Dynamics of Turbine Flowmeter

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Dynamics of turbine flow meters


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Copyright c2007 P.W. StoltenkampCover 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 924Trefwoorden: stromingsleer / pulserende stromingen / debietmeters / meetfouten

Subject headings: flow of gases / volume flow measurements / turbine flow meters /

pulsatile flow / systematic errors


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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


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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


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Contents

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . viii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 General description of a gas turbine flow meter . . . . . . . . . . . 1

1.3 Ideal rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Parameter description . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Reynolds dependency of turbine flow meter readings . . . . . . . . 6

1.6 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. Turbine flow meters in steady flow . . . . . . . . . . . . . . . 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Theoretical models of turbine flow meters . . . . . . . . . . . . . . 102.2.1 Momentum approach . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Airfoil approach . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3 Equation of motion . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Effect of non-uniform flow . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Boundary layer flow . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Velocity profile measurements . . . . . . . . . . . . . . . . 20

2.3.3 Fully turbulent velocity profile in concentric annuli . . . . . 21

2.3.4 Comparison of the different velocity profiles . . . . . . . . 23

2.3.5 Effect of inflow velocity profile on the rotation . . . . . . . 24

2.4 Wake behind the rotor blades . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Wind tunnel experiments . . . . . . . . . . . . . . . . . . . 282.4.2 Effect of wake on the rotation . . . . . . . . . . . . . . . . 29

2.5 Friction forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.1 Boundary layer on rotor blades . . . . . . . . . . . . . . . . 32

2.5.2 Friction force on the hub . . . . . . . . . . . . . . . . . . . 33


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vi

2.5.3 Tip clearance . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.4 Mechanical friction . . . . . . . . . . . . . . . . . . . . . . 37

2.6 Prediction of the Reynolds number dependence in steady flow . . . 38

2.6.1 Turbine meter 1 . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6.2 Turbine meter 2 . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6.3 Effect of tip clearance . . . . . . . . . . . . . . . . . . . . 45

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3. Response of the turbine flow meter on pulsations with main flow . . 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Theoretical modelling . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 A basic quasi-steady model: A 2-dimensional quasi-steady

model for a rotor with infinitesimally thin blades in incom-

pressible flow . . . . . . . . . . . . . . . . . . . . . . . . 503.2.2 Practical definition of pulsation error . . . . . . . . . . . . 52

3.3 Experimental set up . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Determination of the amplitude of the velocity pulsations at the loca-

tion of the rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Acoustic model . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.2 Synchronous detection . . . . . . . . . . . . . . . . . . . . 59

3.4.3 Verification of the acoustic model . . . . . . . . . . . . . . 59

3.4.4 Measurements of velocity pulsation in the field . . . . . . . 64

3.5 Determination of the measurement error of the turbine meter . . . . 65

3.6 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6.1 Dependence on Strouhal number . . . . . . . . . . . . . . . 693.6.2 Dependence on Reynolds number . . . . . . . . . . . . . . 72

3.6.3 High relative acoustic amplitudes . . . . . . . . . . . . . . 73

3.6.4 Influence of the shape of the rotor blades . . . . . . . . . . 74

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4. Ghost counts caused by pulsations without main flow . . . . . . . 794.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Onset of ghost counts . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.1 Theoretical modelling of ghost counts . . . . . . . . . . . . 80

4.2.2 Experimental setup for ghost counts . . . . . . . . . . . . . 87

4.2.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 894.2.4 Comparing measurements with results of the theory . . . . . 91

4.3 Influence of vibrations and rotor asymmetry . . . . . . . . . . . . . 93

4.3.1 Vibration and friction . . . . . . . . . . . . . . . . . . . . . 93

4.3.2 Rotor blades with chamfered leading edge . . . . . . . . . . 93


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vii

4.4 Flow around the edge of a blade . . . . . . . . . . . . . . . . . . . 94

4.4.1 Numerical simulation . . . . . . . . . . . . . . . . . . . . . 94

4.4.2 Experimental set up for flow around an edge . . . . . . . . 95

4.4.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4.4 Comparing measurements with results of the numerical sim-

ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Stationary flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3 Main flow with pulsations . . . . . . . . . . . . . . . . . . . . . . . 110

5.4 Pulsations without main flow . . . . . . . . . . . . . . . . . . . . . 111

5.5 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Appendix 113

A. Mach number effect in temperature measurements . . . . . . . . 115

B. Boundary layer theory . . . . . . . . . . . . . . . . . . . . 117

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

B.2 Blasius exact solution for boundary layer on a flat plate . . . . . . . 119

B.3 The Von Kármán integral momentum equation . . . . . . . . . . . . 120

B.4 Description laminar boundary layer . . . . . . . . . . . . . . . . . 121B.5 Description turbulent boundary layer . . . . . . . . . . . . . . . . . 122

C. Measurements . . . . . . . . . . . . . . . . . . . . . . . 125

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

C.2 Pulsation frequency of 24 Hz . . . . . . . . . . . . . . . . . . . . 126





D. Force on leading edge . . . . . . . . . . . . . . . . . . . . 131

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . 133

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 136


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viii

Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . 139

Dankwoord . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . 143


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Nomenclature

Roman symbols lowercase

a quadratic fit parameter equation 3.25

c0 speed of sound m s−1

f frequency Hz

hblade height of a rotor blade m

k wave number m−1

m′ mass flow kg s−1

n normal unit vector

n number of blades

p pressure P a

p′ pressure fluctuations P a

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 m or s

tblade blade thickness m


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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

A cross-sectional area m2

B′ total specific enthalpy m2 s−2

D pipe diameter m

E relative deviation from ideal rotation equation 2.14

E puls relative error caused by periodic pulsations equation 3.11

F bf force imposed on the fluid by the body N

F D drag force N

F e edge force N

F L lift force N

I rotor moment of inertia of the rotor kg m2

K meter factor m3 rad−1

Lblade chord length of a rotor blade m

Lhub length of the hub in front of the rotor m

Q volume flow m3

s−1

R root-mean-square radius

r2in+r

2out

2 m

S pitch or area m or m2


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xi

T temperature or period of the pulsations K or s−1

T mech mechanical friction torque kg m2s−2

T air air friction torque kg m2s−2

T bf torque imposed on the fluid by the body kg m2s−2

T d driving torque kg m2s−2

T f total friction torque kg m2s−2

U mean velocity in the annulus in front of the rotor m s−1

V volume m3

W width of the rotor m

Greek symbols

α angle of attack ◦

αd damping coefficient m−1

β angle of rotor blade with resect to the rotor axis ◦

β av average of the angle of the rotor blades at the root-mean-square radius ◦

δ 1 displacement thickness m

δ 2 momentum thickness m

Φ complex potential m2s−1

φm mass flow kg s−1

Γ circulation m2s−1

γ Poisson’s ratio

µ dynamic viscosity kg m−1s−1

ν kinematic viscosity m2 s−1

ω rotation speed rad s−1

ωid ideal rotation speed rad s−1


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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−1s−2

τ w shear stress at the wall kg m−1s−2

Dimensionless numbers

C D drag coefficient, F D/(12 ρu

2A)

C ′D drag coefficient, F D/(12 ρu

2wt)

C L lift coefficient, F L/( 12 ρu2A)

He Helmholtz number, fLc0

M Mach number, uc0

Pr Prandtl number, ν/a with a the thermal diffusivity

Re Reynolds number, uLν

Sr Strouhal number, fLu


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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 applica-

tions, 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 verylarge, 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 thisdrawing 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 man-

ifolds, 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


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2 1. Introduction

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 ro-

tor (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


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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 Elster-

Instromet. The dynamical response measurements have been carried out at the Eind-

hoven 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 Elster- Instromet).

flow measurement is 0.1% for volume flows in the range from 20 to 400 m3/h. Themeter is designed for pressures ranging from atmospheric pressure up to 20 bar (thistype of meter is also available for work pressures up to 100 bar). The rotor is madeof aluminium and has helical shaped blades (see figure 1.2). We will refer to this me-

ter as turbine meter 1. Additional steady flow experiments have been performed byElster-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.


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4 1. Introduction

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. Underthese 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 tothe 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 rotoris 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:

ωidr

uin= tan β , (1.2)


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1.4. Parameter description 5

with ωid the angular velocity of the rotor for the ideal situation considered and uin isthe velocity of the flow entering the rotor. The angular velocity in this ideal situation

is

ωid = uin tan β

r =

2πuinS

. (1.3)

Because the volume flow, Q, is equal to the inflow velocity multiplied by the cross-sectional area of the rotor, i.e. Q = uinA, we find a relationship between the volumeflow and the rotational speed:

Q = AS

2π ωid . (1.4)

This relationship is applied in an actual turbine flow meter in the form:

Q = K ωid , (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 dimen-sionless parameters such as:

• the Reynolds number Re = uinL

ν

• the Mach number M = uinc0

• the ratio of mechanical friction torque, T mech, to the driving fluid torque T mechR3ρu2inwhere L is a characteristic length such as the blade chord length, ν is the kinematicviscosity 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 atdifferent 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 = fLuin


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6 1. Introduction

• the amplitude of the perturbations |u′in|uin

• The ratio of fluid density, ρ, and rotor material density, ρm, i.e. ρρmwhere f is the characteristic frequency of flow perturbations and |u′in| is the ampli-tude 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 bythe 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 aim-ing 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 inves-

tigated 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 the-

oretical 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 cali-bration 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


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1.6. Thesis overview 7

theory predicting these errors, is discussed. Measurements are performed to investi-

gate the applicability of this model. A detailed description is given of the measure-

ment 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.


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8 1. Introduction


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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 un-

derstand 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 fluidproperties. Rather than considering a numerical method we will consider an exten-

sion 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, T d, 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 aboundary 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


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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 pro-

vided 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 two-

dimensional 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 momen-

tum 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 approxima-

tion of the flow inside an annulus between r and r + dr. The x-direction refers to theaxial direction. The y-direction refers to the azimuthal direction (see figure 2.2). Theradial velocity is neglected and constant rotation with a rotational angular velocity ωis 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.


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2.2. Theoretical models of turbine flow meters 11

Figure 2.1: The rotor of the turbine flow meter. We assume that the flow in an annulus be-

tween r and r + d r behaves as the flow in a two-dimensional infinitely longcascade 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:

d

dt CV ρdV + CS ρv · ndA = 0 , (2.1)d

dt

CV

ρvdV +

CS

ρv (v · n) dA = −

CS

pndA +

CS

τ ndA + F bf ,(2.2)

applied to a fixed control surface C S enclosing the rotor, this surface has an outernormal 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 arethe 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 incompressibleflow and applying the conservation of mass (equation 2.1) to a volume element of

height dr (figure 2.1), we get:

uin,xdAin = uout,xdAout , (2.3)


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Figure 2.2: Flow entering and leaving the cascade representing the rotor in an annulus be-

tween r and r + d r.

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, respec-tively. If the inflow and outflow area are assumed to be equal and the flow is incom-

pressible, dAin = dAuit = 2πrdr, so that the x-component of the incoming velocityis 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 throughan element dr becomes:

ρ ((uout,y + ωr) uout,xdAout − uin,xωrdAin) = dF bf,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)


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Substituting equations 2.3 and 2.5 in equation 2.4, the y-component of the force

imposed by the rotor on the fluid, dF bf,y is found:

dF bf,y = ρu2out,x tan β dAout − uin,xωrdAin . (2.6)

The force of the fluid on the rotor is opposite and equal to the force of the rotor on the

fluid, dF bf,y = −dF fb,y. The torque exerted by the fluid element on the rotor axis,dT d, is estimated to be:

dT d = rdF fb,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:

T d = − rtiprhub

ρu2out,x(tan β )rdAout + rtiprhub

ρuin,xωr2dAin . (2.8)

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 aninfinite 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, F L, acts perpendicularto the relative inlet velocity, uin,rel = (uin,x, ωr), and the drag force, F D acts parallel

to this inlet velocity. The y-component of the force of the flow on the blade can nowbe expressed in terms of lift, F L, and drag, F D;

F y = n (−F L cos φ + F D sin φ) , (2.9)

where φ = β −α = arctan

ωruin,x

, with β the angle of the rotor blade (with respect

to the x-axis), n is the number of blades and α the angle of attack of the incomingflow. The lift- and drag coefficient are defined as:

C L = F L

12 ρu

2in,relLblade

,

C D = F D12 ρu

2in,relLblade

,

(2.10)

where Lblade is the chord of the blade. The lift and drag coefficients are functionsof the angle of attack, α, depend weakly on Reynolds number and on Mach number.


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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:

T d =

rtiprhub

1

2nρu2in,relLblade (−C L cos φ + C D sin φ) rdr . (2.11)

2.2.3 Equation of motion

The driving torque, T d, is known from equation 2.8 or 2.11. To determine the angularvelocity, ω, of the rotor, the equation of motion of the rotor is used:

I rotordω

dt = T d − T f , (2.12)

where I rotor is the moment of inertia of the rotor and T f is the friction torque onthe rotor, assuming a quasi-steady flow through the rotor. Using equation 2.8 or 2.11


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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:

T d = 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 thisrotation 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 tur-bine meter can be estimated.

In the following sections the analysis will be applied using the momentum ap-

proach (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 mused 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 turbinemeter 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 ge-

ometry 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:

Lblade(r) = W

cos β (r) , (2.15)

with β = arctan

2πrS

the angle of the blade relative to the rotor axis. The blades

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 metercan be written as:

Lblade(r) = Lblade(rhub) + Lblade(rtip)− Lblade(rhub)

hblade(r − rhub) . (2.16)

In the following sections the effect of non-uniform flow, the blade drag and otherfriction forces are investigated separately, the deviation from the ideal rotation is

calculated for several flows up to Qmax as indicated for the meter. Two scenarioswere 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


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turbine meter 1 turbine meter 2

pipe diameter, D (m) 0.1034 0.2030

blade thickness, t (mm) 1.6 4

number of blades, n 16 14

rhub/D 0.360 0.250

rout/D 0.500 0.500

S/D 2.704 3.941

W/D 0.213 0.148

hblade/D = (rtip − rhub)/D 0.140 0.240Lhub/D 0.763 1.049

Table 2.1: Dimensions of the two turbine flow meters used in the calculations, where rhub isthe 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, hbladeis 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 , allvalues 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 tothe test conditions used by Elster-Instromet. The resulting deviation, E , is plottedagainst the Reynolds number, Re = U Lblade/ν , where Lblade is the length of a rotorblade 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 approxi-

mation, 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. How-

ever, 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 tan-

gential 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


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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 im-

portant 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). Thiswould 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 Reynoldsnumber the boundary layer is turbulent from the start, ignoring the effect of transition.

The boundary layer thickness is calculated using the von Kármán integral mo-

mentum 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:

d

dx

U 2δ 2

+ δ 1U

dU

dx =

τ wρ

, (2.17)


30/155


Figure 2.4: The three different regions of the velocity profile in the turbine meter

with U the velocity outside the boundary layers, δ 1 the displacement thickness (fordefinition see equation B.3), δ 2 the momentum thickness (for definition see equationB.4) and τ w the shear stress at the wall. For the calculation of the laminar boundarylayer, a third order polynomial description of the boundary layer profile is used in

combination with Newton’s law for τ w (see Appendix B). This was found to be anaccurate description of a laminar boundary layer by Pelorson et al. (1994) and Hof-

mans (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, δ 1, the momentum thickness, δ 2, and the shear stress at the wall, τ w, arecalculated 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. Usingthe definition of displacement thickness, δ 1, this velocity can be written as:

U (x; Q, δ (x)) = Q

π ((rout − δ 1)2 − (rhub + δ 1)2) , (2.18)

where Q is the volume flow, rout is the radius of the outer wall and rhub is the radiusof 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 di-

mensions equal to the turbine meter 1, is calculated. This meter has a radius of theouter wall, rout = 0.050 m and a radius of the hub rhub = 0.037 m. The hub lengthin 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 tur-

bulent boundary layers the velocity profile is plotted in figure 2.5(b). As expected the


31/155


0.7 0.75 0.8 0.85 0.9 0.95 10

0.2

0.4

0.6

0.8

1

r/rout

u / u m a x

ReLhub

= 1.4 X 105

ReL

hub

= 3.9 X 104

ReL

hub

= 1.2 X 104

ReL

hub

= 3 X103

rhub

(a) laminar boundary layers

0.7 0.75 0.8 0.85 0.9 0.95 10

0.2

0.4

0.6

0.8

1

r/rout

u / u m a x

ReL

hub

= 3 X103

ReL

hub

= 1.2 X 104

ReL

hub

= 3.9 X 104

ReL

hub

= 1.4 X 105

rhub

(b) turbulent boundary layers

Figure 2.5: Velocity profile entering the rotor for turbine meter 1 with a diameter 0.1034 mcalculated using boundary layer theory. The velocity, u , divided by the maxi-mum 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 ve-locity profile with laminar boundary layers, (b) the velocity profile with turbulent

boundary layers.


32/155


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 ad-

equate 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 massflow 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 for-

ward 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 connectedto 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 ata sample frequency of f s = 10 kHz. Before measuring the velocity profile just infront 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 2.6is 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 for-ward 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 = ULhub/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 measurementsshown in figure 2.7, it can be seen that the velocity profile is asymmetric. Theasymmetry is increasing with increasing Reynolds number. It has a maximum ve-

locity 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


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−0.5 0 0.50.40.30.20.1−0.1−0.2−0.3−0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/D

u / u m a x

ReD = 1.5 X 10

4

ReD = 2.8 X 10

4

ReD = 7.1 X 10

4

ReD = 1.1 X 105

Figure 2.6: The velocity profile in the pipe just upstream of the turbine flow meter, measured

at four different Reynolds numbers, ReD = umaxD/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 andthe hub. This can be seen in figure 2.7. The velocity is measured 1 mm downstreamof 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, wherein 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 experi-mentally 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.


34/155


0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.050

0.2

0.4

0.6

0.8

1

r/rout

u / u m a x

Re = 3.1 X 103

Re = 5.6 X 103

Re = 1.2 X 104

routrhub

(a) Re < 2 × 104

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.050

0.2

0.4

0.6

0.8

1

r/rout

u / u m a x

Re = 2.0 X 104

Re = 3.9 X 104

Re = 9.2 X 104

Re = 1.4 X 105

routr

hub

(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.


35/155


Figure 2.8: Schematic drawing of the position of the hot wire during the velocity measure-

ments.

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 mea-

sured 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 bound-

ary 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 outerwall. 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


36/155


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.

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, T d, thedriving 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 torquepredicted 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:

T d = − rtiprhub

ρuin (uin tan β + ωr) 2πr2dr . (2.19)

For steady flow and in absence of friction the equation of motion of the rotor (equa-

tion 2.13) reduces to: T d = 0. For a given geometry of the rotor and a knownincoming 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 deviationof the rotation speed from ideal rotation speed for a uniform inflow, E (see equation2.14), 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-


37/155

2.4. Wake behind the rotor blades 25

103

104

105

106

0

1

2

3

4

5

6

7

8

ReL

hub

E = ( ω − ω i d ) / ω i d * 1 0 0 ( % )

velocity profile measurementsturbulent boundary layers

laminar boundary layers

Figure 2.11: The deviation of the rotation speed, ω , from the rotation speed for a uniforminlet velocity profile, ωid versus Reynolds number, Re = U Lhub/ν . Turbulent boundary layer approximation (solid line), laminar boundary layer approxima-

tion (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 flowalso 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


38/155


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).

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 figure2.12. 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 freestream velocity, u∞ = uin, this equation reduces to:

F D = ρ

wake

uout (u∞ − uout) dy = ρu2∞δ 2,wake , (2.20)

where the integral can be limited to the wake, because uout = u∞ outside the wakeand δ 2,wake is the momentum thickness of the wake. With this equation the drag

coefficient of a blade of length Lblade and thickness t, C ′D = C D

Lbladet

= F D12ρu2

in,relt,

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 wakeequal to the pressure of the uniform main flow, the drag coefficient vanishes C ′D = 0(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


39/155


complex. We will therefore consider the wake at some distance from the trailing edge

of the blade.

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 C ′

D = 0.64.

Hoerner (1965) (see also Blevins (1992)) found experimentally that a blade witha rounded nose and a squared edged base, with the dimensions Lblade/t = 6 (seefigure 2.13) has a drag coefficient C ′D = 0.64 for ReLblade > 10

4. 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, sharpedge 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.

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 effectof the skin friction compared to the pressure drag, the drag coefficient caused by lam-

inar 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,


40/155


C ′D, caused by the skin friction on both side of the blades is:

C ′D,friction = 2 Lbladex=0 τ wdx

12 ρU

2t=

1.328 ReLblade

Lbladet

(2.21)

For turbulent boundary layers the drag coefficient is found empirically (Schlichting,

1979) to be:

C ′D,friction = 0.148 Re− 1

5

Lblade

Lbladet

(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

C ′D ≈ 0.05 for laminar boundary layers and C ′D ≈ 0.25 for turbulent boundary layersfor turbine meter 1. For turbine meter 2 we find C ′D ≈ 0.03 for laminar boundarylayers and C ′D ≈ 0.18 for turbulent boundary layers. As the total drag C ′D = 0.64 forthe 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 thevelocity in the wake is half the mainstream velocity, uwake =

12 uin, using equation

2.20, we can calculate that the rotor blade has a drag coefficient C ′D = 0.5. In caseof the rotor blade with a chamfered trailing edge, we will also assume a wake with a

velocity uwake = 1

2 uin. The wake thickness, wwake will be tuned in order to match

the measured values of C ′D for a two-dimensional model of the rotor blade. Theexperiments 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 ma 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 48.9cm. 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, F D inducedby the flow around the blade. The second one is a Mettler PJ400 with a 1.5 kgranges and measures the lift force, F L. Both mass balances have an accuracy of 0.1g. Measurements were carried out for Reynolds numbers, Re = uLblade/ν , basedon the blade length, ranging from ReLblade = 4 × 104 up to 3 × 105 at blade angles,


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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).

In figure 2.16 the drag coefficient, C ′D = C DLblade

t = F D/

12 ρu

2wbladet, is

plotted against the Reynolds number, Re = uLblade/ν for a blade angle, α = 0.3◦.

The measurements show a drag coefficient, C ′D, between 0.1 to 0.35, much lowerthan C ′D = 0.64 found for the similar geometry with blunt trailing edge by Hoerner(1965).

Figure 2.16 also shows the estimatedskin friction for laminar and turbulent bound-

ary 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 rotorblade, is that at zero incidence, α = 0, the blade generates a lift force. This can beseen 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). Measure-

ments 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-cient, C ′D, values are close to the value obtained for a single blade.

2.4.2 Effect of wake on the rotation

The model described above is included in the momentum equation. The reduced ve-

locity in the wake can be described with the displacement thickness, δ 1,wake, and the


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0 0.5 1 1.5 2 2.5 3

x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

ReL

blade

C ’ D = F D / ( 1 / 2 ρ u i n , r e l

2

w b l a d e

t )

turbulent

laminar

Hoerner (1965)

Figure 2.16: The drag coefficient, C ′D , as a function of Reynolds number, ReLblade for flat plate with round nose and 45◦ chamfered trailing edge measurements at anangle 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.

−3 −2 −1 0 1 2 3

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

α (o)

C L = F L / ( 1 / 2 ρ u i n , r e l

2

w b l a d e

L b l a d e )

Figure 2.17: The lift coefficient, C L , is plotted at various angles of attack, α , for Reynoldsnumber, ReLblade > 3 10

5. The dashed line is a linear fit through the data

points. We observe a net lift coefficient C ′L(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.


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2.5. Friction forces 31

momentum thickness, δ 2,wake (see Schlichting (1979) and Appendix B). Applyingmass conservation and using the displacement thickness, we find:

uout,x = 1

1− nδ1,wake2πr cosβ uin,x , (2.23)

where n is the number of blades of the rotor. Using both the displacement thick-ness, δ 1,wake, the momentum thickness, δ 2,wake, and equation 2.23 in the momentumbalance, the driving torque (equation 2.8) becomes:

T d = − rtiprhub

ρu2in,x(tan β )r1− nδ1,wake2πr cosβ

2

2πr − n(δ 1,wake + δ 2,wake)cos β

dr

+ 2π rtiprhub

ρuin,xωr3dr .

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 δ 1,wake = 12 wwake and the momentum thickness is δ 2,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

in Hoerner (1965) of C ′D = 0.64. To account for this, the thickness of the wake canbe 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 calculationsNeglecting 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 wakethickness 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. Thereare 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


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thickness of the wake deviation, E = ω−ωidωid × 100%turbine meter 1 turbine meter 2

0 0 012 t 1.6 1.7

t 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 asthe 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 lengthof Lblade (the length of the blade) and a width of hblade (the height of the blade). Thedistance between two successive blades is 2πr

n − tcosβ . We consider two cases: the

case of a laminar boundary layer and the case of a turbulent boundary layer. The

displacement thickness, δ 1,bl, the momentum thickness, δ 2,bl, of the boundary layer

formed in this channel is calculated using the Von Kármán equation (2.17). For thelaminar 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


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flow, the out-going velocity component in the x-direction, uout,x, becomes:

uout,x = 2πr

2πr − n(2δ1,bl+δ1,wake)cosβ uin . (2.24)

Using the definition of the displacement thickness, δ 1,bl, and the momentum thick-ness, δ 2,bl, (see Appendix B), for the boundary layer thickness at the end of the chan-nel (the trailing edge of the blade), the equation for the driving torque, T d, becomes:

T d = − rtiprhub

ρu2in,x(tan β )r

1− n(δ1,wake+2δ1,bl)2πr cosβ 2 ×

2πr − n(δ 1,wake + δ 2,wake + 2(δ 1,bl + δ 2,bl))cos β

dr + 2π

rtiprhub

ρuin,xωr3dr ,

where δ 1,wake is the displacement thickness caused by the wake and δ 2,wake is themomentum 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 displace-

ment 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 rhubis the radius of the hub, n is the number of blades and t is the blade thickness. Thevelocity outside the boundary layer will be assumed constant for simplicity reasons

and equal to the relative velocity:

urel =

U 1 + nt2π cosβ hub

2+ (ωrhub)2 , (2.25)


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103

104

105

106

−2

0

2

4

6

8

10

12

14

16

ReL

blade

E = ( ω − ω i d ) / ω i d * 1 0 0 ( % )

air at 1 bar, turbulent

air at 1 bar, laminarnatural gas at 9 bar, turbulent

natural gas at 9 bar, laminar

(a) turbine meter 1

103

104

105

106

−1

0

1

2

3

4

5

6

7

ReL

blade

E = ( ω − ω i d ) / ω i d * 1 0 0 ( % )

air at 1 bar, turbulentair at 1 bar, laminar

natural gas at 9 bar, turbulentnatural gas at 9 bar, laminar

(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 displace-ment 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, τ w, caused by

this boundary layer, two limits are considered. The first case is the upper limit for theshear stress; the boundary layer starts at the entrance of the rotor. The flat plate has

a length of W cosβ hub, where W is the width of the rotor and β hub 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


47/155


and continues at the rotor. In this case the flat plate model of the flow has a length of

Lhub + W

cosβ hub. 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, τ w, becomes:

τ w = 0.0288 ρu9

5

relν 1

5 x−1

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, T fr,hub = F fr,hubrhub sin β hub, is:

T fr,hub wrhub sin β hub

Lhub+ W cosβhubLhub

τ wdx

=0.036 ρν 1

5 u9

5

relwrhub sin β hub

Lhub +

W

cos β hub

45

− L4

5

hub

.

(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. Thisimposes 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).


48/155


103

104

105

106

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

ReL

blade

E = ( ω − ω i d ) / ω i d * 1 0 0 ( % )

air at 1 bar, lower limit

air at 1 bar, upper limit

natural gas at 9 bar, lower limitnatural gas at 9 bar, upper limit

(a) turbine meter 1

103

104

105

106

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

ReLblade

E = ( ω − ω i d ) / ω i d * 1 0 0 ( % )

air at 1 bar, lower limitair at 1 bar, upper limit

natural gas at 9 bar, lower limitnatural gas at 9 bar, upper limit

(b) turbine meter 2

Figure 2.19: The upper an lower limits of the deviations from ideal flow caused by the bound-

ary 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 consid-

ered to be similar to the drag in a journal bearing. This results in friction torque

caused by the tip clearance, T tc of

T tc = 0.0782 Re0.43tip

ρu2rel,rtiprtipLbladetn . (2.29)

Here the Reynolds number is defined as Retip = urel,rtip(rout − rtip)/ν , with routthe radius of the pipe wall of the turbine flow meter.


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50/155


103

104

105

106

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

ReL

blade

E = ( ω − ω i d ) / ω i d * 1 0 0 ( % )

air at 1 barnatural gas at 9 bar

(a) turbine meter 1

103

104

105

106

−14

−12

−10

−8

−6

−4

−2

0

ReL

blade

E = ( ω − ω i d ) / ω i d * 1 0 0 ( % )

air at 1 barnatural gas at 9 bar

(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 figure2.21 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 ro-

tor is non-uniform, is compared to the calibration measurements of the two turbine

8/16/2

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Dynamics of Turbine Flowmeter