Self-diagnosis techniques and their applications to error reduction for ultrasonic flow measurement

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(1):. DOCTORAL T H E S I S. Self-Diagnosis Techniques and Their Applications to Error Reduction for Ultrasonic Flow Measurement. Jonathan Berrebi. Luleå University of Technology Department of Computer Science and Electrical Engineering, Division of EISLAB :|: -|: - -- ⁄ -- .

(2) Self-Diagnostic Techniques and Their Applications to Error Reduction for Ultrasonic Flow Measurement Jonathan Berrebi June 30, 2004.

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(4) Abstract Flow metering plays a major role in modern life. In the process industry, flow metering is critical in industries ranging from food processing to cosmetics. It is also essential in custody transfer or billing, as flow meters are present in gas pumps and district heating substations. In the district heating industry, the ultrasonic flow meter has become the desired meter in many of its applications because it has a low cost while being accurate. This accuracy is however sensitive to installation effects and other sources of errors. This thesis stems from research that addresses the recognition of these installation effects, informs when they are unacceptable and considers reducing the measurement errors. To present these concepts, the thesis details the estimation of the mean flow velocity, the calibration of the meter and the measurement noise properties. Once installed, any kind of meter provides larger errors than in the facility where it has been calibrated and compensated. It is particularly true for ultrasonic flow meters as they are very sensitive to installation effects. Installation effects can either be static or dynamic. Special attention is paid to errors generated by temperature and velocity profile variations. Velocity profile variations can be due to pipe bends or flow pulsations. Such disturbances often induce a bias error and change the properties of the measurement noise. It is therefore with help of the change in noise that velocity profile disturbances can be detected. The detection of such abnormal behaviour of the measurement process constitutes a diagnosis. A diagnosis of the sensitivity of the meter to installations effects would allow for compensations for the errors. Signal analysis allows detection of specific noise properties, characteristic of installation effects. An example of self-diagnosis showing the detection of real pulsations in a flow is described in details. The detection of the flow pulsations and the estimation of their frequency allow to reduce the error of estimation on the flow rate. This technique is confirmed by the simulations of a pulsating flow. To empower one with the decision whether a flowmeter performance is normal or abnormal, a study of the relative error as a function of flow rate and temperature has been conducted.. i.

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(6) Acknowledgement I wish to thank Pr. Jerker Delsing, dean of the engineering faculty, for having received me at Lule˚ a University of Technology. As he is one of my two supervisors, his knowledge on flowmeters has also guided my research during the period of Ph.D. studies. I am grateful to the Swedish District Heating Association (Svensk Fjärrvärme) and the Swedish Energy Agency (Energimyndigheten) for financing the project I have been working on. I wish to thank particulary the members of the reference group, G¨ ote Ekstr¨ om (Fjärrvärme), Kristian Arkesten (Tekniska Verken i Link¨ oping), Jan Eliasson (G¨ oteborg Energi), and Hans Engström (Lule˚ a Energi) for the interesting discussions during the meetings at Lule˚ a. Before being employed as a usual Ph.D. student, I have worked on this Ph.D. thesis for 14 months as a volunteer in the context of the cooperation between France and Sweden. Therefore, I wish to thank the French Ministry of Foreign Affairs for co-financing the Ph.D. studies during these 14 months, and for giving this opportunity for young engineers to work abroad in good conditions. This thesis has been the occasion for me to build a link between what I have learned at school and real life. All the things we learn at school up to the master thesis are very woolly until they are confronted to practical cases. This link would have been impossible to make without the help of my supervisor Jan van Deventer. He has inspired me during these years, and given me lots of useful critical comments. I also wish to thank Carl Carlander (D-Flow), Pär-Erik Martinsson, Michel Cervantes, Torbj¨ orn L¨ ofquist and Johan Carlson, Yassin Jomni, and Kimmo Yliniemi for their interesting discussions about ultrasonic flow metering, fluid dynamics and signal processing. More Generally, I am very grateful to my coworkers at EISLAB with whom I have spent great moments. All these people have become friends more than coworkers. However, I have friends who don’t work with me but give a lot of support. I wish to thank them as well, and I hope that they will recognise themselves. Finally, I want to express my gratitude to my parents and my sister to whom I dedicate this thesis for their constant support, even if it was not so easy for them to know me so far away from Paris.. iii.

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(8) Contents Abstract. i. Acknowledgement. iii. I. 1. General discussion. 1 Introduction 1.1. 1.2. Motivations and Goals . 1.1.1 Motivations . . . 1.1.2 State of the Art 1.1.3 Goals . . . . . . Plan of the thesis . . . .. 3 . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 2 Ultrasonic flow metering 2.1. 2.2 2.3. 5. Flow metering . . . . . . . . . . . . . . . . . . . 2.1.1 What is a flow rate? . . . . . . . . . . . 2.1.2 Various flow metering techniques . . . . Different ultrasonic techniques . . . . . . . . . The transit-time ultrasonic flowmeter . . . . . 2.3.1 Operating Principle . . . . . . . . . . . 2.3.2 Acoustic propagation in a moving media 2.3.3 Sampling the flow rate . . . . . . . . . . 2.3.4 Calibration . . . . . . . . . . . . . . . . 2.3.5 Measurement error . . . . . . . . . . . . 2.3.6 Measurement noise . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 3 Different sources of errors 3.1. 3.2. 3 3 3 4 4. 5 5 6 6 7 9 9 10 13 15 21 27. Influence of temperature on the flow rate measurement error 3.1.1 Influence of temperature on the velocity profile . . . . 3.1.2 Influence of temperature on the acoustic propagation . Velocity profile variations due to installation effects . . . . . . v. . . . .. . . . .. 28 28 29 30.

(9) vi 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5. Installation effect . . . . . . . . . . . . . . . . . . . . . . . Static installation effects . . . . . . . . . . . . . . . . . . . Dynamic installation effects . . . . . . . . . . . . . . . . . Experimental obtention of a pulsating flow . . . . . . . . characterisation of a pulsating flow for detection purpose. 4 Self-diagnosis 4.1 4.2 4.3 4.4. 43. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different communities . . . . . . . . . . . . . . . . . . . . . . . . Application to our case . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Waiting for stationary flow . . . . . . . . . . . . . . . . . 4.4.2 Detection of a static installation effect . . . . . . . . . . . 4.4.3 Detection of a pulsating flow as a dynamic installation effect. 5 Summary of papers and contributions 5.1 5.2 5.3 5.4 5.5. Paper Paper Paper Paper Paper. A B C D E. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. II. 55 55 56 56 57 59. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7 Appendix A 7.1. 43 43 44 44 44 52 52 55. 6 Conclusion 6.1. 30 32 36 38 38. 59 61. Skewness and Kurtosis . . . . . . . . . . . . . . . . . . . . . . . .. Papers. 61. 67. A Diagnostic of the error generated by a single elbow on an ultrasonic flow meter 69 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 A.2 Theory . . . . . . . . . . . . . . . . . . . . A.2.1 Detection with likelihood ratio test A.2.2 General case . . . . . . . . . . . . A.2.3 decision by the variance . . . . . . A.3 Results . . . . . . . . . . . . . . . . . . . . A.4 Conclusion . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 70 71 73 73 74 74.

(10) vii B Detection of pulsating flows in an ultrasonic flow meter B.1 Introduction . . . . . . . . . . . . . . . B.2 Theory . . . . . . . . . . . . . . . . . . B.2.1 Description of a pulsating flow B.2.2 Detection of the pulsation . . . B.3 Experimetal set-up . . . . . . . . . . . B.4 Results . . . . . . . . . . . . . . . . . . B.5 Discussion . . . . . . . . . . . . . . . . B.6 Acknowledgement . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 77 . . . . . . . .. 77 78 78 78 79 79 82 82. C Ultrasonic Flow Metering Errors due to Pulsating Flow C.1 Introduction . . . . . . . . . . . . . . . . . . . . . C.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . C.2.1 Description and principle of the ultrasonic C.2.2 The prediction Error . . . . . . . . . . . . C.2.3 Zero-crossing error . . . . . . . . . . . . . C.3 Results/Discussion . . . . . . . . . . . . . . . . . C.3.1 Prediction error . . . . . . . . . . . . . . . C.3.2 Zero-crossing in water applications . . . . C.3.3 Zero-crossing in gas applications . . . . . C.4 Conclusion . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . flow meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. 87 . 87 . 89 . 89 . 91 . 92 . 97 . 97 . 97 . 97 . 100. D Reducing the Flow Measurement Error Caused by Pulsations in Flows D.1 Introduction . . . . . . . . . . . . . D.2 Theory . . . . . . . . . . . . . . . . D.2.1 Model of a stationary flow . D.2.2 Model of a pulsating flow . D.2.3 The error of estimation . . D.2.4 Detection of flow pulsations D.3 Simulations . . . . . . . . . . . . . D.4 Conclusion . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 103 103 104 104 106 107 111 111 114. E Temperature and Flow Rate Effects on Ultrasonic Flowmeter Performance using 23 Factorial Design. 117 E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 E.2 Possible factors of variations of the noise level . . . . . . . . . . . 118 E.2.1 The flow rate as a factor of the flow measurement error . 119 E.2.2 The temperature as a factor of the flow measurement error 120 E.2.3 Interactions between flow rate and temperature . . . . . . 121 E.3 Experimental method . . . . . . . . . . . . . . . . . . . . . . . . 121 E.3.1 The random error . . . . . . . . . . . . . . . . . . . . . . 121 E.3.2 Experimental design . . . . . . . . . . . . . . . . . . . . . 122.

(11) viii E.3.3 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.4.1 Analysis of variance of the two-factor factorial experiment E.4.2 Regression analysis of the random error . . . . . . . . . . E.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 122 123 123 125 126.

(12) Part I. General discussion. 1.

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(14) Chapter 1. Introduction 1.1 1.1.1. Motivations and Goals Motivations. The accuracy of a measurement system is often crucial. It is also clear that the accuracy of a measurement system depends closely of its environment. When the environment changes, it thus induces a measurement error. For obvious reasons it is of interest to introduce the knowledge of these errors to people interested. It is thus of great interest to device technology capable of diagnosing measurement systems, i.e. self-diagnoses, in such a way that errors giving an economical impact can be reported. The idea of self-diagnosing technology forms the core of this thesis. The application chosen is district heating and the measurement of energy transferred to customers. Since a flow meter is the critical component of such measurement I have chosen to work on the self-diagnoses of error to flow metering. In district heating ultrasonic technology now is the most interesting new technology coming on. Today, more than 90 % of the flowmeters used in district heating industries in Denmark are of the ultrasonic type. In Sweden, the proportion of transit-time ultrasonic flowmeters used in district heating applications is about 30 % and keeps on increasing. I have thus chosen to use ultrasonic flowmeters a the technology platform for my work on self-diagnostic flowmeters, still aiming at results transferable to other technologies.. 1.1.2. State of the Art. In 2001, Carl Carlander presented his Ph.D thesis on: Installation effects and self-diagnosis of an ultrasonic flowmeter [1]. His work led to the following conclusions: • the turbulence intensity is affected by installation effects. 3.

(15) 4 • it is possible to detect effects due to installation by analysing the measurement noise • preliminary data conditioning is necessary to avoid false alarm • pulsating flow generate harmonic structure in the frequency spectrum of the measured flow velocity. The conditions to be fulfilled by the measured mean flow velocity signal were not yet established. Besides, if the characterisation of pulsating flows by its harmonic structure was found, their detection was not implemented.. 1.1.3. Goals. The present project is a continuation of Carlander’s project and aims to improve the characterisation and the automatic detection of both temperature and installations effects. Special attention is paid to static and dynamic installation effects causing velocity profile disturbances. The conditions required for applying well-known detection methods are also investigated in chapter 4.. 1.2. Plan of the thesis. The introduction ends in the next section with the definition of a flow rate, necessary for the rest of the thesis. Then, the transit-time ultrasonic technique [2] is explained in chapter 2. Chapter 3 deals with the main drawback of transittime ultrasonic flowmeters that is their sensitivity to installation effects, and especially to velocity profile disturbances. In chapter 4, diagnostic methods for detecting installation effects are presented. Finally, the papers included at the end of the thesis give details about the different installation effects, the diagnostic detection methods, and how they can be used to reduce the estimation error on the flow rate..

(16) Chapter 2. Ultrasonic flow metering 2.1 2.1.1. Flow metering What is a flow rate?. The expression ”flow rate” has been used in the introduction before any mathematical definitions of a flow rate was given. Indeed, human beings measured flow rates thousands of years ago, i.e. even before the notion of flow rate was defined. In the antic Rome, individual consumers were taxed according to the size of the nozzle installed in their houses [3], but the flow rate was neither measured nor defined. In order to comfort the reader with the context used in the thesis, the following sections define the notion(s) of flow rate. In the different chapters, two different entities related to notion of flow rate are used: • the mean flow velocity • the volumetric flow rate. It is important for the readers to get familiar with them. But, first of all, it is necessary to introduce the notion of point velocity, even if it is rarely mentioned in this thesis. Point velocity The point velocity is the velocity of an infinitesimal fluid element in a flow. A point velocity measurement is called anemometry, and is performed with devices such as Pitot-tube, hot wire, or Laser-Doppler velocimeter (L.D.V.) [4]. The notation for the velocity at point M and time t is v(M, t).. 5.

(17) 6 Mean flow velocity With the latter definition, it is possible to define, at time t, the average of the point velocity over the cross-section area A of a pipe: Z 1 v¯(t) = v(M, t) · ds(M ), (2.1) A M ∈A where v¯(t) is called the mean flow velocity, and ds(M ) is a infinitesimal element of surface around point M . This notion is essential in this work for two reasons. Firstly, an ultrasonic flowmeter physically measures the mean flow velocity in the ultrasonic path. Secondly, the volumetric flow rate is constantly proportional to the mean flow velocity in a pipe, since the cross-section area is constant. Flow rate The volumetric flow rate Q, commonly called flow rate, is simply given by: Q(t) = v¯(t) A.. (2.2). The notion of (volumetric) flow rate is more intuitive then the notion of mean flow velocity. It tells directly an information on the whole pipe flow. The units used are m3 /s, m3 /h, l/s, etc.. 2.1.2. Various flow metering techniques. Historically [3], flow measurements were already performed in Egypt in 1500 B.C., but the first flowmeters that included the notion of time appeared in the late 1700s. Of course, mechanical flowmeters, such as turbine flowmeters, or positive displacement flowmeters appeared first. Table 2.1 recapitulates the date of apparition of most types of flowmeters. It is obvious that it took centuries between the discovery of the ground principle and the industrial use. One can also notice from table 2.1 that the ultrasonic technique is recent and in development. Its expansion is mainly due to its good accuracy and its low price.. 2.2. Different ultrasonic techniques. There are at least three different ways to measure a flow rate with help of ultrasounds that are the Doppler technique ([5] and [2]), the correlation technique, and the transit-time technique ([6] and in [2]). This thesis focuses on the transittime technique only, because of its dominant industrial usage. Besides, it has the best cost-efficiency and does not requires any tagging (bubbles, particles, etc)..

(18) 7 Type of flowmeter Turbine Positive displacement Differential pressure. Electromagnetic Ultrasonic (tt) Coriolis Vortex shedding. Fundament Archimede (1500 B.C.) 1800 Torrecelli Newton Venturi (1808) Faraday (1832) Galileo Coriolis da Vinci (1513) Strouhal (1878) Von Karman (1912). First patent 1790 1815 1896. Industrial use 1938 1820 1909. 1917 1931 1953 1954. 1952 1980 1980 1958. Table 2.1: Date of apparition of different types of flowmeters from their ground principles to their industrial use.. 2.3. The transit-time ultrasonic flowmeter. Two implementations of transit-time ultrasonic flowmeters are depicted in figures 2.1 and 2.2. In the longitudinal, or axial, configuration, the ultrasonic path is parallel to the flow. In the diagonal configuration, there is an angle between the ultrasonic path and the flow. The axial configuration is most often used for diameters smaller 25 mm. The diagonal configuration allows diameters varying in the range of several millimetres up to 10 metres. Diagonal meters are less intrusive than axial meters. A transit-time flow meter can have more than one pair of transducers and also use reflectors in order to make the ultrasonic signal travelling through the whole flow profile. The transducers are generally piezoelectric and allow signals whose bandwidths are often over 1 MHz. They can be wetted or clamped-on. The accuracy of transit-times flow meters is usually of about 2%, or better. As for all flowmeters, this accuracy varies with the flow rate. In my work, axial or diagonal transit-time flow meters with one pair of ultrasonic transducers have been used. All experimental and theoretical work has been performed on such meters. Nevertheless the principle is valid for all kind of transit-time ultrasonic flowmeters. At first, the principle of the transit-time ultrasonic flowmeter is explained in simple terms in 2.3.1. Then, a more complete description of the measurement process is given. The physical principle uses the propagation of acoustic waves in moving media, that is examined in 2.3.2. Then, the sampling of the mean flow velocity is detailed in 2.3.3. Section 2.3.4 looks carefully at the calibration issue. Finally, descriptions of the measurement error and noise are necessary in sections 2.3.5 and 2.3.6 to define the normal behaviour of the measurement process..

(19) 8. DOWNSTREAM. UPSTREAM FLOWMETER BODY FLOW. Transducer. Figure 2.1: A D-Flow ultrasonic flow meter with a longitudinal configuration used for tests reported in this thesis. One can se the upstream transducer in red. The diameter in the metering section is 9 mm.. 48 mm 23,5 mm. 18,9 mm. flow. inlet. outlet. 10 mm. ultrasonic beam downstream transducer. upstream transducer. Figure 2.2: The diagonal configuration. This meter is a D-Flow transit-time ultrasonic flowmeter actually used during tests. The diameter in the metering section is 10 mm..

(20) 9. 2.3.1. Operating Principle. The present paragraph aims to briefly explain the principle of a transit-time ultrasonic flowmeter. Schemes of the longitudinal and diagonal configurations are shown in figures 2.1 and 2.2. One estimation of the mean flow velocity v¯, or the flow rate Q, requires two measurements in order to cancel the speed of sound. As the electronic required to perform both measurements simultaneously is too expensive, the mean flow velocity estimation process is divided in two steps. In the first step, the upstream transducer sends an ultrasonic pulse at speed c¯0 + v¯ received by the downstream transducer after time Tdw . The propagation of the pulse is then fostered by the flow, and its velocity can be written as: c¯0 + v¯ cos α =. L , Tdw. (2.3). where c¯0 is the spatial-mean speed of sound over the metering area, L is the distance between the two transducers, and α is the angle made between the beam and the flow direction (c.f. figure 2.2). In the second step, the roles of the transducers are inverted and the upstream transducer receives a pulse coming after time Tup with velocity: c¯0 − v¯ cos α =. L . Tup. (2.4). Then, equations (2.3) and (2.4) form a system of two linear equations with two unknowns. Its solution gives an estimation of the mean flow velocity: vˆ ¯= ∧. 1 L 1 ( ∧ − ∧ ), 2 cos α T dw T up. (2.5). ∧. where T dw and T up are the estimations of the downstream transit-time Tdw and the upstream transit-time Tup respectively. Expression (2.5) is intuitively simple to understand, but the addition of the mean flow velocity and the speed of sound requires actually some assumptions that are not always fulfilled as it is now shown in 2.3.2.. 2.3.2. Acoustic propagation in a moving media. As the media where the ultrasonic wave propagates is moving and inhomogeneous, the behaviour of the ultrasonic pulse is not in agreement with the usual acoustic wave equation of D’Alembert: ∇2 p −. 1 ∂2p = 0, c20 ∂t2. (2.6). where p is the pressure of the fluid, and c0 the phase velocity. Indeed, the pulse’s propagation is rather governed by Pierce’s (approximative) equation valid in.

(21) 10 moving and inhomogeneous media: 1 1 ∇ · (ρ∇Φ) − Dt ( 2 Dt Φ) = 0, ρ c0. (2.7). where Φ is the velocity potential of the irrotational flow, ρ is the density, and Dt denotes the particular derivative ∂/∂t + v · ∇. By adding the assumption that the wave is locally plane, the velocity c of such a pulse, at point M and time t, is the sum of the speed of sound in water c0 and the contribution of the flow velocity v at point M and time t (equation (43b) in [7]): c(M, t) = c0 + v(M, t) · n(M, t),. (2.8). where n(M, t) is the unit vector normal to the surface of constant phase at point M and time t. The latter equation is the ground principle of the transit-time ultrasonic flowmeter. As the central frequency f of the ultrasonic pulse is high (about 1 MHz), the corresponding wave length, given by: λ0 =. c0 , f. (2.9). is smaller than the diameter of the transducer. This allows the generation of a beam and consequently the approximation of plane wave. Hence, if the flow is stationary during the propagation of the ultrasonic pulse, the integration of c(M, t) over the beam volume (a cylinder) gives an averaged expression for c: c¯ = c¯0 + v ¯ · n,. (2.10). where v¯ is the (spatial-) mean flow velocity. This justifies the use of equations (2.3) and (2.4) that were so far only intuitive. The beam assumption (diffraction neglected) is a fairly good approximation, whereas the stationary and irrotational assumptions can lead to some interrogations since the beam propagates through the boundary layer and the transducers cannot be perfectly non-intrusive nor non-invasive.. 2.3.3. Sampling the flow rate. Usually, a discrete-time representation of a measured signal requires a transducer, an anti-aliasing filter, and an analogue-digital converter (c.f. figure 2.3.3). For example, a Pt-100 thermometer delivers an electric analogue signal imaging the temperature. This signal is low-pass filtered and sampled at a rate respecting the Shannon-Nyquist criterium. In the case of the transit-time ultrasonic flowmeter, the transducer delivers an electric signal imaging the pressure variations at its surface. But equation (2.5) reveals that the pressure is not used for the determination of the mean flow velocity. Instead, it is the upstream and downstream transit-times, determined by the pressure variations in the vicinity of the receivers, that are used for the.

(22) 11 metering. Nevertheless, the output vˆ¯(n) of a transit-time ultrasonic flowmeter is a discrete-time sequence of samples of the continuous mean flow velocity signal v¯. The estimation of the transit-times is described in 2.3.3 and the question of how to represent the ”sampling” process is discussed in 2.3.3. Transit-time estimation The estimation of transit-times Tdw and Tup (i.e. the times of arrival of the ultrasonic pulses) can be done in two ways. The first way is to sample the received pulse with an analogue-digital converter and to implement a match-filter with a copy of the signal sent. This method requires expensive components such as an analogue-digital converter and a processor to compute the output of the matchfilter. Another method, called ”zero-crossing”, uses a peak-detector to detect the pulse. The latter is obviously more sensitive to the noise, but also much cheeper than the match-filter. It is therefore chosen for most flowmeters with industrial applications. In this thesis, all flowmeters used the ”zero-crossing” for measuring the transit-times. As the mean flow velocity v¯ is much lower than the speed-of sound, Tdw and Tup will always be of the order of the zero-flow transit-time T0 : T0 =. L . c¯0. (2.11). Hence, with L = 15 cm and c0 = 1500 m/s, the time of travel is of the order of 100 µs. Depending on the application, the path length can reach up to 1 or 2 meters, and the transit time can then reach up to 1 ms. In order to be able to distinguish non-zero flow transit-times from T0 , the time resolution ∆T has to be much lower than T0 . Generally, modern electronics is capable of a time measurement resolution, ∆T , of 100 ps. By differentiating expression (2.5), one obtains the velocity resolution as a function of the time resolution: ∆vˆ¯ '. L ∆T . cos α T02. (2.12). Hence, a typical velocity resolution is of 1.6 mm/s corresponding to a flow rate resolution of 0.12 ml/s = 5·10−4 m3 /h for a metering section with diameter D = 1 cm. Such a resolution is sufficient for number of industrial applications. The resolution can be artificially improved by using a technique, called SingAround ([8]), that consists in sending another pulse immediately after the first pulse has reached the receiver. By iterating the process, a number N of pulses can be periodically sent to the receiver, for example the downstream transducer. The transit-time TN dw for N pulses to travel downstream to the receiver leads to an estimation of Tdw : ∧ TN dw . (2.13) T dw = N When the N th pulse has reached the receiver, the downstream transducer can send N other pulses upstream. An expression equivalent to (2.13) is found for.

(23) 12 Tup : ∧. TN up . (2.14) N The time resolution is still the same (100 ps), but the velocity and flow rate resolutions are divided by N . However, one can picture an imaginary improved time-resolution of 100 ps/N . Hence, by taking N = 10, the flow rate resolution is 5 · 10−5 m3 /h, and by taking N = 1000, it becomes 5 · 10−7 m3 /h. The main drawback of the Sing-Around method is that it requires steadiness of the flow parameters during the sending of the N pulses upstream and the N pulses downstream. T up =. Representation of the sampling process Figure 2.3 represents the sampling process in the transit-time flow meter used in our experiments. A sampling interval of length Ts is composed of a waiting time, whose duration is about 10 ms, and the time required for sending N pulses, N T0 . The latter becomes preponderant when N is large. What is clearly illustrated in figure 2.3 is the alternation between the measurements of Tdw and Tup that are performed at rate 2 fs = 2/Ts . Each sample of Tdw and Tup is used twice for the estimation of vˆ ¯. This allows a sampling rate of fs that ranges between 1 Hz and 100 Hz depending on L and N . The alternation between the measuring of Tdw and Tup is not in agreement with relation (2.5) that assumes these measurements to be done simultaneously. Now, for the aim of simplicity, let us focus on the estimation of v¯(n + 1) = v¯((n + 1)Ts ). The same thought processes can be applied to the sampling of v¯(n + 2), etc. The estimation of v¯(n + 1) is performed as soon as Tup has been sampled (c.f. figure 2.3). But the sampling of Tdw occurred at time n Ts . A very simple predictor is used to estimate Tdw at time (n + 1) Ts , since it is: Tˆdw (n + 1|n) = Tdw (n).. (2.15). Hence, the flow should be as steady as possible between times n Ts and (n+1) Ts . For this reason, the value of N and the duration of the waiting times should not be too large. Otherwise, the following error: δTdw (n + 1|n) = Tˆdw (n + 1|n) − Tdw (n). (2.16). on the transit-time estimation has repercussions on the estimation of the mean flow velocity v¯(n + 1): Es (n + 1|n) '. L δTdw (n + 1|n). 2 T0 cos α. (2.17). If Ts is of the order of 0.1 s and the flow rate has monotone variations under the time interval [nTs , (n + 2)Ts ], the following interpolation is to be preferred to the usual prediction presented in expression (2.15): 1 Tˆdw (n + 1|n + 2, n) = (Tdw (n) + Tdw (n + 2)) . 2. (2.18).

(24) 13. Sampling of Tdw. TNdw= N Tdw. Sampling of Tup wait time # 10 ms. 0,1 ms < TNup= N Tup < 0,1 s v(n). v(n+1). v(n+2). v(n+3). v(n+4). Sampling of v. Ts= 1/fs. Figure 2.3: The sampling process: The alternation between measurements of downstream and upstream transit-times is visible. Each measurement of Tdw or Tup is used for two consecutive estimations of the mean flow velocity. In the left part of the figure, the flow is decreasing, so that Tdw is increasing and Tup is decreasing.. The process of the interpolation is explained in figure 2.4 and delays the estimation of v¯(n + 1) by Ts . Such an interpolation reduces the measurement error made on the mean flow velocity when the flow is non-stationary.. 2.3.4. Calibration. In practice, the estimation of the mean flow velocity as it is written in 2.5 has to be corrected by a calibration factor K. This factor is generally expressed as a function of the Reynolds number and depends on the flow meter body, the installation effects, the velocity profile, etc. The shape of the curve plotted in fig.2.7 representing the K-factor as a function of the Reynolds number is globally explained by the velocity profile changing. At laminar flows (low Reynolds number), the velocity profile is parabolic (c.f. figures 2.5 and 2.6). The velocity of particles of fluids moving along the central axis of the flow meter is then much higher than the velocity of particles of fluids close to the wall. A weighing has to be performed over the different particles of fluids in order to estimate correctly the mean flow velocity. An appropriate.

(25) 14. v. sampling. interpolation sampling n Ts. (n+1) Ts. (n+2) Ts. t. Ts. Figure 2.4: The interpolation of the transit-time according to relation (2.18). If the flow rate is increasing or decreasing, this interpolation reduces the error due to alternation..

(26) 15 weighing would give the same weight to concentric annuli of equal area [5]. At higher Reynolds number, the boundary layer undergoes a laminar to turbulent transition. A hump is then more (figure 2.8) or less (figure 2.7) visible on the K-curve. The latter figures correspond to two diagonal designs. The meter shown in figure 2.2 presents a smaller hump due of its improved design. Besides, by reducing the diameter in the metering section, the transition from laminar to turbulent flow can be delayed. The reason why the hump has to be avoided is that the K-factor is chosen from the K-curve accordingly to the primary mean flow velocity measurement. Then an error on this primary measurement can be amplified by a bad choice of the K-factor on the sharp slope leading to the hump (c.f. figure 2.8). The slope actually starts (and is maximal) already in the laminar domain, where the errors are reach their maxima. Nevertheless, erasing the hump leads to better meter performances as it is shown in figures 2.10 and 2.9. When the flow become turbulent, the velocity profile becomes more and more flat. Therefore, the K-factor is almost equal to unity at turbulent flow. The cavities unavoidable in the diagonal configuration of the meter play an important role in the calibration process. The average velocity in the cavities is zero. The ultrasonic path extension due to the cavities is the reason why the K-factor reaches values larger than one at turbulent flow-rates in the diagonal configurations. In practice, the flow meter is calibrated in a facility or in situ and the K-factor that is determined integrates compensations for the velocity profile and the path extension and for some more specific details specific to the meter and its installation. Moreover, the final aim is to estimate the flow rate Q that is related to the mean flow velocity by: Q = v¯ S,. (2.19). where S is the cross-section area of the meter. The cross section area can also be either calculated or integrated to the K-factor. Some ”intelligent” flow meters are able to calculate the cross-section area variations against the fluid temperature [9]. Indeed, the dilatation of flow meter bodies changes slightly the dimensions of the meter. This dilatation can be compensated and the compensated cross-section area can be integrated into the K-factor.. 2.3.5. Measurement error. The accuracy of transit-time ultrasonic flowmeters is typically between 1 % and 2 % of the flow rate. Some meters reach better accuracy, up to 0.1 %. Numbers of standards have different requirements on the accuracy of flowmeters. The one that is the focus in the present work is the European standard EN-1434-(1) [10] for heat meters, that defines the lower and the upper bounds of the interval of flow rates where a certain accuracy has to be respected. The lower limit of the flow rate, qi , is the lowest flow rate, above which the heat meter shall function without the maximum permissible errors being exceeded..

(27) 16. Figure 2.5: Simulation of the velocity profile (arrows) inside the longitudinal flowmeter for Re ' 10 (laminar), and when the flow is not disturbed upstream from the flowmeter.. The upper limit of the flow rate, qs , is the highest flow rate, at which the heat meter shall function for short periods (¡1 h/day; 200 h/year), without the maximum permissible errors being exceeded. In real life, and especially in district heating applications, it is not unusual to see flow rates lower than qi . Inside the interval [qi , qs ], there is one flow rate, called permanent flow rate, and noted qp , that has a particular importance, since it is ”the highest flow rate at which the meter should function continuously without the maximum permissible errors being exceeded”. The maximum permissible error (M.P.E.), Ef , for meters of class 2 follows an hyperbolic function starting at 5 % for flow rates lower than qp , and ending at 2 % for higher flow rates. More exactly, it is defined in [10] as follows: Ef = ±(2 + 0.02qp /Q), but not more than 5 %,. (2.20). where Q is the flow rate. Figures 2.9 and 2.10 show the measurement error of the flow meters used in my thesis, as well as the limits of the standard. The experimental set-up for obtaining these k-curves consisted in having a long straight pipe upstream from the flowmeters. The length of the pipe was more than 110 times the diameter in the metering section. We will see in chapter 3 that ultrasonic flowmeters are subject to many errors when the installation differs from that quasi-ideal configuration..

(28) 17. Figure 2.6: Simulation of the velocity profile (arrows) inside the diagonal flowmeter for Re ' 10 (laminar), and when the flow is not disturbed upstream from the flowmeter. One can easily see that the velocity profile is parabolic at the inlet of the flowmeter, and slightly disturbed when entering in the metering area..

(29) 18. 2.5. 2. K factor. 1.5. 1. 0.5. 0 1 10. 10. 2. 3. 10 R eynolds number. 10. 4. 10. 5. Figure 2.7: The K-curve at 180 C obtained for the diagonal meter shown in figure 2.1. The curve is plotted as a function of the Reynolds number. One can see that for laminar flows, the K-curve is a linear function of the Reynolds number. The transitional and turbulent domains correspond to the hump and the flat area respectively. The constant value for turbulent flow is equal to 1.8 and not to 1 since the cavities where the transducers are located play a role.

(30) 19. 16 14 12. K factor. 10. Error in the choice of the K-factor. 8 6 4 2 0. Re measured 10. 1. 3. Re true 10 R eynolds number. 10. 5. Figure 2.8: The K-curve at 200 C obtained for the diagonal meter shown in figure 2.1. The curve is plotted as a function of the Reynolds number. The hump is larger than for the meter shown in figure 2.2 and is susceptible to generate larger errors. A small error on the primary (before applying the k-factor) Reynolds number estimation can lead to a large error on the final estimation of the flow rate because of a bad choice of the k-factor..

(31) 20. 5. meas ured flow rate E N 1434 1 (clas s 2). 4 3. E rror (%). 2 1 0 1 2 3 4 5 10. 3. 10. 2. 10. flow rate (l/s ). 1. 10. 0. Figure 2.9: The relative error on the flow rate measurements performed by the diagonal flowmeter (improved design according to figure 2.2).. 10 0 10. E rror (%). 20 30 40 50 60 70 80 90. meas ured flow rate E N 1434 1 (clas s 2) 10. 2. 10. 1. 10. 0. flow rate (l/s ). Figure 2.10: The relative error on the flow rate measurements performed by the diagonal flowmeter (non-improved design). One can clearly see that the area corresponding to the hump in figure 2.8 is more sensitive to errors..

(32) 21. mean flow velocity v(t). noise due to turbulence ~ b(t) = v(t) - w(t). FLOW in the pipe and the flowmeter. observation noise w(t) +. +. measured mean flow velocity ^ v(t). Figure 2.11: The measurement noise is the sum of the additive observation noise w(t) and perturbations due to turbulence b(t).. 2.3.6. Measurement noise. As for any measurement system, the measurement error obtained from a flowmeter, and more specifically an ultrasonic flowmeter, can be decomposed as the sum of a bias error and a random error. The random error is also called measurement noise. It is generally seen as the additive observation noise w(t) according to figure 2.11. In this work, the measurement noise is defined as the non-constant part of the measured mean flow velocity vˆ¯. That means that if we decompose the mean flow velocity as follows: vˆ¯ = vˆ¯0 + v˜,. (2.21). the measured signal is vˆ ¯0 and the measurement noise is v˜. In chapter 4, we will see that v˜ is the signal used for the diagnosis since it contains all the necessary information except vˆ ¯0 . In its turn, the measurement noise v˜ can be decomposed, in several ways, as the sum of a signal and a noise. Three experiments corresponding to three different water temperatures (15o C, o 50 C, and 80o C) have been run on a diagonal transit-time ultrasonic flowmeter at the calibration facility in Lule˚ a in order to look at Reynolds number, temperature and velocity dependence of the noise. In figure 2.12, the Reynolds number is approximately equal to 1500 (laminar): the P.S.D. (power spectral density) seems to be equivalent for all three temperatures and does not exceed 10−6 . In figures 2.13 and 2.14, the flow is turbulent: the P.S.D. is much larger and reaches easily 10−3 . This shows us the Reynolds number dependance, of.

(33) 22. x 10. 7. o. 15 C Re=1.5e3 50 oC Re=1.6e3 80 oC Re=1.7e3. 12. 10. P.S.D.. 8. 6. 4. 2. 0.5. 1. 1.5. 2 2.5 Frequency (Hz). 3. 3.5. 4. 4.5. Figure 2.12: The measurement noise at Reynolds number 150.. the noise v˜. v˜ = v˜(Re).. (2.22). The noise v˜ cannot simply be the usual additive observation noise w encountered on all measurement systems and that is assumed to be independent (and therefore uncorrelated) from the observations. Figure 2.15 shows the zero-flow measurement noise w that has the same level as the noise of the laminar case (figure 2.12). The difference in noise levels observed in figures 2.13 and 2.14 are due to the contributions of the velocity fluctuations. In chapter 4, we will see that the standard deviation and the Gaussian assumption are often invoked for the purpose of the diagnosis. It is therefore interesting to look at the statistical properties of the measurement noise v˜. The normalised standard deviation is plotted in figure 2.16 as a function of the Reynolds number for three transit-time flowmeters used. The adjective ”normalised” means that v˜ has been divided by vˆ0 before the standard deviation was estimated. Otherwise, the curve would be increasing with Reynolds number. Globally, the standard deviation is higher at low Reynolds numbers, because of the resolution problem evoked in 2.3.3. One peak appears in the transition domain as it was discussed in 2.3.4. The standard deviation seem to increase at Reynolds numbers about 105 , but the calibration facility could not provide higher flow rates to confirm this evolution for fully turbulent flows. The skewness and kurtosis (flatness) factors , (see appendix A) are also plotted against Reynolds number in figures 2.17 and 2.18 respectively. Important deviations from the Gaussian case, where the skewness and the kurtosis are.

(34) 23. 8. x 10. 3. 15 oC Re=2.75e4 o 50 C Re=2.54e4 80 oC Re=2.63e4. 7 6. P.S.D.. 5 4 3 2 1 0 0. 1. 2. 3. 4. 5. Frequency (Hz). Figure 2.13: The measurement noise at Reynolds number 25000.. 0.06. 15 oC Re=1.12e5 Q=0.88 l/s 50 oC Re=2.03e5 Q=0.89 l/s 80 oC Re=3.01e5 Q=0.86 l/s. 0.05. P.S.D.. 0.04. 0.03. 0.02. 0.01. 0 0. 1. 2. 3. 4. 5. Frequency (Hz). Figure 2.14: The measurement noise at Reynolds number 250000..

(35) 24. 1.2. x 10. 6. o. 20 C Q=0 o 50 C Q=0 80 oC Q=0. 1. P.S.D.. 0.8. 0.6. 0.4. 0.2. 0 0. 1. 2. 3. 4. 5. Frequency (Hz). Figure 2.15: The observation noise w(t) (measurement noise when vˆ = 0).. equal to 0 and 3 respectively, are observed for Reynolds numbers higher than 104 , and are probably due to the non-Gaussian behaviour of turbulent signals (c.f. [4] and [11])..

(36) 25. diagonal 1 diagonal 2 longitudinal. standard deviation. 10. 10. 1. 0. 1. 10. 10. 2. 10. 1. 3. 10 R eynolds number. 10. 5. Figure 2.16: The relative standard deviation of the mean flow velocity in normal behaviour against Reynolds number.. 1.5. diagonal 1 diagonal 2 longitudinal G aus s ian cas e. skewness. 1. 0.5. 0. 0.5. 1. 10. 1. 3. 10 R eynolds number. 10. 5. Figure 2.17: The skewness factor of the mean flow velocity in normal behaviour against Reynolds number..

(37) 26. 7. 6. diagonal 1 diagonal 2 longitudinal G aus s ian cas e. K urtosis. 5. 4. 3. 2. 10. 1. 3. 10 R eynolds number. 10. 5. Figure 2.18: The kurtosis (flatness) factor of the mean flow velocity in normal behaviour against Reynolds number..

(38) Chapter 3. Different sources of errors A flow meter is usually selected for the best performance for the given application and the funds available. This performance in use might be degraded by installation effects and environmental factors. Installation of the flow meter in a network of pipe alters the flow profile, which perturbs the estimation of the flow. Environmental factors like temperature and humidity also influences the perception of the average flow rate. Besides, the nature of the perturbation also plays an important role. In the case of ultrasonic flow metering, errors can be caused by as many different sources as E.M.C. disturbances on the embedded electronics, fouling of the transducers, variations of water properties (temperature, pressure, density, viscosity, etc), air-bubbles (scattering), velocity profile perturbations, ambient temperature, etc. A measurement system like a flowmeter is often represented according to figure 2.11 that shows the input (the flow), the output (the flow measurement), and the disturbances. It is not an easy task (and often not possible) to identify if an error is due to perturbations caused by an installation effects or by irregularities of the inputs. The expression ”irregularities of the inputs” means any deviation of the fluid’s parameter from when the meter was calibrated. Even though inputs and perturbations interact with each others, table 3.1 tries to put a distinction between errors caused by installation effects and errors due to variations of the fluid’s properties. Major effects on the error are caused by the irregularities of the velocity profile, the variations of the fluid’s properties, and the fouling of the transducers. The errors caused by variations, or irregularities, of the velocity profile are common to numbers of flowmeters (e.g. electromagnetic, vortex shedding, etc) and depend both on installation effects and variations of the fluid’s properties. That is the reason why they will constitute the main focus. Section 3.1 deals exclusively with the influence of temperature on the velocity 27.

(39) 28 Source of error Velocity profile variations fouling of the transducers air bubbles ambient air temperature E.M.C. disturbances flowmeter design. installation effect X X X X X. fluid’s properties variations X. Table 3.1: Diverse sources of errors on transit-time ultrasonic flow rate measurement encountered in district heating applications. The errors can be due to installation effects, or to variations of the fluid’s properties. There can be interactions between installation effects and the variations of the fluid’s properties. A third kind of error emphasizes the role played by the design of the flowmeter. profile. Section 3.2 begins by defining, with a short example, what an installation effect is. Then, subsections 3.2.2 and 3.2.3 make distinctions between static and dynamic installation effects that generate velocity profile variations.. 3.1. Influence of temperature on the flow rate measurement error. The flow rate expression 2.3.4 can be rewritten including the estimates as functions of the fluid’s temperature: ˆ ) = vˆ¯(T ) S(T ˆ ). Q(T. (3.1). The last expression shows a dependance on temperature only, but in fact, other parameters have an influence on the estimates. We already saw in chapter 2 that the cross-section area S depends on the temperature because of the dilatation of the flowmeter body. So it is normal that its estimate is also a function of temperature. The estimate of the mean flow velocity depends also, and for several reasons, on temperature. Temperature has an effect on the velocity profile as it is shown in paragraph 3.1.1, and consequently on the calibration curve. But it also has an influence on the ultrasonic interrogation.. 3.1.1. Influence of temperature on the velocity profile. Velocity profile variations are not only the consequences of installation effects. For example, temperature has an effect on variations of the velocity profile. Indeed, by assuming that the flowmeter body is a circular pipe and that the sensors are perfectly non-intrusive, the velocity profile v(r) of a laminar flow is given by Poiseuille’s parabolic solution: v(r) =. 1 ∆p 2 (R − r2 ), 4η ∆x. (3.2).

(40) 29. 2.5. 2. K factor. 1.5. 1. 0.5. 0 1 10. o. 20 C 50 oC o 80 C 10. 2. 3. 4. 10 10 R eynolds number. 10. 5. 10. 6. Figure 3.1: The calibration curve as a function of Reynolds number for three different temperatures. The use of different calibration curve for different temperatures reduces the bias error.. where ∆p is the pressure drop along a portion of pipe of length ∆x (including the flowmeter), η is the dynamic viscosity, R is the pipe radius, and r is the polar radius. The dynamic viscosity η is a function of temperature as well as the pipe radius that is dilating with temperature. Hence, the same velocity profile can be representative of two different flows with different pressure drop and different temperatures. Figure 3.1 shows the calibration curve of the diagonal meter for three different temperatures. The performances of a flowmeter are then improved by measuring the temperature and choosing thereafter the right k-curve. The dilatation of the pipe diameter induces also an error when passing from the mean flow velocity to the flow rate, since the cross-section area is a square function of the pipe radius.. 3.1.2. Influence of temperature on the acoustic propagation. Now, it is also important to consider that temperature has an effect on the ultrasonic wave propagation, since the fluid’s density is a function of temperature. This effect exists at both small and large scales. At large scales, the main effect of temperature is on the speed of sound. As the speed of sound is, both for gases and liquids, a function of temperature, temperature variations induce a variation in the transit time. In principle, the contribution of the speed of sound is canceled in the solution presented.

(41) 30 in2.5. If the sampling frequency is high enough, the repercussions on the flow rate measurement error are minor, since the speed of sound is almost constant between upstream and downstream interrogations. If the sampling rate is not high enough to guaranty that the speed of sound is constant, Delsing’s algorithm presneted in [12] presented can be used to compensate for the change in speed of sound between upstream and downstream interrogations. At smaller scales and when the flow is turbulent, temperature fluctuations, as well as velocity fluctuations, bend the acoustic ray pathes in different directions, and induce therefore changes in phase and amplitude on the ultrasonic transducers [4]. Ray tracing algorithms [13], similar to those encountered in optics, have been used by Iooss, Lhuillier, and Jeanneau [14] to simulate the measurement uncertainty due to large and small scales. Their conclusion was that temperature heterogeneities at small scales contribute significatively to a measurement uncertainty of about 1%.. 3.2. Velocity profile variations due to installation effects. Installation effects such as pipe bends and pumps generate flow disturbances taking the form of distortions and swirls in the velocity profile. Recalling the decomposition of the measured mean flow velocity established in 2.21: vˆ¯ = vˆ¯0 + v˜, we can see that the consequences of flow disturbances will be a bias error on the estimation of the mean flow velocity vˆ¯0 , and a change in the noise v˜. The noise term, v˜, is the signal that will be used for the characterisation (chapter 3) and the detection (chapter 4) of installation effects. In other words, the characterisation of the change in v˜ will help to detect bias errors on vˆ¯0 . It seems necessary to define in 3.2.1 what are installation effects in a more general context before putting any difference between static and dynamic installation effects. Then, detailed descriptions of both static and dynamic installation effects are given in 3.2.2 and 3.2.3 respectively, in order to discriminate them vis-` a-vis the normal behaviour. Hence, their detection will be easier to perform in chapter 4.. 3.2.1. Installation effect. The performances of any meter are optimal in an environment similar to the one where it has been calibrated. There is a considerable difference between the accuracy of a meter in laboratory conditions and its accuracy when installed in real life. For example, thermometers indicating the outside temperature are usually placed close to the house in order to be readable from the house. The proximity of the house influences the thermometer. In winter, the house can be.

(42) 31 seen as a heat source and its neighbourhood is warmer than any other place in the street. The temperature indicated by the meter is then higher than it should be, since the aim of the metering is to know the temperature in the street. The bias generated by the house seen as a heat source can be compensated by a calibration in situ of the thermometer with help of a reference thermometer correctly placed in the street. Another example of installation effect is given in [15] where a thermocouple is placed in an engine exhaust pipe in order to measure the exhaust gas temperature. A bias error is induced by the radiation heat loss from the probe to the pipe wall. Installation effects on flow metering errors constitute a well-known issue [16] [17]. The source of disturbances on the velocity profile can be a pipe bend, a pump, or a diameter reduction placed at the inlet of the flowmeter. Delsing, Holm, and Stang [18] have studied the influence of static installation effects, such as pipe bends, on transit-time ultrasonic flowmeters. A few years later, Delsing and H˚ akansson [19] [20] investigated the influence of pulsating flows on transit-time ultrasonic flowmeters. Generally, studies on the subject conclude that the flowmeter should be installed in the middle of a long straight pipe in order to avoid flow disturbances. An alternative, published by Knapp in 1964, consists in using multiple off-diameter paths for interrogating the velocity profile. This method estimates the mean flow velocity independently of the velocity profile. It is used for diameters over 100 mm, and is rather expensive compared to the usual single path technique chosen in our project. The effects due to installations are more or less important depending of the type of flowmeter. For example, positive displacement flowmeters are not sensitive to velocity profile disturbances, while ultrasonic flowmeters are very sensitive to them. Obviously, positive displacement flowmeters have some drawbacks. They generate flow pulsations, work only in a small range of flows, are easily damageable, and are expensive. That is the reason why it is worth to solve problem due to installation on ultrasonic flow meters. Now, it would be interesting to classify installation effects and to describe how they disturb the measurements. The details of installation effects susceptible to generate errors on ultrasonic flow meters are enumerated in [1]. It is more particulary essential to describe installation effects that disturb the flow. For this purpose, let us recapitulate the ”normal” behaviour of the velocity profile. In chapter 2, we saw in figures 2.5 and 2.6 laminar velocity profiles inside the metering area as they are supposed to be during the calibration of the flowmeters, when a long straight pipe is placed upstream. The shape of the velocity profile is then parabolic at low Reynolds numbers, a mix of parabolic and logarithmic for Reynolds numbers a bit higher, and almost flat for fully turbulent flows. The shape in the transitional domain is more difficult to define. The calibration curve takes into account such variations of the velocity profile, as well as temperature variations (3.1) and if the conditions are optimal, the right calibration factor is chosen when measuring the flow rate. But what happens when distortions, swirls and turbulent structures are.

(43) 32. 0.135. mean flow velocity (m/s). 0.13. 0.125. 0.12. 0.115. 0.11 0. 20. 40. time (s ). 60. 80. 100. Figure 3.2: Unstable flow rate measurement due to a cylindrical thermometer placed at the inlet of the ultrasonic flowmeter.. present in the flow because of some installation effect (elbow, bluff body, etc)? As previous work done by Delsing, Holm [21], and Carlander [1] distinguishes static installation effects from dynamic ones, subsections 3.2.2 and 3.2.3 keep this distinction. Both parts present a list of the different installation effects frequently encountered in real life. They also present their experimental obtention at the calibration facility in Lule˚ a. Then, the influence of each kind of installation effect on the flow rate error is discussed. Finally, characterisations of both static and dynamic installation effects are performed for the purpose of detection (c.f. chapter 4).. 3.2.2. Static installation effects. Frequently encountered static installation effects A static installation effect is all permanent installation that differs from the long straight pipe placed upstream from the flowmeter during its calibration. Notice that in the last sentence, the adjective ”static” refers the permanence of the installation, and not the effect on the measured signals. Some static installation effects, like for example a thermometer placed at the inlet of the flow meter, can create unstable flows, that are non-stationary (c.f. 3.2). However, in order to simplify the classification task, the present work will try to characterise only the stationary effects on the measurements. Figure 3.3 shows the main static installation effects present in industrial applications:.

(44) 33. Figure 3.3: (from [22]) Examples of flow disturbances causing distortions and swirls. Figures f, and a to d, are static installation effects, while figure e designs a dynamic installation effect.. • single and double elbows • diameter reduction • partially open valve • bluff bodies (thermometers, etc). Elbows and diameter reductions placed at the inlet of an ultrasonic flow meter are among the most encountered examples of static installation effects. Both distort the velocity profile, and swirls occur in the water at the outlet of an elbow (c.f. figure 3.4). A good principle is to avoid placing flowmeters at the vicinity of such static installation effects, but it is not always possible. Another technique consists in using straighteners (c.f. [17]) to remove swirls, and sometimes even distortions of the velocity profile. But straighteners can be expensive, generate head loss and are not appropriate for small size meters. A district heating substation is composed of many elements and has to be as compact as possible. A consequence is shown in figure 3.5 where one can clearly see a pipe bend at the inlet of the ultrasonic flowmeter. The calibration factor cannot be suitable with such installation effects, since it is estimated in laboratory conditions where a long straight pipe of 110 times the meter’s diameter is placed at the inlet of the flow meter body..

(45) 34. Figure 3.4: 2D simulation showing distortions generated by a single elbow.. Figure 3.5: Part of a district heating substation showing a single elbow at the inlet of a transit-time ultrasonic flowmeter..

(46) 35. Figure 3.6: The flow meter accuracy when a single elbow is placed at the inlet of the ultrasonic flow meter.. experimental obtention of a static installation effects In our experiments, a single elbow, as presented in figure 3.3, was used to disturb the flow. A distance of 11 times the metering section diameter was introduced between the inlet of the flowmeter and the elbow. The error generated by the elbow on the flow rate measurement is plotted in figure 3.6 against Reynolds number and compared to the maximum permissible error according to the European standard EN 1434-1 [10]. The error is often out of the limits drawn by the standard (can reach 10 %), and the repeatability of the error lets presume that a new calibration, including the elbow, should be performed. Hence, calibrations in situ or at least including the district heating substation would improve the meter performances. Characterisation for detection purpose In his thesis ([1]), Carlander already mentioned a global change in the noise measurement when installation effects (both static and dynamic) were placed upstream from the flowmeter. This change can be observed in figure 3.7 where the normalised standard deviation of the mean flow velocity is plotted against Reynolds number for the case with elbow. A comparison between this figure and figure 2.16 leads to the conclusion that the disturbances generated by the elbow erase the effect due to the transitional hump. The statistical properties at orders 3 and 4 are also plotted in figures 3.8 and 3.9, in order to verify that the assumption of a Gaussian distributed mean flow velocity is suitable, even.

(47) 36. 10. standard deviation. 10. 10. 10. 10. 1. 0. 1. 2. 3. 10. 2. 10. 0. 2. 10 R eynolds number. 10. 4. 10. 6. Figure 3.7: The relative standard deviation of the mean flow velocity against Reynolds number, when a single elbow is placed at the inlet of the flowmeter.. when an elbow is placed at the inlet of the flowmeter. Now, if the skewness factor is still different from zero for Reynolds numbers over 104 , it is not the case for the Kurtosis that is (with an estimation error) constantly close to 3.. 3.2.3. Dynamic installation effects. Pulsating flow The most encountered dynamic installation effect in flow measurements is pulsating flows [20]. This problem has been well-known for one century [17] and is caused by pumps, compressors and reciprocating valves. These flow pulsations generate different errors on different flowmeters. Differential pressure flowmeters are affected by the square-root error that is rather an estimation error than a measurement error on the mean flow velocity. Other well-known errors caused by pulsating flows are the resonance error and the velocity profile error. Of all these error, transit-time ultrasonic flowmeters are mostly affected by velocity profile errors as it is shown in figure 3.10. In 1956, Uchida found a trigonometric solution, for laminar flow pulsations, that p he presented in [23]. The solution depends on the dimensionless parameter f /ν R where f is the frequency of the pulsations, ν is the cinematic viscosity, and R is the pipe radius. In our experiments, f ' 6 Hz, ν ' 10−6 m2 .s−1 , and R = 5mm, so that the parameter is equal to 12 approximately. As this parameter is larger than 10, the variations are said to be ”rapid”, and it is generally the case in water applications. The.

(48) 37. diagonal 1 Gaussian case 0.5. 0.4. skewness. 0.3. 0.2. 0.1. 0. 0.1 10. 1. 10. 3. 10. 5. Reynolds number. Figure 3.8: The skewness factor plotted against Reynolds number when a single elbow is placed at the inlet of the ultrasonic flowmeter.. 3. diagonal 1 Gaussian case. 2.95 2.9. Kurtosis. 2.85 2.8 2.75 2.7 2.65 10. 1. 3. 10 Reynols number. 10. 5. Figure 3.9: The Kurtosis (flatness) factor plotted against Reynolds number when a single elbow is placed at the inlet of the ultrasonic flowmeter..

(49) 38. Figure 3.10: The flow meter accuracy when flow pulsations perturb the measurements.. expression of the velocity profile is heavy and will not be used in this work, but its integration over the pipe section will be used in chapter 4. However, it is interesting to look at the time-evolution of the velocity profile in figure 3.11 in order to observe the change in the velocity profile compared to the ideal laminar (parabolic) case. One can then easily understand that such variations induce an error on the flow rate measurement. As usual, the turbulent case is (much) more difficult to describe.. 3.2.4. Experimental obtention of a pulsating flow. Experimentally, pulsations in flow were obtained by rotating a butterfly valve (c.f figure 3.12) with help of an A.C. motor. The rotation frequency of the motor was measured with a photo-tachometer sending and receiving light beams that were reflected on the motor’s axis. The accuracy of the tachometer was ± (0.05 % + 1 digit). The distance between the source of pulsations and the flowmeter was more than 110 times the pipe diameter.. 3.2.5. characterisation of a pulsating flow for detection purpose. In [1], one can see that the measured mean flow velocity of pulsating flows has a harmonic structure for all considered Reynolds numbers (102 − 105 ). In fact this harmonic structure is more or less kept for turbulent flows depending on.

(50) 39. Figure 3.11: The time-evolution (over one period of ppulsation) of the velocity profile of a pulsating flow inside a smooth pipe for f /ν R = 10 [23].. Figure 3.12: The experimental set-up used for the pulsating flow experiments..

(51) 40. standard deviation. 10. 10. 0. 1. 10. 10. diagonal 3 (puls ) diagonal 2 (puls ) diagonal 3 (no puls ). 1. 2. 10. 1. 3. 10 R eynolds number. 10. 5. Figure 3.13: The relative standard deviation plotted against Reynolds number when a pulsating flow is generated upstream from the ultrasonic flowmeter.. the design of the flowmeter and on the amplitude and the frequency of the oscillations. Indeed the P.S.D. of the mean flow velocity plotted in figures 4.3 shows a fundamental with tones at approximately 12.6 Hz at Reynolds number 5800, while the P.S.D. at Reynolds number 8.7 · 104 , plotted in figure 4.5 has totaly lost the harmonic structure, certainly because of a low signal-to-noise ratio. In fact, the fundamental in figure 4.3 is at frequency 6.34 Hz, and we will see at the end of chapter 4 that it will be confirmed by a detection tool, named harmogram [24]. The reason why the fundamental seem to be around 12.6 Hz is that the motor rotates at 6.34Hz, but there is an invariance of the butterfly valve by rotation of of an angle of π (c.f. figure 3.12). Hence, the frequency of flow pulsations is 12.6 Hz. The fundamental at 6.34 Hz represents the imperfection of the invariance of the rotation of the disc. Now, looking at the statistical properties, the standard deviation of the measured mean flow velocity seems to be considerably affected by the flow pulsations, especially at low Reynolds numbers (c.f. figure 3.13). The higher order properties deviate slightly from the Gaussian distribution, but as they are subject to estimation errors, the parent population of the noise will be considered as Gaussian for the detection purpose in chapter 4..

(52) 41. 0.8. 0.6. diagonal 3 G aus s ian cas e diagonal 2. 0.4. skewness. 0.2. 0. 0.2. 0.4. 0.6. 1. 10 10 R eynolds number. 3. 10. 5. Figure 3.14: The skewness factor plotted against Reynolds number when a pulsating flow is generated upstream from the ultrasonic flowmeter.. 3 2.9 2.8. K urtosis. 2.7 2.6 2.5 2.4 2.3 2.2 2.1 10. 1. 3. 10 R eynolds number. 10. 5. Figure 3.15: The Kurtosis (flatness) factor plotted against Reynolds number when a pulsating flow is generated upstream from the ultrasonic flowmeter. The Kurtosis is around 2.2. But this result has to betaken carefully, because of the error on the estimation.

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(54) Chapter 4. Self-diagnosis 4.1. Definition. Coming from the Greek roots dia:”to look through, to distinguish from” and gnosis:”knowledge”, the word diagnosis means hypothesis, prediction, or identification based on signs or symptoms. That term is mainly used in a medical context to describe the process of determining a disease by symptoms. By extension, with the endless development of processors, computers and signal processing tools, a diagnosis can be performed to predict an earthquake, to detect anomalies in a car, etc. When examining an electrocardiogram, a physician is actually ”looking through” some ”symptoms”. The physician does not look directly at the original signals coming from some transducers but uses the images of these original signals through transformations like convolution, Fourier transform, filters, etc. These transformations are used in order to raise the readability of the data. They help the scientist to take a decision about the presence of failures.. 4.2. Self-diagnosis. A diagnosis made on an electrocardiogram is both complex and important. The physician can take the time to use his knowledge and his intuition to look carefully at the data. The final decision can and should be taken by a human being, whereas in real-time applications (as for our flow meter), the interpretation must be done automatically and endlessly at a frequency that could reach several Hertz. Moreover, the detection of disturbances in a flow is not as crucial as the detection of a heart attack. The diagnosis can then be done automatically and become a self-diagnosis. A self-diagnosis requires the normal behaviour of the flow metering to be modeled. The coherence between the observations and the model guides the system in taking a decision. Algorithms and programs for the decision rule are incorporated in a processor.. 43.

(55) 44. 4.3. Different communities. The way of modelling the normal behaviour of the system and of testing the coherence can be different depending on the system itself. Two communities of research on diagnosis have emerged: The FDI (Fault Detection and Isolation) community and the DX (Artificial Intelligence) community. Traditionally, The FDI approach applies when the system has a dynamic behaviour that can be represented by a state-space model. Then a diagnosis can be performed by looking at the residuals [25], [26]. The DX approach [27], [28], [29], [30] is the result of research on artificial intelligence, neural networks and fuzzy logic. The DX approach allows a localisation of the faults and does not require preliminary hypothesis. Hybrid systems using both approaches are now developed. The self-diagnosis problem studied in this work cannot be described by the FDI approach and hardly be described by the DX approach. However, the word diagnosis is suitable since the flowmeter is looking for abnormal behaviour of the flow metering process. But in fact, the problem is more to detect installation effects than to perform an analysis of the whole measuring system.. 4.4. Application to our case. A self-diagnostic flowmeter should indicate when disturbances or important measurement errors occur. The description and the characterisation of static and some dynamic installation effects are outlined in 3.2. Their detection is necessary for the purpose of a diagnosis. As the signal processing methods employed to detect the installation effects are suitable for stationary flows only, the next part will define the notion of stationary signals and verify that the measured flow rate obey to that definition. The methods for detecting static and dynamic installation effects are described in parts 4.4.2 and 4.4.3 respectively.. 4.4.1. Waiting for stationary flow. time-invariant mean The notion of non-stationary signals refers to variations in time of some properties of the signal. The first idea that comes into mind is that the global pattern, notably the time-average, of a stationary signal should be constant in time. In fact, this idea is in contradiction with the strict definition of stationary signals that is given in the next paragraph. The flow measured by a flowmeter is more or less constant depending on the application. For example, the flow of oil in pipelines does not vary as much as the flow of water inside a house. If the time-averaged flow rate is supposed to be constant in a certain application, a test ([31]) can be performed in order to check for the constance of the flow rate. Denoting by vˆ0− the time average on N1 samples of the mean flow velocity for times before t = 0 and by vˆ0+ the time-average on N2 samples of the mean flow velocity vˆ(n) for times between 0 and tN2 = N2 Ts , the test’s hypotheses are:.

(56) 45 H0 : v¯0 = v¯m H1 : v¯0 6= v¯m If the assumption of a Gaussian distributed mean flow velocity with variance σ 2 is valid (or if the number of samples is large enough so that the central limit theorem can be applied), a decision rule based on the following test statistic: t0 = r. vˆ0− − vˆ0+ q 2, S12 S2 N1 + N2. (4.1). can be used. As the true variance σ 2 is unknown, the test uses the sample variances S12 and S22 corresponding to v¯0− and v¯0+ respectively. The decision rule is then: H0 : |t0 | < t0.025,ν H1 : |t0 | > t0.025,ν where t0 is the test statistic to be compared to the threshold t0.025,ν designing the upper 2.5 % point of the t-distribution with ν degrees of freedom: 2 2 S22 S1 + N1 N2 . (4.2) ν = (S 2 /N )2 (S22 /N2 )2 1 1 + N1 −1 N2 −1 This t-test has been applied to a sequence of flow rate samples measured by an ultrasonic flowmeter. The time-averaged flow rate was equal to 0.87 l/s before t = 0 and to 0.88 l/s after t = 0. N1 and N2 were chosen equal to 3000 and 100 respectively. The execution of the test gave a value of t0 equal to 1.21 to be compared to t(0.025, 106.9) = 1.98. The decision taken was then that the mean flow velocity was constant. The choice of N1 = 3000 provided a good estimation of the variance of v¯0− . In fact, assuming that S1 = σ would have led to a much simpler expression of the t-test. When applying the same test to only N1 = 100 of v¯0− , the test statistic took the value 1.01 that was to be compared to 1.97. It seems that the value of v¯0 has slowly varied in time. That is the reason why the test statistic is lower when N1 = 100 than when N1 = 3000. The same t-test has been applied at the opening of the valve. The time-averaged flow rate was equal to 0.54 l/s before t = 0 and to 0.70 l/s after t = 0. The test statistic t0 was then equal to 18.8 what is much larger than the threshold of 1.97. The test hypothesis H1 was retained, i.e. that the flow is not constant and that any of the diagnostic methods presented in the following parts had to be postponed. Instead of simply detecting non-constant flows, it could be tempting to reject trends (i.e. large evolutions in time of the flow rate) that can make the detection of small perturbations due to installation effects impossible. Carlander [1] uses a differentiating MA-filter (MA for moving average) in order to eliminate trends and outliers: v¯dif f (n) = v¯(n) − v¯(n − 1), (4.3).

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