal gas metering stations NPD. Section 3.052 % 95 % (normal) A 0.8 and 5. If another equation of state is used for one or both of Z and Z0.1 % 95 % (normal) A 0. For the model uncertainty. 5.9. k Eρ 0. or if the “detailed level” setup does not fit sufficiently well to the densitometer at hand. in-line).4. the user specifies the uncertainty figures of the online installed vibrating element densitometer in question.8. in case of a different installation of the densitometer (e. 5. in case a different method for density measurement is used (e. 5. The “Density” worksheet in the program EMU . EMU .1 Overall level When the “overall level” is chosen for specification of input uncertainties to calculation of the density measurement uncertainty. see Fig.2 Detailed level When the “detailed level” is chosen for specification of input uncertainties to calculation of the density measurement uncertainty.4 and Section 3. CMR (December 2001) 161 5.62 kg/m³ Relative Expanded Uncertainty (95% confidence level.2.080000 kg/m³ Expanded Uncertainty (95% confidence level. the user specifies only the relative expanded uncertainty of the density measurement.USM Fiscal Gas Metering Station Density measurement Ove ra ll input le ve l Select level of input: De ta ile d input le ve l OVERALL INPUT LEVEL Given Confidence Level Type of Standard Sensitivity Input variable Uncertainty (probability distr. calculation from GC analysis). Cf.g. This option is used e. shown for the “overall level” option. Table 3. Table 3.USM fiscal gas metering stations NPD. uc(ρ) 0.16 kg/m³ 95 % (normal) B 0. uc(ρ) 0.0064 (kg/m³)² 2 Density Measurement Sum of variances. k=2).08 kg/m³ 1 0.0064 (kg/m³)² Combined Standard Uncertainty. and the accompanying confidence level and probability distribution. k=2).) uncertainty Uncertainty Coefficient Variance Density measurement 0.8. The user must himself document the input uncertainty values for the density measure- . and its confidence level and probability distribution. The user must himself document the input value used for the relative expanded uncertainty of the density measurement (the “given uncertainty”).g.7.4.7. NFOGM. in addition to the accompanying confidence levels / probability distributions.Handbook of uncertainty calculations . Cf.1960 % Fig. k uc(ρ) 0. when the user does not want to go into the “detailed” level of the density measurement.16 kg/m³ Operating Density.g. ρ 81. 5.USM Fiscal Gas Metering Station. CMR (December 2001) 162 ment. (Corresponds to Tables 4. Fig.4 for details. NFOGM. see Section 4.24.Handbook of uncertainty calculations . 5. 5.2. These are the same as used in Table 4. Table 4. cf.g.11.USM fiscal gas metering stations NPD.1. shown for the “detailed level” option. 1999] have been used. a calibration certificate. on basis of a manufacturer data sheet.11.4 and 4. A blank field denoted “type of in- . 5. and Fig.9 the uncertainty figures specified for the Solartron Model 7812 Gas Den- sity Transducer [Solartron. e.USM Fiscal Gas Metering Station. The “Density” worksheet in the program EMU .) In Fig. or other manufacturer information.9. 1500 % Fig. Cf. CMR (December 2001) 163 strument” can be filled in to document the actual instrument being evaluated. The input uncertainty of the line temperature (T).4. together with its confidence level and probability distribution.) uncertainty Uncertainty Coefficient Variance Gross Calorific Value. 5.Handbook of uncertainty calculations . EHs 5. for reporting purposes.625E-07 Relative Combined Standard Uncertainty.10. Table 4.4 for details. The user must himself document the input value used for the “miscellaneous uncertainty” of the densitometer measurement.2.USM Fiscal Gas Metering Station Calorific value at standard reference conditions OVERALL INPUT LEVEL Relative Relative Given Relative Confidence Level Type of Standard Sensitivity Relative Input variable Uncertainty (probability distr. 5. the user must specify four gas densitometer constants. and the “densitometer VOS”. k=2). The “Hs” worksheet in the program EMU . is calculated by the program. cc.USM fiscal gas metering stations NPD. 5. In addition to the input uncertainty values. a “blank cell” has been defined.4.000750 Relative Expanded Uncertainty (95% confidence level. k EHs 0.8 Calorific value uncertainty The worksheet “Hs” for evaluation of the expanded uncertainty of the superior (gross) calorific value estimate is shown in Fig. cf. are taken directly from the “Gas parameters” worksheet. K19.4 and Section 4.075 % 1 5.2. the densitometer temperature (Td). .USM Fiscal Gas Metering Station. The “calibration VOS”. Kd and τ. are taken from the “T” and “P” worksheets. defined in Section 2. Section 4. where the user can specify miscellaneous uncertainty contributions to the density measurement not covered by the other input cells in the worksheet. and the line pressure (P). In addition to the “usual” densitometer input uncertainties given in the worksheet.625E-07 2 Calorific Value Measurement Sum of relative variances. The uncertainty due to pressure difference between line and densitometer conditions (∆Pd). NFOGM. EHs 0.10. Hs 0. cd. K18.15 % 95 % (normal) B 0. EMU . the user specifies the relative expanded uncertainty of the superior (gross) calorific value estimate. for the 1 m/s flow velocity. Flow data can then be entered either as (a) flow velocity or (b) volumetric flow rate at line conditions.13.5 for some more details.Handbook of uncertainty calculations .10. 5. (Cor- responds to Table 4. NFOGM.2. That means. Cf. the user is to specify the M flow velocities (or flow rates) for which flow calibration has been made.11.9 Flow calibration uncertainty The worksheet “Flow cal.) In the first column from the left. The user must himself document the input value used for the relative expanded uncertainty of the superior (gross) calorific value estimate (the “given uncertainty”). 5. in the range 4 . with M = 6 flow velocities specified.11 the example discussed in Section 4. Fig. 5. . 5. 5.5 and 4.” for evaluation of the expanded uncertainty of the USM flow calibration is shown in Fig. and its confidence level and probability distribution. the number (M) of flow calibration points (calibration flow rates) is chosen. The “Flow cal.” worksheet in the program EMU .USM fiscal gas metering stations NPD. In Fig.2. Sections 3.3 has been used.10. and flow velocities corresponding to the flow rates given in Table 4.USM Fiscal Gas Metering Station.11. see Fig. CMR (December 2001) 164 Only the “overall level” is available. and the accompanying confidence level and probability distribution.12. First. 14 (for two of the six flow rates). On basis of these input data. This uncertainty contribution may be specified to be flow rate dependent.12 and in Fig. but in Table 4. 4. Sections 3.2.1 and 4. M. flow calibration”. together with the accompanying confidence level / probability distribution. defined by Eq.3. Finally. Cf. NFOGM. 5.3. The two parts of the worksheet are described separately below. 5.11. CMR (December 2001) 165 In the second column from the left. the user is to specify the “Deviation (corrected)” at the M calibration flow rates.j. the repeatability of the USM in flow calibration is to be specified at the M calibration flow rates.2.3 and 4. the expanded uncertainty of the USM flow calibration is calculated as shown in Fig. In the third column from the left.3.3. in the fourth column from the left. although simplified). This uncertainty contribution may also be specified to be flow rate dependent. Note that this is the deviation after flow calibration correction using the correction factor K.3. Cf. That is. 5. together with the accompanying confidence level / probability distribution. as described in Section 2.18 below. Cf.11 it has been taken to be constant over the flow range (which may be a common approach.10 USM field uncertainty The worksheet “USM” for evaluation of the expanded uncertainty of the USM in field operation is basically divided in two main parts: • The “USM field repeatability”. .2 and Table 4. Sections 3. the corrected relative deviation DevC. j = 1.1 for details. but in Fig. cf. although simplified). and • The uncertainty of uncorrected “USM systematic deviations re.3. Both of these can be specified at an “overall level” and a “detailed level”.3 for details. 5.12 for details.Handbook of uncertainty calculations .12 -5. the expanded uncertainty of the flow calibration laboratory is to be specified at the M calibration flow rates. in a similar approach as shown in Tables 4.13 and 4.USM fiscal gas metering stations NPD.10). (2. 5.11 it has been taken to be constant over the flow range (which may be a common approach. Sections 3. …. Figs. 1 USM field repeatability The “USM field repeatability” can be specified at an “overall level” and a “detailed level”. for the USM in field operation. NFOGM. The user must himself document the input value(s) .12 shows this option. Fig.1. The two options are described separately below. respectively. Fig.) The flow rate repeatability can be specified to be flow rate dependent. 5. the standard uncertainty of the flow rate repeatability may simply be taken to be the standard deviation of the flow rate measurements.15.4. at the M flow rates chosen in the worksheet “Flow cal. together with the accompanying confidence level / probability distribution. specification of flow rate repeatability).2. and (2) the repeatability of the measured transit times. related to the USM repeatability in field operation (shown for the “overall level” option. (Corresponds to Table 4.12 it is taken to be constant over the flow rate (which in practice may be a common approach. although in Fig. for the same example as shown in Table 4. CMR (December 2001) 166 5. Cf. 5. also Table 3.4.15.10. For a given flow rate.8 and Sections 3. Both can be given to be flow rate dependent. The user specifies the relative expanded uncertainty of the flow rate repeatability.USM fiscal gas metering stations NPD. Part of the “USM” worksheet in the program EMU .”.1. although simplified).Handbook of uncertainty calculations . 5.10. corresponding to specifying (1) the repeatability of the indicated USM flow rate measurement. 5. this corresponds to specification of the repeatability of the measured flow rate for the USM in field operation.12.1 Overall level When the “overall level” is chosen for specification of the “USM field repeatability”. 4.USM Fiscal Gas Metering Station. 13 shows this option. 5.2 Detailed level When the “detailed level” is chosen for specification of the “USM field repeatability”.1.8 and Sections 3. although in Fig. The user specifies the expanded uncertainty of random transit time variations (repeatability). 5.2).4.4. related to the USM repeatability in field operation (shown for the “detailed level” option. for the USM in field operation.13.Handbook of uncertainty calculations . on basis of the USM manufacturer data sheet or other manufacturer information.13 it is taken to be constant over the flow rate (which is a simplified approach. The user must himself document the input value(s) used for the uncertainty of the transit time repeatability. specification of transit time repeatability).16.1. Fig. 4. together with its confidence level and probability distribution.) . 5.USM Fiscal Gas Metering Station. Fig. e. the standard uncertainty of the random transit time variations may simply be taken to be the standard deviation of the transit time measurements. The transit time repeatability can be specified to be flow rate dependent.g. for the same example as shown in Table 4. together with its confidence level and probability distribution.”. 5. this corresponds to specification of the repeatability of the mesured transit times for the USM in field operation. cf. Section 3. Part of the “USM” worksheet in the program EMU . NFOGM. together with the accompanying confidence level / probability distribution. CMR (December 2001) 167 used for the relative expanded uncertainty of the flow rate repeatability. at the M flow rates chosen in the worksheet “Flow cal.2. At a given flow rate. also Table 3.USM fiscal gas metering stations NPD. on basis of USM manufacturer information.10. Cf.4.16. (Corresponds to Table 4. 10. The “USM” worksheet in the program EMU .14.USM fiscal gas metering stations NPD. for an example corresponding to the example given in Table 4. flow calibration”. flow calibration The uncertainty of uncorrected “USM systematic deviations re. Fig. Cf.20.8. 5.USM Fiscal Gas Metering Station. NFOGM. 5. Fig. . shown for the “overall level” options both for (1) “USM field repeatability” and (2) “USM systematic deviations re.14 shows this option.1 Overall level For specification of the “USM systematic deviations re.2 USM systematic deviations re. The two options are described separately below. flow calibration” at the “overall level”. CMR (December 2001) 168 5. flow calibration” can be specified at an “overall level” and a “detailed level”.Handbook of uncertainty calculations . 5. see Table 3.10.4. the user specifies the relative expanded uncertainty of all uncorrected systematic deviations of the USM relative to flow calibration.2. also Section 3. on basis of possible USM manufacturer information (cf.g.10. no P and T correction used). • Uncertainty related to uncorrected systematic transit time effects.17 (i. flow calibration”. flow calibration” at the “detailed level”.USM fiscal gas metering stations NPD. for the same example as given in Table 4. Handbook of uncertainty calculations . the user specifies whether correction for pressure and temperature effects is used by the USM manufacturer or not.4.4.8. Sections 3. to be used in case the user does not want to go into the “detailed level” of uncertainty input with respect to “USM systematic deviations re. Fig. or other information.2 Detailed level For specification of the “USM systematic deviations re. e.2.15 shows this this part of the “USM” worksheet. on basis of a USM manufacturer data sheet. together with their accompanying confidence level and probability distributions: • Uncertainty related to systematic USM meter body effects.2. 4. Table 3. CMR (December 2001) 169 This option is implemented as a simplified approach. 5. These three input types are discussed separately below. Meter body With respect to the “USM meter body uncertainty” part of the worksheet. the user specifies three types of input uncertainties. The user must himself document the input value used for the relative expanded uncertainty of the uncorrected systematic deviations of the USM relative to flow calibration. or other information. together with the associated confidence levels and probability distributions.1 and 4. The user must himself document the input uncertainty values used for the pressure and temperature expansion coefficients.g. e. Table 3.4. and the relative expanded uncertainties of the pressure and temperature expansion coefficients.8 and Sections 3. and • Uncertainty related to integration method (installation conditions). . cf. 5. Chapter 6). cf. NFOGM.e.4. Cf. together with the accompanying confidence level and probability distribution. related to the “USM systematic deviations re.4. cf. Part of the “USM” worksheet in the program EMU .2 and 4. Cf. subsection “USM meter body uncertainty”). flow calibration” (for the “detailed level” option. together with the accompanying confidence levels / probability distributions. The actual uncertainty figure is preferably to be specified by the USM manufacturer.4. Sections 3.USM fiscal gas metering stations NPD. for the same example as used in Table 4. Chapter 6. on basis of USM manufacturer information (cf. the user specifies the input expanded uncertainty of uncorrected systematic transit time effects on the measured upstream and downstream transit times (deviation from flow calibration to field operation).8. NFOGM. Examples of such effects are given in Tables 1.15. The user must himself document the input uncertainty values used for “USM transit time uncertainties (systematic effects)”.USM Fiscal Gas Metering Station. for the case of no temperature and pressure correction. e.16 shows this part of the “USM” worksheet. CMR (December 2001) 170 Fig.19. 5.g.Handbook of uncertainty calculations . Fig. 5.3. . (Corresponds to Table 4.17.) Systematic transit time effects With respect to the “USM transit time uncertainties (systematic effects)” part of the worksheet. Chapter 6).4 and 3. Fig.4. subsection “USM integration uncertainty (installation effects)” and “Miscellaneous effects”).Handbook of uncertainty calculations . 5.) Integration method (installation effects) With respect to the “USM integration uncertainty (installation effects)” part of the worksheet.4. Cf. flow calibration” (for the “detailed level” option. on basis of extensive testing/simulations/experience with different installation conditions for the meter in question (e.USM Fiscal Gas Metering Station. 5. or more general). 5.19.USM Fiscal Gas Metering Station. flow calibration” (for the “detailed level” option. for the same example as given in Section 4. CMR (December 2001) 171 Fig. related to the “USM systematic deviations re. The user must himself document the input uncertainty values used for the “USM integration uncertainty (installation effects)”. subsection “USM transit time uncertainties (systematic effects)”).17 shows this part of the “USM” worksheet. e. for a specific type of installation.17.g.USM fiscal gas metering stations NPD.4 and 3. together with the accompanying confidence level and probability distribution. the user specifies the relative expanded uncertainty of uncorrected installation effects (due to possible deviation in conditions from flow calibration to field operation).3. (Corresponds to Table 4.8. also Section 3. Part of the “USM” worksheet in the program EMU .4. Examples of such effects are given in Tables 1. The actual uncertainty figure is preferably to be specified by the USM manufacturer. NFOGM. Part of the “USM” worksheet in the program EMU . .16. related to the “USM systematic deviations re. on basis of information provided by the USM manufacturer. Fig.g. 5.20. 5. 5.USM Fiscal Gas Metering Station.19. Fig.17.18.19. CMR (December 2001) 172 Summary . i. The values used here correspond to the example given in Tables 4. The “Computer” worksheet in the program EMU . 5.21 and Fig. together with the accompanying confidence levels / . 5. NFOGM. is shown in Fig. Fig.26. for the same example as given in Section 4. Figs. summarizing the USM field uncertainty calculations (example). 5.15-5.Handbook of uncertainty calculations .18 shows the part of the “USM” worksheet which summarizes the uncertainty calculations for the USM field uncertainties.e. (Corresponds to Tables 4. 4.12 and 5. due to signal communication and flow computer calculations. This display is common to the “overall level” and “detailed level” options (cf.5.USM Fiscal Gas Metering Station.) 5.20 and 4. 5.Expanded uncertainty of USM in field operation Fig.14).USM fiscal gas metering stations NPD. The user specifies the relative expanded uncertainty of signal communication effects and flow computer calculations.11 Signal communication and flow computer calculations The worksheet “Computer” for evaluation of the expanded uncertainty of “Flow computer effects”. Fig.21 (for two of the six flow rates). Part of the “USM” worksheet in the program EMU . qe.USM fiscal gas metering stations NPD.g. .. also Sections 3. • Actual volume flow (i. • Actual volume flow (i. qv. • Mass flow rate.USM Fiscal Gas Metering Station to plot and display the calculation results.e. qm.. qe. conditions). qv. The user must himself document the input uncertainty values used for the “Flow computer effects”. Q.1 Uncertainty curve plots Plotting of uncertainty curves is made using the “Graph” worksheet. NFOGM. 5. CMR (December 2001) 173 probability distributions. • Standard volume flow (i. Q. and • Energy flow rate. qe. e. conditions). Cf. • Mass flow rate. 5.e. qm.. Editing of plot options is made using the “Graph menu” worksheet (“curve plot set-up”). These worksheets are described in the following. and • Energy flow rate. • Standard volume flow (i. These can be plotted as a function of (for the set of M flow velocities/rates chosen in the “Flow cal. the volumetric flow rate at standard ref.e. the volumetric flow rate at line conditions). Q.e.e.12. qm.5 and 4. the volumetric flow rate at line conditions).5. qv. The relative expanded uncertainties above can be plotted together with the following measurands: • No curve. on basis of information provided by the USM manufacturer.12 Graphical presentation of uncertainty calculations Various worksheets are available in the program EMU . conditions). the volumetric flow rate at line conditions). and • Energy flow rate. • Flow velocity.e. Plotting of the relative expanded uncertainty can be made for the following four “measurands”: • Actual volume flow (i. such as curve plots and bar-charts. the volumetric flow rate at standard ref. the volumetric flow rate at standard ref.Handbook of uncertainty calculations . • Standard volume flow (i. • Mass flow rate.” worksheet): • Flow velocity. 27)110. plotted vs. Editing of bar chart options is made using the “Graph menu” worksheet (“bar-chart set-up” section). the default method implemented in Microsoft Excel 2000 is used. The example used here is the same as used in the text above. line between points and smooth curve109) are available.3 (cf. . Tables 4.26 and 4. mass flow rate. and in Section 4. 110 The front page shows the same evaluation example. with k = 2.USM Fiscal Gas Metering Station (example).20. Bar 109 For the “smooth curve” display option.Handbook of uncertainty calculations . and various options for curve display (points only. Fig. 5. cf. CMR (December 2001) 174 Axes may be scaled according to user needs (automatic or manual).3) is plotted together with the mass flow rate itself.USM fiscal gas metering stations NPD. NFOGM. The “Graph” worksheet in the program EMU .6. as a function of flow velocity.12. Fig.20 shows an example where the relative expanded uncertainty of the mass flow rate (at a 95 % confidence level and a normal probability distribution. 5. 5.2 Uncertainty bar-charts Plotting of bar charts is made using the “NN-chart” worksheets. Section B. The example used here is the same as the example given in Table 4. EMU .Handbook of uncertainty calculations . level) [bar] Fig. CMR (December 2001) 175 charts are typically used to evaluate the relative contributions of various input uncertainties to the expanded uncertainty of the “measurand” in question. 5.USM fiscal gas metering stations NPD.1 Pressure The pressure-measurement bar chart is given in the “P-chart” worksheet. • USM flow calibration (“FC-chart” worksheet).4. Such bar-charts are available for the following seven “measurands”: • Pressure measurement (“P-chart” worksheet).USM Fiscal Gas Metering Station Contributions to the expanded uncertainty of pressure measurement Transmitter uncertainty Stability. NFOGM.21 shows an example where the contributions to the expanded uncertainty of the pressure measurement are plotted (blue). and • Gas metering station (“MetStat-chart” worksheet).1 0. These bar charts are described separately in the following. • Temperature measurement (“T-chart” worksheet). 5. The “P-chart” worksheet in the program EMU . 5.USM Fiscal Gas Metering Station. • Density measurement (“D-chart” worksheet). • Compressibility factor measurement / calculation (“Z-chart” worksheet). axes may be scaled according to user needs (automatic or manual).6 and Fig.12.15 0. 5. 5. Fig. As for the "Graph" worksheet. k = 2 (95 % conf.2 Expanded uncertainty. transmitter Atmospheric pressure Miscellaneous Pressure measurement 0 0.) . • USM field operation (“USMfield-chart” worksheet).05 0.4.21. together with the expanded uncertainty of the pressure measurement (green). (Corre- sponds to Table 4.2. transmitter RFI effects Ambient temperature effect.6 and Fig. 9. The example used here is the same as the example given in Table 4.4 Density The density-measurement bar chart is given in the “D-chart” worksheet.Handbook of uncertainty calculations . together with the relative expanded uncertainty of the density measurement (green).) 5. CMR (December 2001) 176 5. transmitter Ambient temperature effect.24 shows an example where the contributions to the relative expanded uncertainty of the density measurement are plotted (blue).6. 5. NFOGM.11 and Fig.3 Compressibility factors The compressibility factor bar chart is given in the “Z-chart” worksheet. together with the expanded uncertainty of the temperature measurement (green). transmitter RFI effects. Fig.12.15 0. 5.12.8 and Fig.2.12.2 Expanded uncertainty. . level) [°C] Fig. transmitter Stability.22. 5.23 shows an example where the contributions to the expanded uncertainty of the Z- factor measurements/calculations are plotted (blue). The example used here is the same as the example given in Table 4. (Corre- sponds to Table 4.9 and Fig.USM Fiscal Gas Metering Station Contributions to the expanded uncertainty of temperature measurement Element and transmitter uncertainty Stability.22 shows an example where the contributions to the expanded uncertainty of the temperature measurement are plotted (blue). 5. k = 2 (95 % conf. 5.USM Fiscal Gas Metering Station.USM fiscal gas metering stations NPD.1 0.7. 5. 5. EMU .05 0. 5. Fig. 5.2 Temperature The temperature-measurement bar chart is given in the “T-chart” worksheet.8 and Fig. temperature element Miscellaneous Temperature measurement 0 0. The “T-chart” worksheet in the program EMU .2.6. Fig. together with the expanded uncertainty of the Z-factor ratio (green). The example used here is the same as the example given in Table 4.2. densitometer to line VOS.3 0. densitometer gas Periodic time VOS correction constant. Z Model uncertainty.USM Fiscal Gas Metering Station Contributions to the relative expanded uncertainty of the compressibility ratio. 5.USM Fiscal Gas Metering Station.4 Relative expanded uncertainty.12.24. Kd Temperature correction model Miscellaneous Density measurement 0 0.1 0. Z/Z0 Model uncertainty.2 0. (Corre- sponds to Table 4. level) [%] Fig.23.15 0.USM Fiscal Gas Metering Station Contributions to the expanded uncertainty of density measurement Densitometer accuracy Repeatability Calibration temperature Line temperature Densitometer temperature Line pressure Pressure difference. k = 2 (95 % conf.USM fiscal gas metering stations NPD. calibration gas VOS. NFOGM. CMR (December 2001) 177 EMU .11 and Fig. level) [kg/m³] Fig.) EMU . k = 2 (95 % conf. 5. among the set of . Z0 Analysis uncertainty Compressibility ratio.9.1 0. (Corre- sponds to Table 4.9 and Fig.2 Expanded uncertainty. 5.) 5. The “D-chart” worksheet in the program EMU .USM Fiscal Gas Metering Station.2.Handbook of uncertainty calculations . Z/Z0 0 0. The bar chart can be shown for one flow velocity (or flow rate) at the time.5 USM flow calibration The USM flow calibration bar chart is given in the “FC-chart” worksheet.7. The “Z-chart” worksheet in the program EMU . 5.05 0. for a flow velocity of 1 m/s.18. 5. among the set of M flow velocities chosen in the “Flow cal.25 shows an example where the contributions to the relative expanded uncertainty of the USM flow calibration are plotted (blue). Fig.4 0.20 and Figs.11. 5. 5.” worksheet.USM fiscal gas metering stations NPD.26 shows an example where the contributions to the relative expanded uncertainty of the USM flow calibration are plotted (blue).13 and Fig.13 and Fig.2 0. level) [%] Fig.2. for a flow velocity of 1 m/s. k = 2 (95 % conf.6 0. NFOGM.25. The desired flow velocity is set in the “Graph menu” worksheet.6 USM field operation The USM field operation bar chart is given in the “USMfield-chart” worksheet.15-5. 5. CMR (December 2001) 178 M flow velocities chosen in the “Flow cal. 5. The desired flow velocity is set in the “Graph menu” worksheet.) Fig. 5.12.USM Fiscal Gas Metering Station.USM Fiscal Gas Metering Station Contributions to the relative expanded uncertainty of USM flow calibration at flow velocity: 1 m/s Flow calibration laboratory Deviation factor USM repeatability Flow calibration 0 0. The bar chart can be shown for one flow velocity (or flow rate) at the time. . together with the relative expanded uncertainty of the USM flow calibration (green).13 and 5.Handbook of uncertainty calculations .” worksheet.8 1 Relative expanded uncertainty. 5. (Cor- responds to Table 4. The “FC-chart” worksheet in the program EMU .11. together with the relative expanded uncertainty of the USM flow calibration (green). The example used here is the same as the example given in Table 4. The example used here is the same as the example given in Table 4. EMU . 12.2 Relative expanded uncertainty.USM Fiscal Gas Metering Station.) 5.1 0. CMR (December 2001) 179 EMU . flow calibration Flow computer.2. for the mass flow rate at 1 m/s flow velocity.2 0.USM Fiscal Gas Metering Station Contributions to the relative expanded uncertainty of qm (the mass flow rate) at flow velocity: 1 m/s Gas parameters Density Flow calibration Flow calibration laboratory Deviation factor USM repeatability USM field operation USM repeatability Systematic dev.) . 5.26.27. rel.USM fiscal gas metering stations NPD.4 0.31.6 Relative expanded uncertainty.2 0.20.8 1 1. EMU . level) [%] Fig.3 0. 5. 5. Signal communication Flow computer calculations Total for qm 0 0.USM Fiscal Gas Metering Station.20 and Fig. (Corresponds to Table 4. etc. NFOGM. (Corresponds to Table 4.USM Fiscal Gas Metering Station Contributions to the relative expanded uncertainty of the USM field operation at flow velocity: 1 m/s USM repeatability Meter body uncertainty Systematic transit time effecs Integration (installation effects) Miscellaneous effects USM field operation 0 0.18. level) [%] Fig. k = 2 (95 % conf.26 and Figs. 5. k = 2 (95 % conf. 5.4 0.6 0. The “USMfield-chart” worksheet in the program EMU . The “MetStat-chart” worksheet in the program EMU .7 Gas metering station The bar chart for the overall uncertainty of the USM fiscal gas metering station is given in the “MetStat-chart” worksheet.Handbook of uncertainty calculations .5 0. The example used here is the same as the example given in Table 4. depending on the type of input used in the “Flow cal.22.28-5. and the energy flow rate (qe). 5. The examples shown in Figs.28. A report can be prepared for each of the four flow rate measurands in question (qv. 5. for a flow velocity of 1 m/s.28-5. the volumetric flow rate at standard reference conditions (Q). Q. at a given flow velocity (or volumetric flow rate. calculated at the flow velocity 1 m/s. Fig.Handbook of uncertainty calculations .Expanded uncertainty of USM fiscal gas metering station A “Report” worksheet is available in the program EMU . CMR (December 2001) 180 The bar chart can be shown for one flow velocity (or flow rate) at the time.26 and Fig.13 Summary report . respectively. Blank fields are available for filling in program user information and other comments. 5.26 and 4. the mass flow rate (qm). among the set of M flow velocities chosen in the “Flow cal.20. 4. or together with printout of other worksheets in the program.” worksheet. 5. Also some of the settings of the “Gas parameter” and “USM setup” worksheets are included for documentation purposes. The desired flow velocity (or volumetric flow rate) is chosen in the “Report” worksheet.31 show the “Report” worksheet. qm and qe). for the four flow rate measurands in question: the volumetric flow rate at line conditions (qv) . Figs.31 are the same as those given in Tables 4. together with the relative expanded uncertainty of the gas metering station (green).USM Fiscal Gas Metering Station to provide a condensed report of the calculated expanded uncertainty of the USM fiscal gas metering station. The desired measurand (type of flow rate to be evaluated) is also set in the “Graph menu” worksheet.” worksheet. Q. The desired flow velocity is set in the “Graph menu” worksheet. respectively. For documentation purposes.24.” worksheet).27 shows an example where the contributions to the relative expanded uncertainty of the mass flow rate are plotted (blue).USM fiscal gas metering stations NPD. 5. 4. this one-page report can be used alone. among the M calibration flow velocities (or volumetric flow rates) specified in the “Flow cal. NFOGM. One may choose among the four measurands in question. . qv. qm and qe. 28. for the volumetric flow rate at line conditions. 5.) . CMR (December 2001) 181 Fig.USM Fiscal Gas Metering Station. (Corresponds to Table 4. NFOGM.Handbook of uncertainty calculations .22. The “Report” worksheet in the program EMU .USM fiscal gas metering stations NPD. USM Fiscal Gas Metering Station.Handbook of uncertainty calculations .USM fiscal gas metering stations NPD.29. CMR (December 2001) 182 Fig. The “Report” worksheet in the program EMU .24. 5.) . NFOGM. (Corresponds to Table 4. for the volumetric flow rate at standard reference conditions. 5.30.26 and Fig.USM Fiscal Gas Metering Station. The “Report” worksheet in the program EMU . for the mass flow rate. CMR (December 2001) 183 Fig. 5.27. (Corresponds to Table 4.Handbook of uncertainty calculations .USM fiscal gas metering stations NPD.) . NFOGM. for the energy flow rate. The “Report” worksheet in the program EMU . (Corresponds to Table 4.Handbook of uncertainty calculations .28.USM fiscal gas metering stations NPD.31.USM Fiscal Gas Metering Station.) . CMR (December 2001) 184 Fig. NFOGM. 5. of paths. 5. for the chosen pipe diameter and flow rates involved.2. Such a listing may be useful for reporting purposes. cf. Fig..2. For convenience.USM Fiscal Gas Metering Station. However. This effect is included in the uncertainty model and the program through the terms u( t̂ 1systematic i ) and u( t̂ 2systematic i ) . The “Readme” worksheet gives regulations and conditions for the distribution of the Handbook and the program.Handbook of uncertainty calculations . since in the program EMU . giving a listing of all transit time data used for the USM calculations.4. The “About” worksheet. of reflections. a “Transit time” worksheet has also been included. “Ray bending” effects are negligible in this context. . the transit times are used only for calculation of sensitivity coefficients. NFOGM. note that “ray bending” effects may influence on the USM reading at high flow velocities [Frøysa et al.15 Program information Two worksheets are available to provide information on the program. no.USM Fiscal Gas Metering Station. and the transit time difference. and can be activated at any time. 2001]. 5.USM Fiscal Gas Metering Sta- tion to provide listing of data involved in the uncertainty evaluation. cf. These are the “About” and the “Readme” worksheets.33. CMR (December 2001) 185 5. 111 Effects of non-uniform flow profiles and transversal flow (“ray bending”) are thus not included here. and no transversal flow111. diameter. gives general information about the program. and in case the user needs to present the data in a form not directly available in the program EMU . Section 3. The transit time calculations have been made using a uniform axial flow velocity profile. inclination angle and lateral chord positions). which is displayed at startup of the program EMU .32.USM fiscal gas metering stations NPD.USM Fiscal Gas Metering Station. Fig. no. The “Plot data” worksheet gives a listing of all data used and plotted in the “Graph” and “NN-chart” worksheets.. This involves upstream and downstream transit times. etc. 5. Note that the transit time values will change by changing the path configuration setup in the “USM setup” worksheet (i.e.14 Listing of plot data and transit times Two worksheets are available in the program EMU . cf. Note that the contents of the “plot data” sheet will change with the settings used in the “Graph menu” sheet. Handbook of uncertainty calculations . CMR (December 2001) 186 Fig.USM Fiscal Gas Metering Station (example). . NFOGM.USM fiscal gas metering stations NPD.32. 5. The “Plot data” worksheet in the program EMU . The “Transit times” worksheet in the program EMU . CMR (December 2001) 187 Fig.USM fiscal gas metering stations NPD. NFOGM.33.Handbook of uncertainty calculations .USM Fiscal Gas Metering Station (example). . 5. USM fiscal gas metering stations NPD.1)). Typical USM uncertainty data currently specified by USM manufacturers. USM MANUFACTURER SPECIFICATIONS The present chapter summarizes some input parameters and uncertainty data related to the USM.5 % of reference.1. Informa- tion on how the “accuracy” varies with pressure. lateral chord positions ( y i 0 ) and inclination angles ( φ i 0 ) are not provided by all manufacturers).integration weights (wi).Handbook of uncertainty calculations . Data sheets do not specify what types of uncertainties the “accuracy” term accounts for (e.5 % Without flow calibration: Without flow calibration: ≤ 0. temperature and installation con- . Today. and which preferably are to be specified by the USM manufacturer. • Whether pressure and temperature correction of meter body dimensions are used or not.: • USM dimensional data.1.2 % of measured value specified velocity range For evaluation of a USM fiscal gas metering station and its uncertainty.g. and the deviation re.value. CMR (December 2001) 188 6. 2.value. typical data as specified by USM manufacturers today are given in Table 6.2 % of reading in ± 0. . installation effects. the buyer or user of a USM may occasionally end up with some questions in relation to manufacturer data. which should preferably be known to enable an uncertainty evaluation of the USM fiscal gas metering station at a “detailed level” with respect to the USM. systematic transit time effects. • Flow calibration results (e. Fig.g. • USM uncertainty data (“accuracy” and repeatability).5 % of meas.). information normally available from USM manufacturers includes e. Table 6. the number of calibration points (M). Typical “problems” may be: • The “accuracy” specified in the data sheets is usually not sufficiently defined. Repeatability ≤ 5 mm/s < 0. cf. ±0.g. etc. ±0. With flow calibration: With flow calibration: Higher accuracy.Sonic) (SeniorSonic) (MPU 1200) “Accuracy” ≤ 0. reference after applying the correction factor K ( Dev C . NFOGM. • USM path configuration data (to some extent. Instromet (2000) Daniel (2000) Kongsberg (2000) (Q. With respect to the USM uncertainty. j .25 % of meas. β Table 6.4. 5. the user does not know whether the figure specified in the data sheet shall be divided by 1. α and Y. NFOGM.“ ----- of linear temperature expansion for the meter body material. At least it should be specified whether the repeatability figure accounts for both or not. Proposed “USM meter body” data and uncertainties to be specified by the USM manufacturer. Such data can be used directly as input to the Excel uncertainty evaluation program EMU .4. is generally lacking.3. Sections 3.3.3 Average wall thickness w mm 2. . Whether pressure and temperature correc. . such as R0.2.2 gives the proposed “USM meter body” data and corresponding uncertainties to be specified. 3 or 3 (or another number) in order to obtain the corresponding standard uncertainty value.10. 5. cf.3 and 3.4.2. If the repetability is different in flow calibration and in field operation. standard uncertainty of the coefficient u( βˆ ) | βˆ | % ----. 3.3. 5.“ ----- of linear pressure expansion for the meter body material. and deviation from flow calibration to field operation conditions.2. • The repeatability is often not specified as a function of flow velocity (or flow rate). • A single repeatability figure is specified in data sheets. standard uncertainty of the coefficient u( αˆ ) | αˆ | % ----. Chapter 5.1. both may be needed. the confidence level and probability distribution should be specified. and whether pressure and temperature correction . at the "detailed level".2 Uncertainties Rel.4.6.3 Material data Temperature expansion coefficient α K-1 2. some more specific USM uncertainty data are proposed here.3. 5. CMR (December 2001) 189 ditions. 5.USM fiscal gas metering stations NPD.3 Young’s modulus (or modulus of elasiticity) Y MPa 2. Handbook Section Dimensions Inner diameter ("dry calibration" value) 2R0 mm 2. w. 2. 2. α Rel.USM Fiscal Gas Metering Station.Handbook of uncertainty calculations .3. Meter body Quantity Symbol Unit Ref. the confidence level and probability distribution should be specified.1). which may be the case in practice (cf.3 P & T correc. Most of these data are normally available from USM manufacturers today. for uncertainty evaluation of the USM fiscal gas metering station. cf. That means. Tables 6. For improved evaluation of USM fiscal gas metering stations and their uncertainty. Table 6. Note that for all uncertainties. • Confidence level(s) and probability distribution(s) are lacking (both for “accuracy” and repeatability). For uncertainties.3.2-6. tion tion of meter body dimensions is used or not 4. Quantity Symbol Unit Tentative Ref. 2. as also indicated in Table 6. ….USM Fiscal Gas Metering Station: for each of φ i 0 .3. 112 The domains for specification of nominal values of φ i 0 . φ i 0 . for uncertainty evaluation of the USM fiscal gas metering station. y i 0 R 0 and w i the manufacturer may specify a value within a domain around the actual (unavailable) value. In such cases the following compromise approach may be used to run EMU .1 ----. none of the parameters listed in Table 6.3 are only tentative. Tentative domains have been proposed in Table 6.4. cf. i y i0 R 0 ----. w i . The "ideal" situation with respect to uncertainty evaluation . With respect to φ i 0 .3: ± 5 o or better for the inclination angles. cf. N wi . 5. N Nrefl.3 are already available from all USM manufacturers today. Handbook domain112 Section No.2. y i 0 R 0 . i φi0 o ±5 o ----. Some of the data set up in Table 6.3 .3 need to be specified. Table 6. N ± 0. manufacturer data may not always be available. If the “overall level” is used for USM field operation (both with respect to repeatability and systematic deviation relative to flow calibration. but may not be available from others.“ ----- ("dry calibration" value).1 or better for the integration weights. A more systematic analyses with respect to such domains should be carried out. of acoustic paths N . N Integration weight of path no. - No. i = 1.“ ----- Inclination angle of path no. The two relative uncertainty terms u( αˆ ) | αˆ | and u( βˆ ) | βˆ | may be more difficult to specify. based on only a few limited investigations for a 12” USM. …. y i 0 R 0 and w i . Note that the sum of the integration weights w i is to be approximately equal to unity. these are available from some USM manufacturers. i = 1. NFOGM. “USM path configuration” data which may preferably be specified by the USM manufacturer.1 or better for the relative lateral chord positions. How- ever.i ----. and ± 0. …. i.“ ----- ("dry calibration" value). Note that these are needed only if the “detailed level” is used for the USM in field operation. Section 4.USM fiscal gas metering stations NPD. i.10). i = 1. i = 1. i = 1. N. y i 0 R 0 and w i given in Table 6.i.“ ----- N åw i =1 i ≈1 Table 6. ± 0. .would be that the manufacturer data for these were known.3.Handbook of uncertainty calculations .1 - Relative lateral chord position of path no. such as N and Nrefl.at least from a user viewpoint . ± 0. . CMR (December 2001) 190 of the meter body dimensions are used or not.3 gives the “USM path configuration” data which may preferably be specified by the USM manufacturer. …. of reflections in path no. Section 5. ….3. Section 5.10. accounting for all paths (i..3. Table calibration to field operation.2. for uncertainty evaluation of the USM fiscal gas metering station.4. 4.4.…. Table 3. 4. t̂ 1i . the 5.6 (i.4.8 ings) Repeatability of USM transit time readings in field opera.5 gives the proposed “USM field operation” uncertainty data to be specified by the USM manufacturer.4. Proposed “USM field operation” uncertainty data to be specified by the USM manufacturer. For uncertainties.1. Table 6.3. 2. t̂ 1i . i.8 change of conditions from flow calibration to field operation Relative standard uncertainty of the USM integration E I . NFOGM.4. j. no. 4.4. are related to the USM repeatability in field operation. u( t̂ 1systematic ) ns 3. E rept .3. Table 3.e.2.USM fiscal gas metering stations NPD. M (cf. the relative standard deviation of the flow rate read. M.3. j = 1.1.1. Fig.4.8 change of conditions from flow calibration to field operation Standard uncertainty due to systematic effects on the down.e.1. The two first parameters in the table.4. 4. for uncertainty evaluation of the USM fiscal gas metering station.3.8 Standard uncertainty due to systematic effects on the up.…. Table relative standard deviation of the transit time readings) 3. due to 5. i = 1. j = 1.2. j. 5. the confidence level and probability distribution should be specified. DevC. cf. i = 1. Table 3.e.3.j .3.. Quantity / Uncertainty Symbol Unit Ref.8 Table 6. 4. method. 4.10.9. no. Fig. CMR (December 2001) 191 Table 6. u( t̂ 1random i ) tion. Quantity / Uncertainty Symbol Unit Ref.∆ % 3.5.2. Handbook Section No. at the actual flow rate. of flow calibration points M . 2. at flow calibration pt.4. due to 5.10.2.2.2. i stream transit time of path no. Proposed “USM flow calibration” data and uncertainties to be specified by the USM manufacturer.Handbook of uncertainty calculations . 2. 4. reference (after using the correction factor K). the actual flow rate.4 gives the proposed “USM flow calibration” data and uncertainties to be specified by the USM manufacturer.2.. N.2.2.1.…. i.3. the relative standard deviation of the flow rate reading) Table 6.10. the confidence level and probability distribution should be specified. 3.4. ----. u( t̂ 2systematic ) ns 3.2.1 Deviation re.4.10. 5. 2. j % 3.10. For uncertainties.…. Table (i. They represent the repeatability of .1) Repeatability of USM flow rate reading. Handbook Section Repeatability of USM flow rate reading in field operation. E rept and u( t̂ 1random i ) . i stream transit time of path no.“ ----- at flow calibration pt.4. at E rept % 3. N. These data are normally available from the manufacturers. due to change of installation conditions from flow 5. ns 3. 3. gas composition. The last three parameters included in Table 6.4. Fig. NFOGM.4. However. 5. 5.5. Proposed “Flow computer” uncertainties to be specified by the USM manufacturer. the confidence level and probability distribution should be specified.13). Sectiuons 3. for uncertainty evaluation of the USM fiscal gas metering station. this is not likely to be the situation in practice.5) so that the user of the program could himself choose which one to use.2 and 4. respectively.6 gives the proposed “Flow computer” uncertainties to be specified by the USM manufacturer. u( t̂ 1systematic i ) . Fig. temperature. Preferably.11. 5. E rept is needed if the “overall level” is used for the USM repeatability in field operation (cf. communication with flow computer Table 3.USM fiscal gas metering stations NPD. and there is no drift in the transducers.6. u( t̂ 2systematic i ) and E I . For uncertainties.10. Handbook Section Relative standard uncertainty of the estimate q̂ v due to signal E comm % 3. Note that if the field conditions and flow calibration conditions are identical (with respect to pressure. Uncertainty Symbol Unit Ref. These data should be readily available from the manufacturers.∆ . both parameters should be specified by the USM manufacturer (as indicated in Table 6.5.USM Fiscal Gas Metering Station. Preferably. Cf. respectively. Table 6. Both types of data should be readily available from USM flow computers.7 Relative standard uncertainty of the estimate q̂ v due to flow E flocom % ----. They are needed if the “detailed level” is used for the systematic USM effects in the program EMU . 4.2.1 for a discussion.“ ----- computer calculations Table 6. cf. and u( t̂ 1random i ) is needed if the “detailed level” is used (cf. are related to systematic deviations of the USM relative to flow calibration. They represent the systematic effects on the upstream and downstream transit times.Handbook of uncertainty calculations . CMR (December 2001) 192 the flow rate and the transit times. these three terms would be zero. all three parameters should be specified by USM manufacturers. flow profiles (axial and transversal)). Sec- tion 5.12).5 . . and installation effects. and • Multipath ultrasonic gas flow meter (USM). qe. In spite of that. NFOGM. CONCLUDING REMARKS A Handbook of uncertainty calculation for fiscal gas metering stations based on a flow calibrated multipath ultrasonic gas flow meters has been worked out.. such as the GUM [ISO. CMR (December 2001) 193 7.e.USM Fiscal Gas Metering Station for calculation of the expanded uncertainty of such metering stations. cf. including a Microsoft Excel program EMU .after a period of practical use of the Handbook and the program . The uncertainty model for USM fiscal gas metering stations presented in this Hand- book is based on present-day “state of the art of knowledge” for stations of this type. It is expected that the most important uncertainty contributions have been accounted for. • Compressibility factor calculation (from GC gas composition measurement). The uncertainty evaluation is made in conformity with accepted international standards and recommendations on uncertainty evaluation. Evaluation of the effects of these factors on the uncertainty of the metering station should be possible with the uncertainty model and the program EMU . the volumetric flow rate at line conditions). Section 2.the uncertainty model pre- . • Temperature measurement.Handbook of uncertainty calculations . qv. the volumetric flow rate at standard reference conditions). and is not expected to be complete with respect to description of effects influencing on such metering stations. • Standard volume flow (i. • Calorific value measurement (calorimeter). The following metering station instrumentation has been addressed (cf. It is the intention and hope of the partners presenting this Handbook that .1): • Pressure measurement. 2000].USM fiscal gas metering stations NPD. • Mass flow rate. the uncertainty model does account for a large number of the important factors that influence on the expanded uncertainty of metering stations of this type. qm. Q.e. The uncertainty of the following four flow rate measurements have been addressed (cf. Chapters 2 and 3): • Actual volume flow (i. Appendix B. 1995a] and ISO/CD 5168 [ISO.USM Fiscal Gas Metering Station developed here. • Density measurement (vibrating element densitometer). and • Energy flow rate. and others with interest in this field.USM Fiscal Gas Metering Station should constitute a useful basis for implementation of upgraded descriptions of the uncertainty model.3 are covered at the “overall level”.a useful and accepted method for calculation of the uncertainty of USM fiscal gas metering stations can be agreed on.e. as discussed briefly in the following. NFOGM.: (A) The uncertainty of alternative instruments and/or calculation methods. With respect to measurement of the energy flow rate (qe). (B) The USM field uncertainty. (C) The functionality of the Excel program.USM fiscal gas metering stations NPD. without correlation to other measurements involved (gas chromatography). However. Table 2. CMR (December 2001) 194 sented here will be subject to necessary comments and viewpoints from users and developers of USMs. Table 2. cf. Table 2. 2 different approaches are accepted by the NPD regulations [NPD. the following topics may be of relevance: • For the volumetric flow rate at standard reference conditions (Q).3. With respect to possibilities for improvements. That means. method no.2. This may concern e. in the present approach the calorific value may implicitely be assumed to be measured using a calorimeter (i. at least 3 different approaches are used by different gas companies. cf. Only selected methods are covered at the “detailed level”. An update of the Handbook and . cf.Handbook of uncertainty calculations . Only one of these is addressed here (method no.2).1).1-2. Only one of these is addressed here (method no. the Handbook and the program EMU . 5 of Table 2. as a basis for a possible later revision of the Handbook. as described above. In the present Handbook and Excel program all methods described in Tables 2.g. (A) With respect to possible upgraded uncertainty descriptions of the instruments and/or calculation methods involved in USM gas metering stations. For the mass flow rate (qm). the calorific value uncertainty is only addressed at an “overall level”.in the end . at least 5 different approaches to measure the energy flow rate may be used. 3 of Table 2. 1 of Table 2.3). in the Norwegian metering society as well as internationally. The overall objective of such a process would of course be that . 2001].1. 3 at a “detailed level”113. • Here the instantaneous values of the respective flow rates are addressed.19). • For on-line vibrating element densitometers. (2. and all of these represent simplifications. NFOGM. (B) The Handbook and the program should also constitute a useful basis for implementation of possible upgraded uncertainty descriptions of the USM field uncertainty.20)-(2. with respect to measurement of Q.4 may be included in the uncertainty model. (3.USM Fiscal Gas Metering Station on such points.Handbook of uncertainty calculations . CMR (December 2001) 195 the program EMU .e. as options. (2. to cover several or all methods indicated in Tables 2.: • With respect to pressure expansion of the meter body.6.3. such as e.USM fiscal gas metering stations NPD. at which the actual value of β becomes essential. Eq. 113 Such an upgrade would need to address possible correlation between the compressibility factors. The program can be updated to account for the accumulated flow rates (i.4. several methods are in use for VOS correction. a single model for the coefficient of radial pressure expansion β has been implemented in the present version of the program.2. Hs.22) are used for pressure correction of the meter body. based on the functional relationship for ρ u . would need to be addressed. qm and qe. a description of the various uncertainty contributions listed in Section 3. including the uncertainty due to the integration of the instantaneous flow rates over time). Eq. especially in connection with high pressure differences between flow calibration and field operation. Several models for β are used in current USMs. (2. . In a possible future revision of the Handbook other methods for VOS correction may be implemented than the method used here.14). Also possible correlation between Z and Z0 and the superior calorific value. Table 2. Z and Z0.23).g. cf. and the molar weight. as described in Section 2. might represent a useful extension to cover a broader range of metering methods used in the gas industry. m. • Implementation (in the program) of an option with automatic calculation of the possible error made if Eqs. Im- plementation of a choice of various models for β may thus be of interest. With respect to the contribution to u ( ρˆ misc ) in Eq. cf.1-2. • For the input uncertainties of the compressibility factors. which would provide a possibility of entering the USM input uncertainties in a physically more "correct" way. the need for a more detailed level of input uncertainties may also grow. such as: (1) ”Overall level” (completely meter independent.2.Handbook of uncertainty calculations . Such exchange may result in changed time delay and ∆t-correction (“dry calibration” values). That means.2. preferably over a relatively short time period. this does not influence on the uncertainty of the flow calibrated USM or the metering station. • In case of transducers exchange. The program can be extended to optional input of all four formulations A. as today's "overall level").2). such as e.: • The repeatability may vary between paths. as described in Section 3. Table 2. with respect to: . Section 3. 114 In the present version of the program EMU . 115 Entering of input uncertainties for individual paths. and what is available today from USM manufacturers.3. this may be done for only one or two paths.USM fiscal gas metering stations NPD. CMR (December 2001) 196 • Further with respect to pressure and temperature expansion of the meter body. However.g. • Improved flexibility with respect to levels of complexity for entering of input uncertainties can be implemented114. e. for various gas compositions of relevance. but may be more convenient for the user.3. NFOGM. This is done to avoid a too high "user treshold" with respect to specifying USM input uncertainties. An option of several levels of complexity for input uncertatinties may be convenient. and (3) ”Detailed level 2”: More detailed input uncertainties can be given than in today’s ”detailed level”. cf. this may occur at only one or two paths (either upstream or downstream. the analysis uncertainties of Z and Z0 can be evaluated statistically using a Monte Carlo type of method. may be very useful in many cir- cumstances. if the USM manufacturer uses another formulation of the functional relationship than A.g. • In situations with a path failure. the additional effect of transducer expansion / contraction can be evaluated. C and D. USM manufacturers may use historical flow profile data to keep the meter “alive”. as the USM technology grows more mature.input uncertainties entered for individual paths (not only average over all paths)115. and description of the propagation of these uncertainties to the metering stations’s expanded uncertainty.USM Fiscal Gas Metering Station. • In case of erroneous signal period detection (cf.2.5. (2) ”Detailed level 1” (as today’s ”detailed level”). for the . • At present Formulation A is used for input of the geometrical meter body quantities.4. or in both directions). As discussed in Section 2.3. the "detailed level" in the "USM" worksheet is definitely a compromise between what ideally should be specified as input uncertainties. B. 4). Group. That means.meter independent USM technologies (as in today’s version). • The uncertainty model and the program EMU . That is. liquid. and . specific path configurations / integration techniques. and differently for different paths. 2001]) are averaged over all paths.Handbook of uncertainty calculations . There are thus systematic timing uncertainties associated with such procedures. transit time detection methods. and reading such stored data files into the Excel program. • PRV noise have been reported to be detected differently by different paths. (C) With respect to functionality of the Excel program. used in a gas metering station. such as an ordinary data file). and input uncertainties may thus be difficult to quantify in practice. it would be convenient to enable saving the uncertainty evaluation data in another format (less space demanding. Upgrading the program with an additional option for specification of input uncertainties at individual paths (“Detailed level 2”) would enable a more realististic description. and differently by the upstream and downstream transducers within a path. to account for e. In the present version of the program such effects are accounted for by input uncertainties which (for the convenience of the user [Ref.accounting for both correlated and uncorrelated effects between paths (cf. In many cases the specification of input uncertainties for individual paths may also be simpler to understand for the user of the program. Such extension(s) may be of particular interest for meter manufacturer(s).USM fiscal gas metering stations NPD.4 MB per file. the disadvantages of such an option (the specification of a larger number of input transit time uncertainties) should be balanced with the advantages and improved uncertainty evaluation which can be achieved using a “Detailed level 2”. Table 1. Thus. NFOGM.meter dependent USM technologies. variants of the program can be tailored to option- ally describe the uncertainty of a specific meter (or meters).g. data storage requirements may be an issue. . but also for users of that specific meter type. The user could then choose among these three levels. as it is closer to the practical metering situation. may build up differently at the upstream and downstream transducers. path in question.USM Fiscal Gas Metering Station can be extended to account for both . for a given uncertainty evaluation case.xls format). . CMR (December 2001) 197 . Appendix E). correction methods. the lacking upstream and downstream transit times are effectively substi- tuted with “synthetic” transit times.decomposition of random and systematic transit time effects into their individual physical contributions (cf. the consequences of which should preferably be evaluated at an individual path basis. In the present version. To save storage space. “dry calibration” methods. storing of an executed uncertainty evaluation is made by saving the complete Excel file (. etc. • Possible transducer deposits such as grease. etc. which requires about 1. USM fiscal gas metering stations NPD.Handbook of uncertainty calculations . CMR (December 2001) 198 PART B APPENDICES . NFOGM. 15 K). References to corresponding definitions given elsewhere are included. 2001].1. the control electronics. and gas composition.25 hPa. USM A multipath ultrasonic flow meter for gas based on measurement of transit times. Spoolpiece The USM meter body Meter run A flow measuring device within a meter bank complete with any associated pipework valves. Line conditions Gas conditions at actual pipe flow operational conditions. Standard reference conditions Reference conditions of pressure.Handbook of uncertainty calculations . EMU "Evaluation of metering uncertainty" [Dahl et al. NFOGM. Note that relevant definitions and terminology related to uncertainty calculations are listed in Appendix B. flow straighteners and auxiliary instrumentation [ISO. 288. 1999].g. for a dry. and the CPU unit / flow computer. CMR (December 2001) 199 APPENDIX A SOME DEFINITIONS AND ABBREVIATIONS In the present appendix. qUSM. 1998]. temperature and humidity (state of saturation) equal to: 1 atm. USM functional relationship The set of mathematical equations describing the USM measurement of e. 1998]. Small USM A USM with (nominal) diameter < 12” [AGA-9. The wording USM refers to the com- posite of the meter body (“spoolpiece”).. Section B. some terms and abbreviations related to USM fiscal gas metering stations are defined. the ultrasonic transducers. at the USM installation location. 1995b]. Large USM A USM with (nominal) diameter ≥ 12” [AGA-9. with respect to pressure. temperature. the axial volumetric flow rate at line conditions. . real gas [ISO.USM fiscal gas metering stations NPD. and 15 oC (1013. and calculation of transit time differences. real gas [ISO. "Dry calibration"117 (or more precisely. 1998]. 117 The wording “dry calibration” has come into common use in the USM community today. temperature and humidity (state of saturation) equal to: 1 atm. The "dry calibration" is not a calibration of the meter in the normal meaning of the word. 1998] (Section 5. Percentage deviation is given relative to the reference measurement. but a procedure to determine. this procedure is more correctly referred to as “zero flow verification test”. . NFOGM.4). However. temperature and gas composition. cables and electronics. By [NPD. since error refers to comparison with the (true) value of the flow rate. [AGA-9. 2001].Handbook of uncertainty calculations . CMR (December 2001) 200 Normal reference conditions Reference conditions of pressure. In [AGA-9. angles. 1998]. for a dry. or "zero point control"). pressure. Various corrections and correction factors are typically 116 By [AGA-9. 1998]. Here.15 K). 2001] the wording “zero point control” is used. at one or several specific conditions of pressure. Chapter 5. Deviation The difference between the axial volumetric flow rate (or axial flow velocity) measured by the USM under test and the actual axial volumetric flow rate (or axial flow velocity) measured by the reference meter [AGA-9. 273. Only the reference measurement of the flow calibration laboratory is compared with. Measurement of quantities which are needed for the operation of the USM.USM fiscal gas metering stations NPD. “zero flow verification test”. "Dry calibration" measurements are made typically in the factory. Deviation curve116 The deviation as a function of axial flow velocity over a given flow velocity range. and possibly ∆t-correction [AGA-9. and hence the term “deviation curve” is preferred here. usually in the factory.25 hPa. at a specific installation condition. and 0 oC (1013. a set of correction factors to be used in the meter software (including correction of transit times). Flow calibration Measurement of the deviation curve (cf. temperature and gas composition. which is never known. it should be emphasized that this wording may be mislead- ing. the term “error” will be avoided in this context.4). 1998] the deviation curve is referred to as the “error curve”. and is therefore used also here. transit time delays through transducers. such as relevant dimensions. USM fiscal gas metering stations NPD. i. Numerical technique to calculate the average flow velocity in the pipe from knowledge of the average flow velocity along each acoustic path in the USM. CMR (December 2001) 201 established by the "dry calibration". 1993]. such as for transit times. Gauss-Jacobi quadrature A specific integration method [Abromowitz and Stegun. That is. Ideal flow conditions Pipe flow situation where no transversal (non-axial) flow velocity components are present. 1995a]. PRV Pressure reduction valve. Time averaging period Period of time over which the displayed measured flow velocities and volume flow rates are averaged. Integration method (Or flow profile integration method).Handbook of uncertainty calculations . GUM Abbreviation for the ISO document “Guide to the expression of uncertainty in measurement” [ISO. Transversal flow velocity The non-axial components of flow velocity in the pipe. . 1968]. and where the axial flow velocity profile is turbulent and fully developed. both axial and non-axial flow velocity components are essentially zero [AGA-9. flow conditions in an ideal infinite-length straight pipe.e. NFOGM. VIM Abbreviation for the ISO document “International vocabulary of basic and general terms in metrology” [ISO. Axial flow velocity The component of flow velocity along the pipe axis. VOS Velocity of sound. 1998]. Zero flow reading The maximum allowable flow meter reading when the gas is at rest. 2001] and the NORSOK I-104 [NORSOK.2. The GUM procedure used here for evaluating and expressing uncertainty is summarized in Section B. The uncertainty model and the uncertainty calculations reported here are therefore based primarily on the “GUM”. 1995a]118 as the “accepted norm” with respect to uncertainty analysis. on basis of a request from the CIPM.1. IOML: International Or- ganization of Legal Metrology. Consequently. Nancy. Require- ments to documentation of the uncertainty calculations are described in Section B. Oxford.preferably . in 118 The GUM was prepared by a joint working group consisting of experts nominated by BIPM. UK. A list of important symbols used in the Handbook for expressing uncertainty is given in Section B. as a basis for the description of the uncertainty model and the uncertainty calculations.1. BIPM: Bureau International des Poids et Mesures. France (In- ternational Committee for Weights and Measures). IUPAP: International Union of Pure and Applied Physics. 1998a] standard refer to the GUM (Guide to the expression of uncertainty in measurement) [ISO. the definition of some selected terms regarding uncertainty calculations which are used in the present Handbook.a minimum possibility of misunderstandings. ISO and OIML. IFCC. France (International Bureau of Weights and Measures). IEC. ISO: International Organization for Standardization Genève. A brief outline of the GUM terminology used in evaluating and expressing uncertainty is given in Section B. The abbreviations are: CIPM: Comité International des Poids et Mesures. are summarized in Table B. IUPAC: International Union of Pure and Applied Chemistry.3.4. IUPAC. IFFC: International Federation of Clinical Chemistry. ISO. with reference to the source documents used. IUPAP and OIML. Sévres Cedex. NFOGM.1 Terminology for evaluating and expressing uncertainty Precise knowledge about the definitions of the terms used in the Handbook is important in order to perform the uncertainty calculations with . Switzerland.USM fiscal gas metering stations NPD. Frolunda Sweden. Switzerland. Genève. IEC: Interna- tional Electrotechnical Commission. Paris. CMR (December 2001) 202 APPENDIX B FUNDAMENTALS OF UNCERTAINTY EVALUATION The NPD regulations [NPD. IEC. or are important for using the Hand- book.Handbook of uncertainty calculations . The following seven organizations supported the development. France. which was published in their name: BIPM. B. France. . . VIM. 2. 4. p. GUM comment: p.2. 9 which the output quantity.2. in practice a conventional true value is used (cf. VIM. depends. .. §B. §D. 1995]. 32-33. [NIS 3003. through a functional relationship. . §2. 1999] and [ISO/CD 5168. 9 a functional relationship. §1. §3. xi. y = f(x1. 3. the symbol notation is defined in Section B.USM fiscal gas metering stations NPD. §1. “Conventional true value” is sometimes called assigned value. §1. [Bell.2. 32 VIM note (selected): 1. Measurements Measurand Particular quantity subject to measurement.1.2.7 sult of measurement.g. with the word "true" to be re- dundant.6 GUM. 1. An input quantity. [EAL-R2. Handbook comment: 4.2.. §3. y = f(x1. as having an uncertainty ap. p. x2. 1995a]119..5. §4.10. 31 True value Value consistent with the definition of a given particular VIM. M). (of a quantity) propriate for a given purpose.. VIM notes (selected): p. e.. For definition of terms in which symbols are used. VIM. GUM. Definitions of some terms regarding uncertainty calculations. y. True values are by nature indeterminate.20 true value sometimes by conventions. §B. 34). . Some documents which may be helpful in this respect are [Taylor and Kuyatt. xM).3. xi (i = 1. VIM. Appendices E and F [ISO.1 and and units is determined from M other quantities x1. but that affects the re. 1999]. but GUM. Since a true value cannot be determined.1. and the GUM. GUM.4. NFOGM. xM through §4. p.1.1. 1999]. Conventional Value attributed to a particular quantity and accepted.9 Influence quantity Quantity that is not the measurand. cf. Table B. the VIM [ISO. .3. The term “true value” is not used.. This is a value that would be obtained by a perfect GUM.. The input quantities may themselves be viewed as measurands and may themselves depend on other quantities. Measurement Result of Value attributed to a measurand. 1993].19 (of a quantity) quantity. y In most cases a measurand y is not measured directly. p. p.. §B. [Dahl et al. is a quantity upon GUM.2.1. xM) Input quantity. GUM. pp.1. the GUM. 4. best estimate of the value. obtained by measure. §B. conventional value or reference value. x2. 41. Type of term Term Definition Reference Quantities Output quantity.2. (of a quantity) a unit measurement multiplied by a number. since the terms “value of a measurand” (or of a quantity) and the term “true value of a measurand” (or of a quantity) are viewed as equivalent. §4. 1994].2. . 2000].. 1997].18. §2.. Handbook of uncertainty calculations . x2.2. p. CMR (December 2001) 203 which further details may be given. measurement.. §3.1 119 Note that a number of documents are available in which the basic uncertainty evaluation philoso- phy of the GUM is interpreted and explained in more simple and compact manners. GUM. [EA-4/02. For additional definitions of relevance. for practical use in metrology.. GUM. Value Magnitude of a particular quantity generally expressed as VIM. p. The term “precision” should not be used for “accuracy”. GUM.23.2.the indication. it should be made clear whether it refers to: .USM fiscal gas metering stations NPD.4 error. 2. Since the systematic error cannot be known perfectly. §B. the correction must be included in the functional relationship. 33 1. measurement to compensate for systematic error. . GUM. §3.2. 34 VIM note: 1. When a result is given.11. The expression of this concept by numbers should be associated with (standard) uncertainty.13. . p. NFOGM.the corrected result. . or another quantity to be used in calculating the value of the measurand. and whether several values are averaged. the compensation cannot be complete. 33 Correction Value added algebraically to the uncorrected result of a VIM.15. VIM notes: p. error. Handbook comment: 3. the correction must be included in the functional relationship. Indication Value of a quantity provided by a measuring instrument. 2. If a correction factor is applied. 33 Corrected result Result of a measurement after correction for systematic VIM. §B. The value read from the display device may be called the direct indication.16 measurement is multiplied to compensate for systematic GUM.2. §B.2. §3. §B. §B. Accuracy of Closeness of the agreement between the result of a meas. the compensation cannot be complete. Handbook comment: 3. The correction is equal to the negative of the estimated systematic error. §B. 2. VIM notes: p.5 measurement urement and a true value of the measurand. VIM. Handbook comment: 2. Uncorrected result Result of a measurement before correction for systematic VIM. A complete statement of the result of a measurement includes information about the uncertainty of measurement. Since the systematic error cannot be known perfectly.12.3 error. Correction factor Numerical factor by which the uncorrected result of a VIM. 34 1. §3. it is multiplied by the instrument constant to give the indication. VIM notes: p.2 (of a measuring VIM notes (selected): instrument) 1. Handbook of uncertainty calculations . 2. 33 1. If a correction is made. Accuracy should not be used quantitatively. Accuracy is a qualitative concept. §3.2. a measurement signal.24.14. p. GUM. §3. GUM. §3.2. GUM. and the calculation of the combined standard uncertainty must include the standard uncertainty of the applied correction factor. The quantity may be the measurand. CMR (December 2001) 204 results measurement ment. VIM. and the calculation of the combined standard uncertainty must include the standard uncertainty of the applied correction.the uncorrected result. §3. . . 34.2. VIM. §B. Reproducability Closeness of the agreement between the results of meas. Annex D Error Result of a measurement minus a true value of the meas. VIM.method of measurement. Experimental A quantity characterizing the dispersion of the results. 34 Deviation Value minus its reference value. VIM notes: 1. Random error is equal to error minus systematic error. from an infinite number of measurements of the same GUM. . VIM. 33.2.6 sive measurements of the same measurand carried out un.2.2. CMR (December 2001) 205 Repeatability Closeness of the agreement between the results of succes.21. §3.12.4.the same observer.observer. VIM. associated with the result of a measurement.repetition over a short period of time. 4.measuring instrument. Results are here usually understood to be corrected results.9. (of measurement) urand.USM fiscal gas metering stations NPD.2. p.the same measurement procedure. §B. used under the same conditions. der the same conditions of measurement. it is possible to determine only an estimate of random error.location.7 urements of the same measurand carried out under changed GUM. 2.8 standard deviation a series of measurements of the same measurand. VIM. measurand.2.18.17. 3. Because only a finite number of measurements can be made. GUM. 34. §3.16. 33 VIM notes: 1. A valid statement of reproducability requires specification of the conditions changed. 2. §2. p.11 Relative error Error of a measurement divided by a true value of the VIM. . p.2. §3.principle of measurement. §3.2. .Handbook of uncertainty calculations .the same measuring instrument. VIM.conditions of use. . p. conditions of measurement.19. VIM notes: 1. These conditions are called repeatability conditions. §3. Repeatability may be expressed quantitatively in terms of the dispersion characteristics of the results. 3. § B. Random error Result of a measurement minus the mean that would result VIM. . p. Systematic error Mean that would result from an infinite number of meas. Repeatability may be expressed quantitatively in terms of the dispersion characteristics of the results. §B.15. p.14 . . GUM. . for VIM. 34. NFOGM.20. measurement that characterizes the dispersion of the values that could GUM. p. §3. GUM. 2-3. . Repeatbility conditions include: .10. §3. §B. §B. . The changed conditions may include: .time.13. reasonably be attributed to the measurand. §B. §3. GUM. GUM. p.reference standard. 2. 33 Uncertainty of Parameter.the same location. measurand carried out under repeatability conditions. GUM. USM fiscal gas metering stations NPD. p.3. Accuracy Ability of a measuring instrument to give responses close VIM. and with which is associated a probability p.2. 36. and concepts set of values. 3. 35 random variable) Variance A measure of dispersion. Indication of a measuring instrument minus a a true value VIM. VIM. VIM.22 (of a measuring a specified value of the measurand. 35 distribution. see “bias”. VIM. §C. Measuring range. . GUM. 34. chosen for checking instrument) the instrument. VIM notes: 1. Systematic error is equal to error minus random error. Like true value. GUM. p. Section 3. systematic error and its causes cannot be completely known.4 Working range uring instrument is intended to lie within specified limits Resolution (of a Smallest difference between indications of a displaying VIM. §5. §5. Probability A function giving the probability that a random variable GUM.2.2. Error (of indica. Set of values of measurands for which the error of a meas. instrument VIM note: p.23 (of a measuring instrument) Bias Systematic error of the indication of a measuring instru. §5. CMR (December 2001) 206 urements of the same measurand carried out under repeat.20 tion) of a measuring of the corresponding input quantity. §B.2. NFOGM. 34. §5.25 (of a measuring ment. §3. §B. 2. This concept applies mainly where the instrument is compared to a reference standard. instrument VIM note (selected): p. distribution (of a takes any given value or belongs to a given set of values.2 nal range. §5. §5. §5.2 1. §5. Drift Slow change of metrological characteristic of a measuring VIM. instruments VIM note (selected): 1. this is the change in the indication when the least significant digit changes by one step. the difference between the greatest and smallest value is called range.3 instrument) VIM note: note. §5.22. VIM. For a digital displaying device. VIM note (selected): 1. “Accuracy” is a qualitative concept. §B2. Nominal range is normally stated in terms of its lower and upper limits. §C.19.20. Handbook of uncertainty calculations . §5. which is the sum of the squared GUM. For a measuring instrument. VIM. Zero error Datum error for zero value of the measurand. Statistical terms Random variable A variable that may take any of the values of a specified GUM. The bias of a measuring instrument is normally estimated by averaging the error of indication over an appropriate number of repeated measurements.2.1 of measuring the controls of a measuring instrument.14.12 displaying device) device that can be meaningfully distinguished. Repeatability Ability of a measuring instrument to provide closely simi. Datum error Error of a measuring instrument at a specified indication of VIM. Characterisation Nominal range Range of indications obtainable with a particular setting of VIM. VIM note: 1. deviations of observations from their average divided by p. 33 1. 5 1. In some fields of knowledge.16 instrument.2.18 of a measuring to a true value. p. ability conditions minus the true value of the measurand. §5. GUM. Span Modulus of the difference between the two limits of nomi.2.27 (of a measuring lar indications for repeated applications of the same meas- instrument) urand under the same conditions of measurement.2. §C. GUM. tion of the distribution of values that could reasonably be GUM §6. p. p. §2.. 9-18. Estimated value of a quantity.. 3. Standard Uncertainty of the result of a measurement expressed as GUM. p. but . x̂ M ). Chapter 3. p. 37. Input estimate.6. 38. §5. §C. §G..5.2. p. GUM Chapter 5. The sample standard deviation is an unbiased estimator of the population standard deviation. xi. 59.1.USM fiscal gas metering stations NPD.3. Thus the output estimate. Section B. uncertainty urement that may be expected to encompass a large frac. GUM note: p. and An estimate of the measurand.1. is termed combined standard uncertainty. pp. denoted by ŷ . GUM. is ob.1. numerical values to the parameters of a distribu.3. GUM. p. effect Note: p.Handbook of uncertainty calculations .. x2. from the observations in a GUM.4..3.. other quantities.26. §C. .1. …. . Handbook comment: 1. GUM.. GUM. p.. §3. k: standard uncertainty in order to obtain an expanded un.1. factor. Normal distribution GUM. x̂1 . §C. 20 Coverage Numerical factor used as a multiplier of the combined GUM. p. §2. 6. x̂2 .14. §2.19. §3.. 59-65. 3. xM. pp. GUM.. NFOGM. ciated with the result. .21. GUM note (selected): 1.3. attributed to the measurand. xM) p.5. 10.. The symbols used here are those used in this Hand- book. Annex G. Standard deviation The positive square root of the variance. output estimate tained from the functional relationship y = f(x1.3.2.12. i = 1. cf.1.6. Expanded Quantity defining an interval about the result of a meas. 5. uncertainty standard deviation p. GUM. M. mation. 23-24.. sample. and denoted uc. 23. contributions to uncertainty aris- ing from systematic effects are described as “those that remain constant while the measurement is being made. which is the result of the measurement.2. Handbook comment: 1. and is equal to the positive square root of the combined variance obtained from all variance and covariance components.3.2. GUM. using input estimates. x2. Systematic The effect of a recognized effect of an influence quantity. 3.24.1.3. p. 5 By [NIS 81. §C. pp. §C. . p.2. p. or by other means (such as by calculations). Sensitivity Describes how the output estimate y varies with changes in GUM. y. 36. 34 Estimation The operation of assigning.2. GUM §4. 37. 1994. x̂2 . is given by ŷ = f( x̂1 . coefficient the values of an input estimate.3. CMR (December 2001) 207 one less than the number of observations.3. p. 1. uncertainty when that result is obtained from the values of a number of p. x̂ M for the values of the M quantities x1. Combined standard The standard uncertainty of the result of a measurement. 4].2. the population standard deviation. 10. Estimate The value of an estimator obtained as a result of an esti. The sample standard deviation is a biased estimator of §C. Level of confidence GUM. p. certainty. GUM. 37 tion chosen as the statistical model of population from which this sample is taken.2. It is the estimated standard deviation asso. obtained either by measurement. §4.4. 2.1 and relationship.3. Miscellaneous Functional In most cases a measurand y is not measured directly.g. y = f(x1. the symbols used for expression of uncertainty are those used by the GUM [ISO. 9 a functional relationship...USM fiscal gas metering stations NPD. by p. ŷ . p. 1995a. x2. xi. CMR (December 2001) 208 can change if the measurment conditions. u c ( ŷ ) : the combined standard uncertainty of an output estimate.. ŷ : U ( ŷ ) = k ⋅ u c ( ŷ ) .2. and since it should not cause confusion here. . U ( ŷ ) : the expanded uncertainty of an output estimate. p.1. 2001] straight line.1. . xM through §4. p. f is determined from M other quantities x1. Ey. spatial variations of influence quantities.2.1. method or equipment is altered”. §4. but GUM.. Handbook of uncertainty calculations . §4. Random effect The effect of unpredictable or stochastic temporal and GUM. Type A evaluation Method of evaluation of uncertainty by the statistical GUM. 5 Linearity Deviation between a calibration curve for a device and a [NPD. §2. percentage) standard uncertainties. x2. and relative combined standard uncertainty. §3. xM) B. 3 engineering/scientific judgement. ŷ : u ( ŷ ) Ey = c ŷ With four exceptions (see Table B.e. x̂i . 3 Type B evaluation Method of evaluation of uncertainty by means other than GUM.2 and points (1)-(4) below).e.. i. (of uncertainty) the statistical analysis of series of observations. NFOGM. e. relative standard uncertainty. the same symbol.5 and 120 For simplicity in notatation. (of uncertainty) analysis of series of observations.3. u( x̂i ) : the standard uncertainty of an input estimate. Ex : the relative standard uncertainty of an input estimate.3. y. . x̂i : u( x̂i ) Ex = x̂i Ey : the relative combined standard uncertainty of an output estimate120. is used for both types of relative (i...2.2 Symbols for expressing uncertainty In general. §2. ŷ : an estimated value (or simply an estimate) of an output quantity. In each case it will be noted in the text which type of relative uncertainty that is in question.. the following symbols are used in the present Handbook for expressing quantities and uncertainties: x̂i : an estimated value (or simply an estimate) of an input quantity. respectively (such as "X" and "x". (3. 1995. while in the USM community a transit time is commonly denoted by t (cf. 1997]. other than a notation of the type u c ( ŷ ) ŷ (for the relative combined standard uncertainty of an output estimate. involving both capital and small letters as symbols for input/output quantities. Section 3. e. respectively (cf. [EA-4/02. the "conventions" of the GUM are not followed exactly. However. the notation w( x̂ ) = u( x̂ ) x̂ has been used for the relative standard uncertainty of an estimate x̂ (cf. 2000.2. and has been adopted here122. [ISO. no specific symbol was used in the GUM. p. 1994]. (Cf. pp.. Here. 2000]. 1999]. For example.1.notation for relative uncertainties was used also in [Lunde et al.1. 1997]) and physics in general. “Ey” is the symbol for relative uncertainty used by e.1).. 16-17]. 1997]. in the illustration examples [ISO. [Taylor and Kuyatt. 123 The “Ey” . p. For example.Handbook of uncertainty calculations . both capital and small letters are used for input quantites. Note 3 to §4. This notation is considered to be impractical for the present Handbook. 121 In the GUM [ISO. This notation has been used also in [ISO/CD 5168. [ISO. a quantity and an estimate value for the quantity are denoted by capital and small letters.USM fiscal gas metering stations NPD. or even necessary. to avoid unnecessary complexity in writing the expressions for the relative expanded uncertainty of the USM fiscal gas metering station. u c . respectively) (cf. 20]. [ISO/CD 5168. (2) With respect to relative uncertainties. To distinguish between a quantity and the estimate value of the quantity. mainly for practical reasons121. . for simplicity in notation. NFOGM. Annex H. 7]).4). engineering and elsewhere the temperature is uniformly denoted by T. 2000a]. [EAL-R2. the above defined terminology has thus been chosen (with the symbol “ x̂ ” (the “hat notation”) to denote the estimate value of the quantity “x”). 1995. their Eq.11)). CMR (December 2001) 209 §6. 1994] has proposed to denote relative uncertainties by using a subscript “r” for the word “relative”. and the relative expanded uncertainty. 9-10]. [ISO/CD 5168. 2000].r ( ŷ ) ≡ u c ( ŷ ) ŷ and U r ≡ U ŷ for the relative standard uncertainty.1]. u r ( x̂ ) ≡ u( x̂ ) x̂ .123. e. §5. ŷ ) [ISO. 1997. in order to enable use of common and well-established terminology in USM technology (cf. also [NIS 3003. the same symbol has been used for a quantity and its estimate. 1995a. This is one of several examples where this notation is considered to be impractical.g.6. 1995. 1997]. their §D1.e. in physics. 68]. p.g.1. 122 By [EAL-R2. [ISO 5168:1978]. [ISO. the "GUM conventions" are not used consequently. Moreover. pp. (1) With respect to the symbols used for a quantity and the estimate value of the quantity. for the present document. 1997]). the relative combined standard uncertainty. a simpler symbol than u c ( ŷ ) ŷ has been found to be useful. i. also in the GUM. cf. see also [Taylor and Kuyatt.g. the symbol notation used in the Handbook is summarized and compared with the symbol notation recommended by the GUM. NFOGM. re. These symbols are used also by [ISO/CD 5168. q̂ m and q̂e . respectively. the symbols U ( q̂ v ) . to avoid confusion. 2000]. the use of simply “U” has been recommended by the GUM. Handbook of uncertainty calculations . to avoid confusion with the well established notation c used in acoustics for the sound velocity (VOS). “ ŷ ” denotes the estimate value estimate value of the output quantity spectively ("Y" and "y") of the output quantity “y” Standard uncertainty of an input u( x i ) u( x̂ i ) No estimate Combined standard uncertainty of an uc ( y ) u c ( ŷ ) No output estimate Relative standard uncertainty of an u ( xi ) u( x̂i ) input estimate E xi = Yes xi x̂i Relative combined standard u c ( y) uc ( ŷ ) uncertainty of an output estimate Ey = y ŷ Expanded uncertainty U U ( ŷ ) Yes / No Relative expanded uncertainty U U ( ŷ ) No y ŷ Dimensional (absolute) sensitivity ci si coefficients Yes Dimensionless (relative) sensitivity c * s * i i coefficients . In the present document that would lead to ambiguity. re. In Table B. the GUM has recommended use of the symbols ci and c*i . Table B. Hence. However. Q̂ . Symbol notation used in the Handbook in relation to that recommended by the GUM. U ( Q̂ ) . “ x̂ i ” denotes the estimate value Yes estimate value of the input quantity spectively (" X i " and " x i ") of the input quantity “ x i ” Output quantity & Capital and small letters. Chapters 2 and 3.2. (4) With respect to the symbols used for dimensional (absolute) and dimensionless (relative) sensitivity coefficients.2. since the expanded uncertainties of four output estimates are considered in the USM uncertainty model: q̂v . U ( q̂ m ) and U ( q̂ e ) are used for these. the symbols si and si* are used here for the dimensional (absolute) and dimensionless (relative) sensitivity coefficients of the output estimate ŷ i to the input estimate x̂i . CMR (December 2001) 210 (3) With respect to the symbol “ U ( ŷ ) . cf.USM fiscal gas metering stations NPD. Term GUM symbol Handbook symbol Deviation ? Input quantity & Capital and small letters. other symbols are used. xi. 1999]. Chapter 7. on which y depends: y = f(x1. 1999]). U ( x̂i ) .g. the “hat” symbol. the estimated value of the input quantity. which are all based on (and are claimed to be consistent with) the GUM. the text given in Chapter 7 of the GUM is authoritative. Section B. x̂i . the GUM is considered as the authoritative text. The function. Chapter 7. xM). given as125: 1. . [Bell.2. but the wording may be different in some cases. either as Type A evaluation of standard uncertainty (for an input estimate obtained from a statistical analysis of observations). the symbols “y” and “xi” are used for the output and input quantities. or by other means (in accordance with the GUM. is determined. 1995]. x̂i .3 Procedure for evaluating and expressing uncertainty The procedure used here for evaluating and expressing uncertainty is the procedure recommended by the GUM 124 [ISO.. If the uncertainty of the input estimate x̂i is given as an expanded uncertainty. to distinguish between these.. xi. [NIS 3003. Cf. 126 In the general overview given in Appendix B. CMR (December 2001) 211 B. In case of possible inconsistency or doubt. also the GUM. Chapter 7.Handbook of uncertainty calculations . NFOGM. in accordance with the GUM. including all corrections and correction factors. 125 The GUM procedure is here given in our formulation. or as Type B evaluation of standard uncertainty (for an input estimate obtained by other means). In Chapters 2-4. 2. and the input quantities. is used here. Chapter 7]. The standard uncertainty u( x̂i ) of each input estimate x̂i is evaluated. §1)126. k: 124 Other documents of interst in this context are e.USM fiscal gas metering stations NPD. §2)127. 3. The substance is meant to be the same. respectively. should preferably contain every quantity. The (mathematical) functional relationship is expressed between the measurand. x2. 127 With respect to the estimated value of a quantity xi (input or output). 1995a. this expanded uncertainty may be converted to a standard uncertainty by dividing with the coverage factor. that can contribute significantly to the uncertainty of the measurement result. Chapter 3. f. 1997]. [EA-4/02. §3 (cf. which are more in agreement with the general literature on USMs (see also Section B. either on the basis of a statistical analysis of a series of observations. where M is the number of input quantities (in accordance with the GUM. 1994]. 2000]. However. . y.. [EA-4/02. [EAL-R2.2). [EAL-R2. 1999] and [ISO/CD 5168. 1997]. [Taylor and Kuyatt. When an expanded uncertainty is given as a specific number of standard deviations.USM fiscal gas metering stations NPD. u c ( ŷ ) is given as the positive square root of the combined variance u c2 ( ŷ ) . in accordance with the GUM. N. y.3) i =1 è ∂ x i i =1 j = i + 1 ∂ x i ∂ x j c ø where N is the number of input esitmates x̂i . using for the input quanties the estimates x̂i obtained in Step 2. 2 N æ ∂f ö 2 N −1 N ∂f ∂f u ( ŷ ) = å çç 2 ÷÷ u ( x̂i ) + 2 å å u( x̂i . ŷ . §6 (cf. (B. Chapter 7. x̂ j ) = 0 represents uncorrelated quantities. the covariance is given as u( x̂i . converting to standard uncertainty is done by using k = 3 . For two input estimates x̂i and x̂ j . The value is a number between -1 and 1. x̂ j ) (i ≠ j). The combined standard uncertainty. (B. x̂ j ) . ŷ . . If the confidence level is 100 %. u c ( ŷ ) . Chapter 4). i = 1. si. the estimate. The result of the measurement is to be calculated in accordance with the GUM. and a rectangular probability distribution is used. x̂ j ) | = 1 represents fully correlated quantities 5. CMR (December 2001) 212 U ( x̂i ) u( x̂i ) = (B. and the partial deriva- tives are the sensitivity coefficients. where r( x̂i . That is. k = 2. ….1) k For example. x̂ j ) may be determined by engineering judgement or based on simulations or experi- ments. Covariances are evaluated associated with input estimates that are correlated. in accordance with the GUM. NFOGM. 4. §5. is evaluated from the standard uncertainties and the covariances associated with the input estimates. §4 (cf. Chapter 7. 6. of the measurement result (output estimate). and | r( x̂i . If the confidence level is 99 %. f. also the GUM.Handbook of uncertainty calculations .2). x̂ j ) . if U ( x̂i ) is given at a 95 % confidence level. x̂ j ) = u( x̂i )u( x̂ j )r( x̂i . the standard uncertainty is achieved by dividing the given expanded uncertainty with the specific number of standard deviations. i. the correlation coefficient between x̂i and x̂ j (where i ≠ j and |r| ≤ 1). also the GUM.2) where the degree of correlation is characterised by r( x̂i . Section 4. Chapter 7. The value of r( x̂i . of the measurand. is to be calculated from the functional relationship. and a normal probability distribution is used. k = 3. and a normal probability distribution is used.e. 5) on basis of the level of confidence required for the uncertainty interval ŷ ± U . CMR (December 2001) 213 ∂f si ≡ . (B. 4.14. together with its expanded uncertainty. §7 (cf.45 % in case of a normal probability distribution of the output estimate.8. also the GUM. 4. ŷ . it is therefore recommended to use k = 2 which is assumed to produce an interval having a level of confidence of approximately 95 %.3-5.13. k = 3. This includes documentation of the value of each input estimate. This is also in accordance with NPD regulations [NPD. to obtain U = k ⋅ u c ( ŷ ) (B. 4. cf. 2001]. in accordance with the GUM.g.11. Chapter 7. ŷ 128.USM fiscal gas metering stations NPD. u c ( ŷ ) .31 in Chapter 5. 128 Note that a coverage factor of k = 2 produces an interval corresponding to a level of confidence of 95. also the GUM. 4. k = 2 has also been used 8. intervals having levels of confidence of say 94.4) ∂x i 7. Tables 4. k. Chapter 5 and An- nex G). The result of the measurement (the output estimate). . Chap- ter 6).29 in Chapter 4. The GUM justifia- bly emphasises that for most cases it does not make sense to try to distinguish between e.28-5. U. in accordance with the GUM. 95 or 96 %.Handbook of uncertainty calculations . The calculation of intervals having specified levels of confidence is at best only approximate. cf. NFOGM. the coverage factor k is set to k = 2. as well as Figs. 5. Chapter 7. is to be reported. If the confidence level is 100 %. show typical uncertainty budgets used for documentation of the calculated expanded uncertainties. x̂i . by a coverage factor.96 corresponds to a level of confidence of 95 %.USM Fiscal Gas Metering Station.9.2. and the evaluation method used to obtain the reported uncertainties of the output estimate as summarised in steps 1 to 7. and a normal probability distribution is assumed. If the confidence level is 99 %. and the method by which U has been obtained. In the uncertainty caclulations of Chapters 5 and 6. §7 (cf. if U is to be given at a 95 % confidence level. The expanded uncertainty U is determined by multiplying the combined standard uncertainty.45 % while that of k = 1. 4. the individual uncertainties u( x̂i ) which contribute to the resulting uncertainty. Annex G of the GUM. In the program EMU . In practice. For example. 4. corresponding to a level of confidence of 95. and a rectangular probability distribution is supposed.19 and 5. and a normal probability distribution is used.17-4.6. k = 3 . k = 2. Section 1. and the program EMU . In the formulas which are implemented in the program. (3) present the data analysis in such a way that each of its important steps can be readily followed and the calculation of the reported result can be independently repeated if necessary. may be used in a documentation of the uncertainty evaluation of the metering station. . recommended by the GUM. together with the “Report” worksheet and the mathematical expressions given in Chapters 2 and 3. Chapter 6]. the uncertainty calculations reported in Chapters 4. combined standard uncertainties.6) x̂i ŷ ŷ respectively. serves as a basis for the USM uncertainty model described in Chapter 3. A printout of the worksheets used for uncertainty evaluation of the measurand in question. . at a 95 % confidence level (assuming a normal probability distribution). NFOGM. 1995a. (2) list all uncertainty components and document fully how they were evaluated.1.4 Documentation of uncertainty evaluation According to the GUM [ISO. and the expanded uncertainties. all the information necessary for a re-evaluation of the measurement should be available to others who may need it. . 2001] the uncertainties are specified as relative expanded uncertainties. are in many cases expressed as relative uncertainties. In Chapter 6.USM Fiscal Gas Metering Station should fill essential parts of the documentation requirements (1)-(4) above.USM Fiscal Gas Metering Sta- tion described in Chapter 5. defined as u( x̂i ) u c ( ŷ ) U . CMR (December 2001) 214 The above procedure (given by steps 1-8).4 of the GUM it is stated that one should: (1) describe clearly the methods used to calculate the measurement result and its uncertainty from the experimental observations and input data. (B. with k = 2. (4) give all corrections and constants used in the analysis and their sources. the input standard uncertainties.Handbook of uncertainty calculations . In the NPD regulations [NPD.USM fiscal gas metering stations NPD. The present Handbook together with the program EMU . B. NFOGM. containing the uncertainty evaluations and copies of (or at least reference to) the documentation described above. Such documentation may be calibration certificates. and the uncertainty of uncorrected systematic transit time effects. data sheets. and its uncertainty. Furthermore. CMR (December 2001) 215 In addition. such as the integration uncertainty (installation conditions).Handbook of uncertainty calculations . manufacturer information or other specifications of the metering station. For uncertainty calculations on fiscal metering stations this requires that every quantity input to the calculations should be fully documented with its value (if needed). especially with respect to some of the USM input uncertainties. 129 In many cases this is a difficult point. An uncertainty evaluation report should be generated. the user of the program must himself document the uncertainties used as input to the program129. it must be documented that the functional relationships used in the uncertainty calculation programs following the Handbook are equal to the ones actually implemented in the metering station. together with the confidence level and probability distribution. .USM fiscal gas metering stations NPD. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 216 APPENDIX C SELECTED REGULATIONS FOR USM FISCAL GAS METERING STATIONS The present Handbook relates to USM fiscal gas metering stations designed and operated according to NPD regulations [NPD, 2001]. For fiscal metering of gas using ultrasonic meters, the regulations refer to the NORSOK I-104 national standard and the AGA Report No. 9 as recognised standards (“accepted norm”). As a basis for the uncertainty calculations of the present Handbook, a selection of NPD regulations [NPD, 2001] and NORSOK requirements [NORSOK, 1998a] which apply to USM metering stations for sales metering of gas, are summarized in the following (with citations from the respective documents). Only selected regulations are included here. That is, regulations which are related to uncertainty evaluation and to fiscal gas metering stations based on USMs. The selection is not necessarily complete. For the full regulations it is referred to the above referenced documents. C.1 NPD regulations (selection) The following selection of regulations of relevance for USM fiscal gas metering stations are taken from [NPD, 2001]. • Section 8, Allowable measuring uncertainty. Measurement system, Gas metering for sale and allocation purposes: ±1.0 % of mass, at 95 % confidence level (expanded uncertainty with coverage factor k = 2). It shall be possible to document the total uncertainty of the measurement system. An uncertainty analysis shall be prepared for the measurement system within a 95 % confidence level. In the present regulations a confidence level equal to ±2 σ, i.e. a coverage factor k = 2, is used. This gives a confidence level slightly higher than 95 %. In respect of the measurement system's individual components the following maximum limits apply: Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 217 Component Circuit Linearity limits Uncertainty limits Repeatability uncertainty limits (band) component limits (band) Ultrasonic 1 pulse of 100 000, 1.0 % in working ± 0.70 % in work- 0.50 % in working flow meter gas 0.001 %, range (20:1). De- ing range (20:1) range (20:1) after (sales - allocation) at pulse transmis- viation from refer- after zero point zero point control sion of signal ence in calibration control shall be less than ±1.50 % in working range (20:1) before use of calibration factor Pressure ± 0.30 % of NA ± 0.10 % of NA measuring measured value in measured value in working range working range Temperature ± 0.30 oC NA ± 0.20 oC NA measuring oil, gas Density ± 0.25 % of NA ± 0.20 % of NA measuring gas measured value measured value Calorific value gas NA NA ± 0.15 % of NA calorific value Uncertainty NA NA ± 0.001 % NA computer part for oil and gas • Re. Section 8, Allowable measuring uncertainty. The basic principles for uncertainty analysis are stated in the ISO "Guide to the expression of uncertainty in measurement" (the Guide). • Section 10, Reference conditions. Standard reference conditions for pressure and temperature shall in measuring oil and gas be 101.325 kPa and 15 oC. • Section 11, Determination of energy content, etc. Gas composition from continuous flow proportional gas chromatography or from automatic flow proportional sampling shall be used for determination of energy content. With regard to sales gas metering stations two independent systems shall be installed. • Re. Section 11, Determination of energy content, etc. Recognized standard for determination of energy content will be ISO 6976 or equivalent. Reference temperature for energy calculation should be 25 oC / 15 oC (oC reference temperature of combustion / oC volume). When continuous gas chromatography is used, recognized standard will be NORSOK I-104. • Section 13, Requirements to the metering system in general. The metering system shall be planned and built according to the requirements of the present regulations and in accordance with recognized standards for metering systems. On sales metering stations the number of parallel meter runs shall be such that the maximum flow of hydrocarbons can be measured with one meter run out of service, whilst the rest of the meter runs operate within their specified operating range. If necessary, flow straighteners shall be installed. In areas where inspection and calibration takes place, there shall be adequate protection against the outside climate and vibration. The metering tube and associated equipment shall be installed upstream and downstream for a distance sufficient to prevent temperature changes affecting the instruments that provide input signals for the fiscal calculations. • Re. Section 13, Requirements to the metering system in general. If, on an allocation metering station with ultrasonic metering a concept based on only one metering tube is selected, there should Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 218 be possibilities to check the meter during operation and to have the necessary spare equipment ready for installation in the metering tube. In gas metering the maximum flow velocity during ultrasonic metering should not exceed 80 percent of the maximum flow rate specified by the supplier. • Section 14, The mechanical part of the metering system. It shall be documented that surrounding equipment will not affect the measured signals. • Re. Section 14, The mechanical part of the metering system. Re. design of the metering system for hydrocarbons in gas phase, recognized standards are NORSOK I-104, ISO 5167-1, AGA Re- port no. 9 and ISO 9951. • Section 15, The instrument part of the metering system. Pressure, temperature, density and composition analysis shall be measured in such way that representative measurements are achieved as input signals for the fiscal calculations (cf. Section 8). • Re. Section 15, The instrument part of the metering system. Recognized standards are NORSOK I-104 and I-105. The signals from the sensors and transducers should be transmitted so that measurement uncertainty is minimized. Transmission should pass through as few signal converters as possible. Signal cables and other parts of the instruments loops should be designed and installed so that they will not be affected by electromagnetic interference. When density meters are used at the outlet of the metering station, they should be installed at least 8D after upstream disturbance. When gas metering takes place, density may be determined by continuous gas chromatography, if such determination can be done within the uncertainty requirements applicable to density measurement. If only one gas chromatograph is used, a comparison function against for example one densitometer should be carried out. This will provide independent control of the density value and that density is still measured when GC is out of operation. Measured density should be monitored. • Re. Section 16, The computer part of the metering system. Recognized standards are NORSOK I-104 and I-105. In ultrasonic measuring the computer part should contain control functions for continuous monitoring of the quality of the measurements. It should be possible to verify time measurements. • Re. Section 19, General. When equipment is taken into use, the calibration data furnished by the supplier may be used, if they are having adequate traceability and quality. If such is not the case, the equipment should be recalibrated by a competent laboratory. By competent laboratory is meant a laboratory which has been accredited as mentioned in recognized standard EN 45000/ISO 17025, or in some other way has documented competence and ensures traceability to international or national standards. • Section 20, Calibration of mechanical part. The mechanical parts critical to measurement uncertainty shall be measured or subjected to flow calibration in order to document calibration curve. The assembled fluid metering system shall be flow tested at the place of manufacture and flow meter calibration shall be carried out. • Re. Section 20, Calibration of mechanical part. The checks referred to will for example be measuring of critical mechanical parameters by means of traceable equipment. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 219 The linearity and repeatability of the flow meters should be tested in the highest and lowest part of the operating range, and at three points naturally distributed between the minimum and the maximum values. • Section 21, Calibration of instrument part. The instrument loops shall be calibrated and the calibration results shall be accessible. • Section 23, Maintenence. The equipment which is an integral part of the metering system, and which is of significant importance to the measuring uncertainty, shall be calibrated using traceable equipment before start of operation, and subsequently be maintained to that standard. • Section 25, Operating requirements for flow meters. In case of ultrasonic flow measurement of gas the condition parameters shall be verified. During orifice plate gas measuring or or ultrasonic gas measuring the meter tupes shall be checked if there is indication of change in internal surface. • Re. Section 25, Operating requirements for flow meters. Ultrasonic flow meters for gas should be checked after pressurization and before they are put into operation to verify sound velocity and zero point for each individual sound path. Deviation limits for the various parameters shall be determined before start-up. Recalibration should be carried out if the meter has a poor maintenance history. Sound velocity and the velocity of each individual sound path should be followed up continuously in order to monitor the meter. • Section 26, Operation requirements for instrument part. The calibration methods shall be such that systematic measurement errors are avoided or are compensated for. Gas densitometers shall be verified against calculated density or other relevant method. • Section 28, Documentation prior to start-up of the metering system. Prior to start-up of the metering system, the operator shall have the following documents available: ………., (g) uncertainty analysis. • Section 29, Documentation relating to the metering system in operation. Correction shall be made for documented measurement errors. Correction shall be carried out if the deviation is larger than 0.02 % of the total volume. If measurement errors have a lower percentage value, correction shall nevertheless be carried out when the total value of the error is considered to be significant. If there is doubt as to the time at which a measurement error arose, correction shall apply for half of the maximum possible time span since it could bave occurred. • Section 30, Information. The Norwegian Petroleum Directorate shall be informed about : …, (b) measurement errors, (c) when fiscal measurement data have been corrected based upon calculations, …, etc. • Section 31, Calibration documents. Description of procedure during calibration and inspection, as well as an overview of results where measurement deviation before and after calibration is shown, shall be documented. This shall be available for verification at the place of operation. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 220 C.2 NORSOK I-104 requirements (selection) The NPD regulations for fiscal metering of gas [NPD, 2001] refer to NORSOK I-104 [NORSOK, 1998a] as an accepted norm. The following selection of functional and technical requrements are taken from NORSOK I-104. • §4.1, General. Fiscal measurement systems for hydrocarbon gas include all systems for: - Sales and allocation meassurement of gas, - Measurement of fuel and flare gas, - Sampling, - Gas chromatograph. • §4.2, Uncertainty. Uncertainty limits for sales and allocation measurement (expanded uncertainty with a coverage factor k = 2): ±1.0 % of standard volume (other units may be requested (project specific), e.g. mass, energy, etc.). (Class A.) The uncertainty figure shall be calculated for each component and accumulated for the total system in accordance with the following reference document: Guide to the Expression of Uncertainty in Measurement. • §4.4, Calibration. All instruments and field varibles used for fiscal calculations or comparison with fiscal figures shall be traceably calculated by an accredited laboratory to international/national standards. All geometrical dimensions used in fiscal calculations shall be traceably measured and certified to international/national standards. • §5.1.3, Equipment / schematic. The measurement system shall consist of: - A mechanical part, including the flow meter, - An instrument part, - A computer part performing calculations for quantity, reporting and control functions. • §5.1.7, Maintenance requirements: §5.1.7.2, Calibration. If it is impossible to calibrate the meter at the relevant process conditions, the meter shall at least be calibrated for the specific flow velocity range. • §5.1.8, Layout requirements. Ultrasonic flow meters shall not be installed in the vicinity of pressure reduction systems (valves, etc.) which may affect the signals. • §5.2.2, Mechanical part: §5.2.2.1, Sizing. The measurement system shall be designed to measure any expected flow rate with the meters operating within 80 % of their standard range (not extended). §5.2.2.4, Flow meter designs / ultrasonic meters. The number of paths for ultrasonic meters shall be determined by required uncertainty limits. All geometric dimensions of the ultrasonic flow meter that affect the measurement result shall be measured and certified using traceable equipment, at know temperatures. The material constants shall be available for corrections. In order for the ultrasonic meter to be accepted and considered to be of good enough quality the maximum deviation from the reference during flow calibration shall be less than ±1.50 %. The linearity shall be better than ±1.0 % (band) and the repeatablity shall be better than ±0.5 % (band). for flow velocities above 5 % of the maximum measuring range.USM fiscal gas metering stations NPD. The uncertainty (expanded uncertainty with a coverage factor k = 2) of the complete density circuit. For the ultrasonic flow meter. The computer shall calculate flow rates and accumulated quantities for (a) Actual volume flow. §5. in such a way that the overall uncertainty requirements are exceeded. All calculations shall be performed to full computer accuracy. §5. Computer part: §5.2. by automatic monitoring procedures in the instrument or by connecting external test equipment.3.30 % of measured value. the Pt 100 element and temperature transmitter may be installed as one unit where the temperature transmitter is head mounted onto the Pt 100 element (4. unless it is verified that the ultrasonic meter is not influenced by the layout of the piping upstream or downstream. Thermal insulation. Direct density measurement.2.4. the minimum straight upstream length shall be 10 ID. The density shall be measured by the vibrating element technique. The ultrasonic flow meter with associated meter tube should be thermally insulated upstream and downstream including temperature measurement point.or 3-wire system).4. Alterna- tively. The total uncertainty for the temperature loop shall be better than ±0. • §5.2. It shall be possible to verify the quality of the electric signal. Density shall be measured by at least two densitometers in the metering station. Calculations. For fiscal measurement applications the smart temperature transmitter and Pt 100 element should be two separate devices where the temperature transmitter shall be installed in an instrument enclosure connected to the Pt 100 element via a 4-wire system. Pres- sure and temperature measurement shall be measured as close as possible to the density measurement.15 oC. (c) Mass flow.2.5.5. The Pt 100 element should as minimum be in accordance to EN 60751 tolerance A. etc. Density calculation and calibration shall be in accordance with company practice.2. including drift between calibrations. §5.Handbook of uncertainty calculations . The temperature transmitter and Pt 100 element shall be calibrated as one system where the Pt 100 element’s curve-fitted variables shall be downloaded to the temperature transmitter before final accredited calibration.3. If density is of the by-pass type temperature compensation shall be applied.9.3.3. .2. Flow conditioner of a recognized standard shall be installed. and (d) Energy flow (application specific). The transducers shall be marked by serial number or similar to identify their location in the meter body. For the meter run. critical parameters relating to electronics and transducers shall be determined. §5. shall not exceed ±0. in order to reduce temperature gradients. Ultrasonic flow meter. A dedicated certificate stating critical parameters shall be attached. (b) Standard volume flow.2. Continuous measurement of density is required.1. (No additional truncation or round- ing. • §5. The density shall be corrected to the conditions at the fiscal measurement point.3.7. Location of sensors: Pressure and temperature shall be measured in each of the meter runs. Instrument part: §5.) The interval between each cycle for computation of instantaneous flow shall be less than 10 sec- onds. which represents the acoustic pulse. The density measurement device shall be installed so that representative measurements are achieved. CMR (December 2001) 221 These requirements are applicable after application of zero flow point calibration but before application of any correction factors.7. Temperature loop.2.2. The minimum straight downstream length shall be 3 ID. NFOGM. 4) sin φ i 0 Di0 xi 0 = (D. 2001]. (D. consider a temperature change only. Another objective of the present appendix is to show that Eqs.22).3) Di 0 Li0 = (D. [AGA-9..22) for inclination angles of practical interest for current USMs.12)-(2. D. Having defined the notation and most of the quantities involved in Chapter 2.17).13) and (2. CMR (December 2001) 222 APPENDIX D PRESSURE AND TEMPERATURE CORRECTION OF THE USM METER BODY The present appendix gives the theoretical basis of the model for pressure and temperature correction of the USM meter body given by Eqs. (2.g. take as starting point the temperature and pressure expansion of the inner radius of the meter body given by Eqs. i. the influence of temperature and pressure on the other geometrical quantities of the meter body are examined. for USMs where all inclination angles are ±45o.12)-(2. 1998]) R ≈ K T K P R0 .e. (2.18). 1997.Handbook of uncertainty calculations .60o.17) are practically equivalent to Eqs. and the consequences for Eq.20)-(2. (2. (cf.1) In the following. (2. and to evaluate the error of Eqs. K P ≡ 1 + β∆Pdry .USM fiscal gas metering stations NPD. (2. ∆Tdry.12) addressed. (2. The consequences for the geometrical quantities are illustrated in Fig. 40 o . where K T ≡ 1 + α∆Tdry .16)-(2.20)-(2. [Lunde et al.5) tan φ i 0 . NFOGM. Temperature change First. e.2) Di0 = 2 R02 − y i20 (D.1. From the figure one has y i 0 = R0 cos θ i0 (D. . At the other extreme.10) tan φ i tan φ i0 130 This assumption may be a simplification.as a simplification. depending on the type of pipe support for the meter body. (D. i in the USM. for a temperature change ∆Tdry only.8) Di K D Li = ≈ T i 0 = K T Li 0 . Between these two extremes is expected to be the case of a meter body mounted in a finite pipe section (between bends). (D. Fig.7) Di = 2 R 2 − y i2 ≈ 2 K T2 R02 − K T2 y i20 = K T 2 R02 − y i20 = K T Di 0 .6) Consequently.9) sin φ i sin φ i 0 Di K D xi = ≈ T i 0 = K T xi 0 .1 Geometry for acoustic path no. Now. it is assumed here . and the assumption may be poor. the meter body is free to expand in the axial direction. (D. depending on the type of pipe support for the meter body .Handbook of uncertainty calculations . T (sche- matically). for "free" axial boundary conditions (a finite pipe section with free ends).that the meter body will expand equally in the axial and radial directions130. and subscript “0” is used to denote the relevant geometrical quantity at dry calibration conditons. NFOGM. (D.1. showing changes with temperature.USM fiscal gas metering stations NPD. (D. one obtains yi = R cos θ i ≈ K T R0 cos θ i 0 = K T y i 0 . For instance. for nearly "clamped" axial boundary conditions (such as a meter body mounted in an infinitely long pipe). In this case the angles remain unchanged. cf.e. and the assumption should be a reasonable one. i. z ∆R Ro θio Lio y Dio Di yio yi φio= φi xio xi Fig. CMR (December 2001) 223 where Di0 is the chord length of path no. θ i = θ i0 and φ i = φ i 0 . the meter body is not likely to expand much in the axial direction. i. D. D. For the cylindrical tank model (with ends capped). the different models can be treated here conveniently in one single description. For a uniform internal pressure change.2 Geometry for acoustic path no. where K *P ≡ 1 + β * ∆Pdry .5 − σ ) [Roark.Handbook of uncertainty calculations . (D.2. 131 For the cylindrical pipe section model (with free ends). and these give different results for the axial displacement.e. 2001.6). θ i = θ i0 . one obtains yi = R cos θ i ≈ K P R0 cos θ i 0 = K P y i 0 . NFOGM. ∆Pdry. . different models for the meter body pressure expansion / contraction are in use (cf. P (schemati- cally). i in the USM. xi. (D. 2001. However. i. p. so that β * β = −σ . one has β * = R0 Yw and β * = − R0 σ Yw [Roark.11) z ∆R Li Ro Lio θio y Di Dio yio yi φio φi xi xio Fig. Table 2. showing changes with pressure. D. so that β * β = (1 − 2σ ) (2 − σ ) . and β * is the coefficient of linear pressure expansion for the axial displacement131.USM fiscal gas metering stations NPD. consider a pressure change.for a cylindrical pipe section model (ends free). The consequences for the geometrical quantities are illustrated in Fig. Illustrated here . p.without loss of generality .14) is the correction factor for the for the axial displacement of the USM meter body. one has β * = ( R0 Yw)(1 − σ 2) and β * = ( R0 Yw )( 0.13) With respect to displacement in the axial direction. (D. (D. 593]. Let xi = K *P xi0 . 592]. Consequently.12) Di = 2 R 2 − y i2 ≈ 2 K P2 R02 − K P2 y i20 = K P 2 R02 − y i20 = K P Di0 . D. the angle θ i remains unchanged. CMR (December 2001) 224 Pressure change Next. with axial contraction. 3).Handbook of uncertainty calculations . the ratio of the two correction factors is written as K *P 1 + β ∆Pdry æ β* ö * C≡ = ≈ 1 − (β − β * )∆Pdry = 1 − çç 1 − ÷( K P − 1 ) .17) ( = K P Li 0 1 − 2 cos 2 φ i 0 β − β * ∆Pdry ) and Di K D tan φ i 0 tan φ i = ≈ P i0 = . Eq.22) è C ø For steel (σ = 0. .16) leading to [ ( ) Li = xi2 + Di2 ≈ K P2 C 2 xi20 + K P2 Di20 = K P2 1 − β − β * ∆Pdry xi20 + K P2 Di20 ] 2 P [ ( * ) ] ≈ K 1 − 2 β − β ∆Pdry x + K D = K P Li0 2 2 i0 2 P 2 i0 xi20 Li0 ( ) 1 − 2 2 β − β * ∆Pdry (D. one has yi ≈ K T K P y i0 . Table 2.21) æ tan φ i0 ö φ i ≈ tan −1 ç ÷ . one has β * = 0 so that β * β = 0 . with no axial displacement). The parameters involved are defined in Section 2.14) can be written as xi = K P Cxi0 .19) Di ≈ K T K P Di 0 . where the former model gives axial contracton. The two models thus have different sign. and the ends-capped model gives β * β = 0.18) xi K P Cxi 0 C Combined temperature and pressure change Hence. (D. for a combined temperature and pressure change.3 .4 (cf.3. (D. Consequently. (D. CMR (December 2001) 225 One needs to express xi in terms of K P instead of K *P . NFOGM.15) KP 1 + β∆Pdry è β ÷ø where C is a number very close to unity.20) xi ≈ K T K P Cxi 0 (D. and the latter model gives axial expansion. (D. (D. For the infinitely long pipe model (ends clamped. (D.235 .6).USM fiscal gas metering stations NPD. (D. the ends-free model gives β * β = −0. For this purpose. ( ) (D.i + 1 )L2i0 ( t 1i − t 2i ) qUSM .3.Handbook of uncertainty calculations . as addressed in the following. CMR (December 2001) 226 Li ≈ K T K P Li0 1 − 2 cos 2 φ i0 β − β * ∆Pdry .24) xi 0 1 − B( K P − 1 ) [ ( ) ] 2 R0 L2i0 3 3 = K T K P 1 + ( 1 − 2 cos 2 φ i 0 ) β − β * ∆Pdry xi 0 For USMs for which all inclination angles are ±45o. and Eq. the validity of the commonly used expression for qUSM given by Eqs. (D.USM fiscal gas metering stations NPD.21) and (D. (D. N.22) may be evaluated.4.e.23) should constitute a relatively general model for the effect of pressure and temperature on the USM meter body. …. NFOGM. one has 1 − 2 cos 2 φ i 0 = 0 . (D. (D.26) i =1 2 xi t 1i t 2i where N ( N refl . (D.1). φ i 0 = ±45o.19)-(D. From Eqs. i = 1. i = 1.27) is Formulation C with the “dry calibration” values for the geometrical quantities inserted.24) reduces to 2 R 2 L2i R0 L2i 0 3 3 ≈ KT K P . accounting for any model being used for the radial and axial pressure expansion / contraction of the meter body (i. for β and β * . These expressions form the basis of Section 2.i + 1 )L2i ( t 1i − t 2 i ) qUSM = πR 2 å wi ≈ K T3 K P3 qUSM .20)-(2.5 leads to N ( N refl . …. N.23) Eqs.23) one obtains R 2 L2i ≈ ( K T K P R0 ) (K 2 T ( K P Li 0 1 − 2 cos 2 φ i0 β − β * ∆Pdry ) ) 2 xi K T K P Cxi 0 = R0 L2i0 3 3 1 − 2 cos φ i 0 β − β ∆Pdry 2 KT K P 2 * ( ) (D. (2. Meters with inclination angles equal to ±45o From the above analysis.25) xi xi 0 so that insertion into Formulation C given in Table 2.0 (D.e.0 ≡ πR 2 0 åw i =1 i 2 xi0 t 1i t 2 i (D. respectively). i. . B. for the cylindrical tank model (ends capped).22) apply to all the four formulations. CMR (December 2001) 227 It has thus been shown that for meters with inclination angles equal to ±45o. and β is given in Table 2. given by Table 2. Since formulations A. in combination with Eqs. and Eq. However. (D.3. Consequently. C and D. ∆Pdry = 10 and 100 bar yield relative errors in flow rate of about 6⋅10-5 = 0. Meters with inclination angles different from 45o Current USMs use inclination angles in the range 40 to 60o. (D. The relative error made by using Eq. (2. i.26) is then 1 2 ( 1 + σ )β∆Pdry = 12 ⋅ 1.5. ∆Pdry is given in bar. That means.28) which for the three meter body expansion models becomes ( 1 − 2 cos 2 φ i 0 )β ( 1 + σ )∆Pdry . To evaluate whether Eq. From Eq. (2. such er- .26). such errors can be neglected.6. for large pressure deviations ∆Pdry ( 100 bars or more). where β is given in Table 2. the relationship has been shown for formulation C.22). for moderate pressure deviations ∆Pdry (a few tens of bar).20)-(2. (D.24) this relative error is equal to ( ) ( 1 − 2 cos 2 φ i0 ) β − β * ∆Pdry . respectively. which is the same as Eqs. NFOGM.29) 1+σ ( 1 − 2 cos 2 φ i 0 )β ∆Pdry .26) is investigated. (D.06 %.6 for two of the models considered here (ends free and ends capped).3 ⋅ β∆Pdry . (D. consider the ends-free model. which is typically of the order of 6 ⋅ 10 −6 ⋅ ∆Pdry . (2. 2 −σ respectively. approximately. pressure deviations of e. (D.26) may be used also for inclination angles in the range 40o to 60o. (D. steel (σ = 0. pipe model (ends clamped).20)-(2.26) is a good approximation also for such inclination angles.g. for the infinite cylindr. the relative error made by using Eq. for the cylindrical pipe section model (ends free).Handbook of uncertainty calculations . it follows that Eqs. (D. Formu- lation C of Table 2.17). For example. are equivalent (can easily be derived from one another). and a “worst case” example with φ i 0 = 60o for path no. ( 1 − 2 cos 2 φ i 0 )β∆Pdry .5.13)-(2. (Here.006 % and 6⋅10-4 = 0.3).USM fiscal gas metering stations NPD. lead exactly to Eq.) For the 12” USM data given in Table 4. for such pressure deviations and inclination angles. . 2001] state that “Correction shall be made for documented measurement errors. Hence. section C. It should be noted. CMR (December 2001) 228 rors may not be negligible132. (D.USM fiscal gas metering stations NPD.26) represents an approximation which is not so good for inclination angles approaching 60o.Handbook of uncertainty calculations .20) represents only an approximation (with respect to separation of KP).02 % of the total volume. 132 The NPD regulations [NPD. Eq.1). however.” (cf. (2. NFOGM. and Eq. that KT can be separated out in all cases. Correction shall be carried out if the deviation is larger than 0. 3 the results of Section E. In Section E. T. Chapter 3 is intended to be self-contained in that respect. q v .1. (3. and the energy flow rate. (2. q e . as given by Eqs. . so that the model accounts for the USM flow calibration. The overall uncertainty model for the fiscal gas metering station is derived first. and Eqs. it is not necessary to read Appendix E.7)-(3. NFOGM. respectively (cf. Q.USM fiscal gas metering stations NPD. simply by including the uncertainties of P. Z/Z0 and Hs (for q e ). cf. given by Eqs.4)). and P. the USM field measurement. On basis of the detailed derivations given here.1) and (3.1)-(2. etc. CMR (December 2001) 229 APPENDIX E THEORETICAL BASIS OF UNCERTAINTY MODEL The present appendix gives the theoretical basis and mathematical derivation of the uncertainty model for the USM fiscal gas metering station which is summarized in Chapter 3. the footnote accompanying Eqs.1) and (2. respectively. (3. the present appendix is included essentially for documentation of the theoretical basis of the uncertainty model. The uncertainty models for the volumetric flow rate at standard conditions. Chapter 3 summarizes the expressions which are implemented and used in the program EMU – USM Fiscal Gas Me- tering Station. For practical use of the Handbook and the computer program EMU .2 are combinded with the uncertainty model for the gas metering station.4.9).9) for the functional relationship.6) for the relative expanded uncertainty. For convenience and compactness the detailed derivation of the uncertainty model is given here for the actual USM volumetric flow rate measurement only.Handbook of uncertainty calculations . Section E. (2.USM Fiscal Gas Metering Station. The uncertainty model for the un-flow-corrected USM is then given in Section E. ρ (for q m ).2. q v . are then obtained easily from the uncertainty model derived for the volumetric flow rate measurement. the mass flow rate. the flow calibration laboratory. Z/Z0 (for Q). T. As described in Section 1. q m . (B. [ISO. Since qUSM . j and qUSM are correlated. The method used here (for the purpose of deriving the uncertainty model) is to decompose qUSM . 134 This method of decomposition into correlated and uncorrelated parts (of partially correlated quantities). j and qUSM into their correlated and uncorrelated parts. is different from 0 and ±1 (|r| < 1). j and qUSM are those parts of the estimates qUSM . as given by Eqs. . C (E. as shown in Appendix F.3). 1995]. 1995]. p. However. the functional relationship is given by Eq. This partial correlation has to be accounted for in the uncertainty model. CMR (December 2001) 230 E. That is. such as used in Eqs. such as e. so that the correlation coefficient. while others are uncorrelated.Handbook of uncertainty calculations . (2. qUSM . qUSM . The evaluation of r may then in practice be difficult.2.2)-(B. This includes the approach used here.g. However. There are several ways to account for the partial correlation between qUSM . (E.e. i.1) qUSM = qUSM U + qUSM C . r. C C Here. 1997]. [ISO/CD 5168. the method was applied succesfully in [Lunde et al.1)-(E. They are associated with standard uncertainties 133 Three possible approaches to account for the partial correlation between qUSM . j and qUSM are measurement values obtained using the same meter (at the flow laboratory and in the field. respectively).USM fiscal gas metering stations NPD.1 Basic uncertainty model for the USM fiscal gas metering station For the volumetric flow rate measurement. 23 (D5)]. which is equivalent to Eq. (2. some contributions to qUSM .. (E. In fact. j and qUSM which are assumed to be mutually correlated. 1999. [Taylor and Kuyatt. j and qUSM 133. has not been found to be mentionened in other documents on uncertainty evaluation. [NIS. the inclusion of the correlation coefficient r in the evaluation of the standard uncertainty of the measurand is avoided. Appendix G.2)-(B. and a variant of this approach (although for a different type of problem) has been used in [EA-4/02. The various quantities involved are defined in Section 2. NFOGM. Eqs. and its relationship to and equivalence with the approach recommended by the GUM. [EAL-R2. (B. j = qUSM U . j and qUSM are partially correlated. 1997].1). involved in the ”covariance method” recommended by the GUM (cf. j . 1994].: qUSM .2). by using the ”decomposition method”. the ”decomposition method” used here and ”covariance method” recommended by the GUM are equivalent. This method has its main advantage when partial correlations are involved. j + qUSM .9). j and qUSM are addressed in Appendix F. 2000].2) respectively134.3)). This type of description. CMR (December 2001) 231 U U which are due to systematic effects. u c ( K̂ dev . at test flow rate no. j and qUSM which are assumed to be mutually uncorrelated.j ûú [ ] [ 2 + u( q̂ comm ) + u( q̂ flocom ) ]2 where u c ( q̂v ) ≡ combined standard uncertainty of the output volumetric flow rate estimate. j . (E. a correlated and an uncorrelated term. j . j . . NFOGM.4) 2 é ∂q̂ ∂q̂ ù + ê C v u c ( q̂USM C ) + C v u c ( q̂USM C )ú ëê ∂q̂USM ∂q̂USM . qUSM . j (representing the uncertainty of the flow calibration laboratory).3). j ) ≡ combined standard uncertainty of the reference measurement.9) can be written as U qUSM + qUSM C q v = 3600 ⋅ K dev . Formally135. which are uncorrelated with flow calibration). with such a decomposition of the flow rate measurements. (E. j úû 2 2 é ∂q̂ ù é ∂q̂ ù + ê U v u c ( q̂USM U )ú + ê U v u c ( q̂USM U )ú ë ∂q̂USM êë ∂q̂USM . j + qUSM .USM fiscal gas metering stations NPD. is used only for the purpose of developing the uncertainty model. is of course never used in any gas metering station or USM algorithm. (2. 135 Note that the decomposition of each of the flow rate measurements qUSM . j . j and qUSM into two terms. (E. the combined variance of the estimate q̂ v is given as 2 2 é ∂q̂v ù é ∂q̂ v ù u c2 ( q̂v ) = ê u c ( K̂ dev . Eq. j ) ≡ combined standard uncertainty of the deviation factor estimate.Handbook of uncertainty calculations . j U C qUSM From Eq. q̂v . j . j )ú êë ∂K̂ dev . j q ref . j and qUSM are those parts of the estimates qUSM .1)-(E.2). as made in Eqs. U u c ( q̂USM )≡ U combined standard uncertainty of the estimate q̂USM (representing the USM uncertainty contributions in field operation. q̂ ref .j û úû (E.1)-(E. j )ú + ê u c ( q̂ ref . K̂ dev . by inserting Eqs. u c ( q̂ ref . thus. (E.3) . j úû êë ∂q̂ ref .3). j ∂q̂ ref . j ∂q̂USM . To obtain Eq. j U q̂USM . . C u c ( q̂USM )≡ C combined standard uncertainty of the estimate q̂USM (representing the USM uncertainty contributions in field operation. j . = v . This assumption may be questioned. u( q̂ comm ) ≡ standard uncertainty of the estimate q̂v . the symbol “ x̂ ” (the “hat notation”) has been used to denote the estimated value of a quantity “x”.USM fiscal gas metering stations NPD. The other terms represent the contributions from the uncorrelated input estimates. u( q̂ flocom ) ≡ standard uncertainty of the estimate q̂v . j C q̂USM . j (representing the USM uncertainty contributions in flow calibration. j ) ≡ combined standard uncertainty of the estimate q̂USM . As explained in Section B. it has been assumed that the deviation factor estimate K̂ dev .Handbook of uncertainty calculations . j is uncorrelated with the other quantities appearing in Eq. j .g. j .4) represent the contributions C C to the combined variance from the correlated input estimates q̂USM . CMR (December 2001) 232 . j q̂ ref . which are correlated with flow calibration). j K̂ dev . Now. use of analog frequency or digital signal output). as K̂ dev . at test U U u c ( q̂USM flow rate no.5) ∂q̂USM q̂USM U ∂q̂USM q̂USM C ∂q̂ v q̂ ∂q̂ v q̂ =− v . (E.j ) ≡ C C u c ( q̂USM combined standard uncertainty of the estimate q̂USM . NFOGM. (E. due to flow computer calculations. j and q̂USM . ∂q̂USM .3).2. in flow calibration and field operation (assembled in on term). j may be correlated with possible systematic contributions to q̂ ref . (E. due to signal communication between USM field electronics and flow computer (e.3) that ∂q̂v q̂ v ∂q̂v q̂ = . in flow calibration and field operation (assembled in on term). but is not addressed further here. The contributions appearing in the third line of Eq. (E. (E. ∂K̂ dev . in order to distinguish between a quantity and the estimated value of the quantity. j . which are uncorrelated with field operation). since it follows from Eq. =− v .4). = v . at test flow rate no. j ∂q̂v q̂ ∂q̂v q̂ = v . which are correlated with field operation). j (representing the USM uncertainty contributions in flow calibration. E qC ≡ . j + ê c USM ú + ê + − . j ) u( q̂ ref .6) can be written more conveniently as E q2v = E K2 dev .Handbook of uncertainty calculations . (E. . q̂ m u( K̂ dev .j ) E qC ≡ . j U U u c ( q̂USM ) u c ( q̂USM .j ) E qU ≡ . j q̂ ref . q̂ ref .6) ë q̂USM û êë q̂USM . j q̂USM . j [ + EqC − EqC USM USM .USM fiscal gas metering stations NPD. j u( q̂ comm ) u( q̂ flocom ) E comm ≡ . q̂v q̂ v have been used. CMR (December 2001) 233 Eq.7) where the definitions u c ( q̂ v ) E qv ≡ . E qU ≡ . j ≡ relative standard uncertainty of the reference measurement. j úû êë q̂USM q̂USM . j )ù ê ú = ê ~ ú +ê ú ë q̂ v û êë K dev . (E. j ≡ . j + E q2ref . j ] +E 2 2 comm + E 2flocom (E. E K dev . K̂ dev . j )ù 2 é u c ( q̂ v ) ù é u c ( q̂ ref . K̂ dev . j + E q2U + E q2U USM USM . and E qv ≡ relative combined standard uncertainty of the output volumetric flow rate estimate. j ≡ relative standard uncertainty of the deviation factor estimate. NFOGM. j q̂USM . j úû êë q̂ ref . j C C u c ( q̂USM ) u c ( q̂USM . j . (E. j ≡ . E qref . j úû 2 2 é u( q̂comm ) ù é u( q̂ flocom ) ù +ê ú +ê ú ë q̂ v û êë q̂ v úû To simplify the notation.4) becomes 2 2 é u c ( K̂ dev .8) USM q̂USM USM . j úû 2 2 é u c ( q̂USM )ù é u c ( q̂USM )ù 2 é u ( q̂ U ) ù U C C ) u c ( q̂USM . j (representing the uncertainty of the flow calibration laboratory). j ) E K dev .j ú ê ú (E. E flocom ≡ . E qref . q̂v . USM q̂USM USM . Eq. E comm ≡ relative standard uncertainty of the estimate q̂v .2 Uncertainty model for the un-flow-corrected USM Before proceeding with the full uncertainty model for the gas metering station.2).Handbook of uncertainty calculations . To evaluate this. For this purpose it serves to be useful to first use an expression for the uncertainty of a un-flow-corrected USM. (E. j flow rate no.USM fiscal gas metering stations NPD. NFOGM. j (the USM uncertainty contributions in flow calibration. at test USM . j need to be associated with particular physical effects and uncertainty contributions in question for the USM. E qU and E qC . In the calculation program EMU .g.7). E qC ≡ C relative combined standard uncertainty of the estimate q̂USM (the USM USM uncertainty contributions in field operation. which are correlated with flow calibration). CMR (December 2001) 234 E qU ≡ U relative combined standard uncertainty of the estimate q̂USM (the USM USM uncertainty contributions in field operation. which are uncorrelated with flow calibration). E qref .USM Fiscal Gas Metering Station. the relative uncertainty contributions E qU . and next see what uncertainty terms which are .E qC USM USM . due to signal communication between USM field electronics and flow computer (e. in which “all” (or at least most of) the uncertainty contributions are described (Section E. j . j . given by Eq. E qU ≡ relative combined standard uncertainty of the estimate q̂USM U . j . E comm and E flocom are input relative uncertainties. E K dev . use of analog frequency or digital signal communication). j flow rate no. E. which are correlated with field operation). a model for the uncertainty of the USM is needed. in flow calibration and field operation (assembled in on term). in flow calibration and field operation (assembled in on term). j USM USM . E qC ≡ relative combined standard uncertainty of the estimate q̂USM C . E flocom ≡ relative standard uncertainty of the estimate q̂v . The terms related to the USM measurement are analysed in more detail in the following. due to flow computer calculations. j (the USM uncertainty contributions in flow calibration. which are uncorrelated with field operation). j . at test USM . (E. This means that the analysis has to be kept on a relatively general level. not i =1 è i =1 ø only is the correlation between upstream and downstream propagation in path no. Cf. That has not been done here.g. i accounted for.3). but also the correlation between the different paths. 139 136 However.1 Basic USM uncertainty model From [Lunde et al. However. (E. by ac- countinfg for both types of uncertainty terms. 138. The uncertainty model of the USM is to be meter independent.Handbook of uncertainty calculations . q̂USM . as well as possible correlation between flow calibration and field operation. in the sense that the analysis does not rest on specific meter dependent technologies which possibly might exclude application to some meters.USM fiscal gas metering stations NPD. etc. so that only deviations relative to flow calibration conditions are accounted for in the model. “dry calibration” methods. That means. It should be noted that this approach represents a simplification. The model can easily be extended to account for both correlated and uncorrelated effects between paths. 138 One modification has been made here relative to the expression given in [Lunde et al. Cf.3) and (5.USM Fiscal Gas Metering Station on a more generic level.9) is used (with the modification described in the footnote above). can be modelled as137. In the present Handbook... since some effects will be correlated between paths (electronics/transducer delay and ∆t-correction (P & T effects. etc.C * t 2i has ben replaced by çç (n) t 2 i . etc. 1997 (Eqs.18) of that report). to avoid a too complex user interface (difficulty in specifying USM input uncertainties). 1997]. meter dependent integration techniques.9).C s t 1i E t 1i . transit time detection methods.18)] and [Lunde et al.. these more detailed expressions are not used. such as specific integration techniques. an expression corresponding to Eq. after some time. Section E. and since the detailed derivation is available in [Lunde et al. variants of the program can be tailored to described the uncertainty of a specific meter (or meters). (7. (E.).USM Fiscal Gas Metering Sta- tion. (5.C + s *t 2 i E t(2ni . it should be noted that the uncertainty model and the program EMU . Such extension(s) may be of particular interest for meter manufacturer(s). terms of both types mentioned above should be accounted for in the uncertainty model. In addition. but also for users of that specific meter. .). used in a gas metering station. “dry calibration” methods. to avoid a too high "user treshold" (difficulty in specifying USM input uncertainties). expressions are given for the various uncertainty terms appearing in Eq.USM Fiscal Gas Metering Station can be extended to account for meter dependent technologies. while others may be uncorrelated between paths (transducer deposits.2)]. CMR (December 2001) 235 cancelled when applying this model to a flow-corrected USM (cf. are not to be accounted for136.. Chapter 6.. 1997] (Eqs.9) has not been included here since that would further increase the present appendix. in order to keep the program EMU . That means. Chapter 6. 2000a (Eq. In the last term of Eq. E.)C ÷÷ .2. NFOGM. etc. it may be more relevant to include both effects in possible future updates of the Handbook and the program EMU . (5.3) and (5. 137 The derivation of Eq. 139 In the GARUSO model. so that e. (E. Ideally.9) the expression 2 2 å[ ] å[ ] N æ N * (n) ö * s E t 1i (n) +s E t 1i . as the USM technology grows more mature. drift). it is known that the relative combined standard uncertainty of the un-flow-corrected USM volumetric flow rate reading. transit time detection methods. CU ) ) 2 (E.UU ) ) 2 + å ( s t*1i Et(1ni .U ≡ ..U ) t 1i . USM integration method.U ) ) E ( n . etc. For example. in GARUSO the integration uncertainty EI is calculated from more basic input (e.U. ŷi .U ≡ 1i . Similar situations apply to e. Nave is the number of time averagings used for a transit time measurement.U ) t 2 i . .UC ) ) 2 + ( s *t 2 i Et(2ni. u c ( R̂ ) ≡ combined standard uncertainty of the pipe radius estimate.U.)C ÷ .C ) ) E ( n . t̂ 1i t̂ 2 i have been used for the respective relative standard uncertainties. if that should is of interest (e. however. E ( n . due to approximating the integral expression for qUSM by a numerical integration (i.9) N ave i =1 i =1 2 æ N [ ö + ç å s *t 1i E t(1ni .U .Handbook of uncertainty calculations .). cf.. and u( q̂USM .9). NFOGM. CMR (December 2001) 236 [ ] N E q2USM ≈ E I2 + ( s *R E R ) 2 + å ( s *yi E yi ) 2 + ( sφ*i Eφi ) 2 i =1 å [( s ] [ ] N N 1 + * t 1i Et(1ni .g.U ≡ .g. for USM manufacturers).C ) t 2 i .UU ) ) 2 + ( s *t 2 i Et(2ni.USM Fiscal Gas Metering Station. the transit time uncertainties of Eq. EI is to be specified and documented by the USM manufacturer.e. It may be noted. that such more detailed expressions can be implemented in the program as well. Eφi ≡ c i . ŷ i φˆ i ( n . (E.)C + s *t 2 i Et(2ni . R̂ . a finite sum over the number of acoustic paths).U ≡ .)C ≡ .. flow velocity profiles (analytical or based on CFD modelling).U. u c ( ŷ i ) ≡ combined standard uncertainty of the lateral path position estimate. E ( n .g.I ) u c ( R̂ ) EI ≡ . In the program EMU . ] è i =1 ø where the definitions u( q̂USM .10) t̂ 1i t̂ 2i u c ( t̂ 1(in.I ) ≡ standard uncertainty of the estimate q̂USM . Here.U ) u c ( t̂ ) u c ( t̂ 2( in. ER ≡ .C) ) u c ( t̂ 2( in.C) ) Et(1ni . t̂ 1i t̂ 2i u c ( t̂ 1(in.C ) ) u c ( t̂ 2( in.USM fiscal gas metering stations NPD.C ) t 1i . Chapter 6. (E. Et(2ni ..)C ≡ . q̂USM R̂ u c ( ŷ i ) u ( φˆ ) E yi ≡ . and correlated with respect to the Nave upstream “shots” of path no. with respect to different “shots” (for a given propagation direction at path no.U ) ) ≡ combined standard uncertainty of t̂ 1(in.C) (the part of the upstream transit time estimate t̂ 1(in ) which is correlated with respect to downstream propagation. with respect to upstream and downstream propagation (for “shot” no. 141 With respect to transit times and their uncertainties. φ̂ i . N.Handbook of uncertainty calculations . NFOGM. due to use of the finite- sum integration formula instead of the integral. u c ( t̂ 1(in. respectively. and uncorrelated with respect to the Nave upstream “shots”140 of path no.U. ….C ) ) ≡ combined standard uncertainty of t̂ 2( in. u c ( t̂ 1(in.U. i.U. Superscripts “U” and “C” refer to uncorrelated and correlated estimates. i).U.U.U. The superscript “n” refers to “shot” no.USM fiscal gas metering stations NPD. i). E I accounts for effects of installa- 140 For simplicity.C) ) ≡ combined standard uncertainty of t̂ 2( in.U ) (the part of the downstream transit time estimate t̂ 2( in ) which is uncorrelated with respect to upstream propagation. and correlated with respect to the Nave downstream “shots” of path no. subscripts “U” and “C” refer to uncorrelated and correlated estimates. and correlated with respect to the Nave upstream “shots” of path no. i)141.U. i). Here.e. E I is the relative standard “integration uncertainty”. CMR (December 2001) 237 u c ( φˆ i ) ≡ combined standard uncertainty of the inclination angle estimate. i). the contribution to the relative combined standard uncertainty of the estimate q̂USM . u c ( t̂ 2( in. . u c ( t̂ 2( in. n in the sequence n = 1. n. i).U. and uncorrelated with respect to the Nave downstream “shots” of path no.C) ) ≡ combined standard uncertainty of t̂ 1(in. and correlated with respect to the Nave downstream “shots” of path no.C) (the part of the downstream transit time estimate t̂ 2( in ) which is correlated with respect to upstream propagation. respectively. i). u c ( t̂ 1(in.U ) ) ≡ combined standard uncertainty of t̂ 2( in. i).C ) (the part of the upstream transit time estimate t̂ 1(in ) which is uncorrelated with respect to downstream propagation. a “shot” refers here to a single pulse in a sequence of N pulses being averaged. at path no.U ) (the part of the upstream transit time estimate t̂ 1(in ) which is uncorrelated with respect to downstream propagation.C ) ) ≡ combined standard uncertainty of t̂ 1(in. u c ( t̂ 2( in.C ) (the part of the downstream transit time estimate t̂ 2( in ) which is uncorrelated with respect to upstream propagation. i. and out-of-roundness.UU ) are the relative combined standard uncertainties of those contributions to the upstream and downstream transit time estimates t̂1(in ) and t̂ 2( in ) . i. They represent e. Et(1ni . etc. and pressure / temperature effects on ŷ i (including possible pressure / temperature corrections). i)..)C and Et(2ni . but correlated with respect to the Nave “shots” (for a given propagation direction of path no.. Et(1ni . ŷ i .Handbook of uncertainty calculations . flow conditioners. for path no. and coherent noise. which are uncorrelated with respect to upstream and downstream propagation. Et(1ni .UU ) and Et(2ni. for path no. respec- 142 By random effects are here meant effects which are not equal or correlated from "shot" to "shot". E R is the relative standard uncertainty of the meter body inner radius estimate. such as pipe bend configurations.)C are the relative combined standard uncertainties of those contributions to the upstream and downstream transit time estimates t̂1(in ) and t̂ 2( in ) . pressure / temperature effects on R̂ (including possible pressure / temperature corrections).UC ) and Et(2ni. i). E R accounts for the instrument (measurement) uncertainty when measuring R̂ at “dry calibration” conditions. respectively.CU ) are the relative combined standard uncertainties of those contributions to the upstream and downstream transit time estimates t̂1(in ) and t̂ 2( in ) .USM fiscal gas metering stations NPD. E yi accounts for the instrument (measurement) uncertainty when measuring ŷ i at “dry calibration”. Eφi is the relative standard uncertainty of the inclination angle estimate φ̂ i . respectively. such as incoherent noise and turbulence effects (random velocity fluctuations. They represent random time fluctuation effects142.g. and random temperature fluctuations). pipe roughness.. CMR (December 2001) 238 tion conditions influencing on flow profiles.. and also uncorrelated with respect to the Nave “shots” (for a given propagation direction of path no. which are uncorrelated with respect to upstream and downstream propagation. and which therefore will be affected significantly by averaging of measurements over time. and temperature effects on φ̂ i (including possible temperature corrections). R̂ . . NFOGM. Eφi accounts for the instrument (measurement) uncertainty when measuring φ̂ i at “dry calibration”. effects of the clock resolution of the time detection system in the meter. E yi is the relative standard uncertainty of the lateral chord position estimate. )C and Et(2ni . st*1i = i . and with respect to the Nave “shots” (for a given propagation direction of path no. is thus accounted for in the uncertainty model through this last (sum) term in Eq. both with respect to upstream and downstream propagation. where for convenience in notation.11) these are about equal in magnitude but have opposite signs. s *R . They represent systematic effects due to cable/electronics/transducer /diffraction time delay correction (including finite beam effects).t̂ . possible transducer deposits. the definititions N P̂T0 Ẑ 0 ( N refl + 1 ) R̂ 2 − ŷ i2 ( t̂ 1i − t̂ 2i ) Q̂ ≡ å Q̂i .Handbook of uncertainty calculations . (E. CMR (December 2001) 239 tively. st*2i are the relative (non-dimensional) sensitivity coefficients for the sensitivity of the estimate Q̂ to the input estimates R̂ . The last (sum) term appearing in Eq. The partial cancelling effect obtained in USMs through the transit time difference t̂ 1i .11a) R Q̂ i =1 çè ( 1 − ŷ i R̂ ÷ø 2 ) Q̂ 1 − ŷ i R̂ ( ) 2 Q̂i 2 φ i ˆ Q̂ t̂ 2 i Q̂ t̂ 1i sφi = − * . . (E. From Eqs. i. s t*1i and st*2i . respectively. Thus. possible cavity delay correction.t̂ . sφ*i . (E. (E.9). equal relative combined standard uncertainties Et(1ni . st*2i = − i .t̂ 2i .9) involves the upstream and downstream relative sensitivity coefficients of path no. i i 1i 2i given as 1 N æ ç 1 ö ÷ Q̂i ŷi R̂ ( 2 ) s = * å Q̂i 2 + .φˆ .that means. s yi = − sgn( ŷ i ) * . s t*1i . possible beam reflection at the pipe wall (for USMs using reflecting paths) and sound refraction (profile effects on transit times). Q̂i ≡ 7200πR̂ 2 wi .12) i =1 P0 T̂Ẑ t̂ 1i t̂ 2 i sin 2φˆ i have been used.11b) Q̂ tan 2φ i ˆ Q̂ t̂ 1i − t̂ 2i Q̂ t̂ 1i − t̂ 2 i respectively. the resulting relative combined variance due to such drift will be comparably small and nearly negligible relative to the other (uncorrelated) terms. (E. for equal drift in the upstream and downstream transit times . which are correlated. s *yi . (E.USM fiscal gas metering stations NPD. NFOGM. ŷ . i).)C . i.CU ) ≈ Et(1ni .. Similarly.14) This assumption is motivated by the fact that Et(1ni . see Section E. finite clock resolution.U )2 .UU ) and Et(2ni..UU ) ≡ EtU1i .1.U ... (E. Note that this is a simplification relative to the GARUSO model [Lunde et al. it is assumed that Et(1ni .13) This assumption is motivated by the fact that Et(1ni . electronics stability (possible random effects).U . erronous signal .2.U ) 2 + ( EtC1i .15) N ave This single relative uncertainty term Et 1i .U ≡ ( EtU1i ..2.UU ) and Et(2ni. without much loss of generality. and without loss of generality. so that they can be replaced by a single relative uncertainty term.USM Fiscal Gas Metering Station. so that they can be replaced by a single relative uncertainty term. NFOGM.UC ) and Et(2ni.UC ) ≡ EtC1i .2. here denoted as EtU1i ...U thus accounts for the effects of turbulence (random velocity fluctuations.e.UU ) are about equal.. noise (incoherent and coherent).2. with respect to specification of USM input uncertainties. (E.CU ) are about equal. (E.. n).U .USM fiscal gas metering stations NPD. it is assumed that Et(1ni . the USM uncertainty model given by Eq. to be denoted by EtC1i ..UC ) and Et(2ni. n).CU ) are associated with random and systematic transit time effects on two acoustic signals that are propagating in opposite directions almost simultanously in time (upstream and downstream “shot” no.Handbook of uncertainty calculations .1 Uncorrelated transit time contributions In the following. it is then natural to define 1 Et 1i .1.. CMR (December 2001) 240 E. and random temperature fluctuations). Et(2ni.9) can be further simplified with respect to transit time uncertainties. (E. see Section E.g.UU ) ≈ Et(1ni . Et(2ni. That means.2 Simplified USM uncertainty model In order to avoid a too high “user treshold” of the program EMU .. 1997]. E..U . To further simplify the notation.2. possible random effects in signal detection/processing (e.UU ) are associated with random transit time fluctuation effects on two acoustic signals that are propagating in opposite directions almost simultanously in time (upstream and downstream “shot” no. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 241 period identification), and power supply variations, on the upstream and downstream transit times of path no. i (cf. Section E.2.1). E.2.2.2 Correlated transit time contributions For the correlated uncertainties, it is in the following assumed that Et(1ni ,)C are about equal for all Nave upstream “shots” of path no. i, and simply denoted Et 1i ,C , i.e. Et(1ni ,)C ≡ Et 1i ,C . (E.16) Similarly, it is assumed that Et(2ni ,)C are about equal for all Nave downstream “shots” of path no. i, and simply denoted Et 2 i ,C , i.e. Et(2ni ,)C ≡ Et 2 i ,C . (E.17) These assumptions are motivated by the fact that each of Et(1ni ,)C and Et(2ni ,)C is associated with systematic transit time effects on Nave acoustic signals that are propagating in the same direction almost simultanously in time, see Section E.2.1. Such systematic effects are e.g. cable/electronics/transducer/diffraction time delay correction, possible cavity delay correction, possible transducer deposits, possible beam reflection at the pipe wall (for USMs using reflecting paths) and sound refraction (profile effects on transit times), cf. Section E.2.1. E.2.2.3 Simplified USM uncertainty model Insertion of Eqs. (E.13)-(E.17) into Eq. (E.9), and using the fact that st*1i ≈ st*2i , yields a simplified expression for the relative standard uncertainty of the un-flow-corrected USM, [ ] N E q2USM ≈ E I2 + ( s *R E R )2 + å ( s *yi E yi ) 2 + ( sφ*i Eφi ) 2 + 2( s *t 1i Et 1i ,U ) 2 i =1 2 (E.18) æ N [ ] ö + ç å s *t 1i Et 1i ,C + s*t 2 i Et 2 i ,C ÷ è i =1 ø As a basis for the identification of uncertainty contributions to be made in Section E.3, a description of the various relative uncertainties appearing in Eq. (E.18) is summarized in Table E.1. Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 242 Table E.1. Relative uncertainties involved in the uncertainty model of the un-flow-corrected USM. Uncertainty Description term ER Relative standard uncertainty of the meter body inner radius estimate, R̂ , due to e.g.: • instrument uncertainty when measuring R at “dry calibration” conditions, • P & T effects on R̂ (including possible P & T corrections), • out-of-roundness. E yi Relative standard uncertainty of the lateral chord position estimate, ŷ i , due to e.g.: • instrument uncertainty when measuring ŷ i at “dry calibration” conditions, • P & T effects on ŷ i (including possible P & T corrections), Eφi Relative standard uncertainty of the inclination angle estimate, φ̂ i , due to e.g.: • instrument uncertainty when measuring φ i at “dry calibration” conditions, • P effects on φ̂ i (including possible P correction), Et 1i ,U Relative standard uncertainty of those contributions to the transit time estimates t̂ 1i and t̂ 2 i which are uncorrelated with respect to upstream and downstream propagation, such as: • turbulence (transit time fluctuations due to random velocity and temperature fluctuations), • incoherent noise (RFI, pressure control valve (PRV) noise, pipe vibrations, etc.), • coherent noise (acoustic and electromagnetic cross-talk, acoustic reverberation in pipe, etc.) • finite clock resolution, • electronics stability (possible random effects), • possible random effects in signal detection/processing (e.g. erronous signal period identification), • power supply variations. Et 1i ,C Relative standard uncertainty of those contributions to the upstream transit time estimate t̂ 1i (downstream transit time estimate t̂ 2 i ) which are correlated with respect to ( Et 2 i ,C ) upstream and downstream propagation, such as: • cable/electronics/transducer/diffraction time delay, including finite-beam effects (line P and T effects, ambient temperature effects, drift, effects of possible transducer exchange), • ∆t-correction (line P and T effects, ambient temperature effects, drift, reciprocity, effects of possible transducer exchange), • possible systematic effects in signal detection/processing (e.g. erronous signal period identification) • possible cavity delay correction, • possible deposits at transducer front (lubricants, liquid, wax, grease), • sound refraction (flow profile effects on transit times), • possible beam reflection at the meter body wall. EI Relative standard “integration uncertainty”, due to effects of installation conditions influencing on flow velocity profiles, such as: • pipe bend configurations upstream of USM, • in-flow profile to upstream pipe bends, • meter orientation relative to pipe bends, • initial wall roughness, • changed wall roughness over time (corrosion, wear, pitting, etc.), • possible wall deposits / contamination (lubricants, liquid, grease, etc.). Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 243 E.3 Combining the uncertainty models of the gas metering station and the USM In the following, the uncertainty model for the un-flow-corrected USM, Eq. (E.18), is to be used in combination with the uncertainty model for the gas metering station, Eq. (E.7), in which flow calibration of the USM is accounted for. That means, one is to associate each of the relative uncertainty contributions E qU , E qU and E qC - USM USM , j USM E qC with specific physical effects and uncertainty contributions in the USM. For USM , j this purpose, Eq. (E.18) and Table E.1 is used. However, Eq. (E.18) needs some modification to account for flow calibration effects. E.3.1 Consequences of flow calibration on USM uncertainty contributions The method used here to associate the various uncertainty contributions of Table E.1 with E qU , E qU and E qC - E qC , is to evaluate qualitatively each of the uncer- USM USM , j USM USM , j tainty contributions of Table E.1 with respect to (1) which effects that are (practically) eliminated by flow calibration, and (2) for those effects that are not eliminated by flow calibration, whether they are correlated or uncorrelated with respect to field operation re. flow calibration. Table E.2 summarizes the results of this evaluation. E.3.2 Modified uncertainty model for flow calibrated USM In Eqs. (E.9) and (E18), the uncertainty terms related to R̂ , ŷi and φ̂i , i = 1, …, N, have been assumed to be uncorrelated. This is a simplification, as commented in [Lunde et al., 1997]. In practice, they will be partially correlated. For example, instrument (measurement) uncertainties are typically uncorrelated, as well as the out- of-roundness uncertainty of R̂ . However, P and T effects are correlated, cf. Table E.1. By flow calibration, the uncorrelated contributions to E R , E yi and Eφi are practically eliminated (instrument uncertainties, out-of-roundness), cf. Table E.2. The remain- ing uncertainty terms are those related to P and T effects, which are correlated. For application to a flow calibrated USM, thus, the uncertainty model given by Eq. (E.18) needs to be modified, to accont for such contributions. That means, in this case the term Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 244 [ ] N ( s *R E R ) 2 + å ( s *yi E yi ) 2 + ( sφ*i Eφi ) 2 i =1 appearing in Eq. (E.18) is not valid and needs to be modified (replaced by one of the terms in Eq. (E.24), see below). Table E.2. Evaluation of uncertainty contributions in the uncertainty model for the un-flow-corrected USM, Eq. (E.18), with respect to effects of flow calibration. Uncertainty Eliminated Correlated or Contribution (examples) term in by flow uncorrelated effect ? USM model calibration? (flow calibration re. field) ER • instrument uncertainty (at “dry calibration”), Eliminated • P & T effects (including possible P & T corrections) Correlated • out-of-roundness Eliminated E yi • instrument uncertainty (at “dry calibration”) Eliminated • P & T effects (including possible P & T corrections) Correlated Eφi • instrument uncertainty (“dry calibration”) Eliminated • P effect (including possible P correction) Correlated Et 1i ,U • turbulence (random velocity and temperature fluct.) Uncorrelated • incoherent noise Uncorrelated • coherent noise Uncorrelated • finite clock resolution Uncorrelated • electronics stability (possible random effects) Uncorrelated • possible random effects in signal detection/processing Uncorrelated • power supply variations Uncorrelated Et 1i ,C • cable/electronics/transducer/diffraction time delay Correlated • ∆t-correction Correlated • possible cavity delay correction Correlated • possible systematic effects in signal detection Correlated • possible deposits at transducer front Correlated • sound refraction (flow profile effects on transit times) Correlated • possible beam reflection at the pipe wall Eliminated Et 2 i ,C • cable/electronics/transducer/diffraction time delay Correlated • ∆t-correction Correlated • possible cavity delay correction Correlated • possible systematic effects in signal detection Correlated • possible deposits at transducer front Correlated • sound refraction (flow profile effects on transit times) Correlated • possible beam reflection at the pipe wall Eliminated EI • pipe bend configurations upstream of USM Correlated • in-flow profile to upstream bends Correlated • meter orientation relative to pipe bends Correlated • initial wall roughness (corrosion, wear, pitting, etc.) Eliminated • changed wall roughness over time Correlated • possible wall deposits / contamination Correlated • possible use of flow conditioners Eliminated For this purpose, one needs to revisit the basic USM uncertainty model derivation given in [Lunde et al., 1997]. However, it is necessary here to consider only the terms related to R̂ , ŷi and φ̂i , i = 1, …, N, and account for correlations between Handbook of uncertainty calculations - USM fiscal gas metering stations NPD, NFOGM, CMR (December 2001) 245 those. With respect to these parameters, Eq. (5.5) in [Lunde et al., 1997] is then rewritten as (using Eq. (B.3)) 2 2 2 æ ∂Q̂ ö N æ ö N æ ö ∂Q̂ ∂Q̂ u ( Q̂ ) = çç 2 u c ( R̂ )÷÷ + å çç u c ( ŷi )÷÷ + å çç u c ( φˆ i )÷÷ è ∂R̂ i =1 è ∂ŷ i ∂φ c ˆ ø ø i =1 è i ø N ∂Q̂ ∂Q̂ N ∂Q̂ ∂Q̂ + å 2 r( R̂ , ŷ i ) u c ( R̂ )u c ( ŷ i ) + å 2 r( R̂ ,φˆ i ) u c ( R̂ )u c ( φˆ i ) i =1 ∂R̂ ∂ŷ i i =1 ∂R̂ ∂φˆ i (E.19) N N ∂Q̂ ∂Q̂ N N ∂Q̂ ∂Q̂ + åå 2 r( ŷ i ,φˆ j ) u c ( ŷ i )u c ( φˆ j ) + åå 2 r( ŷ i , ŷ j ) u c ( ŷ i )u c ( ŷ j ) i =1 j =1 ∂ŷ i ∂φˆ j i =1 j =1 ∂ŷ i ∂ŷ j N N ∂Q̂ ∂Q̂ + åå 2 r( φˆ i ,φˆ j ) u c ( φˆ i )u c ( φˆ j ) + other terms i =1 j =1 ∂φ i ∂φ j ˆ ˆ where r( â ,b̂ ) is the correlation coefficient of the estimates â and b̂ . Now, let r( â , b̂ ) = sign( â ) ⋅ sign( b̂ ) , (E. 20) where sign( â ) = 1 for â ≥ 0, and -1 for â < 0. This is valid for the correlation between R̂ and ŷi , and between ŷi and ŷ j , i, j = 1, …, N. For the correlation between R̂ and φ̂i , between ŷi and φ̂i , and between φ̂i and φ̂ j , i, j = 1, …, N, however, Eq. (E.20) represents an approximation, since φ̂i is not temperature dependent whereas R̂ and ŷi are (cf. Eqs. (D.1), (D.19) and (D.22)). However, the approximation may be reasonably valid since the φ̂i -dependence of the meter body uncertainty is very weak (cf. Table 4.17 and Fig. 5.15). Eqs. (E.19) and (E.20) lead to 2 2 2 æ ∂Q̂ ö N æ ö N æ ö ∂Q̂ ∂Q̂ u ( Q̂ ) = çç 2 u c ( R̂ )÷÷ + å çç u c ( ŷ i )÷÷ + å çç u c ( φˆ i )÷÷ ∂ ∂ i =1 è ∂φ i c ŷ ˆ è R̂ ø i = 1 è i ø ø N ∂Q̂ ∂Q̂ N ∂Q̂ ∂Q̂ + å 2 sign( R̂ )sign( ŷ i ) u c ( R̂ )u c ( ŷ i ) + å 2 sign( R̂ )sign( φ i ) u c ( R̂ )u c ( φˆ i ) i =1 ∂R̂ ∂ŷ i i =1 ∂R̂ ∂φ i ˆ (E.21) N N ∂Q̂ ∂Q̂ N N ∂Q̂ ∂Q̂ + åå 2 sign( ŷ i )sign( φˆ j ) u c ( ŷ i )u c ( φˆ j ) + åå 2 sign( ŷ i )sign( ŷ i ) u c ( ŷ i )u c ( ŷ j ) i =1 j =1 ∂ŷ i ∂φˆ j i =1 j =1 ∂ŷ i ∂ŷ j N N ∂Q̂ ∂Q̂ + åå 2 sign( φˆ i )sign( φˆ i ) u c ( φˆ i )u c ( φˆ j ) + other terms i =1 j =1 ∂φˆ i ∂φˆ j which can be written as (since sign( R̂ ) = 1 ) (E.Handbook of uncertainty calculations . field operation (repeatability) In Eq.1 Uncorrelated contributions. Section E. (E. Et 1i . Thus.3.24) is the expression used in the following for the uncertainty of the flow calibrated USM. from of the second column in Table E.C ÷ i =1 è i =1 ø Eq.7).3.24) æ [ ö ] N N + 2å ( s *t 1i Et 1i .. cf.22) ∂ ∂ φ c R̂ ŷ ∂ ˆ c i è i = 1 i i = 1 i ø Consequently. it is convenient to define .25) USM i =1 has been made here.2. NFOGM. the term E qU represents the USM uncertainty contributions in field USM operation.)C + s t*2 i Et(2ni .UC ) ) 2 + ( st*2i Et(2ni.U ) 2 + ç å s*t 1i Et 1i ..)C ÷ . the identification N E q2U = 2å ( s *t 1i Et 1i .1. (E.C + s *t 2 i Et 2 i .USM fiscal gas metering stations NPD.3. which are uncorrelated with flow calibration. Eq. Hence. at the flow rate in question. Moreover.. for a flow calibrated USM.CU ) )2 (E. E.23) 2 æ N [ ö ] + ç å s*t 1i Et(1ni . at the flow rate in question.U ) 2 (E. è i =1 ø and it follows that Eq. CMR (December 2001) 246 2 æ ∂Q̂ N ∂Q̂ N ∂Q̂ ö u ( Q̂ ) = çç 2 u c ( R̂ ) + å sign( ŷ i ) u c ( ŷ i ) + å sign( φˆ i ) u ( φˆ )÷÷ + other terms .3 Identification of USM uncertainty terms E. (E. E qU represents the repeatability of USM the USM measurement in field operation. Consequently. on basis of the fourth column in Table E.2 and Eq. for flow calibration re. (E.U is to be associated with the repeatability of the transit time measurements in field operation. (E.18) is to be replaced by [ ] 2 æ N ö E 2 qUSM ≈ E + ç s *R E R + å sign( ŷ i )s *yi E yi + sign( φˆ i )sφ*i Eφi ÷ 2 I è i =1 ø 2 .UU ) )2 + ( s*t 2i Et(2ni.. (E.9) is to be replaced by [ ] 2 æ N ö E 2 qUSM ≈ E + ç s *R E R + å sign( ŷ i )s *yi E yi + sign( φˆ i )sφ*i Eφi ÷ 2 I è i =1 ø [ ] [ ] N N 1 + å ( s*t 1i Et(1ni .24).UU ) )2 + å N ave i =1 i =1 ( st*1i Et(1ni . at test flow rate no. and the repeatability of the flow laboratory reference measurement). j). (E. for flow calibration re.28) USM .7). j where E rept . i. Section E. standard deviation) of the flow calibration measurement (volumetric flow rate).1. cf. On basis of the fourth column in Table E. j )2 (E. (E.26) USM where E rept ≡ repeatability (relative combined standard uncertainty.24). at test flow rate no. which are correlated with flow calibration. E. on basis of the third and fourth columns in Table E. j ≡ E qU . j tions in field operation. Consequently.1.Handbook of uncertainty calculations . NFOGM. (E. Hence. j.2. it is convenient to define E rept .e. i. j = 1. j is to be associated with the repeatability of the transit time measurements in flow calibration. and vice versa.2 Correlated contributions. j ≡ repeatability (relative combined standard uncertainty. (E.U .U . Section E. field operation In Eq. From the second column in Table E. j. j measurement in flow calibration. cf. …. M (due to random transit time effects on the N acoustic paths of the USM.7) the term E qU represents the USM uncertainty contributions USM . at the flow rate in question (due to random transit time effects).2 and Eq. the terms E qC and E qC represent those USM uncertainty contribu- USM USM .2. which are uncorrelated with field operation. the identification N E q2U = 2å ( st*1i Et 1i . standard deviation) of the USM measurement in field operation (volumetric flow rate). in Eq. E qU is to represent the repeatability of the USM USM . Similarly. Thus. at test flow rate no.3. CMR (December 2001) 247 E rept ≡ E qU . Et 1i . (E. j in flow calibration (at test flow rate no. j.e.27) USM . the identification .USM fiscal gas metering stations NPD. j i =1 has been made here.3. ŷ i . t̂ 1i . define E rad . systematic effects which are (practically) eliminated at flow calibration (cf.C ≡ relative standard uncertainty of uncorrected systematic transit time effects on downstream propagation of acoustic path no.C ÷ è i =1 ø has been made here. (E.29) denotes that only deviations relative to the conditions at the flow calibration are to be accounted for in the expressions involving this sub/superscript. due to possible deviation in pressure and/or temperature between flow calibration and field operation.30) . Table E.∆ ≡ relative standard uncertainty of the USM integration method due to change of installation conditions from flow calibration to field operation. i. NFOGM. i.∆ . due to possible deviation in pressure and/or temperature between flow calibration and field operation. t̂ 2i .∆ ≡ relative combined standard uncertainty of the lateral chord position of acoustic path no. Now. due to possible deviation in pressure and/or temperature between flow calibration and field operation. Eφi .Handbook of uncertainty calculations . E R .29) æ [ ] ö N + ç å st*1i Et∆1i .∆ + å sign( ŷ i )s *yi E yi . That means. i.∆ USM .C ≡ relative standard uncertainty of uncorrected systematic transit time effects on upstream propagation of acoustic path no. E yi .C + s t*2 i Et∆2 i . CMR (December 2001) 248 [E ] ≈E [ ] 2 æ N ö + ç s*R E R . are not to be accounted for in this expression. (E. due to possible deviation in pressure and/or temperature between flow calibration and field operation. Et∆2 i . ŷ i . where E I . due to possible deviation in pressure and/or temperature between flow calibration and field operation.∆ ≡ relative combined standard uncertainty of the meter body inner radius.∆ ÷ 2 q CUSM − EqC 2 I . R̂ .∆ ≡ relative combined standard uncertainty of the inclination angle of acoustic path no. Et∆1i . for convenience in notation.USM fiscal gas metering stations NPD.∆ ≡ s*R E R . Note that the sub/superscript “∆” used in Eq. j è i =1 ø 2 (E. i.2).∆ + sign( φˆ i )sφ*i Eφi . due to uncertainty of the lateral chord positions of the N acoustic paths.∆ ≡ å sign( ŷ i )s *yi E yi .31) i =1 N E angle . ….∆ ≡ å s *t 1i Et∆1i . (E.32) i =1 ( ) N Etime . ….C + s *t 2i Et∆2i .∆ . …. N. Etime .∆ ≡ relative combined standard uncertainty of the estimate q̂v . due to systematic effects on the transit times of the N acoustic paths. caused by possible deviation in pressure and/or temperature between flow calibration and field operation. ŷ i . it may be useful to rewrite Eq.∆ + E angle . E angle .Handbook of uncertainty calculations . i = 1. ∆ +E 2 comm +E 2 flocom Since most conveniently. CMR (December 2001) 249 N E chord .∆ ≡ relative combined standard uncertainty of the estimate q̂v . t̂ 1i and t̂ 2i .26)-(E. j + E rept 2 (E. i = 1.∆ ≡ å sign( φˆ i )sφ*i Eφi . the relative combined standard uncertainty of the USM volumetric flow rate measurement becomes E q2v = E K2 dev .USM Fiscal Gas Metering Station may be given at different levels. E chord .∆ ≡ relative combined standard uncertainty of the estimate q̂v . (E. j + E q2ref . caused by possible deviation in pressure and/or temperature between flow calibration and field operation. due to uncertainty of the meter body inner radius. N. R̂ .USM fiscal gas metering stations NPD. NFOGM.33).7) and (E. (E. caused by possible deviation in pressure and/or temperature between flow calibration and field operation.3.∆ ≡ relative combined standard uncertainty of the estimate q̂v . i = 1.∆ ) 2 +E 2 time .34) +E 2 I . N. due to uncertainty of the inclination angles of the N acoustic paths.4 Uncertainty model of the USM fiscal gas metering station From Eqs.34) as .33) i =1 where E rad . E. caused by possible deviation in pressure and/or temperature between flow calibration and field operation. (E.∆ . φ i .∆ ( + E rad . input to the program EMU .C .∆ + Echord . j + E rept 2 . (E. (3. R̂ .j . j + E rept 2 .38) Ebody .16). …. That is. (E.Handbook of uncertainty calculations .∆ ≡ E rad . related to field operation of the USM. Ebody . The corresponding relative expanded uncertainty of the volumetric flow rate measurement is given from Eq.22). the lateral chord positions of the N acoustic paths. N.∆ + Echord . (E. (3.35) where 2 E cal ≡ E K2 dev .∆ + E I . caused by possible deviation in pressure and/or temperature between flow calibration and field operation.∆ 2 2 2 2 EUSM . (E.39) Here. (3.39) are the same as Eqs. j + E q2ref . (E.36) 2 EUSM ≡ E rept 2 + EUSM 2 . EUSM .∆ ≡ relative standard uncertainty of the estimate q̂v .∆ + E time . Eqs.35)-(E.USM fiscal gas metering stations NPD.6).∆ + E angle . NFOGM. (E. and the inclination angles of the N acoustic paths.40) q̂v q̂ v .35) to be U ( q̂ v ) u ( q̂ ) = k ⋅ c v = k ⋅ E qv . (E. CMR (December 2001) 250 E q2v = E cal 2 + EUSM 2 + Ecomm 2 + E 2flocom .19).20) and (3. related to field operation of the USM. E cal ≡ relative combined standard uncertainty of the estimate q̂v . (E. φ i . EUSM ≡ relative combined standard uncertainty of the estimate q̂v .∆ . i = 1. uncertainty of the meter body inner radius.∆ ≡ relative combined standard uncertainty of the estimate q̂v . ŷ i .37) . (E. related to flow calibration of the USM.∆ ≡ E body . (3. due to change of the USM meter body from flow calibration to field operation.∆ . 1995. a coverage factor k = 2 is recommended [ISO.USM fiscal gas metering stations NPD. For specification of the expanded uncertainty at a 95 % confidence level. NFOGM.Handbook of uncertainty calculations . EAL-R2. CMR (December 2001) 251 where U ( q̂ v ) is the expanded uncertainty of the estimate q̂v . . 1997]. j [ ] [ 2 + u( q̂ comm ) + u( q̂ flocom ) ] 2 where r( q̂USM . j )ú + ê u c ( q̂ ref . and (3) a ”variance method”.2 the ”decomposition approach” used in Appendix E is shown to be equivalent with the ”covariance method” recommended by the GUM. q̂USM . CMR (December 2001) 252 APPENDIX F ALTERNATIVE APPROACHES FOR EVALUATION OF PARTIALLY CORRELATED QUANTITIES As mentioned in Section E. given by Eqs.3). From the functional relationship given by Eq. there are several ways to account for the partial correlation between qUSM .1. In Section F.3).1 The “covariance method” The “classical” way to account for partially correlation between quantities is the “covariance method” recommended by the GUM [ISO. Three possible approaches to account for such partial correlation are addressed briefly here. j ë ∂q̂USM c ûú ûú û 2 é ∂q̂ v ù ∂q̂v ∂q̂v +ê u c ( q̂USM . 1995]. (2. F. the combined variance of the estimate q̂ v is then given as 2 2 é ∂q̂v ù 2 é ∂q̂v ù é ∂q̂ v ù u ( q̂v ) = ê 2 u c ( K̂ dev .2)-(B.1). j úû ∂q̂USM ∂q̂USM . j (|r| < 1). j ) u c ( q̂USM )u c ( q̂USM . 1995] (Section F. j ) (F. j and qUSM . j ) is the correlation coefficient for the correlation between the estimates q̂USM and q̂USM . j êë ∂q̂ ref .2). (2) the”decomposition method” used in Appendix E (Section F.9). j )ú + 2r( q̂USM . This includes (1) the ”covariance method” recommended by the GUM [ISO. but which for the present application (with a rather complex functional relationship) has been more complex to use than the method chosen (Section F.1) ëê ∂q̂USM .Handbook of uncertainty calculations . q̂USM . j )ú + ê u c ( q̂USM )ú ëê ∂K̂ dev . which is often useful. for evaluation of partially correlated quantities. and . (B. The reason for choosing the ”decomposition method” for the analysis of Appendix E is addressed. NFOGM.USM fiscal gas metering stations NPD. CMR (December 2001) 253 u c ( q̂USM ) ≡ combined standard uncertainty of the estimate q̂USM (representing the USM uncertainty in field operation). as described in Section E. the physical effects which are influencing them (cf. Table E.2 The “decomposition method” The “decomposition method” which has been used in the present Handbook is described in detail in Appendix E.1). each contribution is evaluated.1).e. Use of the “covariance method”. j (representing the USM uncertainty in flow calibration). j ) is involved. as described in Section F. j ø ∂q̂v ∂q̂v +2 C u c ( q̂USM C )u c ( q̂USM 2 [ . F. Eq. However. since the “covariance method” is the method recommended by the GUM. (E.Handbook of uncertainty calculations . i. j ) ≡ combined standard uncertainty of the estimate q̂USM .j )] è ∂q̂USM ø è ∂q̂USM . so that evaluation of a numerical value for this coefficient is avoided. Table E. Eq. Hence. NFOGM. q̂USM . j ) . j )ú êë ∂K̂ dev . as r is different from 0 and ±1. first the physical contributions to q̂USM and q̂USM . which may be difficult in practice. (E. j úû êë ∂q̂ ref . to justify its use. i. the “decomposition method” is to be shown to be equivalent to the “covariance method”.USM fiscal gas metering stations NPD. u c ( q̂USM . Eq. Identifi- cation of terms is then made.4) is to be related to Eq. j úû 2 æ ∂q̂ v ö 2 U 2 æ ∂q̂ v ö 2 U + çç ÷ [ ÷ u c ( q̂USM ) + u c2 ( q̂USM C ) + ç ç ] ÷ [ ÷ u c ( q̂USM . j ) + u c2 ( q̂USM C .4) can be rewritten as 2 2 é ∂q̂v ù é ∂q̂ v ù u c2 ( q̂v ) = ê u c ( K̂ dev . j . For this reason an alternative (and equivalent) method has been used. involves evaluation of the correlation coefficient r( q̂USM .e. (F. whether it is correlated or uncorrelated between flow calibration and field operation. (F. Instead.3. Here some more general aspects and properties of the method are addressed.1). j ) + u( q̂ comm ) + u( q̂ flocom ) ] [ 2 ] ∂q̂USM ∂q̂USM . j are identified. The main advantage of this method is that no correlation coefficient r( q̂USM . q̂USM . For this purpose. Next. based on skilled judgement and experience (cf.2. The other terms are defined in Appendix E. j )ú + ê u c ( q̂ ref .2). (F. (F.j ) r( q̂USM . j )ú + ê u c ( q̂ ref . cf.6) = 2 æ u c ( q̂USM )ö 2 æ u c ( q̂USM )ö U U ç 1+ç ÷ 1+ ç .j ÷ C è u c ( q̂USM ) ÷ø ç u ( q̂ C è c USM .3) and Eqs.2) (obtained by the “decomposition method”) becomes identical to Eq. j ) = u c2 ( q̂USM C ) + u c2 ( q̂USM U ) u c2 ( q̂USM C . q̂USM . j ∂q̂USM . (F. j )ú (F.5) u c ( q̂USM )u c ( q̂USM . j ) + u c ( q̂USM . Its value ranges from 0 to +1 depending on the numerical values of the two ratios appearing in the last expression of Eq. j úû ∂q̂ v ∂q̂ v +2 ∂q̂USM ∂q̂USM .1) (obtained by the “covariance method”) by defining C C u c ( q̂USM )u c ( q̂USM . j ) 2 C (F.Handbook of uncertainty calculations . (F. j úû 2 2 é ∂q̂ v ù é ∂q̂ v ù +ê u c ( q̂USM )ú + ê u c ( q̂USM . NFOGM. j ) 2 U 1 (F. CMR (December 2001) 254 2 2 é ∂q̂ v ù é ∂q̂ v ù =ê u c ( K̂ dev . j ) = . Eq. C C u c ( q̂USM )u c ( q̂USM . j ) + u( q̂ comm ) 2 [ + u( q̂ flocom ) 2 ] [ ] where the identities ∂q̂v ∂q̂ ∂q̂ v ∂q̂ v ∂q̂ ∂q̂v = Cv = . j U and u c2 ( q̂USM ) = u c2 ( q̂USM U ) + u c2 ( q̂USM C ) .2) ë ∂q̂USM û êë ∂q̂USM . j ) = u c2 ( q̂USM U . j . . j ∂q̂USM . (E. j C u c ( q̂USM C )u c ( q̂USM . The “decomposition method” given by Eq.1)-(E. j úû êë ∂q̂ ref . equivalently. respectively.4) have been used.6). j ) + u c ( q̂USM . Eq.USM fiscal gas metering stations NPD. u c2 ( q̂USM .2).3) ∂q̂USM ∂q̂USM ∂q̂USM U ∂q̂USM . j )ú êë ∂K̂ dev . = Cv = (F. j ) or. q̂USM . (E. (E. (F. j ) ÷ø for the correlation coefficient between the estimates q̂USM and q̂USM .4) is thus equivalent to the “covariance method” given by Eq.1).j ) r( q̂USM . USM fiscal gas metering stations NPD.13)-(2. In some cases (for a relatively simple functional relationship) this approach is very useful. In the ”variance method” approach. . In this case the ”variance method” has been considered to be considerably more complex to use than the method chosen. the covariance term (the last term in Eq. it could be said that r by this method is evaluated more indirectly. j ) would thus be avoided. CMR (December 2001) 255 In the “decomposition method”. (F. by the identification procedure described above and used in Appendix E.9).1 and F. u c ( q̂ v ) would then be evaluated as the square root of the sum of input variances.e.2. j ) would be written out as a single expression.12) and (2.5)-(F.1)) and the correlation coefficient r( q̂USM . involving the combination of Eqs. numbers for u c ( q̂USM ) . (2. F. Eqs. In fact. j ) and u c ( q̂USM . In this method. which is here referred to as the ”variance method”. and r can be calculated from Eq. r is usually not evaluated as a number (i.Handbook of uncertainty calculations . in the present application. is perhaps the most ”intuitive” of the methods discussed here for evaluation of the uncertainty of partially correlated quantities. u c ( q̂USM ). the full functional relationship for q v is considered to be too complex to be practically useful.6) are not used).6). However. it may be noted that by this method. (2. U C u c ( q̂USM . r can be evaluated quantitatively. q̂USM . (F. (F. and leads to (or should lead to!) the same results as the methods described in Sections F. the full functional relationship for q v (including the USM measurements qUSM and qUSM . j ) are in principle available.3 The “variance method” Another method.18). When U C the identification described above has been done. NFOGM. in terms of the basic uncorrelated input quantities. On the other hand. This method is briefly addressed here for completeness. and hysteresis.28). the functional relationship is given by Eq.USM fiscal gas metering stations NPD. NFOGM. K̂ 18 . From Eqs. u( T̂d ) ≡ combined standard uncertainty of the gas temperature estimate in the densitometer. the combined standard uncertainty of the gas density measurement is given as 2 2 2 é ∂ρˆ ù é ∂ρˆ ù 2 é ∂ρˆ ù é ∂ρˆ ù u ( ρˆ ) = ê 2 u( ρˆ u )ú + ê u c ( T̂ )ú + ê u( T̂d )ú + ê u( T̂c )ú ë ∂ρˆ u ë ∂T̂ ∂ ∂ c û û ë d T̂ û ë c T̂ û 2 2 2 é ∂ρˆ ù é ∂ρˆ ù é ∂ρˆ ù é ∂ρˆ ˆ ù 2 +ê u( K̂ 18 )ú + ê u( K̂ 19 )ú + ê u( K̂ d )ú + ê u( τ )ú ë ∂K̂ 18 û ë ∂K̂ 19 û ë ∂K̂ d û ë ∂τˆ û 2 (G. The various quantities involved are defined in Section 2. u( T̂c ) ≡ combined standard uncertainty of the estimate of the densitometer calibration temperature. T̂ . . (3. (B. u( K̂ 18 ) ≡ standard uncertainty of the estimate of the temperature correction calibration coefficient.3) and (2. u c ( T̂ ) ≡ combined standard uncertainty of the line gas temperature estimate. describing the uncertainty model for the gas density measurement.1) 2 2 é ∂ρˆ ù é ∂ρˆ ù é ∂ρˆ ù é ∂ρˆ ù 2 +ê u( ĉc )ú + ê u( ĉ d )ú + ê u( ∆P̂d )ú + ê u c ( P̂ )ú ë ∂ĉ c û ë ∂ĉ d û ë ∂∆P̂d û ë ∂P̂ û 2 é ∂ρˆ ù +ê u c ( Ẑ d Ẑ )ú + u 2 ( ρˆ rept ) + u 2 ( ρˆ misc ) ë ∂( Ẑ d Ẑ ) û where u( ρˆ u ) ≡ standard uncertainty of the indicated (uncorrected) density estimate. CMR (December 2001) 256 APPENDIX G UNCERTAINTY MODEL FOR THE GAS DENSITOMETER The present appendix gives the theoretical basis for Eq. T̂d . For the gas density measurement. including the calibration laboratory uncertainty. (2.Handbook of uncertainty calculations . VOS and by- pass installation corrections.28).4. the reading error during calibration. including temperature.14). T̂c . ρ̂ u . u( ∆P̂d ) ≡ standard uncertainty of assuming that Pd = P. ĉ d . (2. .variations in power supply.flow in the bypass density line. K̂ d . u( ĉ d ) ≡ standard uncertainty of the densitometer gas VOS estimate. CMR (December 2001) 257 u( K̂ 19 ) ≡ standard uncertainty of the estimate of the temperature correction calibration coefficient.possible corrosion of the vibrating element.Handbook of uncertainty calculations . Eq. ĉc . . . shift between calibrations).stability (drift.possible deposits on the vibrating element.neglecting possible pressure dependency in the regression curve. . .USM fiscal gas metering stations NPD.possible gas viscosity effects. such as due to: . ∆P̂d . u( K̂ d ) ≡ standard uncertainty of the estimate of the densitometer transducer constant. (2. . . P̂ . u( τˆ ) ≡ standard uncertainty of the periodic time estimate. .25). u( ĉc ) ≡ standard uncertainty of the calibration gas VOS estimate.23). . u c ( Ẑ d Ẑ ) ≡ combined standard uncertainty of the gas compressibility factor ratio between densitometer and line conditions.reading error during measurement (for digital display instruments). NFOGM. τ̂ .mechanical (structural) vibrations on the gas line.possible liquid condensation on the vibrating element. u( ρˆ rept ) ≡ standard uncertainty of the repeatability of the indicated (uncorrected) density estimate. u c ( P̂ ) ≡ combined standard uncertainty of the line gas pressure estimate. . Ẑ d Ẑ .self-induced heat. K̂ 19 . ρ̂ u .model uncertainty of the VOS correction model. accounting for miscellaneous uncertainty contributions. ρ̂ u . due to possible deviation of gas pressure from densitometer to line conditions. u( ρˆ misc ) ≡ standard uncertainty of the indicated (uncorrected) density estimate. Eq. . . 5) ∂ρˆ ρˆ é K̂ 18 ρˆ u ( T̂d − T̂c ) ù ρˆ * = ú≡ s ρ .3) ∂T̂ T̂ T̂ ∂ρˆ = ρˆ é ê1 + [ T̂d ρˆ u K̂ 18 + K̂ 19 ] ù ρˆ * ú≡ s ρ .Handbook of uncertainty calculations .11) ∂ĉ d ĉ d ë K̂ d + ( τˆĉ d ) 2 û ĉ d d d .τ ≡ s ρ . as well as the estimates P̂ and ∆P̂d .K (G.τ (G.28).K ê [ ] ∂K̂ 19 K̂ 19 ë ρˆ u 1 + K̂ 18 ( T̂d − T̂c ) + K̂ 19 ( T̂d − T̂c ) û K̂ 19 19 19 (G. one obtains ∂ρˆ = ρˆ é [ ρˆ u 1 + K̂ 18 ( T̂d − T̂c ) ] ù ρˆ * ú≡ s ρ ≡ s ρu ê [ ] ∂ρˆ u ρˆ u ë ρˆ u 1 + K̂ 18 ( T̂d − T̂c ) + K̂ 19 ( T̂d − T̂c ) û ρˆ u u (G.7) ∂ρˆ ρˆ é 2 K̂ d2 2 K̂ d2 ù ρˆ * = ê 2 − 2 ú ≡ s ρ .K ê [ ] ∂K̂ 18 K̂ 18 ë ρˆ u 1 + K̂ 18 ( T̂d − T̂c ) + K̂ 19 ( T̂d − T̂c ) û K̂ 18 18 18 (G. From the functional relationship of the gas densitometer.Tc ∂T̂c T̂c ê [ ] ë ρˆ u 1 + K̂ 18 ( T̂d − T̂c ) + K̂ 19 ( T̂d − T̂c ) û T̂c (G.10) ∂ĉc ĉc ë K̂ d + ( τˆĉc ) 2 û ĉc c c ∂ρˆ ρˆ é 2 K̂ d2 ù ρˆ * = ê 2 ú≡ s ρ .K ≡ s ρ . NFOGM.Td [ ∂T̂d T̂d ë ρˆ u 1 + K̂ 18 ( T̂d − T̂c ) + K̂ 19 ( T̂d − T̂c ] ) û T̂d (G.6) ∂ρˆ = ρˆ é ( K̂ 19 T̂d − T̂c ù) ρˆ * ú≡ s ρ .2) ∂ρˆ ρˆ ρˆ = − ≡ s *ρ .USM fiscal gas metering stations NPD.c ≡ s ρ .T ≡ s ρ . (2.1).K ≡ s ρ .8) ∂K̂ d K̂ d ë K̂ d + ( τˆĉ c ) 2 K̂ d + ( τˆĉ d ) û K̂ d 2 3 3 ∂ρˆ ρˆ é 2 K̂ d2 2 K̂ d2 ù ρˆ * =− ê 2 − 2 ú ≡ s ρ . the estimates T̂ .c (G.9) ∂τˆ τˆ ë K̂ d + ( τˆĉc ) 2 K̂ d + ( τˆĉ d ) û τˆ 2 ∂ρˆ ρˆ é 2 K̂ d2 ù ρˆ * =− ê 2 ú≡ s ρ .c (G.c ≡ s ρ . CMR (December 2001) 258 As formulated by Eq.4) ∂ρˆ =− ρˆ é [ T̂c ρˆ u K̂ 18 + K̂ 19 ] ù ρˆ * ú ≡ s ρ .T (G.Tc ≡ s ρ . (G. Eq.Td ≡ s ρ . T̂d and T̂c are assumed to be uncorrelated.K ≡ s ρ . rept ≡ relative standard uncertainty of the repeatability of the indicated (uncorrected) density estimate.Tc ) 2 ET2c + ( s *ρ .13) ∂P̂ P̂ ë P̂ + ∆P̂d û P̂ ∂ρˆ ρˆ ρˆ = ≡ s *ρ . E ρ . (G. NFOGM.1) through with ρ̂ . the reading error during calibration.misc where E ρu ≡ relative standard uncertainty of the indicated (uncorrected) density estimate.24)). T̂ .12) ∂∆P̂d ∆P̂d ë P̂ + ∆P̂d û ∆P̂d ∂ρˆ ρˆ é ∆P̂d ù ρˆ * = ê ú ≡ s ρ . one obtains E ρ2 = ( s *ρu ) 2 E ρ2u + E ρ2 .τ ) 2 Eτ2 + ( s *ρ . Eq. X are the relative (non-dimensional) and absolute (dimensional) sensitivity coefficients of the estimate ρ̂ with respect to the quantity "X". ρ̂ u .Ẑ d (G. define 2 2 é ∂ρˆ ù é ∂ρˆ ù u ( ρˆ temp 2 )=ê u( K̂ 18 )ú + ê u( K̂ 19 )ú (G.∆Pd (G.2)-(G. P ) 2 E P2 + ( s *ρ . (G.rept + ( s *ρ .Handbook of uncertainty calculations . Now.∆Pd ≡ s ρ . (2.cc ) 2 Ec2c + ( s *ρ . s *ρ . and using Eqs. ρ̂ u . ET ≡ relative combined standard uncertainty of the line gas temperature estimate. CMR (December 2001) 259 ∂ρˆ ρˆ é ∆P̂d ù ρˆ * =− ê ú≡ s ρ .15).Td ) 2 ET2d + ( s ρ* .15) ë ∂K̂ 18 û ë ∂K̂ 19 û where u( ρˆ temp ) ≡ standard uncertainty of the temperature correction factor for the density estimate.Ẑ ≡ s ρ .cd ) 2 E c2d (G.16) + ( s *ρ .temp + E ρ2 .T ) 2 ET2 + ( s *ρ .P (G. By dividing Eq.∆Pd ) 2 E ∆2Pd + ( s *ρ . K d ) 2 E K2 d + ( s ρ* . . ρ̂ (represents the model uncertainty of the temperature correction model used.P ≡ s ρ . including the calibration laboratory uncertainty.USM fiscal gas metering stations NPD. and hysteresis. X and s ρ .14) ∂( Ẑ d Ẑ ) Ẑ d Ẑ Ẑ d Ẑ d Ẑ Ẑ Here. Z d Z ) 2 E Z2d Z + E ρ2 . mechanical (structural) vibrations on the gas line. Ẑ d Ẑ . E ∆Pd ≡ relative standard uncertainty of assuming that Pd = P. . EP ≡ relative combined standard uncertainty of the line gas pressure estimate.flow in the bypass density line. . EZd Z ≡ relative combined standard uncertainty of the gas compressibility ratio between densitometer and line conditions. . shift between calibrations). ∆P̂d . .possible corrosion of the vibrating element. due to possible deviation of gas pressure from densitometer to line conditions. ρ̂ (represents the model uncertainty of the temperature correction model used. accounting for uncertainties due to: . E cc ≡ relative standard uncertainty of the calibration gas VOS estimate.24)). ρ̂ u . .possible deposits on the vibrating element.possible liquid condensation on the vibrating element.stability (drift. T̂d . . ĉ d . K̂ d .self-induced heat. . (2. Eτ ≡ relative standard uncertainty of the periodic time estimate. NFOGM. ĉc .misc ≡ relative standard uncertainty of the indicated (uncorrected) density estimate. CMR (December 2001) 260 ETd ≡ relative combined standard uncertainty of the gas temperature estimate in the densitometer.reading error during measurement (for digital display instruments).Handbook of uncertainty calculations . E ρ .variations in power supply. EKd ≡ relative standard uncertainty of the estimate of the densitometer transducer constant. . T̂c . P̂ . .temp ≡ relative standard uncertainty of the temperature correction factor for the density estimate. ETc ≡ relative combined standard uncertainty of the estimate of the densitometer calibration temperature. Eq. τ̂ .possible gas viscosity effects. E ρ . . E cd ≡ relative standard uncertainty of the densitometer gas VOS estimate.USM fiscal gas metering stations NPD. e. Since the equation of state is empirical. NFOGM.ana − E Z .mod )2 + ( E Zd .rept ≡ . EKd ≡ . (2. ET ≡ .model uncertainty of the VOS correction model. 1997]: the model uncertainty (i.18) where E Zd .ana ) 2 . T̂c K̂ d τˆ u( ĉc ) u( ĉ d ) E cc ≡ . CMR (December 2001) 261 .Handbook of uncertainty calculations .mod − E Z . the model uncertainties of the Z-factor estimates Ẑ d and Ẑ have been assumed to be correlated. (G. The argumentation is as follows: Ẑ d and Ẑ relate to nearly equal pressures and temperatures. Eq. ∆P̂d Ẑ P̂ Ẑ d Ẑ u( ρˆ rept ) u( ρˆ temp ) u( ρˆ misc ) E ρ . Eq. For each of the estimates Ẑ d and Ẑ . and the uncertainty of the “basic data” underlying the equation of state). ρˆ ρˆ ρˆ respectively. The model uncertainties are here assumed to be mutually correlated143. (G. .neglecting possible pressure dependency in the regression curve. E ρ . E Z2d / Z = ( E Zd .USM fiscal gas metering stations NPD.23).mod ≡ relative standard uncertainty of the estimate Ẑ d due to model uncertainty (the uncertainty of the equation of state itself. E ρ .25). (G.18). 143 In the derivation of Eq. defined as u c ( ρˆ u ) u c ( T̂ ) u c ( T̂d ) E ρu ≡ . . Eτ ≡ .misc ≡ . two kinds of uncertainties are accounted for here [Tambo and Søgaard.17) ĉ c ĉ d u( ∆P̂d ) u c ( P̂ ) u c ( Ẑ d Ẑ ) E ∆Pd ≡ . ETd ≡ . E cd ≡ . it may be correct to assume that the error of the equation is systematic in the pressure and temperaure range in question. EP ≡ . E Ẑ d ≡ . and the analysis uncertainty (due to the inaccurate determination of the gas composition in the line).temp ≡ . ρˆ u T̂ T̂d u c ( T̂c ) u( K̂ d ) u( τˆ ) ETc ≡ . That means. and so are the analysis uncertainties. the uncertainty of the model used for calculation of Ẑ d and Ẑ ). (2. K d ) 2 E K2 d + ( s ρ* . and implemented in the program EMU . NFOGM.rept + ( s *ρ . .τ u 2 ( τˆ ) + s ρ2 . CMR (December 2001) 262 E Z . and variation in gas composition).18) E Z d Z will in practice be negligible.Td u 2 ( T̂d ) + s ρ2 .T u c2 ( T̂ ) + s ρ2 .∆Pd u 2 ( ∆P̂d ) + s ρ2 .USM fiscal gas metering stations NPD.19) + ( s *ρ .ana ≡ relative standard uncertainty of the estimate Ẑ d due to analysis uncertainty (measurement uncertainty of the gas chromatograph used to determine the gas composition. since the same equation of state is normally used for calculation of Ẑ d and Ẑ .Handbook of uncertainty calculations . Consequently.misc By multiplication with ρ̂ 2 . (G. (3.Td ) 2 ET2d + ( s *ρ .Tc ) 2 ET2c + ( s *ρ .cd u 2 ( ĉ d ) (G.T ) 2 ET2 + ( s *ρ . the corresponding expanded uncertainty of the density measurement is given as u c2 ( ρˆ ) = s ρ2u u 2 ( ρˆ u ) + u 2 ( ρˆ rept ) + s ρ2 .P u c2 ( P̂ ) + u 2 ( ρˆ temp ) + u 2 ( ρˆ misc ) which is the expression used in Chapter 3.τ ) 2 Eτ2 + ( s *ρ . ana ≡ relative standard uncertainty of the estimate Ẑ due to analysis uncertainty (measurement uncertainty of the gas chromatograph used to determine the line gas composition.temp + E ρ2 . E Z . P ) 2 E P2 + E ρ2 .14).cd ) 2 Ec2d (G. and the uncertainty of the “basic data” underlying the equation of state). E Zd .K d u 2 ( K̂ d ) + s ρ2 .mod ≡ relative standard uncertainty of the estimate Ẑ due to model uncertainty (the uncertainty of the equation of state itself. (G. Eq. Eq.cc ) 2 E c2c + ( s ρ* .Tc u 2 ( T̂c ) + s ρ2 .USM Fiscal Gas Metering Station.16) can be ap- proximated by E ρ2 = ( s *ρu ) 2 E ρ2u + E ρ2 . and variation in gas composition).cc u 2 ( ĉc ) + s ρ2 . From Eq.20) + s ρ2 .∆Pd ) 2 E ∆2Pd + ( s ρ* . S. Second edition.A. API.Introduc- tion”. BIPM. A beginner’s guide to uncertainty of measurement”. Part 1 . NFOGM.) API. Chapter 5 . U.. AGA-9. “Manual of petroleum measurement standards. National Physics Laboratory.2.2. Applied Mathematics Series.: “Measurement good practice.Liquid metering.A. API. (Paragraphs 12.1 and 12.2. Personal communication between Norsk Hydro and Per Salvesen. and Stegun. American Gas Association. U.S.Calculation of liquid petroleum quantites measured by turbine or displacement meters”.C.Handbook of uncertainty calculations . A.A.11. 1964). Second edition. 9.G.USM fiscal gas metering stations NPD. First edition.Calculation of petroleum quantities.: Handbook of mathematical functions. AGA-8. (May 1995). “Manual of petroleum measurement standards.S..S. (November 1987). American Gas Association. Vol. November 1992. American Petroleum Institute.1 and 1. UK (August 1999) (ISSN 1368-6550). 2nd printing (July 1994). American Petroleum Institute. 134-144 (1997). Autek. M. Chapter 12 . May 1968. 55 (National Bureau of Standards. American Petroleum Institute.. Norway (Instromet’s Norwegian sales representative) (2001).) Autek. Inc.C. Section 2 . (Paragraphs 1. (September 1981).C. Teddington.5. “Measurement of gas by ultrasonic meters”. Washington D.2. Bell. Section 2 . Section 2 . pp.A. 8. Second edition. I. Washington D. Washington D. “Manual of petroleum measurement standards. Chapter 12 . New York.G.Measurement of liquid hydrocarbons by displacement meters”. Reprinted by Dover Publications. Bureau International Des Poids et Mesures. CMR (December 2001) 263 REFERENCES Abromowitz. Transmission Measurement Committee Report No.2..A.5. (ISBN 92- 822-2110-5) . 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