Dynamics of turbine flow meters Stoltenkamp, P.W. DOI: 10.6100/IR621983 Published: 01/01/2007 Document Version Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the author’s version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher’s website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Stoltenkamp, P. W. (2007). 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Stoltenkamp Cover design by Oranje vormgevers Printed by Universiteitsdrukkerij TU Eindhoven CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Stoltenkamp, P.W. Dynamics of turbine flow meters / by Petra Wilhelmina Stoltenkamp. Eindhoven : Technische Universiteit Eindhoven, 2007. - Proefschrift. ISBN 978-90-386-2192-0 NUR 924 Trefwoorden: stromingsleer / pulserende stromingen / debietmeters / meetfouten Subject headings: flow of gases / volume flow measurements / turbine flow meters / pulsatile flow / systematic errors Dynamics of turbine flow meters PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op maandag 26 februari 2007 om 16.00 uur door Petra Wilhelmina Stoltenkamp geboren te Heino Dit proefschrift is goedgekeurd door de promotoren: prof.dr.ir. A. Hirschberg en prof.dr.ir. H.W.M. Hoeijmakers This research was financed by the Technology Foundation STW, grant ESF.5645 Contents Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1.2 General description of a gas turbine flow meter . . . 1.3 Ideal rotation . . . . . . . . . . . . . . . . . . . . . 1.4 Parameter description . . . . . . . . . . . . . . . . . 1.5 Reynolds dependency of turbine flow meter readings 1.6 Thesis overview . . . . . . . . . . . . . . . . . . . . viii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 4 5 6 6 2. Turbine flow meters in steady flow . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Theoretical models of turbine flow meters . . . . . . . . . 2.2.1 Momentum approach . . . . . . . . . . . . . . . . 2.2.2 Airfoil approach . . . . . . . . . . . . . . . . . . 2.2.3 Equation of motion . . . . . . . . . . . . . . . . . 2.3 Effect of non-uniform flow . . . . . . . . . . . . . . . . . 2.3.1 Boundary layer flow . . . . . . . . . . . . . . . . 2.3.2 Velocity profile measurements . . . . . . . . . . . 2.3.3 Fully turbulent velocity profile in concentric annuli 2.3.4 Comparison of the different velocity profiles . . . 2.3.5 Effect of inflow velocity profile on the rotation . . 2.4 Wake behind the rotor blades . . . . . . . . . . . . . . . . 2.4.1 Wind tunnel experiments . . . . . . . . . . . . . . 2.4.2 Effect of wake on the rotation . . . . . . . . . . . 2.5 Friction forces . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Boundary layer on rotor blades . . . . . . . . . . . 2.5.2 Friction force on the hub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 10 10 13 14 16 17 20 21 23 24 25 28 29 31 32 33 vi . . . . . . . 35 37 38 39 42 45 46 3. Response of the turbine flow meter on pulsations with main flow . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Theoretical modelling . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 A basic quasi-steady model: A 2-dimensional quasi-steady model for a rotor with infinitesimally thin blades in incompressible flow . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Practical definition of pulsation error . . . . . . . . . . . . 3.3 Experimental set up . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Determination of the amplitude of the velocity pulsations at the location of the rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Acoustic model . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Synchronous detection . . . . . . . . . . . . . . . . . . . . 3.4.3 Verification of the acoustic model . . . . . . . . . . . . . . 3.4.4 Measurements of velocity pulsation in the field . . . . . . . 3.5 Determination of the measurement error of the turbine meter . . . . 3.6 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Dependence on Strouhal number . . . . . . . . . . . . . . . 3.6.2 Dependence on Reynolds number . . . . . . . . . . . . . . 3.6.3 High relative acoustic amplitudes . . . . . . . . . . . . . . 3.6.4 Influence of the shape of the rotor blades . . . . . . . . . . 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 49 50 4. Ghost counts caused by pulsations without main flow . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Onset of ghost counts . . . . . . . . . . . . . . . . . . . . . 4.2.1 Theoretical modelling of ghost counts . . . . . . . . 4.2.2 Experimental setup for ghost counts . . . . . . . . . 4.2.3 Experiments . . . . . . . . . . . . . . . . . . . . . 4.2.4 Comparing measurements with results of the theory . 4.3 Influence of vibrations and rotor asymmetry . . . . . . . . . 4.3.1 Vibration and friction . . . . . . . . . . . . . . . . . 4.3.2 Rotor blades with chamfered leading edge . . . . . . 79 79 80 80 87 89 91 93 93 93 2.6 2.7 2.5.3 Tip clearance . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Mechanical friction . . . . . . . . . . . . . . . . . . . Prediction of the Reynolds number dependence in steady flow 2.6.1 Turbine meter 1 . . . . . . . . . . . . . . . . . . . . . 2.6.2 Turbine meter 2 . . . . . . . . . . . . . . . . . . . . . 2.6.3 Effect of tip clearance . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 52 53 57 57 59 59 64 65 67 69 72 73 74 75 vii 4.4 4.5 Flow around the edge of a blade . . . . . . . . . . . . . . . . . . . 4.4.1 Numerical simulation . . . . . . . . . . . . . . . . . . . . . 4.4.2 Experimental set up for flow around an edge . . . . . . . . 4.4.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Comparing measurements with results of the numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusions . . . . . . . . . 5.1 Introduction . . . . . . . . . . 5.2 Stationary flow . . . . . . . . 5.3 Main flow with pulsations . . . 5.4 Pulsations without main flow . 5.5 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 94 94 95 98 102 104 109 109 109 110 111 111 113 A. Mach number effect in temperature measurements . . . . . . . . 115 B. Boundary layer theory . . . . . . . . . . . . . . . . B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Blasius exact solution for boundary layer on a flat plate . . B.3 The Von Kármán integral momentum equation . . . . . . . B.4 Description laminar boundary layer . . . . . . . . . . . . B.5 Description turbulent boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 117 119 120 121 122 C. Measurements . . . . . . . . C.1 Introduction . . . . . . . . . . C.2 Pulsation frequency of 24 Hz C.3 Pulsation frequency of 69 Hz C.4 Pulsation frequency of 117 Hz C.5 Pulsation frequency of 363 Hz C.6 Pulsation frequency of 730 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 125 126 127 128 129 130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Force on leading edge . . . . . . . . . . . . . . . . . . . . 131 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . 133 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 136 viii Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . 139 Dankwoord . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . 143 Nomenclature Roman symbols lowercase a quadratic fit parameter c0 speed of sound f frequency hblade height of a rotor blade equation 3.25 m s−1 Hz m m−1 k wave number m′ mass flow n normal unit vector n number of blades p pressure Pa p′ pressure fluctuations Pa r radius m rhub radius of the hub m rout radius of the outer wall m rtip radius at the tip of the rotor blade m s distance between two subsequent rotor blades m t blade thickness or time tblade blade thickness kg s−1 m or s m x u′ velocity fluctuations m s−1 uac acoustic velocity amplitude m s−1 uin inlet velocity m s−1 umax maximum velocity m s−1 uout outlet velocity m s−1 v velocity vector m s−1 w width m Roman symbols uppercase m2 A cross-sectional area B′ total specific enthalpy D pipe diameter E relative deviation from ideal rotation equation 2.14 Epuls relative error caused by periodic pulsations equation 3.11 m2 s−2 m F bf force imposed on the fluid by the body N FD drag force N Fe edge force N FL lift force N Irotor moment of inertia of the rotor K kg m2 m3 rad−1 meter factor Lblade chord length of a rotor blade m Lhub length of the hub in front of the rotor m Q volume flow R root-mean-square radius S pitch or area m3 s−1 q 2 +r 2 rin out 2 m m or m2 xi T temperature or period of the pulsations K or s−1 Tmech mechanical friction torque kg m2 s−2 Tair air friction torque kg m2 s−2 T bf torque imposed on the fluid by the body kg m2 s−2 Td driving torque kg m2 s−2 Tf total friction torque kg m2 s−2 U mean velocity in the annulus in front of the rotor V volume W width of the rotor m s−1 m3 m Greek symbols ◦ α angle of attack αd damping coefficient β angle of rotor blade with resect to the rotor axis ◦ βav average of the angle of the rotor blades at the root-mean-square radius ◦ δ1 displacement thickness m δ2 momentum thickness m Φ complex potential m2 s−1 φm mass flow kg s−1 Γ circulation m2 s−1 γ Poisson’s ratio µ dynamic viscosity ν kinematic viscosity m2 s−1 ω rotation speed rad s−1 ωid ideal rotation speed rad s−1 m−1 kg m−1 s−1 xii ω0 steady rotation speed without pulsations rad s−1 ρ density kg m−3 ρ′ density fluctuations kg m−3 τ viscous stress tensor kg m−1 s−2 τw shear stress at the wall kg m−1 s−2 Dimensionless numbers CD drag coefficient, FD /( 12 ρu2 A) ′ CD drag coefficient, FD /( 12 ρu2 wt) CL lift coefficient, FL /( 21 ρu2 A) He Helmholtz number, M Mach number, Pr Prandtl number, ν/a with a the thermal diffusivity Re Reynolds number, Sr Strouhal number, fL c0 u c0 uL ν fL u 1 Introduction 1.1 Introduction In industry axial turbine flow meters are used to measure volume flows of gases and liquids. They are considered reliable flow meters and at suitable conditions can attain high accuracies in the order of 0.1% for liquids and 0.25% for gases. An accuracy up to 0.02% can reached for high accuracy meters at ideal flow conditions (Wadlow, 1998). Turbine flow meters of different design are used in a broad variety of applications, for example in the chemical, petrochemical, food and aerospace industry. The internal diameter of these flow meters can vary from very small, e.g. 6 mm, to very large, e.g. 760 mm. In the Netherlands gas turbine flow meters are commonly used to measure natural gas flow. Because the Netherlands transported in 2005 95.2 billion m3 of natural gas, small systematic measurement errors can lead to over- or underestimation of large volumes of natural gas. This makes the accuracy of flow meters crucial at all flow conditions. A new development is the exploration of the possibility to correct flow measurements for non-ideal flow conditions on the basis of a physical model for the response of the meter to deviations from the ideal flow conditions. 1.2 General description of a gas turbine flow meter A schematic drawing of a typical turbine flow meter is shown in figure 1.1. In this drawing the most important elements of a turbine flow meter are given. Turbine flow meters are placed in line with the flow. Sometimes they are placed in measuring manifolds, where several flow meters are placed in parallel streams, in order to increase the overall dynamic range of the set up. Usually the flow passes first through a flow 2 1. Introduction C A B Figure 1.1: Schematic drawing of a turbine flow meter with A) flow straightener and B) rotor. C) shows the position of the mechanical counter straightener or a flow conditioning plate (A) to remove swirl and create a uniform flow. Subsequently, the flow is forced through an annular channel and through the rotor (B), see also figure 1.2. The blades of the rotor are often flat plates or have a helical shape. The shaft and bearings are placed inside the core, which usually is suspended Figure 1.2: Photograph of the rotor of turbine flowmeter, Instromet type SM-RI-X G250. downstream of the rotor. There are several ways to detect the rotation speed of the rotor. The most common detection methods are mechanical detection and magnetic detection. Mechanical detection of the rotor speed is measured by transferring the rotor speed through the rotor axis and via gears to a mechanical counter (C). During 1.2. General description of a gas turbine flow meter 3 magnetic detection a pulse is measured by disrupting a magnetic field every time a designated point on the rotor, for example the rotor blades, passes a measuring point. These pulses can be processed electronically. The experiments in this thesis are performed on gas turbine flow meters of ElsterInstromet. The dynamical response measurements have been carried out at the Eindhoven University of Technology with the gas turbine meter type SM-RI-X G250, see figure 1.3. This meter has an internal pipe diameter of 100 mm. The accuracy of the Figure 1.3: Photograph of the SM-RI-X G250 turbine flow meter (by courtesy of ElsterInstromet). flow measurement is 0.1% for volume flows in the range from 20 to 400 m3 /h. The meter is designed for pressures ranging from atmospheric pressure up to 20 bar (this type of meter is also available for work pressures up to 100 bar). The rotor is made of aluminium and has helical shaped blades (see figure 1.2). We will refer to this meter as turbine meter 1. Additional steady flow experiments have been performed by Elster-Instromet with simplified prototypes which we refer to as turbine meter 2, 3, 4 and 5. Additional experiments with oscillatory flow have been performed by Gasunie with a larger version of the SM-RI-X G250, the SM-RI-X G2500 with a internal pipe diameter of 300 mm. 4 1. Introduction u u in re l u b o u t,x w r = u in b w r Figure 1.4: Steady flow entering and leaving the rotor for an ideal frictionless rotor with infinitesimally thin helical rotor blades with blade angle β. 1.3 Ideal rotation When ideal rotation is considered, it is assumed that the flow through the turbine meter is uniform, incompressible and steady, that the rotor rotates with no friction and that the rotor is shaped as a perfect helix with infinitesimally thin blades. Under these circumstances the rotation speed of the rotor is determined by the pitch of the rotor, S, defined by: S= 2πr , tan β (1.1) with r the radius of the rotor and β the angle of the rotor blades with respect to the rotor axis (see figure 1.4). In an ideal case the pitch corresponds to the axial displacement of the fluid during one revolution of the rotor. For a perfect helicoidal rotor the pitch, S, is constant over the whole radius of the rotor, while the blade angle, β, changes. Because friction is not considered, the flow entering and leaving the rotor is parallel to the blades of the rotor. This means that the inlet velocity and the rotation velocity are related through the angle of the rotor blades, β, as: ωid r = tan β , uin (1.2) 1.4. Parameter description 5 with ωid the angular velocity of the rotor for the ideal situation considered and uin is the velocity of the flow entering the rotor. The angular velocity in this ideal situation is ωid = 2πuin uin tan β = . r S (1.3) Because the volume flow, Q, is equal to the inflow velocity multiplied by the crosssectional area of the rotor, i.e. Q = uin A, we find a relationship between the volume flow and the rotational speed: Q= AS ωid . 2π (1.4) This relationship is applied in an actual turbine flow meter in the form: Q = Kωid , (1.5) where K is called the meter factor, which is determined by calibration. Ideally, K should be a constant. 1.4 Parameter description In principle for steady flow the meter factor K of a specific meter depends on dimensionless parameters such as: • the Reynolds number Re = • the Mach number M = uin L ν uin c0 • the ratio of mechanical friction torque, Tmech , to the driving fluid torque Tmech R3 ρu2in where L is a characteristic length such as the blade chord length, ν is the kinematic viscosity of the fluid, c0 is the speed of sound R is the root mean square radius of the rotor and ρ the fluid density. The manufacturer uses steady flow calibrations at different pressures to distinguish between Reynolds number effects and the influence of mechanical friction. In general the Mach number dependency is a small correction due to a Mach number effect in the temperature measurements at high flow rates (see Appendix A). In this thesis we will consider unsteady flow. In such case the response of the meter will also depend on: • the Strouhal number Sr = fL uin 6 1. Introduction • the amplitude of the perturbations |u′in | uin • The ratio of fluid density, ρ, and rotor material density, ρm , i.e. ρ ρm where f is the characteristic frequency of flow perturbations and |u′in | is the amplitude of the perturbations. 1.5 Reynolds dependency of turbine flow meter readings In the ideal case the rotational velocity changes linearly with the volume flow. In reality friction forces and drag forces cause the rotor to rotate at a rotation speed that differs from the rotational speed of the ideal rotor. The difference between the actual rotor speed and the ideal rotor speed is known as rotor slip. Because the drag forces depend on flow velocity and the viscosity of the medium, the rotor slip depends on Reynolds number, Re. A meter designer tries to make the volume flow measured by the meter to be a function that is as linear as possible in terms of the rotational speed for a dynamic range of at least 10:1. With every meter the manufacturer provides a calibration, that gives the rotor slip as function of the Reynolds number or sometimes as function of the volume flow. This calibration is unique for every meter due to the sensitivity of the meter to small manufacturing differences or differences caused by damage or wear. One of the aims of the designer is to reduce this sensitivity of the meter factor, i.e. the quantity K, for manufacturing inaccuracies, damage or wear. 1.6 Thesis overview In this thesis, the behaviour of turbine flow meters is investigated experimentally aiming at development of physical models allowing corrections for deviations from ideal flow. In chapter 2 the Reynolds number dependence of the turbine flow meter is investigated analytically. The driving torque on the rotor is obtained by using conservation of momentum on a two-dimensional cascade of rotor blades. Using the equation of motion of the rotor, its rotation speed is determined. We use in this chapter a theoretical model developed by Bergervoet (2005) which we extent by considering the influences of non-uniform flow and drag forces. The effect of the inlet velocity profile is investigated using models and measurements. The effect of several friction forces is modelled analytically. The last part of this chapter compares the model with calibration measurements obtained by Elster-Instromet for several turbine flow meters. Chapter 3 studies the effect of pulsations superimposed on main flow. Pulsation can induce large systematic errors during measurements. A simplified quasi-steady 1.6. Thesis overview 7 theory predicting these errors, is discussed. Measurements are performed to investigate the applicability of this model. A detailed description is given of the measurement set up and measurements methods. Finally, the results are discussed. Chapter 4 deals with the extreem case of chapter 3, where the flow is purely oscillatory and there is no main flow. This can induce the rotor to rotate and measure a flow while there is no net flow. We call this ghost counts or spurious counts. The first part of this chapter describes two physical models to predict the onset of ghost counts. The models are compared with experiments. The second part of this chapter investigates the flow around the edge of a rotor blade in pulsating flow. First, this investigation is carried out experimentally. These results are compared with a discrete vortex model. The main results of these thesis are summarised in chapter 5. 8 1. Introduction 2 Turbine flow meters in steady flow 2.1 Introduction In this chapter a model is developed to predict the response of a turbine flow meter in steady flow. The development of a theoretical model describing the behaviour of a turbine flow meter has been endeavoured experimentally and analytically for a long time (Baker (2000), Wadlow (1998), Lee and Evans (1965), Lee and Karlby (1960), Rubin et al. (1965) and Thompson and Grey (1970)). More recent attempts to understand the behaviour of turbine flow meters use numerical methods to compute the flow field in a turbine flow meter (von Lavante et al. (2003), Merzkirch (2005)). A theoretical model allows the investigation of, for example, meter geometry, making it possible to develop better design criteria, or to assess the influence of different fluid properties. Rather than considering a numerical method we will consider an extension of the more global analytical model as proposed by Thompson and Grey (1970). Our global model aims at understanding important phenomena in the behaviour of turbine flow meters. Since in practice deviations in the dependence on Reynolds number of 0.2% are significant, we do not expect to succeed in making such accurate predictions of the deviations. We try to obtain some insight into the problem of the design of a flowmeter. The turbine meter is modelled using the equation of motion for the rotor. The flow passing through the rotor induces a driving torque, Td , on the rotor. First, two approaches to obtain this driving torque will be discussed. Next, the influence of the inlet velocity, uin at the front plane of the rotor will be investigated by using a boundary layer description, actual velocity measurements in a dummy of a turbine flow meter and a model for fully developed flow. Wind tunnel measurements have been performed to investigate the drag forces on the rotor blade. The effect of other 10 2. Turbine flow meters in steady flow friction forces on the rotor is described and discussed in the following section and their individual effect on the rotation speed of the rotor will be shown. In the last part of this chapter the model is applied to different turbine flow meters at different Reynolds numbers and the results are compared to calibration measurements provided by Elster-Instromet. 2.2 Theoretical models of turbine flow meters In general two approaches have been used in literature; the momentum approach (Wadlow, 1998) and the airfoil approach (Rubin et al., 1965). In the momentum approach the integral momentum equation is used to calculate the driving torque on the rotor. One of the main limitations of this method is that full fluid guidance is assumed. It is assumed that there is a uniform flow tangential to the rotor blades at the rotor outlet. This assumption is only true for rotors with high solidity. This implies a gap between successive blades, which is narrow compared to the blade chord length. Weinig (1964) showed, using potential flow theory for a twodimensional planar cascade, that the ratio of the gap between the blades and blade length (chord), s/Lblade should be smaller than 0.7 to allow such an assumption. The airfoil approach on the other hand derives the driving torque on the rotor by using airfoil theory to obtain the lift coefficient of an isolated rotor blade. With this approach there is no assumption of full fluid guidance, but blade interference is ignored. This means that increasing the number of blades would always increase the lift force proportionally. Thompson and Grey (1970) improved this approach by using the two-dimensional planar cascade theory of Weinig (1964) to account for the interference effects. Both the integral momentum method and the airfoil method will be explained in more detail in the following sections. We later actually use only the integral momentum method, which has been used earlier in simplified form by Bergervoet (2005) at Elster-Instromet. 2.2.1 Momentum approach The turbine meter is a complex three-dimensional flow device (see figure 2.1). As an approximation this three-dimensional problem will be treated as a two-dimensional infinite cascade of rotor blades with uniform axial flow, uin , at radius r as approximation of the flow inside an annulus between r and r + dr. The x-direction refers to the axial direction. The y-direction refers to the azimuthal direction (see figure 2.2). The radial velocity is neglected and constant rotation with a rotational angular velocity ω is assumed. To obtain the torque on the rotor we will integrate over the blade length in radial direction. The control volume enclosing the rotor is shown in figure 2.2. 2.2. Theoretical models of turbine flow meters h r r tip 11 b la d e h u b x q r w d r Figure 2.1: The rotor of the turbine flow meter. We assume that the flow in an annulus between r and r + dr behaves as the flow in a two-dimensional infinitely long cascade shown in figure 2.2. To calculate the driving torque on the rotor, the integral mass conservation law and integral momentum equation is used for this two-dimensional cascade of blades: ZZ ZZZ d (2.1) ρv · ndA = 0 , ρdV + dt CV CS ZZ ZZ ZZ ZZZ d ρv (v · n) dA = − pndA + τ ndA + F bf ,(2.2) ρvdV + dt CV CS CS CS applied to a fixed control surface CS enclosing the rotor, this surface has an outer normal n, the fixed control volume within CS is denoted as CV , ρ is the fluid density, v is the velocity vector, p is the pressure, τ is the viscous stress tensor and F bf are the forces imposed on the fluid by the turbine. Full fluid guidance is assumed; the flow leaves the rotor with a velocity parallel to the blades along the whole circumference (or the y-direction in our 2D model, figure 2.2). This implies that we neglect radial velocities and the effect of the Coriolis forces. We assume that the flow enters the rotor without any azimuthal velocity, vθ = 0 (in a two dimensional representation vy = 0). Assuming steady incompressible flow and applying the conservation of mass (equation 2.1) to a volume element of height dr (figure 2.1), we get: uin,x dAin = uout,x dAout , (2.3) 12 2. Turbine flow meters in steady flow w r W b u u u in re l b w r u o u t,x = o u t,y in n re l t n u u L C S b la d e y x Figure 2.2: Flow entering and leaving the cascade representing the rotor in an annulus between r and r + dr. where uin,x and uout,x are x-component of the the incoming and outgoing velocity, respectively, and dAin and dAout are the inflow area and the outflow area, respectively. If the inflow and outflow area are assumed to be equal and the flow is incompressible, dAin = dAuit = 2πrdr, so that the x-component of the incoming velocity is equal to the x-component of the outgoing velocity, i.e. uin,x = uout,x . Using the same assumptions as mentioned above and neglecting the viscous forces, Re >> 1, the momentum equation in the y-direction for a steady flow through an element dr becomes: ρ ((uout,y + ωr) uout,x dAout − uin,x ωrdAin ) = dFbf,y , (2.4) From the velocity diagram in figure 2.2 it can be seen that: uout,y = uout,x tan β − ωr . (2.5) 2.2. Theoretical models of turbine flow meters 13 Substituting equations 2.3 and 2.5 in equation 2.4, the y-component of the force imposed by the rotor on the fluid, dFbf,y is found: dFbf,y = ρu2out,x tan βdAout − uin,x ωrdAin . (2.6) The force of the fluid on the rotor is opposite and equal to the force of the rotor on the fluid, dFbf,y = −dFf b,y . The torque exerted by the fluid element on the rotor axis, dTd , is estimated to be: dTd = rdFf b,y . (2.7) By integrating this equation from the radius of the rotor hub, rhub to the rotor tip, rtip (see figure 2.1), the driving torque on the rotor is: Z rtip Z rtip ρuin,x ωr2 dAin . (2.8) ρu2out,x (tan β)rdAout + Td = − rhub rhub 2.2.2 Airfoil approach An alternative method to obtain the driving torque on the rotor, is the airfoil approach. Again the element of the rotor at radius r and thickness dr is approximated as an infinite two-dimensional cascade of rotor blades (see figure 2.3). In contrast to the momentum approach there is no assumption that flow is attached. The driving torque on the rotor blade is now evaluated by determining the lift and drag forces on the rotor blades in a coordinate system fixed to the blade. The lift force, FL , acts perpendicular to the relative inlet velocity, uin,rel = (uin,x , ωr), and the drag force, FD acts parallel to this inlet velocity. The y-component of the force of the flow on the blade can now be expressed in terms of lift, FL , and drag, FD ; Fy = n (−FL cos φ + FD sin φ) , (2.9) where φ = β − α = arctan uωr , with β the angle of the rotor blade (with respect in,x to the x-axis), n is the number of blades and α the angle of attack of the incoming flow. The lift- and drag coefficient are defined as: CL = FL 1 2 2 ρuin,rel Lblade , FD CD = 1 2 , 2 ρuin,rel Lblade (2.10) where Lblade is the chord of the blade. The lift and drag coefficients are functions of the angle of attack, α, depend weakly on Reynolds number and on Mach number. 14 2. Turbine flow meters in steady flow F a u in ,r e l f u w r in F F D ,y f L ,y D F L b t L b la d e y x Figure 2.3: Lift and drag force acting on a blade of a two dimensional cascade Using these coefficients the driving torque on a rotor with n blades can be written as: Z rtip 1 Td = nρu2in,rel Lblade (−CL cos φ + CD sin φ) rdr . (2.11) rhub 2 2.2.3 Equation of motion The driving torque, Td , is known from equation 2.8 or 2.11. To determine the angular velocity, ω, of the rotor, the equation of motion of the rotor is used: dω = Td − Tf , (2.12) dt where Irotor is the moment of inertia of the rotor and Tf is the friction torque on the rotor, assuming a quasi-steady flow through the rotor. Using equation 2.8 or 2.11 Irotor 2.2. Theoretical models of turbine flow meters 15 for the torque implies that we assume a quasi-steady flow through the rotor. In this chapter we investigate the rotor in steady rotation, for which the equation of motion reduces to: Td = T f . (2.13) The different friction forces will be discussed in the following sections. This equation can be used to predict the steady rotation speed of the rotor, ω. By comparing this rotation speed with the ideal rotation speed, ωid (see equation 1.3), the deviation of the rotation speed of the turbine meter from ideal rotation can be determined as: E= ω − ωid . ωid (2.14) Calculating the deviation at various Reynolds numbers, Re, the dependence of a turbine meter can be estimated. In the following sections the analysis will be applied using the momentum approach (equation 2.8) to two types of turbine flow meters. The first one, referred to as turbine meter 1, is the Instromet SM-RI-X G250 with a diameter of D = 0.1 m used in the experiments at the set up in Eindhoven. The second one is a simplified turbine meter with diameter of D = 0.2 m, this rotor will be referred to as turbine meter 2. The second turbine meter has a simplified geometry. An example of this simplification is the geometry at the rotor tip (see section 2.5.3). This simplified geometry should allow a better comparison of experiment with the theory. Information about the geometry of the two flow meters is given in table 2.1 The chord length of the rotor blades of turbine meter 1 can be calculated using: W , (2.15) cos β(r) the angle of the blade relative to the rotor axis. The blades with β = arctan 2πr S of the second turbine meter, turbine meter 2, are reduced at the tip to a chord length of Lblade (rtip ) = 0.035 m. The chord length of the rotor blades of this turbine meter can be written as: Lblade (r) = Lblade (r) = Lblade (rhub ) + Lblade (rtip ) − Lblade (rhub ) (r − rhub ) . hblade (2.16) In the following sections the effect of non-uniform flow, the blade drag and other friction forces are investigated separately, the deviation from the ideal rotation is calculated for several flows up to Qmax as indicated for the meter. Two scenarios were followed; in the first scenario the calculations were done using the properties of air at 1 bar (absolute pressure), ρ = 1.2 kg/m3 and ν = 1.5 × 10−5 m2 /s, and 16 2. Turbine flow meters in steady flow pipe diameter, D (m) blade thickness, t (mm) number of blades, n rhub /D rout /D S/D W/D hblade /D = (rtip − rhub )/D Lhub /D turbine meter 1 0.1034 1.6 16 0.360 0.500 2.704 0.213 0.140 0.763 turbine meter 2 0.2030 4 14 0.250 0.500 3.941 0.148 0.240 1.049 Table 2.1: Dimensions of the two turbine flow meters used in the calculations, where rhub is the radius of the hub, rout is the radius of the outer wall, rtip is the radius at the tip of the blades, S is the pitch (equation 1.1), W is the width of the rotor, hblade is the height of the blade (span of the blades) and Lhub is the length of the hub in front of the rotor. Except for the blade thickness t and the number of blades n, all values are made dimensionless with the diameter, D. in the second scenario the properties of natural gas at 9 bar (absolute pressure) were used, ρ = 7.2 kg/m3 and ν = 1.5 × 10−6 m2 /s. These conditions correspond to the test conditions used by Elster-Instromet. The resulting deviation, E, is plotted against the Reynolds number, Re = U Lblade /ν, where Lblade is the length of a rotor blade measured at the tip and U the velocity at the rotor. For the calculation in this chapter only the momentum approach is being used. This approach assumes full fluid guidance, i.e. attached flow. This is a good approximation, if the ratio of the distance between the blades and blade length is sufficiently small, s/Lblade < 0.7. In case of the first turbine meter this assumption is valid. For turbine meter 2 this assumption is no longer valid at the tip of the blades. However, the departure from full fluid guidance is expected to be small. Using the theory of (Weinig, 1964), we estimate that the tangential velocity uout,y will be about 5% smaller than the tangential velocity for full fluid guidance. The reduction in the tangential velocity decreases the driving torque exerted by the flow on the rotor and this decreases the rotation speed of the rotor. Because this effect will be small in this case, we will ignore it in our model. 2.3 Effect of non-uniform flow As can be seen from equation 2.8 the driving torque depends on the velocity entering the flow meter. The flow entering the turbine is generally non-uniform. Boundary 2.3. Effect of non-uniform flow 17 layers will form along the walls and in pipe systems swirl inevitably occurs due to upstream bends. Parchen (1993) and Steenbergen (1995) showed that swirl decays extremely slowly. Swirl can have effect the accuracy of turbine meters (Merzkirch, 2005). Properly designed flow straighteners as designed by Elster-Instroment placed in front of a turbine flow meter reduce the effect of swirl considerably. Therefore in the calculation we assume that there is no azimuthal velocity (no swirl). We limit our discussion to the non-uniformity of the axial velocity, uin (r). Thompson and Grey (1970) predicted that the inlet velocity profile plays an important role in the rotation speed of the rotor. The influence of the velocity profile entering the rotor will be investigated in this section. The shape of the velocity profile entering the rotor is first calculated using boundary layer theory. Velocity profile measurements carried out in a dummy of a turbine meter will be compared with the boundary layer theory and a fully developed turbulent annulus flow assumed by Thompson and Grey (1970). The rotation rate of a rotor for velocity profile based on boundary layer theory and for a measured flow profile will be compared with predictions of the ideal rotation rate. 2.3.1 Boundary layer flow The flow enters the turbine meter, passes a flow straightener and continues through an annular pipe segment of length Lhub around the hub of the turbine meter (see 1.1). Upon entering the annulus, the gas is accelerated because of the area contraction. Due to this acceleration the thickness of the boundary layers is strongly reduced. At the leading edge of the hub a new boundary layer starts to form on the hub and on the outer wall. The velocity profile is assumed axisymmetric and can be divided in three regions (see figure 2.4). The first region is the boundary layer on the hub. The second region is the region between the boundary layers, where the velocity is approximately uniform. The third region is the boundary layer on the outer pipe wall. Calculation are carried out for two cases; laminar and turbulent boundary layers. The transition from laminar to a turbulent flow occurs for flat plates under optimal conditions around a Reynolds number of ReLhub ≈ 3 × 105 (Schlichting, 1979). This would imply that there is a significant laminar part of the boundary layer on the hub even for ReLhub > 3 × 105 . However, we will assume that above a critical Reynolds number the boundary layer is turbulent from the start, ignoring the effect of transition. The boundary layer thickness is calculated using the von Kármán integral momentum equation (see Schlichting (1979)). Appendix B provides a brief discussion of boundary layer theory. The von Kármán equation obtained by integration of the mass and momentum equations over the boundary layer is: dU τw d U 2 δ2 + δ1 U = , dx dx ρ (2.17) 18 2. Turbine flow meters in steady flow o u te r w a ll r II u (r) I d (x ) ro to r s tra ig h te n e r III h u b h b la d e r h u b L o u t W h u b r x Figure 2.4: The three different regions of the velocity profile in the turbine meter with U the velocity outside the boundary layers, δ1 the displacement thickness (for definition see equation B.3), δ2 the momentum thickness (for definition see equation B.4) and τw the shear stress at the wall. For the calculation of the laminar boundary layer, a third order polynomial description of the boundary layer profile is used in combination with Newton’s law for τw (see Appendix B). This was found to be an accurate description of a laminar boundary layer by Pelorson et al. (1994) and Hofmans (1998). For turbulent flow the boundary layer is described using a 1/7th power law description for the velocity profile combined with the empirical law of Blasius for the wall shear stress (see Appendix B). Using these models, the displacement thickness, δ1 , the momentum thickness, δ2 , and the shear stress at the wall, τw , are calculated just upstream of the turbine flow meter. The mean velocity in the annulus, U , is corrected for the boundary layer on the hub as well as on the pipe wall. Using the definition of displacement thickness, δ1 , this velocity can be written as: U (x; Q, δ(x)) = Q , π ((rout − δ1 )2 − (rhub + δ1 )2 ) (2.18) where Q is the volume flow, rout is the radius of the outer wall and rhub is the radius of the hub. The boundary layers on the outer pipe wall and on the hub are assumed to have the same thickness. The velocity profile in front of the rotor of a turbine meter with geometrical dimensions equal to the turbine meter 1, is calculated. This meter has a radius of the outer wall, rout = 0.050 m and a radius of the hub rhub = 0.037 m. The hub length in front of the rotor is Lhub = 0.076 m. For laminar boundary layers figure 2.5(a) shows the calculated velocity profile in the annulus just upstream of the rotor. For turbulent boundary layers the velocity profile is plotted in figure 2.5(b). As expected the 2.3. Effect of non-uniform flow 19 1 0.8 rhub u/umax 0.6 0.4 ReL = 3 X103 4 ReL hub = 1.2 X 10 4 ReL hub = 3.9 X 10 Re hub = 1.4 X 105 0.2 L hub 0 0.7 0.75 0.8 0.85 r/r 0.9 0.95 1 out (a) laminar boundary layers 1 0.8 rhub u/umax 0.6 0.4 ReL = 3 X103 ReL hub = 1.2 X 104 ReLhub = 3.9 X 104 Re hub = 1.4 X 105 0.2 L hub 0 0.7 0.75 0.8 0.85 r/rout 0.9 0.95 1 (b) turbulent boundary layers Figure 2.5: Velocity profile entering the rotor for turbine meter 1 with a diameter 0.1034 m calculated using boundary layer theory. The velocity, u, divided by the maximum velocity, umax is plotted against the radius for different Reynolds numbers (Re = U Lhub /ν = 3 × 103 , 1.2 × 104 , 3.9 × 104 and 1.4 × 105 . (a) shows the velocity profile with laminar boundary layers, (b) the velocity profile with turbulent boundary layers. 20 2. Turbine flow meters in steady flow laminar boundary layers are thinner than the turbulent boundary layers. The velocity profile for turbulent boundary layers is more uniform than that for laminar flow. 2.3.2 Velocity profile measurements To examine whether the boundary layer description of the velocity profile is an adequate approximation of the velocity profile, measurement were carried out with a hot wire anemometer and a Pitot tube in the set up described in section 3.3. In this set up turbine flow meter 1 with a diameter of D = 0.1 m, is placed at the end of a pipe with a length of more than 30 times its diameter. The pipe flow is supplied by a high pressure dry air reservoir (60 bar). A choked valve is controlling the mass flow through the pipe. In order to measure the velocity profile just upstream of the rotor, the flow meter was replaced by a dummy. The dummy is a replica of the forward part of the meter, including the flow straightener, up to the rotor. The remainder of the flow meter, including the rotor, has been removed providing easy access for the measurement probes. The Pitot tube has a diameter of 1 mm and is connected to an electronic manometer, Datametrics Dresser 1400, and a data acquisition PC. The single wire hot wire anemometer (Dantec type 55P11 wire with 55H20 support) is also connected to a PC. More details of the set up can be found in section 3.3. The pressure and velocity are determined by averaging over a 10 s measurement at a sample frequency of fs = 10 kHz. Before measuring the velocity profile just in front the rotor (but in absence of the rotor), the velocity profile in the pipe upstream of the turbine flow meter was measured using the Pitot tube. Measurements were performed at four different velocities in the pipe, 2, 4, 10, 15 m/s. The measured profiles are plotted in figure 2.6. The Reynolds number, ReD , mentioned in figure 2.6 is based on the diameter, D, of the pipe and the maximum velocity measured, umax . The measured velocity profile is symmetric and approaches that of a fully developed turbulent pipe flow. Measurements of the annular flow 1 mm downstream of the dummy of the forward part of the meter were performed at seven different average velocities in the pipe (0.5, 1, 1.5, 2, 4, 10 and 15 m/s), resulting in Reynolds numbers, Re = U Lhub /ν, where Lhub is the length of the hub in front of the rotor (see figure 2.4) and U the mean velocity in the annulus outside the boundary layers (equation 2.18). This Reynolds number ranges from 3.0 × 103 up to 1.5 × 105 . From the measurements shown in figure 2.7, it can be seen that the velocity profile is asymmetric. The asymmetry is increasing with increasing Reynolds number. It has a maximum velocity closer to the outer wall than to the hub. For lower Reynolds numbers near the walls the velocity profile resembles the laminar boundary layer velocity profile, for Reynolds number above 104 the velocity profile resembles more the turbulent 2.3. Effect of non-uniform flow 21 1 0.9 0.8 0.7 u/umax 0.6 0.5 0.4 ReD = 1.5 X 104 0.3 4 ReD = 2.8 X 10 0.2 4 ReD = 7.1 X 10 0.1 0 −0.5 −0.4 −0.3 −0.2 −0.1 5 ReD = 1.1 X 10 0 r/D 0.1 0.2 0.3 0.4 0.5 Figure 2.6: The velocity profile in the pipe just upstream of the turbine flow meter, measured at four different Reynolds numbers, ReD = umax D/ν = 1.5 × 104 , 2.8 × 104 , 7.1 × 104 and 1.1 × 105 . boundary layer profile. It is difficult to determine the exact velocity profile near the wall of the pipe and the hub. This can be seen in figure 2.7. The velocity is measured 1 mm downstream of the dummy of the turbine meter. At this point there is a flow for r/rout > 1, because of entrainment of air in the airjet flowing out of the dummy (figure 2.9). We therefore observe some velocity at the location of the pipe wall, r/rout = 1, where in the pipe the velocity vanishes. 2.3.3 Fully turbulent velocity profile in concentric annuli Fully developed turbulent axisymmetric axial flow in a concentric annulus has been studied in literature, because of the many engineering applications and in order to obtain fundamental insight in turbulence. Brighton and Jones (1964) found experimentally that the position of the maximum velocity of such fully developed flows is closer to the inner wall than to the outer pipe wall. The position depends on Reynolds number and ratio rhub /rout of the inner wall radius, rhub , and the outer wall radius, rout . The results found here differs in that respect. 22 2. Turbine flow meters in steady flow 1 0.8 rout r hub u/umax 0.6 0.4 Re = 3.1 X 103 0.2 Re = 5.6 X 103 4 Re = 1.2 X 10 0 0.65 0.7 0.75 0.8 0.85 r/rout 0.9 0.95 1 1.05 (a) Re < 2 × 104 1 0.8 rout rhub u/umax 0.6 0.4 Re = 2.0 X 104 Re = 3.9 X 104 0.2 4 Re = 9.2 X 10 5 Re = 1.4 X 10 0 0.65 0.7 0.75 0.8 0.85 r/rout 0.9 0.95 1 1.05 (b) Re ≥ 2 × 104 Figure 2.7: Velocity profile at the entrance of the rotor (turbine meter 1, D = 0.1034 m) measured with the hot wire anemometer 1 mm downstream of a dummy of the forward part of the meter. The velocity, u, normalised by the maximum velocity, umax as a function of the radius for four different Reynolds numbers. 2.3. Effect of non-uniform flow 23 p ro b e o u te r w a ll r o u t s tra ig h te n e r u (r) h u b L 1 m m h u b r h u b r x Figure 2.8: Schematic drawing of the position of the hot wire during the velocity measurements. p ip e w a ll flo w e n tra in m e n t m a in (je t) flo w Figure 2.9: The air outside the pipe is entrained in the airjet exiting the pipe 2.3.4 Comparison of the different velocity profiles The velocity profile calculated using boundary layer theory (figure 2.5), the measured profile (figure 2.7) and the profile of a fully developed turbulent flow as found by Brighton and Jones (1964) are quite different. Comparing the result of the boundary layer calculations for turbine flow meter 1 with the measurements in the same meter, the measured profiles show a clear asymmetry dependent on the Reynolds number. Fully developed turbulent flow in an annular channel (e.g. Brighton and Jones (1964)) displays a maximum velocity closer to the inner wall than to the outer wall. However, in our measurements the maximum velocity is closer to the outer wall. This indicates that the measured velocity profile does not resemble the fully developed turbulent flow in an annulus. This is not surprising, since the length of the hub, Lhub , is relatively short, Lhub ≈ 5.5(rout − rhub ). The asymmetry in the 24 2. Turbine flow meters in steady flow measured profile can be caused by flow separation at the front of the hub, resulting in a velocity profile with higher velocity along the outer wall (see figure 2.10). The observed velocity maximum would be due to the flow separation at the sharp edge of the nose of the hub. Similar behaviour is observed downstream of a sharp bend in a pipe. o u te r w a ll r o u t r h u b ro to r s tra ig h te n e r u h u b L h u b r x Figure 2.10: Flow is expected to separate at the leading edge of the hub causing the flow to accelerate close to the outer wall 2.3.5 Effect of inflow velocity profile on the rotation To investigate the effect of the inlet velocity profile on the driving torque, Td , the driving torque is calculated using the predicted velocity profile based on boundary layer theory. The mechanical friction forces, the fluid friction and the thickness of the blades are ignored. The results are compared to the calculation of the driving torque predicted for a uniform velocity. As we assume incompressible flow, the continuity equation gives that the incoming velocity is equal to that of the axial component of the outgoing velocity, uin = uin,x = uout,x . The momentum equation (equation 2.8) reduces to: Z rtip Td = − ρuin (uin tan β + ωr) 2πr2 dr . (2.19) rhub For steady flow and in absence of friction the equation of motion of the rotor (equation 2.13) reduces to: Td = 0. For a given geometry of the rotor and a known incoming velocity profile, the rotation speed of the rotor can then be calculated. As the velocity profile depends on the Reynolds number, Re = U Lhub /ν, the deviation of the rotation speed from ideal rotation speed for a uniform inflow, E (see equation 2.14), is plotted against Reynolds number. In figure 2.11 the deviation in rotation speed has been plotted for the laminar and turbulent boundary layer profiles (figure 2.5) and for the measured profile (figure 2.7). Compared to a uniform flow the rota- 2.4. Wake behind the rotor blades 25 8 velocity profile measurements turbulent boundary layers laminar boundary layers E = (ω − ωid) / ωid * 100 (%) 7 6 5 4 3 2 1 0 3 10 4 5 10 10 6 10 Re L hub Figure 2.11: The deviation of the rotation speed, ω, from the rotation speed for a uniform inlet velocity profile, ωid versus Reynolds number, Re = U Lhub /ν. Turbulent boundary layer approximation (solid line), laminar boundary layer approximation (dashed line) and the measured velocity profile (◦). tion speed of the rotor increases in the order of one percent for a velocity profile based on laminar or turbulent boundary layer theory. The turbulent boundary layer causes the rotor to rotate faster than the laminar boundary layers. The measured velocity profile induces much larger deviations. As we are aiming for an accuracy of 0.2%, it is clear that the velocity profile plays a very significant role in the rotation speed of the rotor, as already observed by Thompson and Grey (1970). In further calculations discussed in this chapter, the boundary layer model is used. We have to keep in mind that the measured profile induces a larger deviation. 2.4 Wake behind the rotor blades The flow around the rotor blades does not only provide a driving torque, but the flow also exerts a drag force on the rotor. The effect of the forces caused by the pressure difference between the pressure and the suction side of the rotor blade and by the friction of the fluid on the solid surface of the blades (described in section 2.5.1) can be included in the momentum conservation balance described in section 2.2.1. To 26 2. Turbine flow meters in steady flow include the pressure drag, a model for the wake is proposed. In this model we will assume that the wake of the blade in the rotor has the same structure as for a single isolated blade in free stream (see figure 2.12). u n in ,r e l u ro to r b la d e A f w a k e m C S o u t,r e l w w a k e A Figure 2.12: Wake behind a rotor blade. Betz, Prandtl and Tietjens (1934) found that it is possible to calculate the drag force on a body in an unbounded uniform flow by applying a momentum balance on a large control surface surrounding the body. The control volume is chosen around the rotor blade, with a control surface CS with a normal vector n as shown in figure 2.12. The control volume has to be chosen far from the body. There, the streamlines in the flow are again approximately parallel and the pressure over the wake can be considered uniform and equal to the pressure of the uniform flow. The rotor blade and the wake cause a displacement of the flow over the sides. We apply the momentum equation on this controle volume for steady incompressible flow. Assuming that outside the wake the velocity, u, can be approximated by the free stream velocity, u∞ = uin , this equation reduces to: Z uout (u∞ − uout ) dy = ρu2∞ δ2,wake , (2.20) FD = ρ wake where the integral can be limited to the wake, because uout = u∞ outside the wake and δ2,wake is the momentum thickness of the wake. With this equation the drag FD ′ = C Lblade = , coefficient of a blade of length Lblade and thickness t, CD D 1 t ρu2 t 2 in,rel can be determined from the velocity distribution in the wake. Note, that if this momentum approach is used for a model of the wake, in which the velocity directly behind the blades is assumed zero and the pressure in the wake ′ =0 equal to the pressure of the uniform main flow, the drag coefficient vanishes CD (Prandtl and Tietjens (1934)). This is not a realistic value for the drag coefficient. Obviously, the pressure at the base of the blade is lower than the free stream pressure and a drag is experienced by the blade. The flow just behind the blade is extremely 2.4. Wake behind the rotor blades 27 complex. We will therefore consider the wake at some distance from the trailing edge of the blade. U t L b la d e Figure 2.13: Rounded edge geometry used by Hoerner (1965). This geometry with ′ Lblade /t = 6 has a drag coefficient with t as reference length of CD = 0.64. Hoerner (1965) (see also Blevins (1992)) found experimentally that a blade with a rounded nose and a squared edged base, with the dimensions Lblade /t = 6 (see ′ = 0.64 for Re 4 figure 2.13) has a drag coefficient CD Lblade > 10 . This geometry is comparable to our rotor blade, except for the geometry of the trailing edge. The ratio of the thickness and the blade length of a rotor blade of turbine meter 1 is Lblade /t ≈ 20 and for turbine meter 2 the ratio is Lblade /t ≈ 8. The chamfered, sharp edge reduces the drag coefficient, because the flow will not separate immediately at the edge, which reduces the thickness of the wake. This is illustrated in figure 2.14. u n in ,r e l ro to r b la d e A f m u w w a k e C S o u t,r e l w a k e A Figure 2.14: Wake behind a rotor blade with chamfered trailing edge. The drag consist of a combination of the pressure drag and of the drag caused by skin friction. The skin friction will be calculated separately. To determine the effect of the skin friction compared to the pressure drag, the drag coefficient caused by laminar and turbulent boundary layers is now estimated by considering the rotor blade as a flat plate. For laminar boundary layers the wall shear stress can be calculated using Blasius’ numerical result (Schlichting (1979) and Appendix B). The drag coefficient, 28 2. Turbine flow meters in steady flow ′ , caused by the skin friction on both side of the blades is: CD ′ CD,f riction = 2 R Lblade τw dx x=0 1 2 2 ρU t 1.328 Lblade =p ReLblade t (2.21) For turbulent boundary layers the drag coefficient is found empirically (Schlichting, 1979) to be: −1 ′ 5 CD,f riction = 0.148 ReLblade Lblade t (2.22) For the rotor blades considered in this chapter, the contribution of the skin friction to the drag coefficient depends on the Reynolds number and whether the boundary layers are laminar or turbulent. For the range of Reynolds numbers used in the present experiments the contribution to the drag coefficient of the skin friction is typically ′ ≈ 0.05 for laminar boundary layers and C ′ ≈ 0.25 for turbulent boundary layers CD D ′ ≈ 0.03 for laminar boundary for turbine meter 1. For turbine meter 2 we find CD ′ ′ = 0.64 for layers and CD ≈ 0.18 for turbulent boundary layers. As the total drag CD the blade geometry with a blunt trailing edge (see figure 2.13 and Hoerner (1965)), we expect that the contribution of the pressure drag will be in the order of 0.5. Assuming that the wake has a thickness equal to the blade thickness, wwake = t, and that the velocity in the wake is half the mainstream velocity, uwake = 12 uin , using equation ′ = 0.5. In case 2.20, we can calculate that the rotor blade has a drag coefficient CD of the rotor blade with a chamfered trailing edge, we will also assume a wake with a velocity uwake = 12 uin . The wake thickness, wwake will be tuned in order to match ′ for a two-dimensional model of the rotor blade. The the measured values of CD experiments used to measure this drag coefficient are discussed in the next section. 2.4.1 Wind tunnel experiments In a wind tunnel with a test section of a height hwt = 0.5 m and width wwt = 0.5 m a two-dimensional wooden model of a single rotor blade is placed. The blade model has a thickness, t, of 1.8 cm, a length, Lblade of 14.6 cm and a width, wblade of 48.9 cm. It has a rounded leading edge and a chamfered trailing edge (see figure 2.15). The angle of the trailing edge is 45◦ . The blade is connected to two balances with rods and ropes. The first balance is a Mettler PW3000 with a range of 3 kg and measures the drag force, FD induced by the flow around the blade. The second one is a Mettler PJ400 with a 1.5 kg ranges and measures the lift force, FL . Both mass balances have an accuracy of 0.1 g. Measurements were carried out for Reynolds numbers, Re = uLblade /ν, based on the blade length, ranging from ReLblade = 4 × 104 up to 3 × 105 at blade angles, 2.4. Wake behind the rotor blades 29 h t a u L b la d e w w t b la d e w w t Figure 2.15: Wind tunnel set up. α from −3◦ to 3◦ . The blade angles are determined using an electronic level meter (EMC Paget Trading Ltd model: 216666). ′ = C Lblade = F / 1 ρu2 w In figure 2.16 the drag coefficient, CD D D blade t , is t 2 plotted against the Reynolds number, Re = uLblade /ν for a blade angle, α = 0.3◦ . ′ , between 0.1 to 0.35, much lower The measurements show a drag coefficient, CD ′ than CD = 0.64 found for the similar geometry with blunt trailing edge by Hoerner (1965). Figure 2.16 also shows the estimated skin friction for laminar and turbulent boundary layers for the wind tunnel model. The contribution of the skin friction to the drag coefficient is significant for turbulent boundary layers. An other consequence of the asymmetric shape of the chamfered edge of the rotor blade, is that at zero incidence, α = 0, the blade generates a lift force. This can be seen in figure 2.17. This effect has not yet been included in the theory described in this chapter, because we expect that the lift coefficient of a blade in a cascade strongly deviates from a single blade in uniform flow as presented here. In a closed wind tunnel cascade measurements are only possible at 0◦ incidence, because the walls prevent deflection of the flow. For measurements at different angles of incidence a special cascade wind tunnel should be used (Jonker, 1995). Measurements obtained for a five blade cascade with typical ratio of distance between the blades and blade length, s/Lblade , of 0.55 indicated that the measured drag coeffi′ , values are close to the value obtained for a single blade. cient, CD 2.4.2 Effect of wake on the rotation The model described above is included in the momentum equation. The reduced velocity in the wake can be described with the displacement thickness, δ1,wake , and the 30 2. Turbine flow meters in steady flow Hoerner (1965) C’D=FD/(1/2 ρ u2in,rel wblade t) 0.6 0.5 0.4 0.3 0.2 turbulent 0.1 laminar 0 0 0.5 1 1.5 Re 2 2.5 3 5 x 10 L blade ′ Figure 2.16: The drag coefficient, CD , as a function of Reynolds number, ReLblade for flat plate with round nose and 45◦ chamfered trailing edge measurements at an angle of attack α = −0.3◦ . The arrow indicates the drag coefficient of 0.64 found in Hoerner (1965), the dashed line is an approximation for the part of the drag coefficient in case of laminar boundary layers and the solid line is the approximation for turbulent boundary layer. 0.5 blade blade 0.2 0.1 0 L C =F /(1/2 ρ u2 w L 0.3 in,rel ) 0.4 L −0.1 −0.2 −0.3 −3 −2 −1 0 α (o) 1 2 3 Figure 2.17: The lift coefficient, CL , is plotted at various angles of attack, α, for Reynolds number, ReLblade > 3 105 . The dashed line is a linear fit through the data points. We observe a net lift coefficient CL′ (0◦ ) = 0.1 at a zero angle of attack, α = 0. This is due to the asymmetry in the blade profile at the trailing edge. 2.5. Friction forces 31 momentum thickness, δ2,wake (see Schlichting (1979) and Appendix B). Applying mass conservation and using the displacement thickness, we find: uout,x = 1 1− nδ1,wake 2πr cos β uin,x , (2.23) where n is the number of blades of the rotor. Using both the displacement thickness, δ1,wake , the momentum thickness, δ2,wake , and equation 2.23 in the momentum balance, the driving torque (equation 2.8) becomes: Z rtip u2in,x (tan β)r n(δ1,wake + δ2,wake ) Td = − ρ dr 2 2πr − nδ cos β rhub 1 − 2πr1,wake cos β Z rtip + 2π ρuin,x ωr3 dr . rhub In the proposed model, in which the velocity in the wake, with a wake thickness wwake , of half of the mainstream velocity, uwake = 12 uout , the displacement thickness is δ1,wake = 12 wwake and the momentum thickness is δ2,wake = 14 wwake . From the wind tunnel measurements described above, it is found that the drag caused by the wake behind the blade is overestimated by using the drag coefficient ′ = 0.64. To account for this, the thickness of the wake can in Hoerner (1965) of CD be changed. If a wake thickness is chosen equal to the blade thickness, wwake = t, the pressure drag of the blunt body is obtained. By reducing the wake thickness, the drag coefficient of the rotor blade can be reduced to the values obtained from the measurements. This will be applied in our calculations Neglecting friction forces and assuming a uniform inflow, the deviation from the ideal rotor speed caused by different drag coefficients, or different wake thickness wwake , has been calculated. For the turbine flow meters 1 and 2 the effect of wake thickness can be seen in table 2.2. In this approximation this effect is not dependent on Reynolds number. We observe a significant effect of the drag on the deviation, E, of the order of 2%. 2.5 Friction forces Although turbine flow meters are designed to rotate with minimum friction, there are several important friction forces that influence the rotation speed of the rotor. There are two different kind of friction forces, the mechanical friction force and the friction forces induced by the flow. Mechanical friction forces are the forces caused by the bearings and the magnetic pick up placed on the meter. Flow induced friction consists of the fluid drag on the blades and on the hub, the fluid friction at the tip clearance 32 2. Turbine flow meters in steady flow thickness of the wake 0 1 2t t id deviation, E = ω−ω ωid × 100% turbine meter 1 turbine meter 2 0 0 1.6 1.7 3.2 3.3 Table 2.2: The effect of the wake drag on the deviation of the rotation speed of the rotor from ideal rotation for turbine flow meter 1 and 2, where t is the blade thickness. and it includes the pressure drag due to the wake behind the blades discussed in the preceding section. To approximate the friction forces on the rotor blades and the hub, boundary layer theory has been used, neglecting centrifugal forces as well as the radial velocity. In recent years numerical studies on turbine flow meters (Von Lavante et al., 2003) show that the flow in the rotor has a complicated 3-dimensional structure invoking secondary flows. It should be realised that the theory presented here is a very simplified approximation of reality. In the following sections the effect of these forces on the deviation from ideal rotation will be investigated and discussed separately for both meters discussed in 2.2.3. 2.5.1 Boundary layer on rotor blades Boundary layers are formed on the rotor blades as a result of friction. The boundary layer thickness can be calculated using boundary layer theory and is included in the momentum equation (equation 2.2). We assume that the cascade of rotor blades can be described as row of rectangular channels with boundary layers at the top and bottom of each channel. We neglect centrifugal forces and assume that there is no radial velocity component. The rotor consists of n rectangular channels with a length of Lblade (the length of the blade) and a width of hblade (the height of the blade). The t distance between two successive blades is 2πr n − cos β . We consider two cases: the case of a laminar boundary layer and the case of a turbulent boundary layer. The displacement thickness, δ1,bl , the momentum thickness, δ2,bl , of the boundary layer formed in this channel is calculated using the Von Kármán equation (2.17). For the laminar case a third order polynomial is used to describe the velocity profile in the boundary layer. For the turbulent case a 1/7th power law approximation is used. The velocity between the blades is corrected for the displacement due to the growth of the boundary layers in the channel. Using the mass conservation for incompressible 2.5. Friction forces 33 flow, the out-going velocity component in the x-direction, uout,x , becomes: uout,x = 2πr 2πr − n(2δ1,bl +δ1,wake ) cos β uin . (2.24) Using the definition of the displacement thickness, δ1,bl , and the momentum thickness, δ2,bl , (see Appendix B), for the boundary layer thickness at the end of the channel (the trailing edge of the blade), the equation for the driving torque, Td , becomes: Td = − Z rtip rhub ρ u2in,x (tan β)r n(δ1,wake +2δ1,bl ) 2πr cos β 2 × 1− Z rtip n(δ1,wake + δ2,wake + 2(δ1,bl + δ2,bl )) 2πr − ρuin,x ωr3 dr , dr + 2π cos β rhub where δ1,wake is the displacement thickness caused by the wake and δ2,wake is the momentum thickness caused by the wake (section 2.4.2). The rotation speed of the rotor can now be found by determining iteratively at which rotational speed the total torque in the equation of motion (2.13) is zero. This is determined numerically with the secant method, a version of the Newton-Raphson method. In figure 2.18 the effect of the boundary layers on the two different types of turbine flow meters for steady incompressible flow with uniform inflow velocity and infinitesimally thin blades and without other friction forces. The laminar boundary layer causes the rotor to rotate faster, because the displacement thickness of the thicker laminar boundary layers. For the range of calculated Reynolds numbers turbulent boundary layers cause less variation in the deviation. 2.5.2 Friction force on the hub Not only is there a friction force from the boundary layers on the rotor blades, but also on the hub of the rotor a boundary layer is formed due to the rotation of the rotor. The shape of this boundary layer is complex and we will approximate this boundary layer as a boundary layer on a long flat plate of a width w = 2πrhub − nt, where rhub is the radius of the hub, n is the number of blades and t is the blade thickness. The velocity outside the boundary layer will be assumed constant for simplicity reasons and equal to the relative velocity: urel v u u =t U 1+ nt 2π cos βhub !2 + (ωrhub )2 , (2.25) 34 2. Turbine flow meters in steady flow 16 air at 1 bar, turbulent air at 1 bar, laminar natural gas at 9 bar, turbulent natural gas at 9 bar, laminar 12 id 10 8 id E = (ω − ω ) / ω * 100 (%) 14 6 4 2 0 −2 3 10 4 5 10 10 6 10 ReL blade (a) turbine meter 1 7 air at 1 bar, turbulent air at 1 bar, laminar natural gas at 9 bar, turbulent natural gas at 9 bar, laminar 5 4 id E = (ω − ω ) / ω * 100 (%) 6 id 3 2 1 0 −1 3 10 4 5 10 10 6 10 ReL blade (b) turbine meter 2 Figure 2.18: The deviation from ideal rotation caused by the boundary layers on the blades for turbulent and laminar boundary layers, assuming a uniform inlet velocity profile, versus Reynolds number, Re = U Lblade /ν. where U is the velocity at the entrance of the turbine meter corrected for the displacement due to the boundary layers (see equation 2.18). Again, we assume that there is no radial velocity and secondary flow. To determine the shear stress, τw , caused by this boundary layer, two limits are considered. The first case is the upper limit for the shear stress; the boundary layer starts at the entrance of the rotor. The flat plate has a length of cosW βhub , where W is the width of the rotor and βhub is the angle of the rotor blades with the rotor axis at the hub of the rotor (see figures 2.1 and 2.2). The second case is the lower limit; the boundary layer starts at the front end of the hub 2.5. Friction forces 35 and continues at the rotor. In this case the flat plate model of the flow has a length of Lhub + cosW βhub . The empirical expression for shear stress for a turbulent boundary in a circular pipe, equation B.20, has been found to be a good approximation for the flat plate (Schlichting, 1979). Using this equation and the equation for boundary layer thickness for turbulent flow for a flat plate, the shear stress, τw , becomes: 9 1 1 5 τw = 0.0288 ρurel ν 5 x− 5 . (2.26) The upper limit of the friction torque relative to the rotor axis due to the boundary layers on the hub of the rotor, Tf r,hub = Ff r,hub rhub sin βhub , is: Tf r,hub <wrhub sin βhub Z W cos βhub τw dx 0 1 5 W cos βhub 4 W Lhub + cos βhub 4 9 5 =0.036 ρν urel wrhub sin βhub (2.27) 5 , and the lower limit is; Tf r,hub >wrhub sin βhub Z Lhub + cosW β hub τw dx Lhub 1 5 9 5 =0.036 ρν urel wrhub sin βhub " 5 4 5 − Lhub # (2.28) . Using this in the equation of motion for the rotor (equation 2.13) and assuming steady uniform flow and infinitesimally thin blades, while neglecting all other friction forces, including the boundary layer on the rotor blades, the deviation from the ideal rotation speed is computed. The result is plotted for different Reynolds numbers in figure 2.19. As expected the friction force on the hub slows down the rotor. For turbine meter 2, the larger flow meter, the effect is relatively small (at most 0.2%), while for turbine meter 1, the effect can reach 1.5%. The ratio between the effect of the upper (equation 2.27) and the lower (equation 2.28) limits is about 1.5. 2.5.3 Tip clearance The tip of the rotor blades moves close to the pipe wall of the meter body. This imposes an additional drag force on the rotor. The force caused by the flow around the tip is complicated and depends on the size of the clearance and the Reynolds number, but also on the shape and length of the blade tip. In some meters, for example turbine meter 1, the tip is enclosed in a slot (see figure 2.20). 36 2. Turbine flow meters in steady flow 0 E = (ω − ωid) / ωid * 100 (%) −0.2 −0.4 −0.6 −0.8 −1 air at 1 bar, lower limit air at 1 bar, upper limit natural gas at 9 bar, lower limit natural gas at 9 bar, upper limit −1.2 −1.4 3 10 4 5 10 10 6 10 Re L blade (a) turbine meter 1 −0.02 −0.06 −0.08 id E= (ω − ω ) / ω * 100 (%) −0.04 id −0.1 −0.12 −0.14 air at 1 bar, lower limit air at 1 bar, upper limit natural gas at 9 bar, lower limit natural gas at 9 bar, upper limit −0.16 −0.18 3 10 4 5 10 10 6 10 ReL blade (b) turbine meter 2 Figure 2.19: The upper an lower limits of the deviations from ideal flow caused by the boundary layers on the hub, assuming a uniform inlet velocity profile, as a function of the Reynolds number, Re = U Lblade /ν. Thompson and Grey (1970) suggested that the tip clearance drag can be considered to be similar to the drag in a journal bearing. This results in friction torque caused by the tip clearance, Ttc of Ttc = 0.078 ρu2rel,rtip rtip Lblade tn . 2 Re0.43 tip (2.29) Here the Reynolds number is defined as Retip = urel,rtip (rout − rtip )/ν, with rout the radius of the pipe wall of the turbine flow meter. 2.5. Friction forces 37 p ip e p ip e ro to r b la d e ro to r b la d e (a ) (b ) Figure 2.20: (a) shows an enclosed blade tip, (b) shows blade tip not enclosed However, if vortex shedding occurs at the front edge of the tip and the blade length is relatively small, the flow is no longer comparable to the flow in a journal bearing. This makes it very difficult to give a reasonable prediction of the tip clearance drag. Therefore, no theory for the tip clearance is incorporated in the present calculations. However, to understand the effect caused by tip friction calibration data was obtained by Elster-Instromet for three turbine meters with identical geometries except for the height of the tip clearance. These Reynolds curves and the calculated Reynolds number dependence of the deviation are discussed in section 2.6.3. 2.5.4 Mechanical friction Turbine flow meters are designed to minimize the mechanical friction. However, mechanical friction will always be present. The main part of the mechanical friction is caused by the bearing friction of the rotor. For a small meter the magnetic pick up and the index counter can cause some additional friction, for larger meters this part can be neglected. The mechanical friction torque is assumed to be constant, independent of the rotation speed, ω. Experiments were done for turbine meter 1 to determine the mechanical friction. Two different approaches were used to determine the mechanical friction torque: dynamic and static friction measurements. These measurements are discussed in more detail in section 4.2.3. The friction torque found in the experiments are of the same order of magnitude as the data provided by the manufacturer. For turbine meter 1 a mechanical friction torque of Tmech = 5.6×10−6 N m is assumed. For turbine meter 2 a torque of Tmech = 5.5×10−5 N m is assumed based on the data of Elster-Instromet. Again, the equation of motion for the rotor was solved for steady uniform flow with a rotor with infinitesimally thin blades and no other friction forces than the mechanical friction torque. The deviation is shown in figure 2.21. The deviation caused by the mechanical friction does not depend on Reynolds number, but depends on the volume flow and density, causing different behaviour at a 38 2. Turbine flow meters in steady flow 0 −2 id −3 −4 id E = (ω − ω ) / ω * 100 (%) −1 −5 −6 −7 air at 1 bar natural gas at 9 bar −8 −9 3 10 4 5 10 10 6 10 ReL blade (a) turbine meter 1 −2 id −4 −6 id E = (ω − ω ) / ω * 100 (%) 0 −8 −10 −12 −14 3 10 air at 1 bar natural gas at 9 bar 4 5 10 10 6 10 Re L blade (b) turbine meter 2 Figure 2.21: The deviations from ideal rotation caused by the mechanical friction assuming a uniform inlet velocity profile as a function of the Reynolds number, Re = U Lblade /ν. certain Reynolds number for air at 1 bar and natural gas at 9 bar. We see from figure 2.21 that the mechanical friction is only important at low flow velocities. 2.6 Prediction of the Reynolds number dependence in steady flow To evaluate the model described above, the results of the model including all friction forces discussed in the previous sections and assuming that the flow entering the rotor is non-uniform, is compared to the calibration measurements of the two turbine 2.6. Prediction of the Reynolds number dependence in steady flow 39 meters; turbine meter 1 (Instromet SM-RI-X G250) and turbine meter 2 (more information about the meters can be found in table 2.1). The measured data is obtained from the manufacturer of the turbine meter Elster-Instromet. The results of the calculations shown in the figures below are found assuming an inlet velocity profile based on boundary layer theory with turbulent boundary layers. For the friction on the hub the upper limit described in section 2.5.2 is used, while tip friction is ignored. For the drag caused by either laminar or turbulent boundary layers the predictions for the rotor blades are shown in the figures presented below. In the figures presented below the prediction for both laminar as well as for turbulent boundary layers on the rotor blade are shown. 2.6.1 Turbine meter 1 Typical calibration data is obtained for air at 1 bar (atmospheric pressure) and natural gas at 9 bar for turbine meter 1 (see figure 2.22). The measured data of this turbine meter is compared to the calculated deviation, E, for different Reynolds numbers, ReLblade , using the properties of air at 1 bar and of natural gas at 9 bar and the dimensions given in table 2.1. Because it is not known whether the boundary layers on the blades are either turbulent or laminar, both situations are plotted in figure 2.22. The effect of the wake is calculated for a ′ = 0.5, based on the experiments of Hoerner (1965) as pressure drag coefficient of CD proposed in section 2.4, for a wake of thickness wwake = 21 t (this leads to a pressure ′ = 0.25) and for no wake (C ′ = 0). drag coefficient of CD D ′ = 0.5 predicts measurement errors, E, almost 4 The pressure drag coefficient CD times lower than the measured data. To be able to obtain data similar to the measured data an unrealistic high drag coefficient would be needed. It is unrealistic that the model shows better agreement with the measured data using a drag coefficient much higher than that of the blunt object measured by Hoerner (1965). Other reasons for the observed high values of E have to be found. Deviation between theory and experiment could for example be due to the blade experiencing a nonzero lift coefficient at a zero angle of attack (see section 2.4.1 and figure 2.17). This additional lift would result in a higher rotor speed and accordingly a higher error, E. Another effect that can account for the deviation between theory and experiment is the inflow velocity profile. The measured inlet velocity profiles described in section 2.3.2 are used to calculate the deviation between actual rotation speed and ideal rotation speed. In figure 2.23 the deviation from ideal flow is plotted using the measured inlet velocity profile in the calculations. The calculations have been carried out for both turbulent and laminar boundary layers on the rotor blades and a pressure drag coefficient ′ = 0.5. The actual velocity profile in the turbine meter at the inlet of the rotor CD induces a larger rotation speed, however, it cannot explain the difference between the 40 2. Turbine flow meters in steady flow 30 air at 1 bar, turbulent air at 1 bar, laminar natural gas at 9 bar, turbulent natural gas at 9 bar, laminar 20 15 id E = (ω − ω ) / ω * 100 (%) 25 measurements id 10 5 C’ =0.5 C’DD=0.25 C’D=0 0 −5 −10 3 10 4 5 10 10 6 10 ReL blade Figure 2.22: The deviations from rotation versus Reynolds number, Re = U Lblade /ν for turbine meter 1 in air at 1 bar and natural gas at 9 bar. The measured data for air at 1 bar are indicated by circles • , the solid symbols represents the data for a meter with standard blades, the open circles are for a blade with a chamfered leading and trailing edge. The data indicated by triangles H are for natural gas at 9 bar. The lines represent the calculated data, the results of the calculations are for turbulent boundary layers as well as laminar boundary layers on the rotor blades at three different pressure drag coefficients (solid, dashed and dotted lines). theory and calibration data by itself. The standard rotor blade has a rounded leading edge and a chamfered trailing as shown in figure 2.24(a). For a calibration measurement with air at 1 bar the standard rotor was replaced by a rotor with blades, where the leading and the trailing edge are both chamfered (figure 2.24(b)). These results are plotted in figure 2.22 as the open dots. By changing the shapes of the blades, the shape of the error curve as function of Reynolds changes. In the present case the effect of the shape of the leading edge of the rotor blade is quite small. To compare the shape of the measured curve with the shape predicted by the model, we shifted the calculated data by 12.7% (see figure 2.25). The small deviations caused by the different blade profile cannot be explained using this global model. The measured data resemble the prediction of the model with turbulent boundary layers on the rotor blades much more than the model with laminar boundary 2.6. Prediction of the Reynolds number dependence in steady flow 41 30 air at 1 bar, turbulent air at 1 bar, laminar natural gas at 9 bar, turbulent natural gas at 9 bar, laminar 20 15 id E = (ω − ω ) / ω * 100 (%) 25 id 10 5 C’ =0.5 D 0 −5 −10 3 10 4 5 10 10 6 10 Re L blade Figure 2.23: The deviations from ideal rotation versus Reynolds number, Re = U Lblade /ν for turbine meter 1 in air at 1 bar and natural gas at 9 bar. The symbols indicate the results of the calculations using the measured velocity profile for laminar boundary layers () and turbulent boundary layers (♦) on the rotor blades for ′ CD = 0.5. The lines represent the calculated data using a turbulent boundary layer model for the inlet velocity profile. The calculations are performed for turbulent boundary layers as well as laminar boundary layers on the rotor blades ′ at CD = 0.5 (solid, dashed, dotted and dashed-dotted lines). The measured data for air at 1 bar are indicated by circles •. The data indicated by triangles H are for natural gas at 9 bar. (a) round leading edge (b) chamfered leading edge Figure 2.24: A schematic drawing of the rotor (a) with rounded leading edge and (b) with a chamfered leading edge used in the measurements 42 2. Turbine flow meters in steady flow 24 20 18 16 14 id id E = (ω − ω ) / ω * 100 (%) 22 12 10 8 6 4 3 10 4 5 10 10 6 10 Re L blade Figure 2.25: The deviations from ideal rotation versus Reynolds number, Re = U Lblade /ν for turbine meter 1 in air at 1 bar and natural gas at 9 bar. The calculated data is shifted upwards by 12.7%. The measured data for air at 1 bar are indicated by circles • , the solid symbols represents the data for a meter with standard blades, the open circles are for rotor blades with a chamfered leading edge. The data indicated by a triangle H have been measured in natural gas at 9 bar. The lines represent the calculated data, the calculations are performed for turbulent boundary layers (dotted: air at 1 bar; dash-dot: gas at 9 bar) as well as laminar boundary layers (solid: air at 1 bar; dashed: gas at 9 bar) on the rotor blades. layers over the entire range of Reynolds numbers considered. For the inlet velocity profile a turbulent boundary model was used. Figure 2.23 shows that the measured velocity profile changes the shape of this curve considerably. We conclude that the manufacturer has used modifications of blade tip geometry (figure 2.20) in order to compensate for this Reynolds number dependence of the main flow velocity profile. 2.6.2 Turbine meter 2 The calibration measurements of turbine meter 2 were compared to the results of the calculations for the model assuming either turbulent or laminar boundary layers ′ = 0.5, on the blade. Again, this is done at different pressure drag coefficients, CD 0.25 and 0. The results are shown in figure 2.26. Again, a drag pressure coefficient 2.6. Prediction of the Reynolds number dependence in steady flow 43 30 air at 1 bar, turbulent air at 1 bar, laminar natural gas at 9 bar, turbulent natural gas at 9 bar, laminar 20 15 id E = (ω − ω ) / ω * 100 (%) 25 10 id measurements C’D=0.5 C’ =0.25 C’D=0 5 0 D −5 −10 3 10 4 5 10 10 6 10 ReL blade Figure 2.26: The deviations from ideal rotation versus Reynolds number, Re = U Lblade /ν for turbine meter 2 in air at 1 bar and natural gas at 9 bar. The measured data for air at 1 bar are indicated by circles • and for natural gas at 9 bar by triangles H. The lines represent the calculated data, the calculations are carried out for turbulent boundary layers as well as laminar boundary layers on the rotor blades at three different drag coefficients (solid, dashed and dotted lines). ′ = 0.5 results in deviations from ideal rotation considerably lower than the of CD measured data, as was the case for turbine meter 1. The deviation between the theory and the experiments could be caused by the nonzero lift coefficient at a zero angle of attack (figure 2.17). This additional lift can cause a higher error, E. A comparison between the shape of the measured curve and the calculated curve is made by adding 6.5% over the whole Reynolds range (see figure 2.27). The deviation measured for turbine meter 2 is in general lower than the deviation obtained for turbine meter 1. It is possible that it is the influence of not having full fluid guidance at the tip of the rotor. As explained in section 2.2.3 a deviation from full fluid guidance reduces the rotation speed and the error, E. The shape of the curve predicted by the model agrees very well with the measured data assuming laminar boundary layers for the air set up (ReLblade < 105 ) and turbulent boundary layers for the natural gas set up (ReLblade > 104 ). This could explain the gap between the two different measurement conditions at the same Reynolds numbers. Since the measurement have been done for two different set ups it is possible that the different conditions in the set up can 44 2. Turbine flow meters in steady flow 16 E = (ω − ωid) / ωid * 100 (%) 14 12 10 8 6 4 2 0 −2 3 10 4 5 10 10 6 10 Re L blade Figure 2.27: The deviations from ideal rotation versus Reynolds number, Re = U Lblade /ν for turbine meter 1 in air at 1 bar and natural gas at 9 bar. The calculated data is shifted upwards by adding 6.5%. The measured data for air at 1 bar are indicated by means of circles •. The data indicated by means of triangles H are for natural gas at 9 bar. The line represents the calculated data, the calculations are carried out for turbulent boundary layers (dotted: air at 1 bar; dash-dot: gas at 9 bar) as well as laminar boundary layers (solid: air at 1 bar; dashed: gas at 9 bar) on the rotor blades. force the boundary layers to be either turbulent or laminar. No measurements were done to ascertain the velocity profile at the inlet of the rotor for turbine meter 2. We cannot determine more accurately the influence of the inflow velocity profile in this case. However, we expect this would influence the deviation, E. Comparison between turbine meter 1 with ”enclosed” blad tips and turbine meter 2 with normal blade tip clearance (figure 2.20) clearly illustrates the importance of the tip clearance on the response of the flow meter. The Reynolds number dependence of the measurement error, E, is much stronger for turbine meter 2 than for turbine meter 1. 2.6. Prediction of the Reynolds number dependence in steady flow diameter, D (m) blade thickness, t (mm) number of blades, n rhub /D rout /D S/D W/D hblade /D = (rtip − rhub )/D Lhub /D 45 0.280 3.0 24 0.359 0.500 2.707 0.136 0.138, 0.137, 0.136 1.532 Table 2.3: Dimensions of the turbine flow meters 3, 4 and 5 with different height of the tip clearance, where rhub is the radius of the hub, rout is the radius of the outer wall, rtip is the radius at the tip of the blades, S is the pitch, W is the width of the rotor, hblade is the height of the blade and Lhub is the length of the hub in front of the rotor. Except for the blade thickness t and the number of blades n, all values are made dimensionless with the diameter, D. 2.6.3 Effect of tip clearance As mentioned in section 2.5.3 some extra attention is given to the effect of tip clearance. Because no satisfactory model has been found to describe the friction caused by the flow through the gap, this effect is not included in our model. To estimate the effect of different tip clearance, our model is compared to a series of turbine meters with the same geometry, but with different height of the tip clearance, htc = rout − rtip = 1, 2, 4 mm (see figure 2.28). As for turbine meter 2 the tip is not enclosed in a wall cavity (figure 2.20). The dimensions of the meter are given in tabel 2.3. p ip e r o u t r tip h tc ro to r b la d e Figure 2.28: Tip clearance of a blade that is not enclosed. 46 2. Turbine flow meters in steady flow The calibration data for all three turbine meters (turbine meter 3, 4 and 5) are plotted in figure 2.29. The measurements show a decrease in E for increasing tip clearance height, htc . The theoretical results are shown in figure 2.30, assuming an inlet velocity profile ′ = 0.5 based on turbulent boundary layers, the drag coefficient is taken to be CD and using the upper limit of the hub friction. Although there is no model for tip clearance drag, there is still an effect, because the height of the blade decreases for increasing gap height, htc . For larger blade heights less of the blade is rotating in the boundary layer at the outer wall of the pipe, this causes the increase of rotation speed of the rotor for larger tip clearance as can be seen in figure 2.30. If the effect of tip clearance drag suggested by Thompson and Grey (1970), equation 2.29, is taken into account, the results do not change significantly. Because the mechanical friction of this particular turbine meter has not been measured, it was assumed to be equal to that of turbine 2. It is however clear from the figures that in reality the mechanical friction is larger than the assumed mechanical friction, therefore the calculated deviation, E, decreases considerably slower for small Reynold numbers than in the measured data. It is clear that the theoretical model described in this chapter is not able to describe the effect of tip clearance drag correctly. The results of the experiments shown in figure 2.29 display exactly the opposite behaviour of the results obtained with our model shown in figure 2.30. As the tip gap increases our model predicts an increasing rotational speed, while the experiments indicate a reduction of rotational speed. 2.7 Conclusions To predict the influence of modifications of the geometry of a turbine meter and of changes in fluid properties, a theoretical model of the behaviour of the turbine flow meter in steady flow is necessary. The flow in a turbine meter is 3-dimensional and very complicated. We consider a very simplified model. The model presented in this chapter makes it possible to calculate for a chosen geometry the Reynolds number dependence of the deviation between the rotor response and that of an ideal rotor. We found that the shape of the velocity profile is important. More accurate measurements should be carried out to provide a better understanding of this effect. In the proposed model we included the effect of the friction due to skin friction and a wake displacement effect. The wake thickness can be tuned to match the drag coefficient measured for two-dimensional models of rotor blades in a wind tunnel. However, a comparison with measured data shows that it is not possible to account for the high rotation speed of the rotor compared to ideal rotation by modifying this wake thickness only. A possible explanation is that the flow also generates a lift force on the rotor blades at zero angle of attack (figure 2.17). The lift causes the rotor to rotate faster. A lift force on a single rotor was measured in wind tunnel experiments. 2.7. Conclusions 47 18 E = (ω − ωid) / ωid * 100 (%) 16 14 12 10 8 tc=1 mm; air at 1 bar tc=1 mm; natural gas at 9 bar tc=2 mm; air at 1 bar tc=2 mm; natural gas at 9 bar tc=4 mm; air at 1 bar tc=4 mm; natural gas at 9 bar 6 4 2 0 3 10 4 10 5 10 Re 6 10 7 10 L blade Figure 2.29: The measurement deviation from ideal rotation for turbine meters with tip clearance height of 1, 2 and 4 mm versus Reynolds number. 18 air at 1 bar, turbulent air at 1 bar, laminar natural gas at 9 bar, turbulent natural gas at 9 bar, laminar E = (ω − ωid) / ωid * 100 (%) 16 14 12 10 8 6 4 2 0 3 10 4 10 5 10 ReL 6 10 7 10 blade Figure 2.30: Calculated deviation of ideal rotation for turbine meters with tip clearance height, htc = 1 mm (solid line), htc = 2 mm (dashed line) and htc = 4 mm (dotted line). Calculation are for laminar as well as turbulent boundary layers on the rotor blades and the solid symbols represent the data using the equation of tip clearance drag suggested by Thompson and Grey (1970), equation 2.29. ′ The drag coefficient used is CD = 0.5. 48 2. Turbine flow meters in steady flow Measurements on a cascade are needed to determine this lift force for a turbine meter type of rotor. The present model gives an adequate prediction of the shape of the Reynolds number dependency of the rotor response. However, the effect of changing the shape of the rotor blades and the effect of the tip clearance cannot be explained. Experiments clearly demonstrate that the flow around the blade tip has a strong influence on the Reynolds number dependence of the response of the rotor. 3 Response of the turbine flow meter on pulsations with main flow 3.1 Introduction Gas turbine flow meters can reach high accuracy, generally of the order of 0.2%. This accuracy can only be attained for optimal flow conditions. Acoustic perturbations can induce significant systematic errors. A theoretical prediction of the error would allow a correction in the volume flow measurement. In recent years research has been carried out to determine the pulsation error during the measurement and correct for this error instantaneously. Atkinson (1992) developed a software tool to solve the equation of motion of the rotor (equation 3.5) and used the magnetic pickup registering the passing of a rotor blade to calculate the real volume flow. This method can only be used if the amplitude of the pulsations can still be detected in the turbine signal. As the amplitude of the pulsations in the turbine meter signal decreases rapidly with increasing frequency, it is difficult to predict the real flow for high frequency pulsations. Another tool was developed described by Cheesewright et al. (1996), called the ’Watchdog System’. This system also uses the equation of motion of the rotor (equation 3.5), but now an accelerometer is used to measure the flow noise of a valve or bend. Watchdog is designed for pulsation frequencies less than 2 Hz. We actually focus on the behaviour of the rotor at high frequencies for which the rotor inertia has integrated fluctuations in rotor speed. The errors we consider are due to non-linearities. Assuming quasi-steady, incompressible flow and neglecting friction forces a relationship can be found between the velocity pulsations and the measurement error (Dijstelbergen, 1966). Experiments have been carried out in the past by Lee et al. 50 3. Response of the turbine flow meter on pulsations with main flow (2004), Jungowski and Weiss (1996), Cheesewright et al. (1996) and McKee (1992). These experiments indicated that this basic theory can be used for low frequency pulsations. To explore the limits of the validity of this theory, a set up was build at the Eindhoven University of Technology. In our experiments care was taken to determine accurately the amplitude of the velocity fluctuations at the rotor. This was found to be a limitation in the experiments reported in literature. With the more accurate determination of the acoustic field it is possible to detect small deviations from the basic theory. With this set up it was possible to measure the influence on the flow meter response of the acoustic perturbation with velocity amplitudes from 2% of the main flow velocity up to twice the main flow velocity for frequencies from a few Hz up to 730 Hz. In this chapter the basic theory is discussed, after this the set up is described and the experimental procedures are discussed. We then show the results of the measurements and discuss these results. 3.2 3.2.1 Theoretical modelling A basic quasi-steady model: A 2-dimensional quasi-steady model for a rotor with infinitesimally thin blades in incompressible flow If the rotor is modelled as a 2-dimensional cascade of infinitesimally thin blades in an incompressible, frictionless, steady flow, the integral mass and momentum equation applied to a fixed control surface CS with an outer normal n (equations 2.1 and 2.2) reduce to ZZ v · ndS = 0 , ρ0 (3.1) CS ZZ (3.2) ρ0 v (v · n) dS = F bf , CS where ρ0 is the fluid density, v is the velocity vector and F bf are the forces imposed on the fluid by the turbine blades. The control volume, CS, is chosen as shown in figure 3.1 and we assume that there is no pressure drop over the cascade. Because infinitesimally thin blades and frictionless flow are assumed, the surface area of inflow is equal to the surface area of outflow. It follows from equation 3.1 that the axial component of the incoming velocity, uin , is equal to the outgoing velocity, uout,x ; uin = uout,x . It is assumed that the flow enters the rotor without any angular momentum and that the flow leaves the rotor with a velocity aligned with the blades (see fig. 3.1). This is a realistic assumption if the chord length of the blades, Lblade , is large compared to the distances between the blades, s, i.e. for cascades the ratio, 3.2. Theoretical modelling 51 w r W b u u in u re l b w r u o u t,x = o u t,y in n re l t n u u L C S b la d e y x Figure 3.1: Two-dimensional representation of the flow entering and leaving the rotor modelled as a cascade s/Lblade , should be smaller than 0.7 (Weinig, 1964). By considering the flow in a reference frame attached to the rotor, this implies that : tan β = ωr + uout,y . uout,x (3.3) We assume further that there is no swirl in the incoming flow, so that uin,y = 0 and that the inflow is uniform. The momentum equation in the y-direction becomes ρ0 Auin (uin tan βav − ωR) = Fbf,y , (3.4) where ω is the angular rotation velocity of the rotor, A is the cross-sectional area of the rotor, βav is the average blade angle and R is the root-mean-square q radius of the inner and outer radius of the meter, rin and rout respectively, i.e. R = 2 +r 2 ) (rin out . 2 52 3. Response of the turbine flow meter on pulsations with main flow The force exerted on the fluid by the blade, Fbf,y , is equal and opposite to the force exerted by the fluid on the blade, Ff b,y . The fluid induces a torque on the rotor, Tf b = Ff b,y R, accelerating the rotor. Using the equation of motion of the rotor, we get: dω = ρ0 Auin (uin tan βav − ωR)R − Tf , (3.5) dt where Irotor is the moment of inertia of the rotor and Tf is torque on the rotor caused by the friction forces. We assume periodic pulsations u′in around an average velocity ūin so that uin = T ūin + u′in . We neglect the friction torque, ρu2 fR3 ≪ 1. We assume that the rotation in of the rotor is constant in spite of the pulsations. The integration time of the rotor is much longer than the period of the imposed acoustic pulsations. If the torque is averaged over one period, equation 3.5 reduces to: Z 1 T ρ0 (ūin + u′in )A (ūin + u′in ) tan βav − ωR Rdt = T 0 i h ) tan β − ū ωR = 0, (3.6) ρ0 AR (ū2in + u′2 av in in Irotor with T is the period of the pulsations. From this equation the angular rotation velocity, ω̄, caused by pulsating flow is obtained ! u′2 ūin tan βav in 1+ 2 ω̄ = . (3.7) R ūin βav for the ideal angular rotation velocity Using equation 1.3, i.e. ωid = ūin tan R without pulsation, the error caused by periodic pulsations becomes: ! u′2 ω̄ − ωid in . (3.8) (Epuls )id = = ωid ū2in This means that sinusoidal pulsations, uin = ūin + |u′in | sin(ωt), induce a systematic error of 1 |u′in | 2 (Epuls )id = . (3.9) 2 ūin 3.2.2 Practical definition of pulsation error In the previous section we defined a deviation (Epuls )id between angular velocity, ω̄, for steady rotation and the ideal angular rotation velocity, ωid , in absence of pulsations: ω̄ − ωid (Epuls )id = , (3.10) ωid 3.3. Experimental set up 53 βav . In experiments we use the steady angular velocity ω0 as where ωid = ūin tan R reference instead of ωid , hence: Epuls = ω̄ − ω0 . ω0 (3.11) In order to illustrate the difference between this ideal pulsation error, (Epuls )id , and the definition of the pulsation error used in the experiments, Epuls , we consider the influence of a constant mechanical friction torque, T̄mech on an ideal rotor. Using equation 3.6 we find in absence of pulsations: ω0 = ωid − T̄mech , ρ0 AR2 ūin (3.12) while due to pulsations we would reach a steady rotation of angular velocity: ! u′2 ūin tan β T̄mech 1 + 2in − . (3.13) ω̄ = R ρ0 AR2 ūin ūin Hence, we would predict a pulsation error, (Epuls )exp : (Epuls )exp = ωid ω0 u′2 in , ū2in (3.14) u′2 corresponding to (Epuls )id = ū2in multiplied by a factor ωωid0 . in Other reasons for a deviation between the ideal pulsation error, (Epuls )id , and the measured pulsation error, Epuls defined by equation 3.11, is the unsteadiness of the flow at high Strouhal numbers. Our aim is to provide quantitative information about this Strouhal number dependence. 3.3 Experimental set up A dedicated set up has been built at Eindhoven University of Technology to study the influence of pulsations on gas turbine meters. A high pressure reservoir with dry air at 60 bar (dew point -40◦ C) is connected to a test pipe of 0.10 m diameter, and a length of 3.2 m. At the open end of this pipe a turbine flow meter (Instromet type SM-RI-X G250) is placed. The flow through the turbine flow meter is controlled by means of a valve placed at the upstream end of the test pipe. By adjusting this valve, the critical pressure at the valve is reached, resulting in a velocity, u∗ , at the valve equal to the local speed of sound, c∗ . We have a Mach number of unity, M = u∗ /c∗ = 1 and a so-called ”choked” flow. This provides a constant mass flow, independent of 54 3. Response of the turbine flow meter on pulsations with main flow A a irre s e rv o ir c h o k e d v a lv e C F s ire n B p 1 p 2 p 3 p 4 p 5 p 6 D te m p e ra tu re p 7 p 8 G p u ls e sh a p e r 8 x h o tw ire m e a su re m e n t h w 1 h w 2 E s ig n a l g e n e ra to r trig g e r (8 ) (4 ) S & H + filte r Figure 3.2: Experimental set up: A high pressure reservoir of dry air (A) is connected with a pipe (B) to a turbine meter (D). The flow is being controlled by an adjustable valve (C) creating choked flow with constant mass flow. Pulsations can be induced by a loudspeaker (E) or a siren (F). The pulsations are measured with six pressure transducers (p1,p2,p3,p4,p5 and p6) along in the pipe (B) and two pressure transducers (p7 and p8) placed within the turbine meter (D). Velocity pulsations can be measured with two hot wires (hw1 and hw2) placed within the turbine meter (D). The rotation of the rotor of the turbine meter is being measured by a probe detecting the passing of a rotor blade (G). perturbations in the flow downstream of the valve. The conditions of the reservoir, p0 and T0 , and the valve opening determine the mass flow. Pulsations in the test pipe downstream of the choked valve can be induced by using a loudspeaker placed at the downstream open end of the set up or by means of a siren placed downstream of the valve. The loudspeaker (SP-250P) is controlled using a signal generator (Yokogawa FG120) driving a power amplifier (AIM WPA 301A). The siren is described by Peters (1993). The siren has a frequency range from a 10 Hz up to 1000 Hz. A bypass allows variations in the ratio, uac /u0 , of acoustic velocity, uac , and the main flow velocity, u0 . The siren is a much more efficient sound source than the loudspeaker, by tuning it to the resonance frequencies of the set up, the ratio of acoustic to main flow velocities, uac /u0 , can reach values up to 2. Between the siren and the valve a volume is placed, a pipe with a length 1.22 m and a internal diameter of 0.21 m. Except for the core of the pipe with a diameter of 0.05 m, this pipe is filled with porous material (Achiobouw acoustic foam D80). To avoid chocking at the siren, the opening of the bypass was increased, while the siren 3.3. Experimental set up pressure transducer in pipe 1 2 3 4 5 6 pressure transducer in turbine meter 7 8 55 distance -1.566 m -1.212 m -0.960 m -0.400 m -0.265 m -0.205 m distance 0.07775 m 0.1075 m Table 3.1: Position of the pressure transducer placed in the set up. The distances are measured from the upstream end of the turbine meter, where the positive direction is the flow direction. was turned off, up to the point at which changes in volume flow could no longer be observed. Only measurements using a significantly larger opening of the bypass than this critical point are performed. Goog agreement between the measurements using the loudspeaker and using the loudspeaker confirm that there was no chocking. The acoustic pressure in the set up is measured by means of eight piezo-electric gauges placed flush at the pipe wall. Six pressure transducers (three Kistler type 7031 and three PCB type 116A) are placed in the pipe upstream of the turbine meter each at randomly chosen distances (see table 3.1). Two other pressure transducers (PCB type 116A) are placed within the turbine meter, 0.010 m and 0.040 m upstream of the rotor of the turbine meter (see figure 3.3). The signals from the pressure transducers are amplified using charge amplifiers (Kistler type 5011). They are acquired by means of a PC using an 8 channel Sample and Hold module (National Instrument SCXI 1180) and a DAC card (PCI MIO-16E-I) controlled by LabView software. The pressure transducer and charge amplifier combinations are calibrated in a different set up. In this calibration set up the transducer is placed next to a reference microphone flush in a closed end wall of a 1.0 m long pipe (diameter 0.07 m). Plane waves are generated by a loudspeaker placed at the opposite end of the pipe. All pressure transducers are calibrated against the reference pressure transducer for frequencies between 24 Hz and 730 Hz, i.e. the frequencies used in our experiments. The acoustic velocity of the pulsations can also be measured using two hot wire anemometers (Dantec type 55P11 wire diameter 5 µm with 55H20 support) placed 0.010 m upstream of the turbine meter (see figure 3.3). Accurate measurements of the amplitude of the velocity pulsations are only possible if the ratio between the acoustic velocity amplitude and 56 3. Response of the turbine flow meter on pulsations with main flow p re s s u re tra n s d u c e r h o tw ire a n e m o m e te rs flo w s tra ig h tn e r p re s s u re tra n s d u c e rs ro to r h o tw ire a n e m o m e te r (a) (b) Figure 3.3: The placement of the pressure transducers in the turbine meter. (a) shows a schematic, simplified drawing of a cross-section of the turbine flow meter and (b) shows a photograph of the turbine meter. One pressure transducer and two hot wires are placed at the same distance upstream from the rotor (1 cm), equally distributed around the perimeter of the meter. the main flow velocity is small enough to avoid flow reversal, uac /u0 < 1. The hot wire makes no distinction between forward and reversed flow. The signals of the hot wire anemometers are processed with a constant-temperature anemometer module (Streamline 90n10) in combination with dedicated Dantec application software. The anemometer can follow velocity fluctuations up to 50 kHz. The signals are recorded on a PC in the same way as the signals of the pressure transducers. The hot wire anemometers are calibrated against a Betz water micromanometer (± 1 P a) in a separate free-jet set up in the velocity range 2 m/s to 40 m/s. The output is fitted using a power law description. This results in accuracies of about 1% for velocities above 8 m/s and 5% for velocities from 2 to 8 m/s. The time-averaged volume flow can be measured using the turbine flow meter (Instromet type SM-RI-X G250), using the calibration data provided by Elster-Instromet for normal flow conditions. In the absence of pulsations the volume flow measured has an accuracy of 0.2 % in the range of 6 × 10−3 m3 /s to 1 × 10−1 m3 /s. The rotation of the rotor of the turbine meter is detected by means of a so-called ”reprox probe”, a magnetic pickup generating an inductive pulse when a rotor blade passes the probe. These pulses are converted to electronic pulses and then modified into proper TTL pulses by means of the signal generator. The time interval between the TTL pulses is registered using a counter board (PCI 6250 NI), inserted in a PC, with an accuracy of 50 ns. These intervals are converted to the rotation period of the rotor by multiplying by the number of rotor blades (n=16). Due to small differences in blade geometry the measured rotor speed is not constant during a rotation. An 3.4. Determination of the amplitude of the velocity pulsations at the location of the rotor 57 average rotor speed is calculated for each rotation. 3.4 Determination of the amplitude of the velocity pulsations at the location of the rotor When the loudspeaker or the siren is turned on, velocity pulsations are generated. The velocity in the set up can be described by the average main flow, ūin , and a periodic fluctuating part, u′in . To investigate the effect of the acoustic perturbations on the flow measurements of the turbine meter, it is necessary to determine the velocity pulsations at the position of the rotor. It is impossible to measure the velocity pulsations exactly at the rotor. The measured data has to be extrapolated to the rotor position. By using the measured pressure fluctuations obtained by the microphones, the acoustic velocity at the rotor is determined by using an acoustic model. 3.4.1 Acoustic model p ip e p tu rb in e flo w m e te r p 1 - 1 + p p 2 - 2 p + p 3 3 + - Figure 3.4: A schematic illustration of the acoustic model used to determine the amplitude of the velocity pulsations at the position of the rotor. For the acoustic model the test pipe including the turbine meter is divided into three parts with different cross-sectional areas (see figure 3.4). The first part is the pipe leading to the turbine meter with a diameter D = 0.10 m. The pipe has a crosssectional area of 8.4 × 10−3 m2 . As can be seen in figure 3.3 the core of the turbine meter has a complicated shape around the rotor. In the acoustical model, the turbine meter will be described as two cylindrical parts changing abruptly in cross-sectional areas. The first part of the turbine meter is the front part of the flow straightener and has a length of 0.037 m and a cross-sectional area of 7.3 10−3 m2 . The second part is the main part of the turbine meter and has a smaller cross-sectional area of 3.8 10−3 58 3. Response of the turbine flow meter on pulsations with main flow m2 . It is assumed that the acoustic field can be described within each segment as plane waves for frequencies up to the critical frequency of the pipe, fc = c0 /(2D) ≈ 1.7 kHz. Harmonic plane waves are described by the d’Alembert solution of the one-dimensional equation: + − i(2πf t−kj x) i(2πf t+kj p′j (x, t) = p̂j (x)ei2πf t = p+ + p− j e j e x) , j = 1, 2, 3(3.15) with p̂j the complex amplitude and f the radial frequency. In this case of an uniform main flow, kj+ and kj− are the complex wave numbers of the waves travelling in positive and negative direction, respectively. The wave numbers are defined as: kj+ = 2πf /c0 + (1 − i)αd , 1+M kj− = 2πf /c0 + (1 − i)αd . 1−M (3.16) The imaginary part of the wave number represents the damping coefficient caused by viscous-thermal effects. In a quiescent flow in smooth cylindrical pipes, damping of plane waves by viscous-thermal damping can be described by a damping coefficient, αd (Kirchhoff (1868), Tijdeman (1975), Pierce (1989)) γ−1 1 p νπf 1 + √ αd = , (3.17) r j c0 Pr where rj is the radius of pipe segment j, ν is the kinematic viscosity, γ is the Poisson’s ratio and Pr is the Prandtl number. For air at room temperature and atmospheric pressure the following values are used: ν = 1.5×10−5 m2 /s, γ = 1.4 and Pr = 0.72. Although this damping coefficient is deduced for quiescent flow, it provides a good approximation of the effect of qdamping for frequencies such that the acoustical visν + cous boundary layer, δ = πf , is thinner than the viscous sublayer 10δ with q δ + = ν τρw (Peters (1993), Ronneberger and Ahrens (1977) and Allam and Åbom (2006)). If we use the δ + for smooth cylindrical pipes, we find that this is valid for our experiments. At the abrupt transitions in cross-section the integral formulation of the conservation of mass flow, m′ , and total enthalpy, B ′ are used for compressible potential flow (Hofmans, 1998); m′1 = m′2 , B1′ = B2′ , Aj + −ikj+ x ikj− x m′j = pj e (1 + Mj ) − p− e (1 − M ) , j j c0 1 + −ikj+ x ikj− x pj e (1 + Mj ) + p− e (1 − M ) . Bj′ = j j ρ0 (3.18) 3.4. Determination of the amplitude of the velocity pulsations at the location of the rotor 59 By introducing a matrix M and a vector [pm ] with the pressures measured at the microphones, the solution of the system of the equations 3.18 and the equations 3.16 at the positions of the microphones can be computed. The system of equation becomes: + −1 T pj T M · [pm ] (3.19) − = M M pj From this system of equation the least-square solution of the plane wave amplitudes − p+ j and pj is determined. Using + uac = − −ik3 xr − p− eik3 xr p+ 3e 3 , ρ 0 c0 (3.20) the velocity pulsations at the position of the rotor, xr , is calculated. 3.4.2 Synchronous detection To analyse the measured pressure signals synchronous detection, ’lock-in’, is being used during post-processing. With synchronous detection, it is possible to measure the phase and amplitude at a certain reference frequency using a reference signal. The reference signal, sref , has to be a well-defined signal: we will use a sine wave, sref (t) = sin (2πf t). When the measurements are carried out by using a loudspeaker to induce pulsations, the signal driving the loudspeaker is used as reference. When the siren is used to induce pulsations, one of the pressure transducers is filtered out digitally using a second order band-pass filter to produce a sinusoidal reference signal. From the sine wave reference signal a cosine wave signal is obtained by shifting the phase by π/2. The Hilbert transform routine of Matlab is used to obtain the shifted reference signal. The transducer signals are multiplied by the sine reference signal and integrated over a integer number of oscillation periods to extract the amplitude of the sin(2πf t) component of the signal. The same procedure is repeated for the cosine reference signal to obtain the amplitude of the cos(2πf t) component of the signal. Integration is done typically over a few hundred periods. 3.4.3 Verification of the acoustic model To investigate the accuracy of the procedure for the determination of the acoustic velocity several approaches are used. The pressure transducers placed in the turbine meter are placed close to the rotor. The rotation of the rotor, the wake of the guiding vanes of the flow straightener and the abrupt transitions in cross-section can cause interference on these pressure measurements. This can be investigated by excluding the two microphones placed within the turbine meter. In figure 3.5 an example is given 60 3. Response of the turbine flow meter on pulsations with main flow 400 300 200 100 position rotor 0 −2 −1.5 −1 −0.5 0 0.5 distance from upstream end of turbine meter (m) (a) all pressure transducers pressure amplitude (Pa) pressure amplitude (Pa) of an experiment in which pulsations were induced at 164 Hz. The figure shows the pressure amplitude, the added upstream and downstream travelling pressure waves. It can be found that the difference between these two models is small, in the order of a few percent in velocity amplitude, depending on frequency and standing wave pattern. 400 300 200 position rotor 100 0 −2 −1.5 −1 −0.5 0 0.5 distance from upstream end of turbine meter (m) (b) pressure transducers in the turbine meter are excluded Figure 3.5: The pressure amplitude of the standing wave in the set up during a measurement at a frequency of f = 164 Hz with a mainstream velocity in the pipe of u0 = 2 m/s. The dots represent the measured pressure amplitude of the pressure transducers. Figure (a) shows a example of a measurement were all eight pressure transducers in the set up are used, (b) shows the standing wave predicted when the two pressure transducers in the turbine meter are not used. The different lines indicate the three different parts of the acoustic model (figure 3.4). The accuracy of the acoustic model depends on the position of the pressure nodes of the standing wave. If the pressure node is located around the rotor position, small deviations in the pressure wave induce small deviations in the velocity amplitude, because the velocity is rather uniform around a pressure node. When the rotor is close to a pressure antinode large errors in velocity amplitude can be induced by small deviations. Measurements are only considered when the acoustical velocity can be determined accurately. In figure 3.6 an example is given of a measurement at 362 Hz, for which the position of the rotor is close to a pressure maximum. The results of this experiment were therefore not used in our analysis. To illustrate this, the velocity amplitude was calculated at the front of the rotor and at the back of the rotor (the rotor has a width of 2 cm). In the case shown in figure 3.6(a), where the rotor position is around a pressure antinode, the velocity amplitude changes over the width of the rotor with 28%, while in the figure 3.6(b) the velocity amplitude changes with 10%. Besides the location of the rotor position in reference to the standing wave, the frequency also plays an important roll. In figure 3.7 examples 3.4. Determination of the amplitude of the velocity pulsations at the location of the rotor 61 400 300 position rotor 200 100 0 −2 −1.5 −1 −0.5 0 0.5 distance from upstream end of turbine meter (m) (a) rotor position around pressure maximum pressure amplitude (Pa) pressure amplitude (Pa) 400 300 200 position rotor 100 0 −2 −1.5 −1 −0.5 0 0.5 distance from upstream end of turbine meter (m) (b) rotor position not in pressure maximum Figure 3.6: The pressure amplitude of the standing wave in the set up for a measurement at a frequency of f = 362 Hz with a mainstream velocity in the pipe of u0 = 2 m/s. The dots represent the measured pressure amplitude of the pressure transducers. Figure (a) shows a example of a measurement where the rotor is located close to a pressure antinode, (b) shows a measurement at the same frequency and main stream velocity with the rotor not as close to the pressure maximum. The different lines indicate the three different parts of the acoustic model (figure 3.4). of a measurement at 24 Hz and a measurement at 730 Hz are shown. If we evaluate the accuracy as mentioned above, for 24 Hz the velocity amplitude changes over the rotor with less than 0.1%, while at 730 Hz this is about 3 percent. In this example (730 Hz) this change in velocity amplitude is still relatively small, because the rotor is positioned at a pressure node. To verify the velocity amplitude found with the acoustical model further, the velocity amplitude is measured 1 cm upstream of rotor by means of two hot wires placed at different positions. Two hot wires were used to account for the complicated flow profile behind the blades of the flow straightener. The local relative velocity pulsations for |u′ |/u0 < 1, can be compared with the relative velocity amplitude calculated with the acoustical model based on the pressure measurements. The measurements of the velocity amplitude with the hot wires are within 10% in agreement with the acoustical model for velocities higher than 2 m/s. Below 2 m/s the calibration of the hot wire is problematic. We will discuss the hot wire measurements further in section 3.4.4. The siren generated block pulses in volume flow, which drives many harmonics of the fundamental frequency. But by using the siren only at resonance frequencies of the pipe, the resonant frequency dominates over other frequencies. In that case we obtain an almost harmonic perturbation. Some overtones will, however, still be present. Using equation 3.9 it is expected that the contributions of the different har- 62 3. Response of the turbine flow meter on pulsations with main flow 300 400 300 200 100 position rotor 0 −2 −1.5 −1 −0.5 0 0.5 distance from upstream end of turbinemeter (m) (a) 24 Hz pressure amplitude (Pa) pressureamplitude (Pa) 500 250 200 150 100 position rotor 50 −2 −1.5 −1 −0.5 0 0.5 distance from upstream end of turbine meter (m) (b) 730 Hz Figure 3.7: The pressure amplitude of the standing wave in the set up. The dots represent the measured pressure amplitude of the pressure transducers. Figure (a) shows a example of a measurement at 24 Hz, (b) shows a measurement at 730 Hz. Both measurements are carried out at a mainstream velocity u0 = 2 m/s. The different lines indicate the three different parts of the acoustic model (figure 3.4). monics add quadratically to the error: # " ′ 2 ′ 2 1 |u2 | |u3 | |u′1 | 2 Epuls = + + + ..... 2 uin uin uin (3.21) where the subscripts indicate the different harmonics. This is checked by inducing pulsations using the loudspeaker and the siren simultaneously at different frequencies. As is shown in table 3.2 the measurement error caused by the frequencies separately accumulates, Eadd , within measurement accuracy to the measurement error caused by the two frequencies simultaneously, Esim . When the difference in frequency is small the induced acoustical velocity will display low frequency beats which the siren can follow. This produces the type of signal shown in figure 3.8. As the frequency obtained by using the siren displays some drift in time, we observe some time dependence in the frequency of the beats for simultaneous measurements with two frequencies close together. As shown in figure 3.8 at t = 20 seconds the loudspeaker is turned on and the turbine meter starts to measure a higher velocity. After another 20 seconds the siren is turned on, the measurement error becomes larger and starts oscillating. This is by the beats. Taking the time average of the error during the beats we still find that Eadd ⋍ Esim (table 3.2). For signals in which other frequencies are present, the contribution from each frequency can be added to predict the total error. As the error depends quadratically on the amplitude the harmonics with higher amplitude will dominate. 3.4. Determination of the amplitude of the velocity pulsations at the location of the rotor 63 |u′ |/u0 f = 24Hz 0.21 0.21 0.21 f = 164Hz 0.09 0.09 0.09 f = 164Hz 0.26 0.26 f = 164Hz 0.18 0.18 |u′ |/u0 f = 164Hz 0.19 0.33 0.57 f = 367Hz 0.07 0.12 0.16 f = 166Hz 0.18 0.25 f ≈ 164Hz 0.28 0.33 Esim −Eadd Esim Eadd Esim 0.032 0.074 0.227 0.033 0.074 0.228 0.12 % -0.18 % 0.64 % 0.005 0.011 0.015 0.005 0.010 0.016 -3.06 % -2.73 % 1.04 % 0.036 0.049 0.040 0.048 6.17 % -1.07 % 0.042 0.052 0.041 0.056 -2.32 % 7.03 % × 100% Table 3.2: Measurements with pulsations at two frequencies. Eadd is the added measurement error of these frequencies separately and Esim is the measurement error for measurement for both pulsations simultaneously. All measurements are carried out at a mainstream velocity of u0 = 2 m/s. 64 3. Response of the turbine flow meter on pulsations with main flow 2 .1 5 s ire n o ff 2 .1 v e lo c ity ( m /s ) 2 .0 5 lo u d s p e a k e r o ff 2 1 .9 5 s ire n o n lo u d s p e a k e r o n 1 .9 0 5 0 1 0 0 tim e ( s ) 1 5 0 2 0 0 2 5 0 Figure 3.8: Mainstream velocity of the flow measured with the turbine meter. After 20 seconds the loudspeaker is turned on at a frequency of 164 Hz, after another 20 seconds the siren is turned on at a frequency of about 164.2 Hz. 3.4.4 Measurements of velocity pulsation in the field Determining the amplitude of the velocity pulsations using eight pressure transducers is not practical for industrial use of turbine flow meters. For this reason the options to measure the velocity amplitude by means of a hot wire or two pressure transducers embedded in the turbine meter has been investigated. A hot wire determines the local velocity as well as velocity fluctuations, in the set up two hot wires are placed. Both hot wires are placed 1 cm upstream of the rotor and 1 cm from the pipe wall in the flow. One hot wire was placed in the wake of a vane of the flow straighteners and the other was placed in between two vanes (see figure 3.3). This was done to account for the effects of the complicated flow profile around the flow straightener. The mainstream velocity measured locally with the hot wires are higher, than the mean velocity measured by the turbine meter. This is caused by hot wire measurements being local measurement, and the flow profile in the annulus is not uniform. The local flow velocity can be higher or lower than the mean velocity depending on the position of the hot wire. As expected the average velocity in the wake of the vane is lower than the velocity measured between the vanes of the flow straightener. The measured amplitude of the velocity pulsations for high velocities is within 10% of the amplitude of the velocity pulsations determined 3.5. Determination of the measurement error of the turbine meter 65 with the acoustical model, for low velocities, however, they are much less accurate. The velocity amplitude measured with the hot wire placed in the wake measures systematically a higher velocity amplitude than the other hot wire. This can probably be explained by the contribution of acoustically induced vortices shedding at the vane. For standing waves when the upstream wave is equal to the downstream wave, p+ = p− , we have a negligible phase difference, we can apply a linear approximation using only pressure transducers at two positions in the turbine meter close to the rotor. By using the excitation frequency and considering only plane waves, we can determine the amplitude of the velocity pulsation. Using the linearised momentum equation for harmonic perturbations it follows that: uac = ∆p′ , 2ρ0 πf Lm (3.22) with ∆p′ the pressure difference and Lm the distance between the two microphones. Taking compressibility into account this becomes: ∆p′ ∆p′ −i π2 2 e (1 − M ) + M uac = , (3.23) 2ρ0 πf Lm ρ 0 c0 where M= u0 /c0 is the Mach number. The applicability of this method is very dependent on frequency and standing wave pattern, comparable to the situation shown in figure 3.6. Our measurements show that the velocity amplitude is predicted within 40%. However, in general we will not observe standing waves. Therefore, the velocity amplitude can be calculated using the acoustical model described in section 3.4.1 using just the two pressure transducers in the turbine meter. However, these results show similar accuracy as the standing wave approximation described above. In general the acoustical signal should be distinguished from the pressure fluctuations induced by vortices. For plane waves this can be done by using more than two microphones. Using a few microphones at one specific distance from the rotor one can find the plane wave contribution by selecting the coherent part of the signal of the microphones in that plane. 3.5 Determination of the measurement error of the turbine meter During a measurement the rotation speed of the rotor is recorded, without flow perturbations and with flow perturbations generated by the loudspeaker or the siren. The effect on the rotation speed, averaged over one revolution, is determined from a visual examination of the plots of the signals as shown in figure 3.9. Using a ruler deviations in the signal of 0.1% can be determined. Epuls = ω − ω0 , ω0 (3.24) 66 3. Response of the turbine flow meter on pulsations with main flow s ire n o ff 3 1 .9 1 2 .8 1 .8 9 1 .8 8 w - w 2 .4 v e lo c ity ( m /s ) v e lo c ity ( m /s ) 2 .6 1 .8 6 w 1 .8 5 2 1 .8 3 0 2 0 4 0 tim e ( s ) 6 0 lo u d s p e a k e r o n 1 .8 2 8 0 0 2 0 4 0 6 0 tim e ( s ) (a) 1 .7 8 1 .9 3 v e lo c ity ( m /s ) v e lo c ity ( m /s ) w - w 1 .7 5 0 1 .9 0 5 1 .9 0 2 0 w 1 .7 3 0 4 0 tim e ( s ) 6 0 1 .7 1 8 0 0 2 0 4 0 6 0 tim e ( s ) 8 0 1 0 0 1 2 0 (d) 1 .8 6 5 1 .9 4 1 .8 6 1 .8 5 5 1 .9 3 5 1 .9 3 w - w 1 .9 2 5 1 .9 2 w 0 w 1 .8 5 0 1 .8 4 5 v e lo c ity ( m /s ) v e lo c ity ( m /s ) 0 1 .7 2 (c) 0 1 .8 4 lo u d s p e a k e r o ff lo u d s p e a k e r o n 1 .8 3 5 1 .8 3 s ire n o ff w - w 1 .8 2 5 1 .9 1 5 1 .9 1 0 lo u d s p e a k e r o n 1 .7 4 1 .9 1 5 w 1 2 0 lo u d s p e a k e r o ff 1 .7 6 1 .9 1 1 0 0 1 .7 7 1 .9 2 5 w - w 8 0 (b) s ire n o ff 1 .9 3 5 1 .9 2 0 0 1 .8 4 w 0 w - w 1 .8 7 0 2 .2 1 .8 lo u d s p e a k e r o ff 1 .9 0 0 1 .8 2 2 0 4 0 tim e ( s ) (e) 6 0 8 0 1 .8 1 5 0 2 0 4 0 6 0 tim e ( s ) 8 0 1 0 0 1 2 0 (f) Figure 3.9: The velocity measured with the turbine meter. The black oscillating line corresponds to the instantaneous reading of the flow meter. The smooth white line represents the measured velocity averaged over one rotation. The left figures show typical measurements for perturbations generated by the siren. The siren is turned on the first 30 seconds and then turned off. The right figures show typical results of measurements for the case in which perturbations are generated with the loudspeaker, starting with the speaker turned off, then turned on, subsequently turned off. The figures show some examples of different pulsation levels from extreme high (a,b) to low (e,f). Due to the long transient in the siren the influence of low pulsation levels cannot be detected (e) while they are still very clearly observable when the loudspeaker is used (f). 3.6. Measurements 67 where ω is the angular velocity of the rotor while measuring pulsating flow and ω0 is the angular velocity of the rotor for flow without pulsations. The variations of the pressure in the reservoir, induces slow mass flow variations during a measurement. This is the main cause of inaccuracies in determining the measurement error of the flow meter. The siren needs some time to reach a constant pulsation frequency, therefore these measurements are started with the siren already turned on. After the siren is turned off, this effect can also be seen. The slowing down of the siren causes the frequency to decay, possibly inducing pulsations that can momentarily cause a large oscillations in the measuring error during the transition. The influence of the pulsations is more accurately determined by using the loudspeaker. The loudspeaker is turned on and off during the measurement without complex transitional behaviour, making it easier to measure the effect of the pulsations. However, the loudspeaker could only be used at low flow velocities, up to 5 m/s in the main pipe. Measurements carried out with the siren do match the corresponding measurements carried out with the loudspeaker. 3.6 Measurements To investigate the effect of velocity pulsation on the flow measurements of the turbine meter, measurements were carried out at resonance frequencies of the set up between 24 Hz and 730 Hz and amplitudes of velocity pulsations ranging from small, uac /u0 ≈ 0.01, to very high amplitudes, uac /u0 ≈ 2. The turbine flow meter used in the set up (Instromet type SM-RI-X G250) has a flow range from 20 to 400 m3 /h (5.6 10−3 m3 /s to 0.11 m3 /s), this corresponds to a velocity in the pipe of u0 = 0.7 m/s to 13.3 m/s. In our measurements velocities were varied from u0 = 0.5 m/s up to 15 m/s. In figure 3.10 and in figure 3.11 measurements are shown for a pulsation frequency of f = 164 Hz for different mainstream velocities. Both figures show exactly the same data set, however, in figure 3.10 the data is shown on a double logarithmic scale and in figure 3.11 the data is shown on a linear scale. From these figures it is clear that for a large range of relative velocity amplitude extending over two decades and the range of main stream velocities, the measurements are still in fair agreement with the quasi-steady theory presented in 3.2.1. We observe less than 40% deviation from the theory. By looking at the data, we can see that the deviation from the quasi-steady theory increases for decreasing main flow velocities. Data obtained for pulsation frequencies of 24, 69, 117, 360 and 730 Hz are shown in Appendix C. In the section below the effect of the Strouhal number and the Reynolds number will be investigated systematically. 68 3. Response of the turbine flow meter on pulsations with main flow 1 10 0 relative measurement error, E puls 10 −1 10 quasi−steady theory u0 = 15 m/s u0 = 1 m/s u0 = 1.3 m/s u0 = 1.5 m/s u0 = 1.7 m/s u0 = 2 m/s u0 = 3 m/s u0 = 5 m/s u0 = 7 m/s u0 = 10 m/s −2 10 −3 10 −4 10 −5 10 −2 −1 10 0 1 10 10 relative acoustic amplitude, |u’|/u 10 0 Figure 3.10: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a frequency of 164 Hz. Plotted using double logarithmic scale 0.5 relative measurement error, E puls 0.45 0.4 0.35 quasi−steady theory u0 = 15 m/s u0 = 1 m/s u0 = 1.3 m/s u = 1.5 m/s 0 0.3 0.25 u0 = 1.7 m/s u0 = 2 m/s u = 3 m/s 0 0.2 0.15 u = 5 m/s 0 u0 = 7 m/s u = 10 m/s 0 0.1 0.05 0 0 0.2 0.4 0.6 0.8 relative acoustic amplitude, |u’|/u 1 0 Figure 3.11: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a frequency of 164 Hz. 3.6. Measurements 3.6.1 69 Dependence on Strouhal number In order to verify the range of validity of the quasi-steady theory, measurements have been carried out for a wide range of Strouhal numbers, Sr = f Lublade , where f is the 0 frequency of the pulsations, Lblade is the length of a rotor blade at the tip and u0 is the main flow velocity at the position of the rotor. It is expected that for low Strouhal numbers the quasi-steady theory is valid. From figure 3.11 (and Appendix C), it is found that measurements for a given fixed frequency, f , and a fixed mainstream velocity, u0 , have a quadratic dependence on the relative velocity amplitude, uac /u0 . To investigate the dependence of the deviation in measured volume flow and actual flow Epuls , a quadratic function: Epuls = a uac u0 2 , (3.25) was therefore fitted through the measured data at a given frequency, f , and flow velocity, u0 using least-square fitting. The parameter a will be referred to as the ”quadratic fit parameter”. This parameter is 21 for the quasi-steady theory. An example is shown in figure 3.12 for measurements at main stream velocity u0 = 1 m/s and at frequency of pulsation f = 164 Hz. We will consider only measurements with a relative amplitude uac /u0 < 1 to obtain the quadratic dependence, because higher amplitudes no longer show the quadratic dependence. Measurements with relative amplitudes uac /u0 > 1 are discussed separately in section 3.6.3. In figure 3.13 the quadratic fit parameter, a, for the measurement at pulsation frequencies of 24, 69, 117 and 164 Hz and mainstream velocities from 1 m/s to 15 m/s are plotted against Strouhal number. The error bar gives the 95% confidence level for the quadratic fit through the measured data. It is an indication for the quality of the quadratic fit. In figure 3.13 the data measured using the siren are solid symbols. The trends in the pulsation error measured with the siren and loudspeaker do not differ from each other. However around SrLblade = O(1), the siren data seems to have a slightly higher quadratic fit parameter than the loudspeaker data. An explanation for this could be that most of the measurements using the loudspeaker are for smaller relative amplitudes compared to the measurements using the siren. This indicates that at low amplitudes the pulsation error, E, is probably not exactly quadratically dependent on the velocity amplitude. The figure shows a clear Strouhal dependence, where the deviation from the actual flow decreases with increasing Strouhal number. However, the deviation from the actual volume flow stays within 40% of the quasi-steady theory for Strouhal number, SrLblade , up to 2.5. Using regression, an equation is obtained to predict the depen- 70 3. Response of the turbine flow meter on pulsations with main flow relative pulsation error, Epuls 0.5 0.4 quasi−steady theory measured data fitted equation, a(uac/u0)2 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 relative acoustic amplitude, u /u ac 1 0 Figure 3.12: A quadratic fit, Epuls = a(uac /u0 )2 with a = 0.35 is shown for the measurement data at a mainstream velocity u0 = 1 m/s and with pulsations of frequency f = 164 Hz. Quasi-steady theory gives a = 1/2. dence of Epuls on the Strouhal number, SrLblade : 1 Epuls = −0.3672 SrL5 blade + 0.7407 , for 0.05 ≤ SrLblade ≤ 2.5 . (3.26) This is a purely empirical relationship between the deviation and the Strouhal number, which cannot be explained theoretically. It is interesting to note that for SrLblade < 0.2 we find a > 21 . We cannot explain this. In figure 3.14 the data measured at the higher frequencies (f = 360 and 730 Hz) are shown separately, because they display different behaviour compared to the low frequency data. The measurements at a pulsation frequency of 360 Hz all have a quadratic fit parameter, a, around the 0.5. These measurements are closer to the deviation, Epuls , found by the quasi-steady theory than the equation found for the lower frequencies. The measurements at a pulsation frequency of 730 Hz show the same quadratic fit parameter for Strouhal numbers of around 6. However, at a Strouhal number of about 10 the data seems to support the empirical relation found using the lower pulsation frequencies. It is possible that these frequencies correspond to mechanical resonant frequencies of the turbine meter causing a different behaviour of the rotor. 3.6. Measurements 71 0.65 24 Hz 69 Hz 117 Hz 164 Hz quadratic fit parameter, a 0.6 0.55 quasi−steady theory 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0 0.5 1 1.5 Sr 2 2.5 L blade Figure 3.13: The quadratic fit parameter a is plotted against Strouhal number for pulsation frequencies of 24, 69, 117 and 164 Hz and mainstream velocities from 1 to 15 m/s. The error bars represent the 95% confidence level for the quadratic equation fitted through the measured data. The solid line shows the quadratic fit parameter, a = 0.5, for the quasi-steady theory, the dashed line is a function fitted through the data found from the present measurements. The solid symbols represent the measurements using the siren, the open symbols the measurements using the loudspeaker. 0.8 360 Hz 730 Hz quadratic fit parameter, a 0.7 0.6 quasi−steady theory 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 Sr 8 10 12 L blade Figure 3.14: The quadratic fit parameter a as a function of Strouhal number for pulsation frequencies of 360 and 730 Hz. The ’♦’ show the data of measurements at pulsation frequencies of 360 Hz, the ’’ show the data of measurements at pulsation frequencies of 730 Hz. The solid symbols represent the measurements using the siren, the open symbols the measurements using the loudspeaker. 72 3. Response of the turbine flow meter on pulsations with main flow Besides this, as explained in section 3.4.3 for high frequencies small errors in the pressure can cause large errors in estimated acoustical velocity. This could explain some of the differences in Strouhal number dependence. Another possibility is that the measurements at these frequencies are difficult is the presence of acoustic resonance. When the length of the constriction caused by the core of the meter matches about half the wave length of the acoustical waves, there will be a resonance in this pipe segment. The length of this constriction is about 25 cm, i.e. this would give resonance frequencies of 680 Hz. For this resonance the rotor is close to a pressure antinode which corresponds to conditions in which the acoustical velocity at the turbine is difficult to determine. To exclude the possibility that the hollow space inside the flow straightener could act as a Helmholtz resonator, this area was filled with foam. This did not change the observed response of the flow meter to pulsations. Care was taken to prevent these problems occurring at high frequencies by excluding measurements for which the rotor was close to a pressure antinode.1 To be able to draw more conclusions for the deviation, Epuls , at Strouhal numbers greater than 2.5, additional measurements are necessary. At high frequencies, 367 Hz and 730 Hz, other strange phenomena can be found; at low pulsation levels, |u′ |/u0 ≤ 0.1. A negative measurement error can be observed (see figure 3.15) for low velocities, up to 2 m/s. These errors do not always reproduce. We suspect here a combination of mechanical vibration and friction. During our tests dust particles were present in the flow and this affected the friction in the rotor. However, tests after cleaning the rotor indicated that this had only a minor effect on most of our data. No significant effect is found for u0 > 2 m/s. 3.6.2 Dependence on Reynolds number To investigate if there is also a dependence of the deviation, Epuls , on the Reynolds number, ReLblade , the residual of the Strouhal number dependence predicted by the empirical relation (equation 3.26) and the quadratic fit parameter found for the measurements is plotted as a function of Reynolds number, ReLblade for pulsation frequencies of 24, 69, 117 and 164 Hz (see figure 3.16). Figure 3.16 shows no significant correlation, between the Reynolds number and the difference between the measurements and the empirical relation for the Strouhal number dependence. We conclude that that there is no significant dependence of the Reynolds number, ReLblade , on the deviation, Epuls . 1 Note that all measurements between 360 Hz and 730 Hz have been rejected because of a very large scatter in the quadratic fit coefficient a, which was related to difficulties in the measurement of the acoustical velocity. 3.6. Measurements 73 0 .9 6 6 0 .9 6 4 lo u d s p e a k e r o ff 0 .9 6 2 v e lo c ity ( m /s ) 0 .9 6 0 .9 5 8 0 .9 5 6 lo u d s p e a k e r o n 0 .9 5 4 0 .9 5 2 0 .9 5 0 .9 4 8 0 .9 4 6 0 2 0 4 0 6 0 tim e ( s ) 8 0 1 0 0 1 2 0 Figure 3.15: Some measurements at low velocities and high frequency show a negative measurement error. In this plot the velocity of the flow measured by the turbine meter is given, the smooth white line represents the velocity averaged over one revolution of the turbine meter. The loudspeaker is turned on after 30 s inducing pulsations of 730 Hz, after another 30 s the loudspeaker is turned off. 3.6.3 High relative acoustic amplitudes Several measurements were carried out at relative pulsation amplitudes larger than unity; uac /u0 > 1. Such high pulsation levels are not likely to occur in practice. However, to investigate the range of the applicability of the quasi-steady theory, it is interesting to look at these results. At these high amplitudes the deviation, Epuls , can no longer be described by the quadratic dependence found for lower amplitudes. In general the measured deviation, Epuls , is smaller than the deviation found by extrapolation of the quadratic dependence found for uac /u0 < 1. The difference with this quadratic dependence is still small for relative acoustic amplitudes uac /u0 ≈ 1 and increases for increasing amplitude. Typical measurement data is shown in figure 3.17. At pulsation levels, uac /u0 ⋍ 2.5, the quasi-steady theory overestimates the effect of pulsations by about a factor 2. While for low amplitudes the effect of pulsations is overestimated by a factor 1.4. 74 3. Response of the turbine flow meter on pulsations with main flow 0.1 0.08 0.06 a−a fit 0.04 0.02 0 −0.02 −0.04 −0.06 0 1 2 3 4 Re L blade 5 6 7 4 x 10 Figure 3.16: The difference between the quadratic fit parameter found for the measured data, a and the quadratic fit parameter found by the empirical relation in section 3.6.1, af it ,is plotted as a function of Reynolds number for pulsation frequencies of 24, 69, 117 and 164 Hz. 3.6.4 Influence of the shape of the rotor blades The influence of the shape of the blade was investigated by replacing the standard rotor with a rotor a with different blade shape. The original rotor has blades with a rounded upstream leading edge and a chamfered trailing edge. The rotor was replaced by a rotor with chamfered leading edges similar to the trailing edges (figure 3.18). To determine the behaviour of the rotor with chamfered leading edges in pulsating flow some of the measurements carried out with the standard rotor are repeated using the new rotor. Figure 3.19 shows the results of the measurements carried out at a pulsation frequency of 164 Hz and mainstream velocities of u0 = 1, 5 and 15 m/s, compared to the measurement data obtained for the standard rotor. Within the accuracy of the measurement no difference was found. To verify this further a quadratic fit as explained in section 3.6.1 was made and this parameter was plotted against Strouhal number, SrLblade for low frequencies (f = 24, 69, 117 and 164 Hz) (figure 3.20). Again, we see that within the accuracy level of the measurements there is no difference between the deviation of the volume flow measurement for the rotor with blades with rounded leading edges and the rotor with blades with chamfered leading edges. 3.7. Conclusions 75 2 relative pulsation error, E puls 1.8 1.6 quasi−steady theory measurement data fitted equation, a(u /u )2 ac 0 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 relative acoustic amplitude, u /u ac 2.5 0 Figure 3.17: The deviation, Epuls as a function of the relative acoustic amplitude, uac /u0 . A quadratic fit, Epuls = a(uac /u0 )2 derived for uac /u0 < 1 (with a = 0.35) is shown for the measurement data at a mainstream velocity u0 = 1 m/s and with pulsations of frequency f = 164 Hz. 3.7 Conclusions The effect of the pulsating flow on a turbine flow meter has been investigated experimentally and results have been compared to the results of a simplified quasi-steady model. A set up was built making it is possible to induce pulsations with a frequency from 24 Hz to 730 Hz, relative acoustic velocity amplitudes, uac /u0 , from 2 × 10−2 up to 2 and volume flows ranging from the minimum to the maximum flow specified by the manufacturer, i.e. from 20 to 400 m3 /h. Multi-microphone measurements have been used to determine the amplitude of the velocity pulsation at the rotor. The error caused by the pulsations is obtained from the comparison of the rotation speed of the rotor in presence of pulsations with the one in the case that there are no pulsations. The measurements show that the simplified quasi-steady theory gives a fair approximation of the error caused by the pulsations. The measurements agree with the theory within 40% for nearly all measurements, even for measurements at high relative acoustical amplitudes. We found that the error caused by pulsations is dependent on Strouhal number. For SrLblade < 2.5 an empirical relation was found for the dependence of the error on the Strouhal number. Globally one expects that the influence of pulsations should decrease with increasing pulsation Strouhal number. 76 3. Response of the turbine flow meter on pulsations with main flow (a) rounded leading edge (b) chamfered leading edge Figure 3.18: A schematic drawing of the rotor (a) with rounded leading edges and (b) with a chamfered leading edges used in the measurements This corresponds to our observations. As of yet no physical explanation is found for this specific dependence. For SrLblade > 2.5 the behaviour of the rotor is still unclear, caused by the difficulties in measuring at higher pulsations frequencies. The measurement error caused by the pulsations is not significantly dependent on the Reynolds number. The shape of the upstream edge of the rotor blades does not influence the Strouhal number dependence of the systematic error induced by the pulsations. This study stresses the importance of determining the acoustical velocity at the rotor for a correction of measurement errors due to pulsations. Measurements with local velocity probes such as hot wires are difficult to use because they do not distinguish between vortical perturbations and acoustical waves. Acoustical waves can be detected by means of microphones mounted flush in the wall. This would however involve multiple microphones at a certain position to allow the detection of the plane waves by cross-correlation method analogous to microphone array techniques. 3.7. Conclusions 77 0.18 relative measurement error, Epuls 0.16 0.14 0.12 quasi−steady theory u = 1 m/s; round l.e. 0 u = 1 m/s; sharp l.e. 0 u0 = 5 m/s; round l.e. u = 5 m/s; sharp l.e. 0 0.1 0.08 u = 15 m/s; round l.e. 0 u = 15 m/s; sharp l.e. 0 0.06 0.04 0.02 0 0 0.1 0.2 0.3 0.4 0.5 relative acoustic amplitude, |u’|/u0 0.6 Figure 3.19: The relative measurement error, Epuls , is plotted against the relative pulsation amplitude, |u′ |/u0 , for measurements with the ’new’ rotor with chamferedleading-edge blades and the standard rotor with rounded-leading-edge blades. The data is for measurements at a pulsation frequency of 164 Hz and mainstream velocities, u0 = 1, 5 and 15 m/s. The solid symbols are the data measured with the rotor with chamfered-leading-edge blades. The plot shows that within the measurement accuracy there is no difference in behaviour for the two rotors. 0.65 quadratic fit parameter, a 0.6 0.55 quasi−steady theory 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0 0.5 1 1.5 2 2.5 SrL blade Figure 3.20: The quadratic fit parameter a is plotted against Strouhal number for pulsation frequencies of 24, 69, 117 and 164 Hz and mainstream velocities from 1 to 15 m/s. The ’o’ represents the measured data of the standard rotor with blades with rounded leading edges and ’*’ represents the data measured with the ’new’ rotor with blades with chamfered leading edges. Again, no difference can be found within measurement accuracies for the two rotors. 78 3. Response of the turbine flow meter on pulsations with main flow 4 Ghost counts caused by pulsations without main flow 4.1 Introduction Turbine flow meters are often placed in measurement manifolds, consisting of several runs (side branches). At low volume flows some of the branches in these manifolds are closed. Closed side branches can form resonators which are driven by vortex shedding at the junction with the main pipe (Peters, 1993). Flow pulsations in these manifolds can then affect turbine meters in open pipes, but can also induce ghost counts. This corresponds to spurious flow measurement of the turbine meter in closed side branches where there is no main flow. These ghost counts start above a critical pulsation level. The aim of this chapter is to obtain a better understanding of these spurious counts. In the first part of this chapter1 a theoretical model is presented which explains the occurrence of these spurious counts in the limit of infinitesimally thin turbine blades. The predicted threshold for the occurrence of spurious counts is compared to experimental data at various gas pressures in the range from 1 to 8 bar. In the second part of this chapter, a numerical and experimental study of the flow around the edge of a turbine blade is presented. Aim of this study is to predict the influence of the thickness and the shape of the turbine blade on the onset of spurious counts. An experimental setup has been built to simulate the flow around a model of a blade edge. The vortex shedding at the edge has been visualised. On the blade the surface pressure has been measured. These measurements have been compared with predictions of a discrete vortex model. 1 The first part of this chapter has been published with some minor changes in the Journal of Fluids and Structures; P.W. Stoltenkamp, S.B. Araujo, H.J. Riezebos, J.P. Mulder and A. Hirschberg (2003), Spurious counts in gas volume flow measurements by means of turbine meters, 18(6):771-781. 80 4.2 4.2.1 4. Ghost counts caused by pulsations without main flow Onset of ghost counts Theoretical modelling of ghost counts The spurious count behaviour of a turbine meter can be explained by considering the forces acting on an aerofoil in an oscillating flow. The blades of the turbine rotor most commonly used in gas transport systems have a rounded leading edge and a sharp trailing edge (see figure 4.1). This difference in edge shape causes the spurious rotation. le a d in g e d g e u ' L b la d e t b la d e tra ilin g e d g e Figure 4.1: Blades of the rotor In our model it is assumed that there is only flow separation at the sharp trailing edge of the blade. The flow separation at the sharp edge in an oscillating flow can be seen in Schlieren visualisations (see figure 4.2). In the following we neglect the interaction between the blades in the rotor. Centrifugal forces in a potential flow around the edge of a plate cause a low pressure at the surface of the blade edge, which results in a force directed along the blade, a ”suction force”, which will be called the edge force. In case of a sharp edge vortex shedding reduces this edge force, while for a rounded edge it remains present as long as the flow remains attached. This leads to a net force on the blade. This force brings about a torque on the rotor. Spurious counts start when this torque is large enough to compensate the torque due to the static friction forces. This analysis is restricted to the case of a harmonic acoustical oscillation with frequency f (Hz) 4.2. Onset of ghost counts 81 Figure 4.2: Schlieren visualisations of the flow separation at the sharp edge at a Strouhal number, Srtblade = O(1) and amplitude uac (ms−1 ) of the particle velocity: u′ = uac cos(2πf t). Furthermore, the turbine is considered at the condition that it does not yet rotate, so that the blades have a fixed position. Important dimensionless numbers for this problem are the Helmholtz number, He = f tblade /c , the Strouhal number, Srtblade = f tblade /uac , the Reynolds number, Re = uac Lblade /ν and the ratio, tblade /Lblade , of the thickness, tblade , of the blade compared with the length, Lblade . In these numbers ν (m2 s−1 ) is the kinematic viscosity and c (ms−1 ) the propagation speed of sound waves. It is assumed that the flow is attached and that the viscous boundary layers are thin (Re >> 1). Hence, the flow around a blade of the turbine meter can be described with potential flow theory corrected for the effect of boundary layers. The flow is assumed to be locally incompressible, because the rotor is small compared to the acoustical wave length and the amplitude, uac , is small compared to the speed of sound (He << 1 and M = uac /c << 1). The Strouhal number, Srtblade gives an indication of the blade thickness compared to the acoustical displacement of particles. If the Strouhal number SrLblade = Srtblade Lblade /tblade is small, the blade length is small compared to the acoustical particle displacement and the flow can be assumed to be quasi-steady. If the Strouhal number is of order unity, Srtblade = O(1) with tblade /Lblade << 1, local vortex shedding occurs at the edges of the blade. Finally, if the Strouhal number is much larger than unity, SrLblade >> 1, vortex shedding is negligible except for very sharp edges. In that case the blade thickness is not the relevant length scale. This is illustrated in figure 4.3. In practice the ratio, tblade /Lblade , of the thickness of the blade compared to the 82 4. Ghost counts caused by pulsations without main flow SrtbladeLblade/tblade<<1 Srtblade = O(1), tblade/Lblade=O(10 -1 ) Srtblade >> 1, tblade/Lblade=O(10 -1 ) u' Figure 4.3: Influence of Strouhal number and blade length to thickness ratio L/tblade on the flow. SrLblade ≪ 1 corresponds to quasi-steady flow. When SrLblade > 1, but Srtblade < 1 we have strong vortex shedding at the edges. For SrLblade ≫ 1 and Srtblade > 1 we have local vortex shedding at the sharp edges. length, is small, tblade /Lblade = O(10−1 ). In this theory, the blade is modelled as a flat plate. In a first model the flow separation at the sharp edge is modelled by means of a single point vortex and by applying the Kutta condition (also called Kutta-Joukowski condition) at the sharp edge. This corresponds to the model of Brown and Michael (1954) for flow separation at the leading edge of a slender delta wing. The force on the blade is found by integration of the pressure distribution on the plate. The singular flow around the sharp leading edge results in a finite so-called edge force which is the result of an infinite low pressure applied on a zero surface (Milne-Thomson, 1952). A second model is considered for the limit case, for which the Strouhal number is very large, Srtblade >> 1. Here a potential flow is considered without flow separation, but the contribution of the sharp trailing edge to the net hydrodynamic force on the blade is removed. The idea is that the vortex shedding at this sharp edge has removed the local flow singularity without affecting the global flow around the blade. Theory for the case SrLblade = Srtblade (Lblade /tblade ) = O(1) The theory for the case in which the Strouhal number is order unity is considered first. In a point vortex model, the vortex sheet generated by flow separation at the sharp edge is represented by a single point vortex of varying circulation. The vortex is assumed to be connected to the sharp edge of the plate by means of a feeding sheet. The circulation of the point vortex is calculated by applying the Kutta condition at the sharp edge. The Kutta condition requires the velocity to be finite at the sharp edge. In a real flow, this means that the flow leaves the edge tangentially, accounting for viscous effects. In a point vortex model this implies that at the edge a stagnation point is assumed. The point vortex moves with the flow. Application of the Kutta condition 4.2. Onset of ghost counts 83 implies then that the circulation changes with time. The convection velocity of the point vortex is calculated by means of potential flow theory. For this calculation a free vortex is assumed. This assumption is in contradiction with the time dependence of the circulation of the vortex. This induces a spurious force that will be neglected further (Rott, 1956). This error will appear not to be critical for our results. ξ-plane z-plane Im[ ξ] u' α α Im[z] Γv Γv A Re[z] Re[ξ] Figure 4.4: Flat plate in the z-plane and transformed to a circle in the ξ-plane The flow potential is calculated using a complex potential and conformal mapping. A circle with radius A, in the ξ-plane, is transformed in a flat plate of length Lblade = 4A, in the z-plane using the transformation of Joukowski: z=ξ+ A2 , ξ (4.1) The complex potential of the flow, Φ, in the transformed plane can now be written as: Φ = u′ ξe−iα + iΓv A2 ξv u′ A2 iα iΓv e − ln(ξ − ξv ) + ln(ξ − ). ξ 2π 2π |ξv |2 (4.2) where α is the incidence angle of the flow with respect to the blade (see figure 4.4). 2 The first and the second term on the right hand side are the acoustic flow potential for a circular cylinder in a parallel flow after applying the Milne-Thomson circle theorem (Milne-Thomson, 1952). The third and the fourth term are the contributions of the vortex and that of the mirror-imaged vortex at ξ = A2 /ξv∗ , also found using 2 Note that using a two-dimensional infinite cascade representation of the rotor we could use conformal mapping from a cascade to a circle proposed by Durant (1963). This would allow to take the interaction between the rotor blades into account. 84 4. Ghost counts caused by pulsations without main flow Milne-Thomson’s circle theorem. Here ξv is the position of the vortex in the transformed plane and Γv the circulation of the vortex. The circulation of the vortex is calculated using the Kutta-condition at ξ = A: dΦ =0. dξ ξ=A (4.3) At the first step the vortex is shed, the position of this vortex is calculated using the self-similar solution, given by Howe (1975) for an impulsively starting flow around a semi-infinite plate. The velocity of the vortex, uv , is calculated in the following steps using the following equation, assuming the vortex is a free vortex: u∗v = . dΦ iΓv dz iΓv d2 z . dz 2 dzv∗ dΦ , = + + lim dt dz z=zv ξ→ξv dξ 2π(ξ − ξv ) dξ 2π dξ 2 dξ where zv is the position of the vortex, ξv the position of the vortex in the transformed plane, ∗ indicates the complex conjugate, Φ the complex potential and Γv the circulation of the vortex. The last term is known as the Routh correction (Clements, 1973). The new position of the vortex is calculated using a fourth-order Runge-Kutta integration scheme (Hirsch, 1988). The new circulation, Γv , of the vortex corresponding to this new position, is calculated using the Kutta condition (equation 4.3). In this model the circulation of the vortex vanishes when the acoustic flow is zero. At the next time step a new vortex is shed. As an example the path of a single point vortex and its circulation is plotted in figure 4.5 for a typical case under conditions for which spurious counts were measured. 0.05 1 ac Γ/(U L) y/L 0.5 0 0 −0.5 −0.05 0.45 0.5 x/L 0.55 −1 0 0.2 0.4 0.6 0.8 1 t/T Figure 4.5: a) Calculated path of a single point vortex and b) calculated vortex strength for SrLblade = 9 4.2. Onset of ghost counts 85 To calculate the torque on the turbine, the force on each blade is calculated numerically by integration of the pressure along the plate. F =− I p · ndS = − Z without edge pdS − Z pdS , (4.4) edge where p (P a) is the pressure and S (m2 ) the surface area. The pressure, p, in the first term is obtained from the potential flow solution using the Bernoulli equation for unsteady potential flow: ρ ∂Φ 1 2 + ρu + p = constant(t) . ∂t 2 (4.5) The integration is carried out by means of the midpoint quadrature rule. For the second term, around the singularity, a quasi steady-approximation is used (MilneThomson (1952) and Appendix D). As input for this theory a Taylor expansion of Φ around the point ξ = −A is used. Limit of the theory for SrLblade = Srtblade (Lblade /tblade ) >> 1 The limit of this theoretical model is considered next for SrLblade >> 1. In such a case the effect of the vortex on the global flow around the blade is negligible except for the flow near the sharp trailing edge. A great advantage of this simplified model is that an explicit expression is obtained for the aerodynamic torque on the rotor without the need to determine the details of the vortex path. As explained above the key of this model is that it is assumed that there is only flow separation at the sharp trailing edge of the flat plate. As a consequence, there is a finite velocity at this sharp edge and therefore no edge force. At the leading, rounded edge there is no flow separation and the velocity becomes infinite and this results in an edge force (figure 4.6). a F e a F e Figure 4.6: Flow separation at the sharp edge for a flat plate in an oscillating flow 86 4. Ghost counts caused by pulsations without main flow Because the velocity becomes infinitely large at the edge, convective flow acceleration is larger than the local time dependent flow acceleration. A quasi-steady approximation can be used. The edge force is directed parallel to the plate and can be calculated with potential flow theory (Milne-Thomson, 1952).3 The magnitude, Fe , of the edge force is: Fe = −πρSblade u2ac sin2 α , (4.6) where Fe (N is the edge force), Sblade (m2 ) is the surface area of the plate, uac (ms−1 ) is the acoustic velocity amplitude and α is the angle between the blades and the direction of the acoustical flow. The flow separation generates a vortex close to the sharp edge. As explained above this vortex is assumed to have solely the effect of removing the edge singularity in the flow field and as a consequence the edge force. Flow separation also implies that the boundary layer vorticity is injected into the main flow. This vorticity is assumed to be of small magnitude and confined to a region close to the edge, therefore there will be no significant change of the global circulation for the flow around the flat plate. If the flat plate is placed in a parallel harmonically oscillating flow, the flow will alternate between the left and right situation in figure 4.6. The force perpendicular to the flat plate will also alternate harmonically. If a harmonically oscillating flow is imposed, the average of the normal force taken over one oscillating period will be zero. Consequently, the resultant averaged force for the flat plate over one acoustic period can be simply calculated using the edge force. With this edge force, it is possible to calculate the average torque, Tav , on the blades. Tav = πρrav Sblade 2 uac sin3 α , 8 (4.7) where rav (m) is the average radius at which the force is applied on the blade. Comparison the results of the models In figure 4.7 the relative difference, (T1 − T2 )/T2 between the critical torque, T1 , calculated with the point vortex model for SrLblade = O(1) and the critical torque, T2 , calculated with the model for the limit case SrLblade >> 1 is plotted against the reciprocal Strouhal number, 1/SrLblade = uac /f Lblade , based on the length of the plate. For typical Strouhal numbers as encountered in our experiments (4 < SrLblade ≤ 20) the difference between the results of the two theories is less than 35% which is negligible compared to the difference between theory and experiments. 3 Again in this case the influence of the interaction between the rotor blades can be taken into account by considering an infinite two-dimensional cascades of thin plates (Durant, 1963). 4.2. Onset of ghost counts 87 0 .4 0 .3 5 0 .3 0 .2 (T 2 -T 1 )/T 2 0 .2 5 0 .1 5 0 .1 0 .0 5 0 -0 .0 5 0 0 .0 5 0 .1 0 .1 5 1 /S r 0 .2 0 .2 5 0 .3 L Figure 4.7: The difference (T1 − T2 )/T2 between the critical torque calculated with the two models plotted against the reciprocal Strouhal number 4.2.2 Experimental setup for ghost counts Acoustical oscillations in a closed side branch can either be induced by a resonant response to compressor pulsations or by flow induced pulsations due to vortex shedding. In the experimental set up these flow oscillations are induced using a loudspeaker mounted within a closed pipe segment (figure 4.8). Two set ups were used at Gasunie (Mulder, 2000), a small set up at atmospheric pressure with a pipe diameter D = 100 mm, and a large set up, with a pipe diameter D = 300 mm. In the large set up we had the possibility to vary the mean static pressure from 1 to 8 bar. In the small set up, the gas turbine meter (Instromet SM-RI-X G250, see table 2.1) is placed between two PVC pipes with diameter D = 100 mm and length Lp1 = Lp2 = 1.8 m. A loudspeaker (Visaton W100S) is placed at the end of the pipe, while the other end is closed by a rigid plate. Four dynamical piezoelectric pressure transducers (Kistler type 7031) are placed at positions distributed along the pipes. The pressure transducers are connected to charge amplifiers (Bruel & Kjaer type 2635). Experiments with the small turbine meter were repeated at Eindhoven University of Technology (TU/e) with the set up described in section 3.3. In the large set up the gas turbine meter (Instromet SM-RI-X G2500) is connected with two pipes with diameter D = 300 mm and lengths of Lp1 = 6 m and Lp2 = 2 m. The end of each pipe is sealed to be able to support a pressure up to 8 bar above atmospheric pressure. In the long pipe, Lp1 , a loudspeaker (Peerless XLS10) is placed. The short pipe, Lp2 , is closed by means of a flat and rigid plate. The position of the loudspeaker can be changed, making it possible to modify the resonance frequency of 88 4. Ghost counts caused by pulsations without main flow closed side branch field conditions main flow laboratory experiment D loudspeaker Lp1 Lp2 x=0 x Figure 4.8: Field conditions compared with experimental set up the system. Nine holes are made for placing dynamical pressure transducers (Kistler type 7031) along the pipes, allowing an optimisation of the choice of the position of the four available transducers. To obtain a harmonic voltage signal, a signal generator (LMS Roadrunner) is connected to a power amplifier (Bruel & Kjaer type 2706) from which the signal is applied to the loudspeaker. The amplitude and the frequency can be adjusted separately. For the acquisition of all the signals a twelve channel data acquisition device (LMS Roadrunner compact) is used. The rotational frequency of the blades is measured using standard Elster-Instromet measuring equipment. The pressure transducers are used to measure the acoustic pressure amplitude. As the acoustic waves in the pipe are plane waves, the pulsation pressure amplitude, p′ (x, t), depends only on the coordinate along the axis parallel to the pipe. This can be assumed if the frequency, f , of the pulsations is much smaller than the cut-off frequency, fc , for non planar waves (in the small set up fc = c0 /(2D) ≈ 1.7 × 103 Hz and for the large set up fc = c0 /(2D) ≈ 5.7 × 102 Hz). The pressure amplitude data is used to calculate the acoustic velocity amplitude, uac , if a full reflection at the closed end wall is assumed and thermal and friction losses in the pipe are neglected. The acoustic velocity at a distance x from the end of the wall of the pipe is calculated 4.2. Onset of ghost counts 89 using the equation: |u′ (x)| = |p′ | sin kx , ρ0 c0 cos(kxm ) (4.8) where |u′ | = uac (ms−1 ) is the acoustic velocity amplitude, |p′ | (P a) is the acoustic pressure amplitude measured at a distance xm (m) from the end of the pipe wall, ρ0 (kgm−3 ) is the density of the propagation medium at ambient temperature, c0 ≈ 344 ms−1 is the acoustical wave propagation velocity in air at ambient temperature, k (m−1 ) is the wave number and xm (m) is the distance between the end wall of the pipe and the pressure transducer. To obtain the acoustic velocity, uac , at the blades of the turbine flow meter, incompressibility is assumed and from the mass conservation equation the following is obtained: uac = |u(Lp2 )| Sm , St (4.9) where uac (ms−1 ) is the amplitude of the acoustic velocity at the turbine blades, Lp2 (m) is the pipe length (see figure 4.8), Sm = πD2 /4 (m2 ) is the surface area of the pipe where the measurement is taken and St (m2 ) is the cross sectional surface area of the turbine flow meter. The flow through the turbine meter can be assumed incompressible, because the length of the turbine meter, Lt , is very small compared to the wave length, λ (Lt ≪ λ). The frontal area of the blades of the turbine meter, which is about 10% of the internal area of the turbine meter, is neglected in calculating St . We used the values St = 5.7 × 10−3 m2 for the small meter (Instrometer G250) and St = 4.9 × 10−2 m2 for the large meter (Instromet G2500). Measurements of uac obtained from the four different pressure transducers agree with each other within 10%. 4.2.3 Experiments Critical friction torque Two different experiments have been carried out to obtain the critical static torque above which rotation occurs: a dynamic experiment and a static experiment. The equation of motion for the turbine flow meter is: Irotor dω = Td + T f , dt (4.10) where Irotor (kg m−2 ) is the moment of inertia of the rotor relative to its axis, ω (rad s−1 ) is the angular velocity, t (s) the time, Td = Tav (kg m2 s−2 ) is the driving torque at the blades of the rotor and Tf (kg m2 s−2 ) the total friction torque. The 90 4. Ghost counts caused by pulsations without main flow approximate driving torque has been derived in the previous section (see equation 4.7). The torque caused by friction can be split into the contribution of air friction, Tair , on the rotor and in the torque caused by the friction, Tmech , in the bearing (the friction in the oil and the friction of the shaft). The friction of the shaft can be divided in static friction and dynamic friction. To obtain the torque due to the dynamic friction, the rotor is accelerated to a steady velocity by means of an acoustic field generated by a loudspeaker. When the loudspeaker is turned off, the decay of the angular frequency is registered and a fourth-order curve is fitted through the data. Subsequently, the torque is calculated from the angular velocity using the relation in equation 4.10. The driving torque, Td is assumed to have a second order dependency on the angular velocity (see equation 4.6). The friction torque of the oil in the bearing has a linear dependency on the rotational velocity . The friction of the shaft is assumed to be constant. In the limit of vanishing rotational speed, the only contribution to the torque is the constant dynamic friction of the rotor of the shaft. The value of this dynamic torque is found by taking this limit to be approximately 6 × 10−6 N m for the small set up and 9 × 10−5 N m for the large set up. The static friction torque is measured using a small piece of adhesive tape fixed at a known radius on the rotor. The tape induces a torque on the rotor, which can be calculated from the weight and the position of the tape. The rotor is restrained and released using a photograph shooter. The torque is increased step by step until the release of the rotor induces rotation. With this method the critical static torque was found to be 5.6 × 10−6 N m for the small set up and 1.0 × 10−4 N m for the large apparatus. These values of the critical static torques agree within the experimental accuracy with the dynamic friction torques in the limit of zero rotational speed. Furthermore, these results agree with the specifications of the manufacturer. Critical acoustic velocity Above a critical value of the amplitude of the acoustic velocity, uac , spurious counts occur. This critical acoustic velocity is measured, by keeping the acoustical excitation frequency, f , constant, while increasing the amplitude slowly. This can be done either manually or by using a special function on the signal generator. When the rotor starts to rotate the acoustic velocity is determined from the acoustic pressure amplitude, p′ , as explained in the previous section. These results show considerable deviations for different frequencies, but also for two consecutive measurements at the same frequencies. The standard deviation is approximately 20% of the mean value. Probable reasons for these deviations can be the varying static friction torque caused by the unevenly distributed oil in the bearing and local roughness of the solid surfaces 4.2. Onset of ghost counts 91 in the bearing. Mechanical vibrations induced by the loudspeaker can also influence the threshold for auto-gyration. It is also possible that they are caused by changes that occur in the flow topology or by difficulties in the determination of the threshold of rotation from the experimental data. The mean critical acoustic velocities are determined for the small and the large set up at their first and third resonance mode. For the small set up these frequencies are f1 = 60 Hz and f3 = 210 Hz respectively, and for the large set up f1 = 30 Hz and f3 = 100 Hz respectively. The critical acoustic velocity amplitude, uac , for the large set up has been measured at four different static pressures, 1 bar, 2 bar, 4 bar and 8 bar. 4.2.4 Comparing measurements with results of the theory The results of the experiments can now be compared to the calculated data and to the value of the critical pressure amplitudes in field conditions. Strouhal number (Sr tblade=t blade f/uac ) 1.2 1 0.8 0.6 Gasunie, large app. 30 Hz 0.4 Gasunie, large app. 100 Hz Gasunie, small app. 60 Hz Gasunie, small app. 210 Hz 0.2 TU/e, small app. 69 Hz TU/e, small app. 165 Hz 0 0 1000 2000 3000 4000 5000 6000 7000 Reynolds number (ReLblade =uac Lblade/ν) Figure 4.9: The critical Strouhal number plotted against the Reynolds number. The data obtained at Gasunie (grey and solid circles and squares) complemented with the data obtained at TU/e (open circles and squares) as described in section 4.3). Figure 4.9 shows the critical Strouhal number, Srtblade = tblade f /uac , based on the blade thickness, tblade , at which the rotation starts, plotted against the Reynolds number based on the blade length, Re = uac tblade /ν, with ν the kinetic viscosity. The critical Strouhal number, Srtblade , is of order unity, which corresponds to our qualitative model of the effect of the blade thickness. When the vortex remains at 92 4. Ghost counts caused by pulsations without main flow distances from the edge smaller than the blade thickness, it is not expected that this affects the flow. Hence there is no rotation for Srtblade >> 1. A significant dependency of Srtblade on the Reynolds number can be observed, which we cannot explain with the used model. We did actually not expect such a dependency. Gasunie, large app. 30 Hz 1.8 ratio of measured and predicted critical torque Gasunie, large app. 100 Hz 1.6 Gasunie, small app. 60 Hz Gasunie, small app. 210 Hz 1.4 TU/e, small app. 69 Hz 1.2 TU/e, small app. 165 Hz 1 0.8 0.6 0.4 0.2 0 0 1000 2000 3000 4000 5000 6000 7000 Reynolds number (ReLblade = uac Lblade/ν) Figure 4.10: Ratio between the measured critical torque and the calculated critical torque plotted against the Reynolds number. The data measured at Gasunie (grey and solid circles and squares) complemented with the data measured at TU/e (open circles and squares) which are described in section 4.3. In figure 4.10 the ratio of the measured critical static torque and the predicted torque calculated using the limit case model (SrLblade >> 1) for both set ups is plotted against the Reynolds number, with the length of the blade as the characteristic length. For the small set up the calculated critical torque is two times the measured critical torque for the first resonance mode and five times the measured critical torque for the third mode. For the large set up the calculated critical torque is 20% lower than measured for the first mode and 1.4-1.7 times larger for the third mode. It would be interesting to investigate whether these difference in behaviour for the first and third acoustical mode of the pipe are related to difference in mechanical vibration level. Such vibrations induced by the loudspeaker could affect the critical torque. Using the results the critical torque can be estimated for field conditions of natural gas transportation with a static pressure, p = 60 bar = 6 × 106 P a and a density of the natural gas, ρg = 14 kg m−3 . The geometries of the pipe and of the turbine flow meter are assumed to be similar to that of the large set up. Using equation 4.7 the acoustical velocity, uac , is calculated, at which the critical static torque is reached and spurious counts occur. This is found to be uac ≈ 7 × 10−2 ms−1 . This corresponds 4.3. Influence of vibrations and rotor asymmetry 93 to acoustical pressure amplitudes of the order of p′ = ρ0 c0 uac ≈ 103 P a with for natural gas c0 ≈ 390 ms−1 . In field conditions measurements show spurious counts with acoustic pressure amplitudes of 4 × 103 P a (Riezebos et al., 2001). This shows that the model can provide a fair indication for conditions at which spurious counts occur. 4.3 4.3.1 Influence of vibrations and rotor asymmetry Vibration and friction To verify the measurements described in section 4.2.3 the same measurements were carried out in the set up built at Eindhoven University of Technology to measure the effect on the turbine meter of pulsation with main flow (section 3.3). By closing the valve and using the loudspeaker to induce pulsation, comparable measurements of the critical acoustical velocity amplitude, uac , can be preformed. The turbine meter in this set up is of the same type (Instromet SM-RI-X G250) as used in the small setup described in section 4.2.2. In this set up, however, the loudspeaker is mechanically disconnected from the pipe so that mechanical vibrations are reduced. The amplitude of the critical acoustic velocity is determined at different frequencies by varying the amplitude step by step and waiting a few moments to observe whether or not the rotor starts rotating. Around the critical amplitude the situation can occur that the rotor begins to move but stops before a full revolution is completed. This shows the dependence of the static friction torque on the rotor position. The dynamic friction torque was measured using the dynamical method described in section 4.2.3 and was found to be 1.5 × 10−5 N m. The results are plotted in figures 4.9 and 4.10, and show a good agreement with the other measurements. It should be noted that is found from this test and a check by the manufacturer that the there is an increase of friction caused by the deterioration of the bearings. This is caused by some corrosion problem within the pressure reservoir, causing pollution of the flow with rust particles. The increased friction causes a need for higher pulsation to compensate the friction torque, the maximum pulsation is induced corresponding with a Strouhal number of Srtblade = 0.048, using equation 4.7 of the limit model, this is equal to a torque of about Td = 2.8 × 10−4 N m. 4.3.2 Rotor blades with chamfered leading edge In order to verify that the difference in shape of the edges of the blades of the rotor causes the ghost counts, the rotor in the turbine meter was replaced with a rotor with a blade profile with a chamfered leading edge (figure 4.11). Again pulsations were induced to investigate the occurrence of ghost counts. As 94 4. Ghost counts caused by pulsations without main flow (a) rounded leading edge (b) chamfered leading edge Figure 4.11: A schematic drawing of a rotor (a) with rounded leading edges and (b) with chamfered leading edges used in the measurements expected, no rotation of the rotor occurred. Also imposing an initial rotation of the rotor is not sufficient to induce ghost counts. An initial rotation was induced by a main flow, but when the main flow was stopped the rotor rotation decayed to zero independently of the imposed acoustical pulsations. 4.4 Flow around the edge of a blade In order to get a more quantitative prediction of the threshold for auto-gyration the influence of the blade thickness and shape of the edge should be investigated further. The influence of the vortex shedding on the flow around the edge will be examined more closely in the present section. 4.4.1 Numerical simulation To model the vortex shedding from the blade edge the so-called ”vortex-blob” method (Krasny, 1986) has been used. This method, solves the vorticity-transport equation for two-dimensional flow neglecting viscous effects. For a more detailed description of this method we refer to Hofmans (1998) and Peters (1993). The ”vortexblob” method requires the specifications of separation points. We can use this in order to identify the contribution of various vortices to the edge force. The vorticity field is modelled by vortex blobs, desingularised point vortices, and by applying the Kutta condition at the separation points new vortex blobs are introduced. The desingularisation of the vortex is used to avoid numerics-induced chaotic behaviour in vortex-vortex interaction. When considering the interaction of a vortex with a wall 4.4. Flow around the edge of a blade 95 we consider the vortex as a point vortex. The wall is represented by means of a panel method. The dipole distribution on the walls is determined such that the normal flow velocity equals zero at the panel centers. 4.4.2 Experimental set up for flow around an edge To verify the numerical method described above, an experimental set up was build to study the local flow around the edge of a single blade of blade thickness tblade = 1.00 cm. This was done by building a wooden box of 48.2 cm x 24.8 cm x 22.3 cm with a wall thickness of 2.5 cm divided in the middle to create two separate compartments. These two compartments are connected by means of a duct creating two connected Helmholtz resonators (see figure 4.12). The volume V1 of the first compartment is V1 = 8.15×10−3 m3 . The volume V2 of the second compartment is V2 = 8.35×10−3 m3 . This implies a small asymmetry in the volumes of the compartments in our set up. The duct cross-sections are S = d x w with d = 4.5 cm and w = 10.0 cm. The edge of the dividing wall is a model for the edge of the turbine blade. The tip of the edge of is placed at a distance of 5.0 cm from the top wall closing the set up. It is possible to change the shape of the blade edge. The first model has a sharp chamfered edge with a bevel angle of 45◦ . The second model has a rounded edge and the last model has a square edge. The experiments for the rounded edge show no vortex shedding at the edge. We present here only the results for the sharp edge. The internal dimensions of the set up are given in figure 4.12. The acoustic flow is determined by using two piezo electrical pressure transducers (PCB 116A), one on each side of the box. The signal of these pressure transducers is amplified be using a charge amplifier (Kistler type 5011). A third miniature pressure transducer (Kulite type XCS-093-140mBARD) is placed in the wall right above the blade edge which we will refer too as ”top wall pressure transducer” (figure 4.12). The vortex shedding from the blade can be visualised by means of Schlieren technique. The pressure can be measured at three positions by means of three miniature pressure transducers (Kulite type XCS-093-140mBARD). The pressure transducers are placed in the edge 5 mm from each other along the width of the edge model. The slanted side of the edge model is a 0.4 mm thick plate covering the pressure transducers. Pressure holes of diameter 0.4 mm are made in this plate in order to measure the pressure. The first hole is placed 2.0 mm from the tip of the edge model, the second one 4.0 mm from the the tip and the third one 7.5 mm from the tip in the middle of the slanted side. This is shown in figure 4.13. The pressures measured at the blade edge, will be compared with the pressures obtained in the numerical simulations. In the Schlieren visualisation refractive index contrast is obtained by heating up the edge with an infrared lamp (Philips Infraphil HP 3608) similar to the technique used by Disselhorst (1978). Before each flow visualisation the top plate of the set 96 4. Ghost counts caused by pulsations without main flow p re s s u re tra n s d u c e rs b la d e e d g e p re s s u re tra n s d u c e r w = 9 .9 5 c m lo u d s p e a k e r 1 9 .8 c m (a) to p w a ll p re s s u re tra n s d u c e rs h o t w ire e d g e p re s s u re tra n s d u c e rs p re s s u re tra n s d u c e rs 5 c m p re s s u re tra n s d u c e r S lo u d s p e a k e r 1 V 2 0 .9 c m u '1 u '2 d = 4 .5 c m d = 4 .5 c m V 1 t b la d e = 1 c m 1 7 .4 c m lo u d s p e a k e r 2 1 9 .8 c m 2 2 1 .3 c m (b) Figure 4.12: Drawing (a) shows a 3d picture of the set up and (b) a cross-section of the set up 4.4. Flow around the edge of a blade 97 0 .4 m m 2 2 p 3 .5 e d g e ,1 p 5 p 0 .4 m m e d g e ,2 e d g e ,3 5 t (a) 3 .0 m m b la d e = 1 0 .0 (b) Figure 4.13: Drawing (a) shows a close-up of the pressure transducers in the edge shows (dimensions in mm) and (b) shows a cross-section of the edge model up was removed to allow heating by means of the infra-red lamp. After heating the edge for a few minutes the set up was closed to allow experiments. A flash light, Flashpac 1100, is used as a light source and is triggered by a trigger unit generating a TTL-pulse. The flash duration is typically 20 µs. This signal is used to determine the frequency of the input signal and delayed to obtain a stroboscopic effect. Pictures were taken by a digital high speed camera, the Philips INCA 311, with accurate external triggering. To calculate the velocity in the duct, u′ , we assume a uniform pressure in volume V1 and V2 , so that: Vi dρ′ = −ρ0 Su′ − φloudspeaker , dt i = 1, 2 (4.11) with Vi the volume in one of the two sections of the box, S = d × w the surface area of the connecting duct (see figure 4.12), ρ′ the density fluctuations, t the time, ρ0 = 1.2kg m−3 the ambient density and φloudspeaker is the mass displacement of the loudspeakers. Measurements have been carried out close to the resonance of the set up at frequency f = 120 Hz. The quality factor determined by means of white noise excitation of the setup is q = 3. We neglect φloudspeaker . Using ρ′ = p′ /c20 , with c0 = 344 m/s the speed of sound and p′ the pressure fluctuations, and assuming harmonic flow, we find: u′ = − 2iπf Vi p′ . ρ0 c20 S (4.12) The velocity fluctuations, u′1 , have also been measured using a hot wire anemometer (Dantec type 55P11 wire diameter 5 µm with 55H20 support) placed 11.9 cm from the top wall in the middle of the left duct. The hot wire measurements of the 98 4. Ghost counts caused by pulsations without main flow velocity fluctuations agree within 25% with pressure fluctuation measurements when using equation 4.12. During other measurements the hot wire was removed. The amplitude of the velocity fluctuations, uac = |u′ |, are made dimensionless by means of the Strouhal number, Srtblade = f tblade /uac , with f the frequency and tblade the thickness of the blade. The time scale is related to the pressure in the reservoir V1 , assuming it has a cos (2πf t) time dependence for sin (2πf t) pressure fluctuations. 4.4.3 Measurements Measurement have been obtained for acoustic flows with Srtblade = 0.2, 0.4, 0.8, 1.6 and 3.2. The flow separates at the tip of the edge generating vortices. In figure 4.14, some of the Schlieren pictures are shown. At t/T = 0 the flow starts flowing from V1 to V2 (left to right) and at t/T = 0.5 the flow changes direction and starts flowing from V2 to V1 . The pictures, taken at t/T = 0.3 and 0.8, give an impression of the size of the vortices for Srtblade = 0.2, 0.8, 3.2. For increasing Strouhal number the size of the vortices decreases. From the Schlieren visualizations, it is also observed that depending on the initial conditions the flow can display two different modes (figure 4.15). In the first mode the first vortex is created above the slanted side of the edge model starting at t/T = 0. A second vortex with a circulation of opposite sign is created when the flow changes direction at t/T = 0.5. Both vortices move as a vortex pair away from the edge (figure 4.15(a)). We will refer to this mode as ”mode 1”. In the second mode the first vortex is created starting at t/T = 0.5 next to the edge model. A second vortex of opposite sign is created starting at t/T = 0 above the slated side of the edge. They move away from the edge as a vortex pair over the slanted side (figure 4.15(b)). This mode will be referred to as ”mode 2”. While it is possible to force the flow in mode 2 (figure 4.15(b)), in most experiments mode 1 (figure 4.15(a)) is dominant. At Srtblade = 1.6 the vortices that are created are small and it is no longer possible to sustain mode 2 behaviour during a measurement. At even higher Strouhal number, Srtblade = 3.2, the mode of vortex shedding changed spontaneously during the measurement from one mode to the other and back. These measurements are not included in this discussion. The pressure is measured with the three pressure transducers at the edge, where p′edge,1 , p′edge,2 and p′edge,3 are the pressure transducer closest to the tip of the edge, the second closest and in the middle of the edge, respectively (figure 4.13), and p′top the top wall pressure transducer right above the edge. The signal used to drive the loudspeaker is used to determine the period of oscillation. The pressure signals are phase averaged over 6 × 103 periods. The absolute mean pressure that is established in the set up depends on leaks and is not reproducible. Because of the uncertainties in the mean pressure within the setup, the mean pressure measured at the top 4.4. Flow around the edge of a blade 99 (a) Srtblade =3.2 t/T=0.3 (b) Srtblade =3.2 t/T=0.5 (c) Srtblade =3.2 t/T=0.8 (d) Srtblade =3.2 t/T=1.0 (e) Srtblade =0.8 t/T=0.3 (f) Srtblade =0.8 t/T=0.5 (g) Srtblade =0.8 t/T=0.8 (h) Srtblade =0.8 t/T=1.0 (i) Srtblade =0.2 t/T=0.3 (j) Srtblade =0.2 t/T=0.5 (k) Srtblade =0.2 t/T=0.8 (l) Srtblade =0.2 t/T=1.0 Figure 4.14: Schlieren visualization of the flow for Srtblade = 3.2, 0.8 and 0.2 at t/T = 0.3, 0.5, 0.8 and 1.0. 100 4. Ghost counts caused by pulsations without main flow (a) ”mode 1” (b) ”mode 2” Figure 4.15: Schlieren visualization of the two modes of vortex shedding: a) first vortex is formed on the right side of the edge, the vortex pair moves to the left away from the edge. b) first vortex is formed on the left, the vortex pair moves to the right. has been subtracted from the signal. Figure 4.16 shows the dimensionless pressure, p′ /(ρ0 c0 uac ), for mode 1 at Strouhal number Srtblade = 0.2, 0.4, 0.8 and 1.6. As a reminder small pictures in the plot illustrates the position of the vortices. The pressure fluctuations at the top wall show that there is no perfect standing wave in the set up, because the pressure is not exactly in phase with p1 . The same measurements are found by turning the top plate around 180◦ . Hence this is not due 4.4. Flow around the edge of a blade 101 0 .0 2 0 .0 4 0 0 .0 2 0 a c /(r c 0 u -0 .0 4 d g e ,2 -0 .0 6 p 'e p 'e d g e ,1 /(r c 0 u a c ) ) -0 .0 2 -0 .0 8 S r= S r= S r= S r= -0 .1 -0 .1 2 0 0 .2 0 .4 t/T 0 .6 0 .2 0 .4 0 .8 1 .6 -0 .0 2 -0 .0 4 -0 .0 6 0 .8 -0 .0 8 -0 .1 1 0 0 .2 (a) transducer 1 S r= S r= S r= S r= 0 .0 4 0 .6 0 .8 S r= S r= S r= S r= 0 .2 0 .4 0 .8 1 .6 1 0 .0 1 ) 0 .0 2 0 .0 1 5 0 .2 0 .4 0 .8 1 .6 0 .0 0 5 0 0 p 'to p / ( r c p 'e d g e ,3 /(r c 0 u u a c ) t/T 0 .2 0 .4 0 .8 1 .6 (b) transducer 2 0 .0 6 a c 0 .4 S r= S r= S r= S r= -0 .0 2 -0 .0 4 -0 .0 6 0 -0 .0 0 5 0 -0 .0 1 0 .2 0 .4 t/T 0 .6 (c) transducer 3 0 .8 1 -0 .0 1 5 0 0 .2 0 .4 t/T 0 .6 0 .8 1 (d) top wall transducer Figure 4.16: Dimensionless pressure fluctuations measured for four different Strouhal numbers (Srtblade = 0.2, 0.4, 0.8 and 1.6) at the pressure transducers in the edge model (a,b,c) for mode 1 vortex shedding and at the top of the set up (d). 102 4. Ghost counts caused by pulsations without main flow to a misalignement of the top wall pressure transducer. This is not surprising, because of the asymmetry of the flow caused by the edge model, the absorbtion of sound by the vortices and because of the leakage in the set up. For 0 < t/T < 0.4 the edge pressure is lower for Srtblade = 0.2 than for Srtblade = 1.6. This corresponds to the behaviour that is expected in a potential flow. In order unsteady potential flow the equation of Bernoulli reads: ρ ∂Φ 1 + ρ|~v |2 + p = const(t) ∂t 2 (4.13) R were the flow potential is given by Φ = ~v · d~x. At high Strouhal numbers, such as Srtblade = 1.6, the quadratic term 1/2ρ|~v |2 is almost negligible and we observe an almost harmonic pressure variation (linear behaviour). At low Strouhal numbers, such as Srtblade = 0.2 , for 0 < t/T < 0.4 the non-linear term, 1/2ρ|~v |2 induces a decrease of the local pressure, when there is only local flow separation. For 0.7 < t/T < 1 we see from the flow visualization that there is a strong flow separation at the edge and the pressure trace indicates that the edge force has been suppressed. Considering the average pressure over a period of oscillation we see that the edge pressure is lower than the pressure at the top wall indicating an edge force. Figure 4.17 show the dimensionless pressure, p′ /(ρ0 c0 uac ) measured at these two transducers, for mode 2 behaviours at Srtblade = 0.2, 0.4 and 0.8. As it was not possible to sustain mode 2 vortex shedding behaviour for one measurement, measurements at Srtblade = 1.6 are not included. We observe that on average the edge pressure is lower than the top-wall pressure indicating a net edge force. 4.4.4 Comparing measurements with results of the numerical simulation To be able to compare the measurements with numerical simulations, the geometry of the duct of the set up is modelled. The geometry of the edge and the duct around the edge is discretised using 2800 panels. Around the tip of the edge the density of the panels is increased. The calculations are carried out using 1000 equal time steps per period. The desingularity parameter in the expressions for the velocity induced by a vortex blob was chosen 10 times the time step. Calculations with the blob method for the configuration considered fail to converge to a steady oscillatory solution if vorticity is not removed after some time. Vortex amalgamation methods are difficult to implement. We decided to use a very crude approach. From the flow visualization it is observed that vortex shedding starts close to t/T = 0. At that time the acoustical velocity is reversing from a flow from V1 to volume V2 towards a flow from V2 to V1 . After the reverse of the acoustical velocity at t/T = 0.5, a second vortex is shed containing opposite vorticity. The first and the second vortex travel away as a vortex pair and seem to have little influence 4.4. Flow around the edge of a blade 0 .0 2 103 0 .0 3 S r= 0 .2 S r= 0 .4 S r= 0 .8 0 .0 1 0 .0 1 a c /(r c 0 u -0 .0 1 d g e ,2 -0 .0 2 p 'e d g e ,1 /(r c 0 u a c ) ) 0 p 'e S r= 0 .2 S r= 0 .4 S r= 0 .8 0 .0 2 -0 .0 3 0 -0 .0 1 -0 .0 2 -0 .0 3 -0 .0 4 -0 .0 4 -0 .0 5 -0 .0 5 0 0 .2 0 .4 t/T 0 .6 0 .8 -0 .0 6 1 0 0 .2 (a) transducer 1 0 .6 0 .8 0 .0 1 5 S r= 0 .2 S r= 0 .4 S r= 0 .8 0 .0 4 1 S r= 0 .2 S r= 0 .4 S r= 0 .8 0 .0 1 0 .0 0 5 ) 0 .0 2 0 0 p 'to p / ( r c p 'e d g e ,3 /(r c 0 u u a c ) t/T (b) transducer 2 0 .0 6 a c 0 .4 -0 .0 2 -0 .0 4 -0 .0 6 0 -0 .0 0 5 0 -0 .0 1 0 .2 0 .4 t/T 0 .6 (c) transducer 3 0 .8 1 -0 .0 1 5 0 0 .2 0 .4 t/T 0 .6 0 .8 1 (d) top wall transducer Figure 4.17: Dimensionless pressure fluctuations measured for three different Strouhal numbers (Srtblade = 0.2, 0.4 and 0.8) at the pressure transducers in the edge model (a,b,c) for mode 2 vortex shedding and at the top of the set up (d). 104 4. Ghost counts caused by pulsations without main flow on the next vortex shedding. This allows to use the ”vortex-blob” method, during one single oscillation period, starting without vortices. Figure 4.18 shows the vortex distribution computed by the vortex-blob method. Figure 4.14 and figure 4.18 show a good resemblance. To compare the pressure measured in the set up and the pressures calculated using the vortex blob method, we calculated the difference between the pressure at the three locations on the slanted side of the edge model and the pressure at the top wall. The results are found in figure 4.19 for all three pressure transducers in the edge as function of time. The left side shows the calculations using the blob method and on the right side the measured pressures are shown. The measurements and the numerical simulation show similarities in shape, however the fluctuations in the pressure obtained for the vortex blob method are larger and the peaks are less wide. A comparison is also done for mode 2 vortex shedding. For these calculations t/T = 0 is defined as the start of the flow going from V2 to V1 , the flow starts with the single vortex left of the edge. The difference between the pressures on the edge at the three locations and the top pressure right above the edge is plotted for Srtblade = 0.2, 0.4 and 0.8 in figure 4.20. Although, we find some similarities in the variation with time, the effect of the vortex pair travelling away from the edge has a large effect on the calculations in mode 2. From the prediction of vorticity distribution using the vortex blob method, we can see that while the vortex pair moves away a part of the vortex remains close to the edge. However, the visualisations do not show this vortex left behind. The same effect takes place for mode 1 behaviour (figure 4.18(c,d,g,h and k)). This vortex has less effect on the edge pressure, because it is not close to the edge as in the case of mode 2. 4.5 Conclusions Representing the rotor blade by a flat plate and the flow separation at the sharp edge of the blade by a point vortex, a model is obtained allowing to predict the driving torque on a rotor placed in an oscillatory flow. A simplified model is proposed for high Strouhal number (SrLblade >> 1) which provides an explicit algebraic expression without the need to determine the details of the flow. Comparison between the two models indicates that they are equivalent within the accuracy of the performed experiments. The results show that the thickness of the plate is an important factor for occurrence of spurious counts. The presence of a thick trailing edge in turbine blades increases the critical acoustical pulsation amplitude above which spurious counts appear. The model provides a prediction of the order of magnitude of the critical torque, and can be used to determine typical conditions for the occurrence of 4.5. Conclusions 105 3 3 t/T = 2.5 0.3 t/T = 2.5 0.5 3 3 t/T = 2.5 2 2 2 1.5 0.8 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 −0.5 −0.5 −0.5 −0.5 −1 −1 −1 −1 −1.5 −1.5 −1.5 −1.5 −2 −2 −2 −1 0 1 2 −2.5 (a) Srtblade =3.2 t/T=0.3 3 −1 0 1 2 0.3 −1 0 1 −2.5 2 (c) Srtblade =3.2 t/T=0.8 3 3 t/T = 2.5 −2 −2.5 (b) Srtblade =3.2 t/T=0.5 t/T = 2.5 0.5 0.8 2 1.5 1.5 1.5 1.5 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 −0.5 −0.5 −0.5 −0.5 −1 −1 −1 −1 −1.5 −1.5 −1.5 −1.5 −2 −2 −2 1 2 −2.5 (e) Srtblade =0.8 t/T=0.3 3 −1 0 1 2 (f) Srtblade =0.8 t/T=0.5 0.3 t/T = 2.5 0.5 t/T = 0 1 −2.5 2 (g) Srtblade =0.8 t/T=0.8 −1 1.5 1.5 1.5 1.5 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 −0.5 −0.5 −0.5 −0.5 −1 −1 −1 −1 −1.5 −1.5 −1.5 −1.5 −2 −2 −2 2 (i) Srtblade =0.2 t/T=0.3 −2.5 −1 0 1 2 (j) Srtblade =0.2 t/T=0.5 −2.5 2 t/T = 2.5 2 1 1 0.8 2 0 0 3 t/T = 2.5 2 −1 1 (h) Srtblade =0.8 t/T=1.0 2 −2.5 2 −2 −1 3 3 t/T = 2.5 −2.5 1 2.5 2 0 0 3 t/T = 2.5 2 −1 −1 (d) Srtblade =3.2 t/T=1.0 2 −2.5 1 2 1 −2.5 t/T = 2.5 1 −2 −1 0 1 2 (k) Srtblade =0.2 t/T=0.8 −2.5 −1 0 1 2 (l) Srtblade =0.2 t/T=1.0 Figure 4.18: Prediction of the vortex distribution of the flow for Srtblade = 3.2, 0.8 and 0.2 at t/T = 0.3, 0.5, 0.8 and 1.0. 106 4. Ghost counts caused by pulsations without main flow 0.02 −0.02 −0.08 edge,3 −p −0.06 −0.08 −0.1 (p −0.1 (p edge,3 −0.04 top 0 −0.04 top 0 ac −0.06 )/(ρ c u ) 0 −0.02 −p 0 ac )/(ρ c u ) 0.02 Sr=0.2 Sr=0.4 Sr=0.8 Sr=1.6 −0.12 −0.14 −0.16 0 0.2 0.4 0.6 0.8 Sr=0.2 Sr=0.4 Sr=0.8 Sr=1.6 −0.12 −0.14 −0.16 0 1 0.2 0.4 t/T (a) blob method, pedge,1 0.04 0.02 0.02 −0.06 0 ac )/(ρ c u ) −p edge,2 top −0.04 −0.08 −0.1 Sr=0.2 Sr=0.4 Sr=0.8 Sr=1.6 −0.12 −0.14 0 0.2 0.4 0.6 0.8 −0.04 −0.08 −0.1 Sr=0.2 Sr=0.4 Sr=0.8 Sr=1.6 −0.12 −0.14 1 0 0.2 0.4 t/T Sr=0.2 Sr=0.4 Sr=0.8 Sr=1.6 ac )/(ρ c u ) 0.03 0.02 0 0 ac 1 0.05 Sr=0.2 Sr=0.4 Sr=0.8 Sr=1.6 0.04 )/(ρ c u ) 0.8 (d) measurement, pedge,2 0.05 top −p 0 edge,1 top 0 edge,1 0.01 −0.01 −0.02 (p −p 0.6 t/T (c) blob method, pedge,2 (p 1 0 −0.02 (p (p edge,2 top 0 ac )/(ρ c u ) −p 0 −0.06 0.8 (b) measurement, pedge,1 0.04 −0.02 0.6 t/T −0.03 −0.04 −0.05 0 0.2 0.4 0.6 0.8 t/T (e) blob method, pedge,3 1 −0.05 0 0.2 0.4 0.6 0.8 1 t/T (f) measurement, pedge,3 Figure 4.19: The dimensionless difference between pressure pedge,1 , pedge,2 and pedge,3 and the top wall right above the edge for Srtblade = 0.2, 0.4, 0.8 and 1.6. The mode of vortex shedding is mode 2. On the left side the pressure difference as predicted using the vortex blob method. On the right side the measured pressure difference is shown. 4.5. Conclusions 0.01 0 0 ac 0 −0.02 −p edge,1 −0.04 Sr=0.2 Sr=0.4 Sr=0.8 −0.05 −0.06 0 −0.03 (p top −0.02 top edge,3 −0.01 0 −0.04 )/(ρ c u )/(ρ c u ) −p −0.03 (p ) 0.01 −0.01 ac 107 0.2 0.4 0.6 0.8 Sr=0.2 Sr=0.4 Sr=0.8 −0.05 −0.06 0 1 0.2 0.4 0.02 0.01 0.01 0 ac )/(ρ c u ) top −p −0.03 (p (p −0.02 −0.04 Sr=0.2 Sr=0.4 Sr=0.8 −0.05 −0.06 0 0.2 0.4 0.6 0.8 −0.04 Sr=0.2 Sr=0.4 Sr=0.8 −0.05 −0.06 0 1 0.2 0.4 0.8 1 (d) measurement, pedge,2 0.04 0.03 0.03 0.02 0.02 ac )/(ρ c u ) 0.04 0.01 0.01 0 0 −0.02 (p −0.03 −0.04 0.2 0.4 0.6 0.8 t/T (e) blob method, pedge,3 −0.03 −0.04 Sr=0.2 Sr=0.4 Sr=0.8 −0.05 −0.06 0 top −0.02 0 −0.01 −p top −0.01 edge,1 0 edge,3 ac )/( ρ c u ) (c) blob method, pedge,2 −p 0.6 t/T t/T (p 1 0 −0.01 edge,2 0 −0.03 −p top 0 −0.02 0.8 (b) measurement, pedge,1 0.02 edge,2 ac )/(ρ c u ) (a) blob method, pedge,1 −0.01 0.6 t/T t/T Sr=0.2 Sr=0.4 Sr=0.8 −0.05 1 −0.06 0 0.2 0.4 0.6 0.8 1 t/T (f) measurement, pedge,3 Figure 4.20: The dimensionless difference between the pressure pedge,1 , pedge,2 and pedge,3 and the top wall right above the edge for Srtblade 0.2, 0.4 and 0.8. The mode of vortex shedding is mode 2. On the left side the pressure difference as predicted using the vortex blob method. On the right side the measured pressure difference is shown. 108 4. Ghost counts caused by pulsations without main flow spurious counts in field conditions. In view of its simplicity, the theory for the limit case, SrLblade >> 1, is a useful engineering tool in the prediction of the occurrence of spurious counts. A model describing details of the flow around the sharp trailing edge of a rotor blade is proposed. This model is based on the vortex blob method. In this model we assume that vortex shedding is initiated at the maximum of the pressure (t/T = 0) and that the vortex shed in the second half of a period annihilates the first vortex after one oscillation period (t > T ). The predicted vortex structure agrees qualitatively with the flow visualization and the model predicts reasonably well the dependence of the edge pressure on the Strouhal number. However, to obtain a prediction of the edge force a more accurate numerical model is necessary. 5 Conclusions 5.1 Introduction When used at ideal conditions, gas turbine flow meters allow measurement of the volume flow with an accuracy of about 0.2%. During use systematic changes in response can be induced by wear or damage of the rotor. Systematic errors can also be induced by perturbations in the flow (non-uniformities, swirl or pulsations). This study is limited to the behaviour of turbine flow meters for three different types of flow: • steady flow (Chapter 2) • main flow with acoustic pulsations (Chapter 3) • acoustic pulsation without main flow (Chapter 4) We exclude the swirl and other vortical perturbations of the flow. The acoustical pulsations induce a fluctuation in velocity which is uniform across the rotor. In all these cases the behaviour of the turbine meter is analysed by comparison of the results of analytical models with the results of experiments. Some of these simplified models can be used to apply corrections or as support for design rules in engineering applications. 5.2 Stationary flow An ideal helicoidal turbine meter without mechanical friction nor fluid drag will have a rotational speed, ω, proportional to the volume flow, Q: ω = KQ. In practice the meter constant K determined by calibration will display a slight dependence on 110 5. Conclusions Reynolds number. The aim of the designer is to obtain a meter with an almost constant K on one side but also to achieve a rotor which is robust. This means that K is not strongly affected by wear and other damage of the meter. When a rotor with high solidity is used the flow leaves the rotor in the directions tangentially to the blades. We call this ”full fluid guidance”. We expect that such rotors are less sensitive to wear or impact than rotors with lower solidity. We therefore focus on rotors with high solidity. Results of a two-dimensional model described in chapter 2 have been compared to the calibration data of Elster-Instromet for two different turbine flow meter geometries. The theory assumes that the flow is similar to that in a two-dimensional cascade with full fluid guidance. The theory predicts global trends. Small changes in rotor blade geometry cannot be explained by the model (section 2.6.1). It is clear from measurements that the velocity profile at the inlet of the rotor is Reynolds number dependent (section 2.3). This appears to have a significant effect id on the deviation, E = ω−ω ωid , of the rotation from the ideal behaviour of a helicoidal rotor. Wind tunnel measurements on a model of a rotor blade provide an indication for the influence of the shape of the rotor blades on the drag of the rotor (section 2.4.1). These measurements also show that for a typical rotor blade profile zero angle of incidence with respect to the flow, α = 0, there is a lift force induced by the asymmetric shape of the trailing edge of the blade. This partially explains why the actual rotation speed is higher than the ideal rotation speed, E > 0. It is found that the geometry of the tip clearance between the blades and the pipe wall has an important effect on the deviations from ideal flow (section 2.6.3). By placing the tip of the rotor blade in a cavity in the pipe wall the manufacturers can obtain a rotor constant K which is almost independent of the Reynolds number. A theoretical model of the tip clearance flow effect should therefore be included to obtain a better prediction of the Reynolds number dependence of the flow meter. 5.3 Main flow with pulsations In chapter 3 pulsating flows are investigated for high pulsation frequency. At these high frequencies the inertia of the rotor is so important that a steady rotation speed of the rotor is achieved , so that dω dt = 0. Assuming a quasi-steady behaviour of the flow the model obtained for steady flow ( chapter 2) can be used to predict the effect of 0 pulsations. This quasi-steady flow model predicts that the deviation Epuls = ω−ω ω0 from the steady flow response depends quadratically on the r.m.s value of the relative velocity amplitude Epuls = |u′in |2 u2in , is compared with measurements, where uin is the velocity at the rotor inlet. While significant departure from this model are found, 5.4. Pulsations without main flow 111 the quadratic dependence of the relative measurement error We therefore remains. |u′in |2 . The error caused introduce a quadratic fit parameter a such that Epuls = a u2 in by the pulsation is dependent on Strouhal number (section 3.6.1). The quadratic fit 1 parameter, a, of the measurement errors decreases as SrL5 blade for SrLblade < 2.5. This particular power law is not yet understood. This quadratic fit parameter is not dependent on Reynolds number (section 3.6.2). Measurements at higher Strouhal number were not accurate enough to confirm this dependence at higher Strouhal numbers. The deviation obtained for the superposition of two harmonic perturbations can be predicted by addition of the deviations caused by the two individual perturbations. Tests with a different rotor indicate that the blade shape does not affect the pulsation error, Epuls (section 3.6.4). Determination of the velocity amplitude at the rotor is critical for correcting the measurements. Local measurements of the velocity either by hot wire anemometers or pressure transducers are not reliable (section 3.4.4). More global measurements using multiple pressure transducers in a microphone array set up are necessary. 5.4 Pulsations without main flow The behaviour of turbine meters in a purely oscillatory flow was studied experimentally in chapter 4, with a simplified theoretical model and a discrete vortex model. Pulsations without main flow can induce rotation of rotors with blades with a rounded leading edge and a sharp trailing edge. Experiments using a rotor with sharp trailing and leading edges show that no ghost counts occur, even when the symmetry is broken by an initial rotation of the rotor (section 4.3.2). An explicit equation calculating the edge force on an isolated flat plate using potential flow theory yields an engineering tool to predict the onset of ghost counts (section 4.2.4). This equation can be easily extended to a cascade of blades. Experiments were carried out to obtain more insight into the influence of the thickness of the rotor blade on the edge force. A comparison of the visualisation of the flow around the edge of a blade with the discrete vortex potential flow solution shows reasonable agreement. Similar characteristics in the pressure at the edge are found in the results of the discrete vortex model and the experiments (section 4.4.4). 5.5 Recommendations The aim of this study was to develop and verify simplified analytical models to be used in predicting the behaviour of turbine flow meters at different flow conditions. For steady flow, the two dimensional model is able to explain global effects on 112 5. Conclusions the deviations from real flow. For a designer the model can identify sensitive points in the design, but a better prediction of the Reynolds number dependence can only be achieved by including realistic tip clearance effects in the theory. The quasi-steady flow model predicting the measurement error during pulsating flow provides a reasonable (within 40%) prediction for Strouhal numbers based on the blade tip chord length, SrLblade up to 10. For Strouhal number pulsations up to 2.5 an empirical relation (equation 3.26) can be used to correct the quasi-steady theory for a more accurate prediction. It is recommended to extend this equation to higher Strouhal numbers. This would involve new experiments with more accurate acoustical velocity measurements at high frequencies. Problem in the use of this model in practice is determining the velocity amplitude of the pulsations at the rotor. Placing an array of microphones within the flow meter appears to be the most promising option. Using blades with a rounded leading edge and sharp trailing edge enhances the steady performance of a rotor, but can result in spurious counts in purely oscillating flow. An analytical expression derived in chapter 4 gives an order of magnitude prediction of the onset of spurious counts in purely oscillating flow (without main flow). Using this tool possible problems of ghost counts can be located. By using the cascade theory of Durant (1963) this model can be extended to include interaction between blades. Furthermore, a more accurate determination of the edge force using a numerical method can improve the accuracy of the prediction. APPENDIX A Mach number effect in temperature measurements Assuming that the temperature sensor placed in the meter measures a temperature close to the adiabetic wall temperature, Tw , for a turbulent boundary layer on a flat plate we have (Shapiro, 1953): √ γ−1 2 Tw ≈ 1 + Pr M , T∞ 2 (A.1) where Pr is the Prandtl number, γ is Poisson’s ratio and T∞ is the main flow temperature. In general the meter is calibrated against a series of three other meters placed in parallel, so that the Mach number at the meter being calibrated is about a factor 3 higher than at the reference meters. Assuming Tw ≈ T∞ induces a calibration error in air of the order 0.18M2 . 116 A. Mach number effect in temperature measurements B Boundary layer theory B.1 Introduction This appendix explains some basic concepts of boundary layer theory. It is by no means a complete analysis, only some simplified boundary layer calculations used in this thesis will be addressed. The flow in the turbine flow meters investigated in this thesis operate at flow conditions with high Reynolds numbers, Re = UνL , with U the main stream velocity, L a characteristic length scale and ν the kinematic viscosity. For high Reynolds number flows, Re >> 1, viscous forces can be neglected, except within a thin layer near the wall. Near the wall viscous forces are dominant, causing the fluid to stick to the wall. This thin layer can be described by boundary layer theory, while for the bulk of the flow an inviscid flow method, such as one based on the Euler equations, can be used. An example of a boundary layer is shown in figure B.1. U U u ( x ,y ) d (x ) y x Figure B.1: Boundary layer on a flat plate in parallel flow In the boundary layer the mass conservation and Navier-Stokes equation (equa- 118 B. Boundary layer theory tions 2.1 and 2.2) can be reduced to, ∂u ∂v + =0, ∂x ∂y ∂u ∂u ∂u 1 ∂p ∂2u +u +v =− +ν 2 , ∂t ∂x ∂y ρ ∂x ∂y 1 ∂p − =0, ρ ∂y (B.1) where the x-axis is along the wall and the y-axis is perpendicular to the wall, u is the velocity in the x-direction and v is the velocity in the y-direction. The pressure variation in streamwise direction in the bulk flow is imposed on the boundary layer and can be found using the Euler equation for the bulk flow. For the steady flow along the flat plate this leads to; ∂U (x) ∂p = −ρU (x) (B.2) ∂x ∂x The thickness of this boundary layer, δ, is difficult to determine, because the velocity increases smoothly from zero at the wall and reaches asymptotically to the free stream velocity, U , giving it no exact limit. A well-defined quantity is the displacement thickness, δ1 . The displacement thickness is the distance at which a solid boundary would be placed in order to keep the mass flux equal to the mass flux of the flow with boundary layers; Z δ Z ∞ u u δ1 (x) = 1− dy = dy , (B.3) 1− U U 0 0 where the upper limit of the integrant is extended to infinity, because for y ≥ δ u = U and the integrand is zero. Another useful and well-defined quantity is the momentum thickness, δ2 . The momentum thickness is defined to account for the loss of momentum for the case where the flow has no boundary layer, but is corrected with the displacement thickness and the actual flow; Z ∞ u u 1− dy . (B.4) δ2 (x) = U U 0 For a uniform flow through a channel of height, H, the total momentum flux is equal to ρU 2 (H − 2 (δ1 + δ2 )). For some simple cases an exact solution of the boundary layer equations B.1 can be found, for example the Blasius solution for flat plates (discussed in section B.2), but for most flow problems there are no exact solutions and approximate or numerical methods have to be used. An example of an approximate method is the Von Kármán integral momentum equation, this equation will be discussed in section B.3. B.2. Blasius exact solution for boundary layer on a flat plate B.2 119 Blasius exact solution for boundary layer on a flat plate Blasius considered the boundary layer along a semi-infinite flat plate (Schlichting, 1979) (see figure B.1). The flow has a constant, steady, velocity, U , parallel to the x-axis and there is no pressure gradient. The boundary layer equations B.1 reduce to: ∂u ∂v + =0, ∂x ∂y ∂u ∂2u ∂u +v =ν 2 . u ∂x ∂y ∂y The boundary condition are given by the no-slip condition at the wall, at y = 0 u = v = 0, and the condition of smooth transition from the boundary layer to the main stream velocity, at y = ∞ u = U . Blasius supposed that the dimensionless velocity, Uu , at various distances from the edge is self-similar, i.e. depends on y/δ(x), making it possible to make the variables non-dimensional, using the boundary layer thickness, δ(x) and the main stream velocity, U . The boundary layer thickness based on the solution for a suddenly accelerated plate as derived by Stokes (Schlichting, y 1979) is found to be of the form; δ ∼ νx U . Using η ∼ δ η=y r U , νx (B.5) and introducing a stream function, ψ; √ ψ = νxU f (η) , (B.6) the second-order partial differential equation can be transformed in a third-order ordinary differential equation; 2 d3 f d2 f + f =0, dη 3 dη 2 (B.7) with boundary conditions; η=0 f =0 η→∞ df =1. dη and df =0, dη More details can be found in Schlichting (1979). To solve this equation analytically is difficult and was done by Blasius using power series expansion. 120 B. Boundary layer theory Howarth (1938) solved this equation numerically. With the velocity profile known, the shear stress, τw , on the plate caused by the viscous flow can be found as: r r ∂u U ν ′′ νU = ρU f (0) ≈ 0.332ρU . (B.8) τw (x) = µ ∂y y=0 x x Using equation B.3 the displacement thickness of a boundary layer on a flat plate becomes; r νx . (B.9) δ1 (x) ≈ 1.7208 U B.3 The Von Kármán integral momentum equation As mentioned above for most problems an exact solution cannot be found. For these problems an approximate method can be used. Integrating the momentum equation for steady flow, a global solution can be found; h Z h ∂u ∂u dU ∂2u u +v −U dy = ν 2 dy . ∂x ∂y dx y=0 y=0 ∂y Z (B.10) where h(x) is outside the boundary layer for all values of x. Using the definition of shear stress, τw ; τw = µ ∂u |y=0 , ∂y (B.11) and replacing the normal velocity component, v, using the continuity equation with; Z y ∂u dy , (B.12) v=− 0 ∂x equation B.10 becomes: Z dU τw ∂u ∂u y ∂u − dy − U . dy = − u ∂x ∂y 0 ∂x dx ρ y=0 Z h (B.13) By integrating the second term on the left-hand side by parts, the equation can be rewritten as; Z h y=0 dU ∂ (u (U − u)) dy + ∂x dx Z h y=0 (U − u) dy = τw . ρ (B.14) B.4. Description laminar boundary layer 121 Using the definitions of displacement thickness, δ1 , (equation B.3) and momentum thickness, δ2 , (equation B.4) we get the Von Kármán equation: dU τw d U 2 δ2 + δ1 U = . dx dx ρ (B.15) To solve this equation a velocity profile, Uu , has to be assumed. With this velocity profile the displacement thickness, the momentum thickness and the shear stress at the wall can be determined and the Von Kármán equation can be solved. The next sections will introduce an approximate laminair and turbulent boundary layer descriptions used in this thesis. B.4 Description laminar boundary layer Pohlhausen (1921) used a fourth-order polynomial to describe a laminar boundary layer, however, Hofmans (1998) and Pelorson et al. (1994) found that a third-order polynomial gives a more accurate description of the boundary layer. The velocity in the boundary layer can than be described as: 3 X y i u ai = . U δ (B.16) i=0 To satisfy the no-slip condition at the wall and a smooth transition to the main stream velocity at the edge of the boundary layer the following four boundary conditions are introduced; for y=0: u=0 for y=δ: u=U ν dU ∂2u = −U , 2 ∂y dx ∂u =0. ∂y Introducing the non-dimensional parameter quantity, λ = of the polynomial can be found; a0 = 0 , 3 λ a1 = + , 2 4 λ a2 = − , 2 1 λ a3 = − + . 2 4 δ 2 dU ν dx , the coefficients 122 B. Boundary layer theory Using the definitions for the displacement thickness, δ1 , (equation B.3) and the momentum thickness, δ2 , (equation B.4), it can be found that; 1 3 − λ δ, δ1 = 8 48 39 1 1 3 δ2 = − λ− λ δ. 280 560 1680 If the pressure gradient is neglected, λ = 0, the exact solution of Blasius can be found. Using this third-order polynomial description the shear stress, τw , at the wall can be calculated and is: ∂u U τw = µ (B.17) = a1 µ . ∂y y=0 δ B.5 Description turbulent boundary layer If fully turbulent flow is considered the velocity and pressure components can be separated into a mean motion and a fluctuation, u = ū + u′ v = v̄ + v ′ p = p̄ + p′ . (B.18) To approximate the turbulent boundary layer, we are only interested in average velocities and one has to realize that there are no exact solutions for turbulent flow. Using Prandtl’s mixing-length theory (Schlichting, 1979) it can be calculated, that the velocity profile for a turbulent flow is quite complex composition of logarithmic function and an additional linear layer. However, for many practical applications, it has been shown experimentally that the velocity profile can be approximated by a simple equation in the form of a 1/7th power (Schlichting, 1979); y 1 u 7 . = U δ (B.19) Using this equation the displacement thickness, δ1 , and the momentum thickness, δ2 are; 1 δ1 = δ , 8 7 δ2 = δ . 72 Although this velocity profile is a good approximation, it has some shortcomings. A problem with this profile is that at the wall the gradient of the velocity becomes B.5. Description turbulent boundary layer 123 infinite and the transition to the main flow. As a consequence of this it is impossible to calculate the shear stress caused by the boundary layer on the wall and a empirical relation has to be found. In this thesis, as an approximation the shear stress found for a fully developed turbulent pipe flow; τw = 0.0225ρU 2 ν 1 4 Uδ (B.20) 124 B. Boundary layer theory C Measurements C.1 Introduction The effect of velocity pulsations on flow measurements has been investigated in this thesis by measuring this error at resonance frequencies between 24 Hz and 730 Hz and relative velocity amplitudes, uac /u0 , ranging from about 0.01 to 2. In this appendix the result are represented for the measurements carried out at pulsation frequencies of 24, 117, 360 and 164 Hz. The data is plotted for every pulsation frequency separately using a double logarithmic scale as well as a linear scale. The mainstream velocity at which the measurement is performed are indicated by the different symbols. 126 C.2 C. Measurements Pulsation frequency of 24 Hz 0 quasi−steady theory u0 = 10 m/s relative measurement error, Epuls 10 u = 2 m/s 0 −1 u0 = 5 m/s 10 −2 10 −3 10 −4 10 −2 −1 10 10 relative acoustic amplitude, |u’|/u 0 10 0 Figure C.1: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 24 Hz. Plotted using double logarithmic scale relative measurement error, Epuls 2 quasi−steady theory u0 = 10 m/s u = 2 m/s 1.5 0 u0 = 5 m/s 1 0.5 0 0 0.5 1 1.5 relative acoustic amplitude, |u’|/u 2 0 Figure C.2: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 24 Hz. C.3. Pulsation frequency of 69 Hz C.3 127 Pulsation frequency of 69 Hz 0 quasi−steady theory u0 = 5 m/s relative measurement error, Epuls 10 u = 0.5 m/s 0 −1 u = 1 m/s 10 0 u0 = 1.5 m/s u = 2 m/s 0 −2 10 u0 = 3 m/s −3 10 −4 10 −2 −1 10 10 relative acoustic amplitude, |u’|/u 0 10 0 Figure C.3: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 69 Hz. Plotted using double logarithmic scale relative measurement error, Epuls 2 quasi−steady theory u0 = 5 m/s u = 0.5 m/s 1.5 0 u = 1 m/s 0 u0 = 1.5 m/s u = 2 m/s 1 0 u0 = 3 m/s 0.5 0 0 0.5 1 1.5 relative acoustic amplitude, |u’|/u 2 0 Figure C.4: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 69 Hz. 128 Pulsation frequency of 117 Hz relative measurement error, E puls C.4 C. Measurements quasi−steady theory u = 3 m/s 0 −1 10 u = 1 m/s 0 u = 2 m/s 0 −2 10 −3 10 −4 10 −2 −1 10 10 relative acoustic amplitude, |u’|/u 0 10 0 Figure C.5: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 117 Hz. Plotted using double logarithmic scale 0.5 relative measurement error, E puls quasi−steady theory u = 3 m/s 0 0.4 u0 = 1 m/s u = 2 m/s 0 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 relative acoustic amplitude, |u’|/u 1 0 Figure C.6: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 117 Hz. C.5. Pulsation frequency of 363 Hz C.5 129 Pulsation frequency of 363 Hz 0 10 0 puls relative measurement error, E quasi−steady theory u = 15 m/s u = 1 m/s 0 −1 10 u0 = 2 m/s u = 5 m/s 0 −2 u = 10 m/s 0 10 −3 10 −4 10 −2 10 −1 10 relative acoustic amplitude, |u’|/u 0 10 0 Figure C.7: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 360 Hz. Plotted using double logarithmic scale 0.5 relative measurement error, E puls 0.45 quasi−steady theory u = 15 m/s 0 0.4 u = 1 m/s 0.35 u0 = 2 m/s 0.3 0.25 0 u = 5 m/s 0 u = 10 m/s 0 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 relative acoustic amplitude, |u’|/u 1 0 Figure C.8: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 360 Hz. 130 Pulsation frequency of 730 Hz relative measurement error, Epuls C.6 C. Measurements −1 10 quasi−steady theory u0 = 15 m/s u0 = 1 m/s u0 =2 m/s −2 10 u0 = 5 m/s u0 = 10 m/s −3 10 −4 10 −2 10 −1 10 relative acoustic amplitude, |u’|/u 0 10 0 Figure C.9: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 730 Hz. Plotted using double logarithmic scale 0.5 relative measurement error, Epuls 0.45 0.4 0.35 0.3 quasi−steady theory u0 = 15 m/s u0 = 1 m/s u =2 m/s 0 u0 = 5 m/s u0 = 10 m/s 0.25 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 relative acoustic amplitude, |u’|/u0 1 Figure C.10: The relative measurement error, Epuls , as a function of the relative amplitude of the pulsations, |u′ |/u0 , for measurements at a pulsation frequency of 730 Hz. D Force on leading edge Using the transformation of Joukowski (see equation 4.1), the value of dξ/dz can be calculated close to the leading (singular) edge. lim z→−2A " # √ 1 z i A dξ = lim + p = lim √ (D.1) z→−2A 2 z→−2A 2 z + 2A dz 2 (z − 2A)(z + 2A) where A is the radius of the circle in the ξ-plane. Because near the edge (z → dξ becomes very large and the second term becomes the dominant term and −2A) dz equation D.1 is obtained. The potential of the flow is given by equation 4.2. With this equation and the Kutta condition imposed at the trailing edge, z = 2A, (see equation 4.3) the circulation, Γv , can be found: Γv = 4πuac sin α A(A − ξv )(ξv∗ − A) ξv ξv∗ − A2 (D.2) where uac is the acoustic oscillation amplitude, α is the incidence of the flow and ξv is the position of the vortex in the transformed plane. Using the flow potential, Φ and the circulation Γv , dΦ/dξ close to the leading edge can be calculated: dΦ dξ ξ→−A 1 ξv∗ iΓv + = uac e − uac e − 2π −A − ξv −Aξv∗ − A2 (A − ξv )(A − ξv∗ ) (D.3) = −2iuac sin α 1 + (A + ξv )(A + ξv∗ ) −iα iα By combining equations D.1 and D.3 it follows that: 132 D. Force on leading edge dΦ dz z→−2A dΦ dξ = dξ ξ→−A dz z→−2A √ (A − ξv )(A − ξv∗ ) 1 √ = uac A sin α 1 + (A + ξv )(A + ξv∗ ) z + 2A (D.4) The force on the edge can be found by utilising Blasius’ theorem for the force, i.e. evaluating an integral around the closed contour ǫ iρ Fx − iFy = lim 2 ǫ→0 I ǫ dΦ(z) dz 2 dz (D.5) where Fx denotes the force parallel to the plate (the edge force) and Fy represents the force perpendicular to the plate. Using the Cauchy integral theorem, the force becomes: Fx = Fe (A − ξv )(A − ξv∗ ) 2 1+ = (A + ξv )(A + ξv∗ ) 2 A2 + ξv ξv∗ 2 2 = −4πρuac A sin α (A + ξv )(A + ξv∗ ) −πρu2ac A sin2 α (D.6) Bibliography Allam, S. and Åbom, M. (2006). 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Proceedings Flomeko, Groningen. Wadlow, D. (1998). Chapter 28.4 turbine and vane flowmeters. In Webster, J., editor, The Measurement, Instrumentation and Sensors Handbook. CRC Press, Boca Raton, FL. Weinig, F. (1964). Theory of two-dimensional flow through cascades. In Hawthorne, W., editor, Aerodynamics of turbines and compressors, volume X of High speed aerodynamics and jet propulsion, pages 13–82. Princeton University Press. Summary Dynamics of turbine flow meters Axial turbine flow meters are used in applications, in which accurate volume flow measurements are desired. When used at ideal conditions, turbine flow meters allow gas flow measurements with an accuracy of about 0.2%. The aim of our research is to develop engineering tools allowing to design robust and accurate turbine flow meters. Ideally, the volume flow is proportional to the rotation speed. However, slight deviations are observed as a function of the Reynolds number. In designing a flow meter, the object is to make the dependence on the Reynolds number as small as possible and to create a robust meter, not sensitive to wear nor damage. In this thesis we describe a two-dimensional analytical model to predict the deviation of the rotation speed of the rotor from the rotation speed of an ideal helical rotor in an ideal flow, without mechanical friction or fluid drag. By comparing the results of the model with calibration data of turbine flow meters provided by Elster-Instromet, we find that the theory can explain global effects of the inlet velocity profile, the pressure drag of the wake and other friction forces. For a more accurate model it is necessary to include realistic tip clearance effects and the additional lift induced by the shape of the trailing edge. Another important source of systematic errors are time dependent perturbations in the flow. We investigate the effect of pulsations at high frequencies on the rotation speed of the rotor. Above a critical frequency, determined by the inertia of the rotor, the turbine meter will not be able to follow the variations in volume flow in time. Instead an average rotation speed will be established. Due to non-linearities of the forces exerted by the flow on the turbine blades this rotation speed corresponds to a higher steady volume flow than the actual time-averaged flow. By assuming quasi-steady behaviour of the flow for the model obtained for ideal steady flow described above, the relative error in volume flow is equal to the root-mean-square of the ratio of acoustical to main flow velocities. The range of validity of this predic- 138 Summary tion has been explored experimentally for harmonic pulsations. Although significant deviations from the quasi-steady model were found, the quadratic dependence on the velocity amplitude appears to remain valid for all measurements. The exact quadratic dependence is a function of the Strouhal number of the pulsations. In the range of Strouhal numbers below 2.5, based on the blade chord length at the tip of the rotor blade and the flow velocity at the rotor inlet plane, we find a slow decrease of the error with increasing Strouhal number following a power 15 of Strouhal. Measurements at high Strouhal numbers were not reliable enough to confirm this dependence for higher Strouhal numbers. The deviation obtained for the superposition of two harmonic perturbations can be predicted by addition of the deviations caused by the two individual perturbations. For the extreme case in which we have pulsations but no time-averaged main flow, it is possible that the rotor starts rotating above a critical amplitude of the pulsations. These surious counts or ”ghost counts” are caused by the shape of the rotor blades. Using potential flow theory for thin rotor blades an explicit equation is developed to predict the onset of the ghost counts. This equation is verified by experiments to be an engineering tool allowing to predict the order of magnitude of the onset of spurious counts. A more accurate prediction of the onset of spurious counts can be found by studying the flow around the rotor blades. Experiments were performed to measure the pressure at three different locations on a scale model of the edge of a rotor blade and to visualise the flow around this edge. These measurements and visualisations show reasonable agreement with the results of a discrete vortex blob model. Samenvatting Dynamica van turbinedebietmeters Wanneer het nodig is om het volumedebiet nauwkeurig te meten, worden vaak axiale turbinedebietmeters toegepast. Onder ideale omstandigheden kunnen turbinedebietmeters voor gasstromingen een nauwkeurigheid bereiken tot 0.2%. Het doel van dit onderzoek is om technologische applicaties te ontwikkelen die het mogelijk maken robuuste en nauwkeurige meters te ontwerpen. In het ideale geval is het volumedebiet rechtevenredig met de rotatiesnelheid van de rotor. In praktijk echter treden er kleine afwijkingen op als functie van het Reynoldsgetal. Bij het ontwerpen van een turbinemeter is het doel om deze afhankelijkheid van het Reynoldsgetal zo klein mogelijk te maken. Tegelijkertijd moet de turbinemeter ook robuust zijn, zodat hij niet gevoelig is voor slijtage en kleine beschadigingen. In dit proefschrift wordt een twee-dimensionaal analytisch model beschreven dat het mogelijk maakt afwijkingen te voorspellen van de rotatiesnelheid van de rotor met de rotatiesnelheid van een ideale rotor, waarbij de bladen de vorm hebben van een ideale helix, die ronddraait zonder mechanische wrijving of stromingswrijving. De resultaten van dit model zijn vergeleken met ijkmetingen die beschikbaar zijn gesteld door Elster-Instromet. Uit deze vergelijking kunnen we concluderen dat het model de globale effecten van het instroomprofiel, van de drukweerstand van het zog en van andere wrijvingskrachten verklaart. Om de invloed van de ruimte tussen de uiteinden van de rotorbladen en de wand te verklaren is een uitgebreider model nodig. Dit geldt ook voor het effect van de additionele liftkracht die opgewekt wordt door de vorm van de achterkant van de rotorbladen. Een andere belangrijke bron van systematische fouten zijn tijdsafhankelijke verstoringen in de stroming. Het effect van pulsaties met hoge frequenties op de rotatiesnelheid van de rotor is onderzocht. Boven een kritische frequentie, die bepaald wordt door de massatraagheid van de rotor, is de rotor niet meer in staat de variaties in het volumedebiet te volgen. In plaats hiervan stelt zich een gemiddelde rotatiesnelheid 140 Samenvatting in. Door niet-lineariteit van de krachten die door de stroming worden uitgeoefend op de turbinebladen, correspondeert deze rotatiesnelheid met een volume debiet dat groter is dan het tijdsgemiddelde debiet. Door aan te nemen dat het gedrag van de stroming quasi-stationair is en door gebruik te maken van het model dat hierboven is beschreven, vinden we dat de fout in het gemeten volumedebiet gelijk is aan de rms-waarde van de verhouding tussen de akoestische snelheid en de hoofdstroomsnelheid. We hebben de grenzen onderzocht waarbinnen dit model voor harmonische pulsaties geldig is. Alhoewel we significante afwijkingen van de quasi-stationaire theorie hebben gevonden, lijkt de kwadratische afhankelijkheid van de snelheidsverhouding geldig te blijven. De exacte kwadratische afhankelijkheid is een functie van het Strouhalgetal. Voor Strouhalgetallen, gebaseerd op de koorde van de bladen aan de uiteinden en de snelheid bij het binnengaan van de rotor, kleiner dan 2.5, vinden we een langzame afname van de fout met toenemend Strouhalgetal met een macht 51 van het Strouhalgetal. Metingen voor hogere Strouhalgetallen bleken niet voldoende betrouwbaar om deze trend te bevestigen voor hogere Strouhalgetallen. De afwijking voor twee gesuperponeerde harmonische verstoringen kan voorspeld worden door de afwijkingen van de twee afzonderlijke verstoringen bij elkaar op te tellen. Voor extreme situaties waar pulsaties optreden, maar er geen tijdsgemiddelde stroming is, is het mogelijk dat de rotor begint te roteren als er pulsaties zijn met een amplitude hoger dan een kritische amplitude. Deze foutieve tellingen of ”spooktellingen” worden veroorzaakt door de vorm van de rotorbladen. Door gebruik te maken van potentiaaltheorie voor een stroming om een oneindig dunne rotorblad, kan een expliciete uitdrukking gevonden worden, die de aanvang van spooktellingen kan voorspellen. Door middel van experimenten is deze uitdrukking geverifiëerd en kan hij worden toegepast om een orde-grootte-voorspelling te doen voor de aanvang van foutieve tellingen. Een nauwkeurigere voorspelling kan gevonden worden door de stroming om een rotorblad te bestuderen. Een experimentele opstelling is gebouwd om de druk op drie verschillende locaties op een schaalmodel van de rand van een rotorblad te meten en om de stroming rondom de rand te visualiseren. Deze metingen en visualisaties laten een redelijke overeenkomst zien met de resultaten van een discreet wervel-model. Dankwoord Het is natuurlijk een cliché te schrijven dat het proefschrift dat hier ligt niet het werk is van alleen één persoon, maar natuurlijk is dat ook in mijn geval zeer zeker waar. En ik ben dan ook blij dat ik op deze plaats personen kan noemen die een bijdrage hebben geleverd aan de totstandkoming van dit proefschrift. In de eerste plaatst mijn promotor Mico Hirschberg. Mico, bedankt voor je begeleiding, je creativiteit en al je ideeën waarmee je me vaak hebt gemotiveerd en soms tot waanzin hebt gedreven. Met erg veel plezier heb ik al die jaren met je samengewerkt. Speciaal voor jou wilde ik dit proefschrift niet afsluiten zonder een varken. Harry Hoeijmakers, mijn 2de promotor, bedankt voor het het grondig lezen van het concept. Dank je wel, Rini van Dongen; voor je interesse en je raad gedurende de afgelopen jaren. Het project waar dit proefschrift uit is voort gekomen is gefinancierd door STW. Ik wil dan ook graag de gebruikerscommissie bedanken voor haar aandacht en input. In het bijzonder Jos Bergervoet van Elster-Instromet. Jos, bedankt voor je medewerking en voor het delen van je jarenlange ervaring met turbine debietmeters. Ook wil ik graag Henk Riezebos bedanken voor zijn medewerking en René Peters, die mij de gelegenheid heeft gegeven om twee maanden bij TNO te komen werken. Het waren een leuke en leerzame twee maanden. Stefan Belfroid, dank je wel; twee blobbers weten meer dan één. Ik ben erg dankbaar voor de technische steun die ik heb gekregen. Mijn gebrek aan experimentele ervaring is uitstekend opgevangen door de jarenlange ervaring van Jan Willems. Bedankt Jan, het was erg prettig gebruik te kunnen maken van al je kennis, waarmee het altijd weer mogelijk was dat in elkaar te zetten wat nodig was. Dank je wel, Freek van Uittert, voor al de tijd die je in de meetsystemen heb gestopt. Dit heeft in ieder geval als resultaat gehad dat onze opstelling de meeste computers had van alle opstellingen. Ad Holten, bedankt voor je hulp als één van al deze com- 142 Dankwoord puters raar deed en Freek niet aanwezig was en voor je hulp met de optica. Ook wil ik Remi Zorge en Herman Koolmees bedanken voor hun medewerking. Daarnaast Gerald Oerlemans; bedankt dat je bereid was alle duct tape te trotseren, toen andere technici niet beschikbaar waren. Natuurlijk ben ik ook het secretariaat dank verschuldigd voor het helpen met het administratieve werk. Dank je wel, Brigitte. Merci, Marjan; ik heb erg veel plezier gehad met het hoteltesten en met het voorbereiden van de borrels. Daarnaast heb ik ook het geluk gehad om samen te werken met veel studenten. Dank je wel, Wendy Versteeg, Sergio Aurajo, Jan Küchel, Arjen Hamelinck, Erwin Engelaar, Martijn de Greef, Bram van Gessel, Floor Souren en Ineke Wijnheijmer. De afgelopen (ruim) vier jaar waren niet zo leuk geweest zonder het gezelschap van al mijn collega’s en oud-collega’s. Het is waarschijnlijk niemand echt ontgaan dat ik een liefhebber ben van koffiepauzes (of eigenlijk theepauzes), met name vanwege de discussies en gezelligheid tijdens die pauzes. Dank je wel, Marleen, Werner, Gerben, Geert, Ralph, Gabriel, Jieheng, Vincent, Dima, Laurens, Thijs, Ruben, Rudie, Rinie, Lorenzo, Andrzej, Alejandro, Matı́as, Jurriën, Daniel, Dennis, John, David, Moasheng, Paul, Elke, Gert Jan, Willem, Herman, Gerard, Gert en de personen die al eerder genoemd zijn. Vrienden en familie zijn voor mij erg belangrijk geweest. Niet alleen vanwege de interesse die zij hebben getoond in mijn onderzoek, maar ook vanwege de nodige afleiding die ze me hebben gegeven. Dank jullie wel, allemaal. In het bijzonder Saskia, Remko, Roy, Yvonne en Marijn, bedankt. Ook wil ik mijn ouders noemen: pap en mam, bedankt voor alle steun door de jaren heen. Als laatste wil ik graag degene bedanken die me dagelijks tot steun is geweest en degene die het meest geleden heeft naast mij als het onderzoek niet goed ging. Lieve Patrick, dank je wel. Curriculum Vitae 13 May 1977 Born in Heino, The Netherlands. 1989 - 1995 Stedelijk Gymnasium, Leeuwarden. 1995 - 2002 Student Mechanical Engineering, University of Twente, Engineering Fluid Dynamics Group. • Traineeship at NASA Langley Research Center, Hampton, Virginia, USA. Application of Vortex Confinement on Unstructured Grids • Master thesis: Aerosol Depositions in Lungs awarded the Unilever Researchprijs. 2002 - 2007 PhD Research at the Gas Dynamics Group, Department of Applied Physics, Eindhoven University of Technology. March - May 2003 Visit at TNO Delft, Department of Fluid and Structural Dynamics.

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