Reynolds number dependence of an orifice plate

Flow Measurement and Instrumentation 30 (2013) 123–132

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Reynolds number dependence of an orifice plate a,n ¨ Oliver Buker , Peter Lau a, Karsten Tawackolian b a b

Technical Research Institute of Sweden (SP), Department of ‘‘Measurement Technology’’, Brinellgatan 4, 50462 Bor˚ as, Sweden Physikalisch-Technische Bundesanstalt (PTB), Department of ‘‘Heat and Vacuum’’, Abbestraße 2-12, 10587 Berlin, Germany

a r t i c l e i n f o

abstract

Article history: Received 22 June 2012 Received in revised form 17 November 2012 Accepted 10 January 2013 Available online 13 February 2013

One of the most important process parameters in power plants is the flow rate that is measured in the secondary or feedwater circuit. To improve our understanding of the behaviour of flow instruments for this use, a work package within the European research project JRP ‘‘Metrology for improved power plant efficiency’’ concerning ‘‘Flow’’ was initiated. It comes under the direction of SP, Technical Research Institute of Sweden. Many power plants have to operate below their licensed rating because of the measurement uncertainty of the flow in the feedwater circuit. For that reason – in the field of traceable flow measurement – four European NMIs (PTB, SP, DTI, BEV) investigated four flow sensors based on different measuring principles. The aim is to find a method to extrapolate low temperature calibrations to high temperatures in order to measure feedwater flow with an uncertainty in the range of 0.3%–0.5%. This paper describes the work undertaken at SP on investigations of an orifice plate. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Energy efficiency Power plants Flow rate measurements Orifice plate Discharge coefficient EMRP Laser Doppler Velocimetry

1. Introduction In times of climate change, increasing scarcity of resources and growing energy demand, the better exploitation of the existing resources can help to save energy and to reduce CO2 emissions and emissions of other greenhouse gases. JRP ENG06 project focuses on metrological research to reduce the measurement uncertainty of the important control parameters of power plants, hence, to increase the level of efficiency. The intention of this research is an improvement of the energy efficiency of 2–3% for all types of large power plants [1]. The determination of power output at nominal thermal conditions requires accurate feedwater flow measurements. Work package no. 3 of the above project covers the influence of flow conditions on flow metering. It is intended to reduce the uncertainty of flow measurement to 0.5% at high flow rates and temperatures. Precise measurements of feedwater enable savings by the use of the same amount of resources and achieving to operate closer to the licensed rating power without exceeding the safety margins. In ordinary power plants and distribution networks the four prevailing flow sensors are the orifice plate flow meter, the ultrasonic flow meter, the electromagnetic flow meter and the Venturi meter. Nearly all types of nuclear, coal-fired, geothermal, solar thermal and natural gas power plants as well as waste incineration plants are thermal power plants. The operating principle of a thermal

n

Corresponding author. Tel.: þ46 10 516 5889. ¨ E-mail address: [email protected] (O. Buker).

0955-5986/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2013.01.009

power plant is shown in Fig. 1 using an example of a nuclear power plant. In the steam boiler water is converted into steam and flows through pipelines to the steam turbine. There the absorbed energy is dissipated into kinetic energy and a generator dissipates this kinetic energy into electric energy. In the condenser the expanded and cooled steam condenses due to heat transfer to the environment and is redirected to the feedwater cycle. The feedwater is pumped to a preheating system. Preheating of the feedwater contributes to an increase of the level of efficiency. Afterwards the water passes through a pipe system to the steam boiler and is evaporated and superheated by this heat absorption. There are two common methods for controlling the reactor power. The first one is the use of the control rods drive mechanisms, which allow decelerating or accelerating the reaction. If more heat is needed, that means – to speed up the reaction – the rods can be lifted out of the core and – to curb the reaction – the rods can be lowered into the core. The second and more convenient method for controlling the power is altering the feedwater flow rate through the reactor core regulated by the feedwater flow meters. Often, these flow sensors were calibrated at ambient conditions on test facilities with known uncertainties but not at the actual process conditions, because there are no calibration facilities suitable. That means measurements at calibration conditions with ambient temperatures and pressures at around 0.2 MPa will be transferred to process conditions in power plants with temperatures at 200 1C–280 1C and pressures above 7 MPa. The thermal expansion and the change in material properties due to temperature are relatively well-known but the influence on the flow parameter

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O. B¨ uker et al. / Flow Measurement and Instrumentation 30 (2013) 123–132

Nomenclature A D R r ReD Tu Tus V_ v w wm ws:m

Area of the cross-section of the pipe (m2) Inner pipe diameter (m) Radius of the pipe (m) Radial measuring position (m) Reynolds number (–) Turbulence intensity (–) Turbulence intensity (standard profile) (–) Volumetric flow rate (L s  1) Tangential velocity (m s  1) Local velocity (m s  1) Velocity in the centre of the pipe (m s  1) Velocity in the centre (standard profile) (m s  1)

and the velocity profile by the temperature increase or, respectively, the interaction with the flow sensor must be investigated. Normally flow sensors are highly dependent on the velocity distribution, but the velocity profile itself is dependent on the temperature of the fluid and on the installation situation. Therefore, the goal is to develop and validate accepted models of the temperature influence for the different flow sensors.

2. Test plan Each of the four National Metrology Institutes (NMIs) performed measurements to generate compareable data, according to two common test plans. These test plans are presented in Tables 2 and 3—all further measurement points for the investigations of the flow sensors were chosen individually by the participants. One main goal of the work package is to find a model for the temperature dependence for each flow sensor with this data. The calibration values of the flow sensors to be examined are mainly dependent on the velocity flow profile. On the other hand the

Reactor pressure vessel

Feedwater flow meter

Steam Electric output

Water

Turbine

Heater Recirculation pumps Control rods Suppression pool

n BEV DTI EMRP ENG06 JRP LDV NMI PTB SP

Local velocity (standard profile) (m s  1) Volumetric velocity (m s  1) Fluctuation velocity (m s  1) Temporal average of the velocity (m s  1) Kinematic viscosity (m2 s  1) Austrian Federal Office of Metrology and Surveying Danish Technological Institute European Metrology Research Programme EMRP Call 2009 – Energy – Number 06 Joint Research Project Laser Doppler Velocimetry National Metrology Institute Physikalisch-Technische Bundesanstalt Technical Research Institute of Sweden

velocity profile depends on the temperature of the medium due to density, viscosity and installation effects. Unfortunately, it is very difficult to separate these effects from each other under process conditions. Only in the case of the fully developed turbulent flow profile, can these effects be separated. But this poses a huge challenge for the installation and process conditions. To obtain comparable conditions, the fully developed turbulent flow profile at the test facilities of the participants is fundamental. According to DIN EN 1434 [3], each flow standard test facility of the participating NMIs, listed in Table 1, should provide these flow profiles which highly depend on the Reynolds number. To prove this requirement, several velocity profiles were measured at each participating NMI by using Laser Doppler Velocimetry (LDV). In order to verify the accuracy of the temperature extrapolation model of each participant, the different flow sensors will be verified at a later stage at the new high temperature test rig (HTPA) at the PTB—Institute Berlin. Finally the temperature extrapolation models can be further refined using the newly acquired high temperature data.

3. Description of SP’s test facility

Primary containment

Core

ws wvol w0 w

Generator Condenser

Coolant Circulation pumps Feedwater pumps

Conditions in a feedwater pipeline: Temperature: 210 °C Pressure: 7 MPa (70 bar) Flow rate: 3000 m3/h Pipe diameter: DN 400 to DN 600 Reynolds number: up to 107 Fig. 1. Principle of a power plant according to Takamoto [2].

SP as the leader of work package no. 3 concentrated on flow and temperature measurements. The NMI holds the national laboratory for volume, flow and temperature and has a variety of flow calibration facilities using water between 15 1C and 85 1C and flow rates up to 200 L s  1. SP has organized and participated in many European intercomparisons and proven its capability. The measurements with the ANSI 400 orifice plate were performed at the primary flow rig VM 4, as shown in Fig. 2. A total of three compact piston provers with 20 L, 60 L and 250 L for the lower flow rates and a ball prover with 0.5–3.5 m3 for the higher flow rates are available as primary standards. VM

Table 1 Overview of the test rigs of the participating laboratories including the temperature range, the highest possible flow rate and the expanded uncertainty Uðk ¼ 2Þ. Temp. (1C)

Flow (L s  1)

Uncertainty

Lab.

15–85 12–80 3–90 90–130 3–90 90–230

200 111 50 50 278 50

0.06 0.10 0.05 0.07 0.04 0.40

SP DTI BEV BEV PTB PTBn

n

Under construction.

O. B¨ uker et al. / Flow Measurement and Instrumentation 30 (2013) 123–132 To compact provers To volume standards

1

P T

125

2

3 4

T

Two-way ball prover 3.5 m3 7 sub-volumes 0.5 m3 each

Master meter 1 Mag meter (DN 100)

8

7

6

5

P T P T Test bench DN 300 P T Test bench DN 150

Heating

T

DN 100 Zanker flow straightener 13 D

Piston prover 60 L

Orifice plate T

P 58 D

Regulation of flow and pressure

Regulation of flow and pressure

T

Master meter 2 Coriolis (DN 40) Metal hose DN 150

Meter package 2 m

Legend

Cooling

open valve in forward direction closed valve in reverse direction

3

Water tank 60 m

forward flow direction reverse flow direction

Fig. 2. Simplified flow scheme of the calibration test rigs VM 3 (test bench DN 300) and VM 4 (test bench DN 150) at SP.

4 offers the additional possibility to calibrate – in comparison to other primary standards – more precisely two volume standards of 0.5 m3 and 3.5 m3. For measurements up to ReD ¼ 8  104 a coriolis flow meter was calibrated as a secondary standard ten times with the 60 L piston prover at the desired test points. The orifice plate was subsequently measured with at least two series with ten repetitions per series in comparison with the calibrated coriolis flow meter. For the measurements from ReD ¼ 8  104 an electromagnetic flow meter (mag meter) was calibrated as a secondary standard at least four times with the 3.5 m3 ball prover at the desired test points. Afterwards the orifice was measured with at least two series with ten repetitions per series in comparison with the calibrated mag meter. The differential pressure of the orifice plate was measured with two differential pressure transmitters connected in parallel to detect possible failed measurements.

4. LDV measurements The fully developed flow profile is fundamental for the comparability of the flow measurements and the Reynolds number theory. For that reason the undisturbed axial flow profile dependence (primary component) at 20 1C, 30 1C and 50 1C was measured by all participating NMIs. Furthermore, it was verified that the maximum swirl angle of the secondary components is smaller than 21 for the undisturbed flow profile at a temperature of 20 1C. An innovative LDV window chamber makes it possible to measure the full velocity profile with all three components over the entire cross-section [4]. In order to compare the measured profiles, theoretical profiles can be used as a reference. These theoretical fully developed flow profiles are mainly based on empirical data.

4.1. Evaluation of the measured velocity profile The evaluation of the measured velocity profile was accomplished according to [5,6] by the use of flow indicators. Yeh and Mattingly have introduced these indicators for investigations of installation effects on flow instruments by means of LDV.

4.2. Velocity profile according to Gersten and Herwig The indicators are normalized to the fully developed turbulent flow condition. The required fully developed turbulent flow profile, the so-called standard turbulent flow profile introduced by Gersten and Herwig [7,8] is the basis of the comparison. One of the main advantages of this analytical, hydraulically smooth profile is the closed mathematical description of the flow profile including the boundary layer and the core region. This means there are, for example, in contrast to the well-known power law, no discontinuities in the middle of the pipe and no infinite gradients in wall proximity. An additional benefit is the possibility for an easy update of this theoretical turbulent flow profile by the adaption of the constants used to the current state of scientific research. The latest constants used here [8,9] refer to the theoretical data provided from the Superpipe Experiment at Princeton University in cooperation with the California Institute of Technology (Caltech). The calculated theoretical flow rate as a function of the Reynolds number ReD is given by [8] wðr=RÞ 2Ret ¼ Z wvol ReD

ð1Þ

ReD is the Reynolds number based on the pipe diameter D, and Ret is the Reynolds number related to the friction velocity wt ReD ¼

D  wvol

n

and

Ret ¼

R  wt

n

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O. B¨ uker et al. / Flow Measurement and Instrumentation 30 (2013) 123–132

Table 2 Fixed test plan including the resulting flow rates for Reynolds number ReD ¼ 5  105 for different temperatures. Temperature (1C)

10

20 1.3081

6

Kin. visc. ð10 m2 s1 Þ Av. flow velocity (m s  1) Flow rate (m3 h  1)

6.40 188.993

30 1.0048

4.92 145.173

50 0.8018

3.92 115.850

70

0.5539

85

0.4133

2.71 80.034

0.3444

2.02 59.721

1.68 49.760

Table 3 Fixed test plan including the resulting flow rates for Reynolds number ReD ¼ 1  106 for different temperatures. Temperature (1C)

10

20

Kin. visc. ð106 m2 s1 Þ Av. flow velocity (m s  1) Flow rate (m3 h  1)

12.80 377.985

1.3081

1.0048 9.83 290.347

30

50 0.8018

70 0.5539

7.85 231.701

85 0.4133

5.42 160.069

0.3444

4.04 119.443

3.37 99.520

Table 4 Constants [8,9] used for the calculations of the theoretical flow distribution for

With the constants a,b the constants a, b can be calculated

Reynolds numbers ReD Z 3  105 .

a¼

2a2 ð1 þ bÞ ab

b¼

2b ð1 þ aÞ ba

Constant

Value

Constant

Value

Constant

Value

k

0.421  0.2714 0.119 1.23

Cþ b

5.60 5.567  0.1656  4.28

A B

6  10  4 0.0011 7.735

2

a

L C

a C

b

Furthermore, Z is a function of the dimensionless wall distance y þ as well as the radial position r=R 2 3   1 61 Ly þ þ 1 1 2Ly þ 1 p 7 pffiffiffi Z ¼ 4 ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffi arctan þ 5 L 3 6 3 3 ðLy þ Þ2 Ly þ þ 1 1 1 a 4 lnð1þ aðr=RÞ2 Þ þ lnð1 þ kBy þ Þ 4k k 2a þ

1

k

lnð1 þ9r=R9Þ

1 b

k 2b

lnð1 þbðr=RÞ2 Þ þ C

The dimensionless wall distance y þ is determined by y

þ

¼y

wt

n

¼

wt

n

For the mathematical description of the velocity profile altogether eleven constants, summarized in Table 4, are needed. Five of these constants (wall region: k, C þ , A, core region: a, b) are obtained from experiments and the others can be calculated from them, as shown below. The two constants L, B can be determined by using k, C þ , A and the following two equations, but in this case B must be solved numerically:

L ¼ ðAþ BÞ1=3 C

C ¼

ð1 þaÞð1 þ bÞ 1 þa ln 1þb kðabÞ

4.3. Profile factor The profile factor [10] is based on indicator P5 as defined by Yeh and Mattingly [5] as the integral of the deviation of the local axial velocity in the centre of the pipe multiplied by the local dimensionless pipe radius related to the double volumetric velocity. The profile factor is calculated for the measured velocity profile K p:m as well as for the reference velocity profile K p:s and both are put in relation Kp ¼

K p:m K p:s

Rð19r=R9Þ ¼ Ret ð19r=R9Þ

The following relation between ReD and Ret based on the friction law can be found in [8]:   1 1 ReD ¼ Ret ln Ret þC þ þ C þ C 2 k

þ

With the constants k,a,b, a, b the constants C ,C can be calculated   1 a b lnð1þ aÞ þ lnð1þ bÞln 2 C¼ k 2a 2b

2p 1 ¼ pffiffiffi þ lnðkBÞ 3 3L 4k

R K p:m ¼ R K p:s ¼

ðwm wÞ dr 1 ¼ wvol D 2wvol

ð2Þ Z

ðws:m ws Þ dr 1 ¼ wvol D 2wvol

1

ðwm wÞ d 1

Z

r  R

1

1

ðws:m ws Þ d

r  R

The profile factor Kp is a measure for the superelevation ðK p 41Þ or, respectively, the flatness ðK p o 1Þ of the measured flow profile in comparison with the theoretical flow profile. 4.4. Asymmetry factor The asymmetry factor [10] is based on Dmn (m ¼1, n ¼0) according to Yeh and Mattingly [5] and describes the relative shift of the centroid of each individual profile from the plane of symmetry. It follows: r

R1 r 1 w d R R Ka ¼ R 1  100% 2 1 w d Rr

ð3Þ

O. B¨ uker et al. / Flow Measurement and Instrumentation 30 (2013) 123–132

127

Durst [11] specified an equation for the turbulent intensity in the middle of a fully developed turbulent channel flow. The core region is defined as the area with 0:2 r r=R r 0:2. Pashtrapanska [12] showed that this relation is also a good approximation for pipe flows

4.5. Turbulence factor The turbulence factor [10] is based on the determination of the maximum intensity of turbulence in the core region of the flow.

r=R ¼ 0:2

K Tu ¼

Tumax 9r=R ¼ 0:2

ð4Þ

Tus

pffiffiffiffiffiffiffiffi w02 Tu ¼ w

and

  1=8 ws:m Tus ¼ 0:13 ReD ws:vol

4.6. Maximum swirl angle For a quantitative determination of the secondary velocities in the flow, the maximum swirl angle [10] is defined by Yeh and Mattingly [6] as a measure for the deviation of the flow vector (secondary components) from the axial direction of flow (primary component)   9v9max Fmax ¼ arctan ð5Þ wvol

4.7. Assessment of the flow indicators The goal is to define acceptance ranges or maximum values for the individual flow indicators for a better description of the velocity flow profile. Comparing the calculated flow indicators in accordance to the defined indicators provides information on the present flow profile. If the indicators are within the accepted limits, a sufficiently developed turbulent flow profile is assumed. For swirl the declaration of a maximum swirl angle is sufficient, the same applies to the turbulence factor. The determination of these acceptable ranges is difficult, because currently no experience or recommendations are available. Only for flow sensors based on differential pressure measurement, a requirement for acceptable flow conditions can be found in ISO 5167 [13]. Thereby, a maximum deviation of the local axial velocity of 5% in comparison with the maximum local axial velocity of a fully developed swirl-free flow profile after a 100 D

-1

-0.5

1.0 0.8 0.6 0.4 0.2 0

0 r/R

0.5

1

KP = 1.067 KA = 0.117 % KTu = 1.673 Theoretical profile Measured profile

-1

-0.5

0 r/R

0.5

1

1.0 0.8 0.6 0.4 0.2 0

-1

-0.5

0 r/R

0.5

1

Θ = 108°

KP = 1.067 KA = -0.139 % KTu = 1.701 Theoretical profile Measured profile

-1

-0.5

0 r/R

1.0 0.8 0.6 0.4 0.2 0

0.5

1

1.0 0.8 0.6 0.4 0.2 0

KP = 1.067 KA = 0.315 % KTu = 1.709 Theoretical profile Measured profile

-1

-0.5

0 r/R

0.5

1

Θ = 126°

KP = 1.035 KA = -0.215 % KTu = 1.669 Theoretical profile Measured profile

-1

-0.5

0 r/R

1.0 0.8 0.6 0.4 0.2 0

KP = 1.071 KA = 0.414 % KTu = 1.701 Theoretical profile Measured profile

-1

0.5

1

1.0 0.8 0.6 0.4 0.2 0

-0.5

0 r/R

0.5

1

Θ = 144°

KP = 1.016 KA = -0.098 % KTu = 1.685 Theoretical profile Measured profile

-1

-0.5

0 r/R

Θ = 72°

w/wvol

w/wvol

w/wvol

Theoretical profile Measured profile

w/wvol

Θ = 90°

KP = 1.029 KA = 0.220 % KTu = 1.693

Θ = 54°

w/wvol

Theoretical profile Measured profile

1.0 0.8 0.6 0.4 0.2 0

Θ = 36°

w/wvol

KP = 1.010 KA = 0.120 % KTu = 1.636

w/wvol

1.0 0.8 0.6 0.4 0.2 0

Θ = 18°

w/wvol

w/wvol

Θ = 0°

1.0 0.8 0.6 0.4 0.2 0

KP = 1.067 KA = 0.378 % KTu = 1.709 Theoretical profile Measured profile

-1

0.5

1

1.0 0.8 0.6 0.4 0.2 0

-0.5

0 r/R

0.5

1

Θ = 162°

w/wvol

Fig. 3. Comparison of the calculated and the measured velocity profile for ReD ¼ 5  105 at a temperature of 20 1C. (a) Calculated velocity profile. (b) Measured velocity profile including the measuring grid.

KP = 1.016 KA = -0.103 % KTu = 1.648 Theoretical profile Measured profile

-1

-0.5

0 r/R

0.5

1

Fig. 4. Segmentation of the measured velocity profile for ReD ¼ 5  105 at a temperature of 20 1C into its individual paths including the calculated flow indicators.

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O. B¨ uker et al. / Flow Measurement and Instrumentation 30 (2013) 123–132

inlet section and a maximum swirl angle of 21 should not be exceeded. In the establishment of these acceptable ranges it should be noted that the flow development along a straight inlet section is not necessarily monotonous towards the fully developed flow profile. The velocity distribution ‘overshoots’ during the development along the inlet section and approaches – finally after a further inlet section – the theoretical fully developed flow profile [5]. Currently, there are ongoing research activities by European NMIs to define acceptance ranges of these flow indicators for calibrations. The current state of knowledge according to [10] provides the following acceptance ranges:

An empirical equation for the discharge coefficient based on measurements by different test facilities is presented by ReaderHarris/Gallagher (RHG) [13] " 2

8

C d ¼ 0:5961þ 0:0261b 0:216b þ 0:000521 " þð0:0188 þ0:0063AÞb

3:5

106 ReD

Kp – Range: 0:8 rK P r 1:3 KA – Maximum value: K A,max ¼ 1% KTu – Maximum value: K Tu,max ¼ 2 Fmax – Maximum value: Fmax ¼ 21

These acceptance ranges apply to ordinary calibration test rigs, for example, gauging offices, manufacturers or suppliers. For the NMIs these limits can certainly be made somewhat stricter. 4.8. Measuring results Measurements at ReD ¼ 5  105 were carried out in accordance with the test plan. Fig. 3 shows exemplarily the measured axial velocity profile at 20 1C in comparison to the calculated velocity profile according to Gersten and Herwig. For these measurements a measuring grid consisting of concentric circles with 321 points (angle division 181) was used. Subsequently, the complete velocity profile was split in its individual paths and the flow indicators were calculated as listed in Fig. 4. It can be seen that the acceptance criteria are fulfilled by the measured profile. Similar flow indicators were measured at the other temperatures. The maximum swirl angle was measured on only two perpendicular paths due to time restrictions. It was found, that the swirl angle Fmax was much smaller than 11 and therefore was also within the acceptance range.

5. Flow meter calibrations The principle of an orifice plate is based on the measurement of the static pressure difference between the upstream and downstream sides. According to ISO 5167 [13] and ASME MFC3M [14], the mass flow qm in pipes by using an orifice plate, nozzle or venturi can be determined with the following equation: Cd p 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6Þ qm ¼ qffiffiffiffiffiffiffiffiffiffiffiffi  e   d  2Dp  r1 4 4 1b

#0:7

ReD

#0:3

þð0:043 þ0:080 e10L1 0:123 e7L1 Þ  ð10:11AÞ 0:031  ðM 02 0:8 M02 1:1Þb

   

b  106

1:3

b4 4 1b ð8Þ

The two values of L1 and L02 – by using an orifice with corner tappings – are given by L1 ¼ 0 and L02 ¼ 0. The other required values, M 02 and A, can also be calculated according to [13] M02 ¼ ð2L02 Þ=ð1bÞ

A¼

 0:8 19000b ReD

With the specified geometry parameters, the discharge coefficient Cd in the RHG equation is only a function of the Reynolds number ReD and the diameter ratio b. Fig. 5 shows the variation of the discharge coefficient in dependence of the diameter ratio for three different Reynolds numbers. As shown in this figure, there are two areas of interest. In the first area, for orifice plates with small diameter ratios of 0:1r b r0:3, it is apparent that there is no dependency on the Reynolds number. However, this range is not interesting for practical applications due to the relatively high pressure loss of such orifice plates. The second area is the range in the diameter ratio of 0:50 r b r0:65, where the discharge coefficient according to RHG has inflection points for the different Reynolds numbers based on the fully developed turbulent flow profiles. That means that variations, for example diameter changes due to deposits on the orifice plate, only affect the discharge coefficient to a small extent. On the other hand, there is a relatively large Reynolds number dependency in this range that needs to be characterized. For example, for b ¼ 0:5, Cd changes about 0.2% between ReD ¼ 1  106 and ReD ¼ 2:2  107 . As is shown in Fig. 5, for high Reynolds numbers Cd converges quickly to an asymptotic value. For that reason the main interest is to determine this asymptotic value.

Here the differential pressure Dp can be measured by the use of a differential pressure transmitter. In addition, the density r1 , the pipe diameter D and the orifice diameter d can be calculated by measuring the fluid temperature and using a linear model for the thermal material expansion. Since the orifice plate and pipe were made from one material the orifice diameter to pipe diameter ratio b remains nearly constant. The expansion factor e for incompressible flow through an orifice is e ¼ 1. 5.1. Calculation of the discharge coefficient The discharge coefficient Cd can be calculated by rearranging equation (6) qffiffiffiffiffiffiffiffiffiffiffiffi 4 4  qm  1b Cd ¼ ð7Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e  p  d  2Dp  r1

Fig. 5. Dependency of the discharge coefficient on diameter ratio at three different Reynolds numbers.

O. B¨ uker et al. / Flow Measurement and Instrumentation 30 (2013) 123–132

5.2. Experiments The participating NMIs measured an ANSI 400 flow meter with a nominal diameter of D ¼102.26 mm, or respectively, a DN 100 flow meter. Based on the conclusions of the previous section, an orifice plate with a diameter ratio of b ¼ 0:5 was chosen by SP for the study. Since the diameter, specifically the diameter ratio is essential for the discharge coefficient, the orifice plate was measured at various locations across the diameter inside with a micrometer caliper at SP’s workshop. Table 5 lists the geometric parameters of the investigated orifice plate as the results of these diameter measurements. Based on the test plan shown in Tables 2 and 3 measurements were performed at disturbed and undisturbed conditions. Besides undisturbed conditions, the Reynolds number dependence of disturbed conditions with two flow disturbers, illustrated in Fig. 6, was also investigated. The measurements were carried out in accordance with the test plan at different temperatures and flow rates but at fixed Reynolds numbers. Four elementary types of axial velocity distributions that can occur in front of orifice plates are illustrated in Fig. 7.

129

5.2.1. Undisturbed tests As a test result, shown in Fig. 8, the theoretical discharge coefficient calculated with the RHG equation has an offset of 0.49% for the lower and 0.33% for the higher Reynolds numbers as compared to the measured discharge coefficient. The measured discharge coefficients are higher than the theoretical values. A higher discharge coefficient could be an indication of an affected edge sharpness [15] or a higher surface roughness [16,17] of the pipes used, due to a resulting superelevated flow profile. A sharp edge of an orifice plate is defined by the edge radius rk according to the ISO 5167 [13] with r k =d r 0:4  103

ð9Þ

In this case the used orifice plate will be considered as sharp if rk is smaller than 20 mm. It was found by Herning [15] that an increase in edge radius rk of 0.5  10  3d leads to an increase of Cd of about 0.33%. It seems reasonable that the definition of Cd according to Eq. (8) is valid for an edge radius rk of 0.2  10  3d which is in the middle of the permissible range according to Eq. (9). The investigated orifice plate might therefore have an edge radius rk of around 0.7  10  3d (rk ¼35 mm).

Table 5 Nominal and measured diameters (mm) of the orifice plate – with a sharp edge, an annular chamber and corner tappings – and the resulting value of the diameter ratio b. Nom.

Meas.

d D1 D2 D2b D3

51.13 102.26 102.26 102.26 102.26

51.13 101.84 102.26 103.49 102.28

b

0.500

0.494

Discharge coefficient Cd

0.620 Measurements 20 °C Measurements 30 °C Measurements 50 °C Measurements 70 °C Measurements 85 °C Theo. discharge coefficient

0.615

0.610

0.605

0.600

0

2×105

4×105 6×105 8×105 Reynolds number ReD

1×106 1.2×106

Fig. 8. Results of the discharge coefficient from the undisturbed measurements.

Fig. 6. Overview of the different flow disturbers. (a) Undisturbed. (b) SP-designed. (c) PTB-designed.

Fig. 7. Measurement principle of an orifice plate.

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Despite this, the differences of the measured values are within the specified uncertainty of the calculated theoretical discharge coefficient of 0.50% for 0:2r b r 0:5. The primary objective of the work is to find a mathematical model for the extrapolation from calibration data at low temperatures to high temperatures. On the basis of the monotone curve characteristic, a theoretical fit can be developed in the simplest form c2 C d,fit ¼ c1 þ pffiffiffiffiffiffiffiffi þ c3  ReD ReD

ð10Þ

The first value c1 is the displacement parallel to the y-axis, the second value c2 describes the curvature of the curve and c3 changes the gradient of the curve mainly in the higher Reynolds number range. The adjustment of the coefficients was carried out with the least square curve fitting method, more precisely with the Levenberg–Marquardt algorithm (LMA). As a side note, also the RHG equation can be adapted by the use of Eq. (10). In a first attempt, the experimental data measured at 20 1C, 30 1C and 50 1C were used to predict the data expected at 85 1C. As a result the difference was 0.047% on average and 0.104% at maximum. Using the data from four temperature measurements (20 1C, 30 1C, 50 1C and 70 1C) for an extrapolation, the difference between the experiment and prediction is even less than 0.004% on average and 0.044% at maximum. From the tested fittings, illustrated in Fig. 9, the curve progression of the second fitting reproduces best the asymptotic behaviour of the theoretical curve. Therefore the second fitting currently seems to give the most reasonable extrapolation result. A further confirmation of this choice will be possible when the measuring results of the new high temperature flow rig (HTPA) at the PTB—Institute Berlin for Reynolds numbers up to 6  106 are available. Since the theoretical curve behaviour is hardly affected for high Reynolds numbers, it will obviously be possible to determine the expected values with an uncertainty of 0.1% at the HTPA. And then, with the values from the HTPA in a Reynolds number range from 1  106 to 6  106 measured later, the extrapolation to 1  107, i.e. the Reynolds number under real power plant operating conditions, will be possible. For the very high Reynolds numbers ðReD Z 1  107 Þ the curve is nearly parallel to the x-axis. For this reason, the given task to find an accepted model of the influence of process conditions to reduce the uncertainty of flow rate measurement by using an orifice plate from around 2% to at least 0.5% seems in consequence to be possible under undisturbed conditions. Furthermore it is

Measurements 20 °C Measurements 30 °°C Measurements 50 C Measurements 70 °C Measurements 85 °C 1st fitting 20 °C - 50 °C 2nd fitting 20 °C - 70 °C 3th fitting 20 °C - 85 °C Theo. discharge coefficient

0.615

0.610

0

1×106

2×106 3×106 4×106 Reynolds number ReD

5×106

A

For a flat velocity profile K d ¼ 1 and for fully developed velocity profiles Kd is larger than 1 and decreases with the Reynolds number. Therefore, compared to a flat velocity profile, the fully developed profiles at low Reynolds numbers have a higher dynamic pressure at the inlet of the orifice plate. If the same total pressure is assumed, the measured static pressure is therefore lower, resulting in a lower measured differential pressure and a higher value of Cd. Therefore, if on the other hand the velocity profile is more uniform, Kd will be lower and the resulting static pressure drop will increase (Cd decreases). This was already found out by Reader-Harris [18], Mattingly [19] and Irving [20]. According to Irving [20], an asymmetric profile develops usually within a small length to a symmetric profile that is flatter than the fully developed profile. In summary, the flatter profile leads to an increase of the pressure on the upstream position, and the downstream jet, illustrated in Fig. 7 will contract further. This in turn leads to a higher downstream jet velocity and a lower downstream pressure. Now for the fully developed profiles, there

0.620

0.605

0.600

5.2.2. SP-designed flow disturber tests This disturber (see Fig. 6(b)) should represent a gasket that projects in the fluid flow or alternatively a partly closed ball valve. The measured values of the disturbed discharge coefficient presented in Fig. 10 are lower than the undisturbed case. Without taking into account the effect of secondary velocities, the influence of the axial velocity distribution on the measured pressure difference can be attributed mainly to its dynamic pressure contribution. The dynamic pressure is proportional to the quantity Z K d ¼ w2 dA=ðAw2wol Þ, for : b-1 ð11Þ

Discharge coefficient Cd

Discharge coefficient Cd

0.620

proposed to use the expected asymptotic value of Cd for all very high Reynolds numbers. In addition to the undisturbed measurements, also measurements with two flow disturbers were performed to identify the influence of installation effects on the discharge coefficient. For the undisturbed measurements the velocity profile was mainly influenced by the temperature change. Hence, the velocity profile was deliberately influenced by the flow disturber. In practice however the temperature and the installation effects cannot be separated easily from each other. The investigations of the PTB-designed flow disturber at a distance of 12D upstream of the orifice plate were part of the test plan. In addition, some further investigations with an alternative distance of 58D upstream and a test with a second flow disturber designed by SP at a distance of 12D upstream were performed.

6×106

Fig. 9. Prediction for the discharge coefficient to the high Reynolds numbers based on the undisturbed measurements.

Measurements 20 °C Measurements 50 °C Undisturbed Theo. discharge coefficient

0.615

0.610

0.605

0.600

0

2×105

4×105 6×105 8×105 Reynolds number ReD

1×106 1.2×106

Fig. 10. Results of the measurements with the SP-designed flow disturber.

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0.620 Discharge coefficient Cd

is a certain value of Kd for every Reynolds number that can be seen as the reference value. If the Kd value of the actual non-ideal flow is below this reference value, e.g. for an asymmetric flow, then Cd will also be lower. For general velocity disturbances, the value of Kd is not obvious and has to be computed from measurements or simulations. For small b ratios the process is more complex as the incoming fluid from the near wall region is accelerated towards the center and the velocities in this region therefore become even more important [21]. Some of the most critical parameters [20] are the number, position and orientation of the pressure taps. In conclusion, orifice plates with corner or flange taps and also orifice plates with higher b values are generally more affected. This particular flow disturber is not specified in the ISO 5167 [13], only the required straight length between the orifice plate and a fully opened valve for b ¼ 0:5 can be found with 12D. In this case, for a distance between 6D and 12D from the fitting to the orifice plate, it is advised to add an additional uncertainty of 0.50% in comparison to the undisturbed calibration. The additional measurements with a distance of 58D between the flow disturber and the orifice plate showed results that were consistent with the undisturbed tests.

131

Measurements 20 °C Measurements 30 °C Measurements 50 °C Measurements 70 °C Measurements 85 °C Undisturbed Theo. discharge coefficient

0.615

0.610

0.605

0.600

0

2×105

4×105 6×105 8×105 Reynolds number ReD

1×106 1.2×106

Fig. 11. Results of the measurements with the PTB-designed flow disturber.

0.50% from the undisturbed measurements in good agreement with this assumption value.

6. Conclusion 5.2.3. PTB-designed flow disturber tests The new flow disturber (see Fig. 6(c)) was developed by PTB [22] on the basis of fluid dynamic considerations and allows the generation of a velocity disturbance that is similar to the flow behind a double bend out of plane. It was found from numerical simulations that the basic features at the outlet of such a disturbance are a single swirl and an asymmetry of the axial velocity distribution that change angular orientation along the axial direction. In this case, the centre of the axial velocity distribution travels on a helical path in the axial direction. For the investigated Reynolds numbers, it was found that about five diameters downstream of the disturbance, Kd is smaller than the fully developed cases. It can therefore be expected that Cd is decreased because of the axial velocity disturbance. On the other hand there is an opposing influence of the swirling flow. Irving [20] gives a physical explanation for the influence of a profile that is affected by swirl on the discharge coefficient. When the flow passes the orifice, the rotational velocity increases and the centrifugal effects lead to a faster expansion of the rotating jet on the downstream side. Therefore, the downstream jet, illustrated in Fig. 7, approaches closer towards the orifice plate and the taps downstream measure a higher pressure. The higher downstream pressure results in a lower differential pressure and consequently in an increased discharge coefficient. For a smaller b ratio the rotational velocity increases to a greater extent and therefore orifice plates with smaller b ratios have higher discharge coefficients in swirling flows. According to [20,23], the swirl effect is opposed to the axial velocity profile effect and the b ratio dictates which effect is dominant. As shown in Fig. 11 the discharge coefficient for the disturbed case is higher than for the undisturbed case. The measured deviation of 2.8% between the disturbed and the undisturbed case seems low for this strong disturbance. This result is certainly influenced positively by the use of an orifice plate with an annular chamber in contrast to individual holes. According to ISO 5167 [13], the required straight length between the orifice plate and two 901 bends in perpendicular planes, represented by the PTB-designed flow disturber, for b ¼ 0:5 is 75D. For a distance between 34D and 75D from the fitting to the orifice plate, it is necessary to add an additional uncertainty of 0.5% in comparison to the undisturbed calibration. The additional measurements with a distance of 58D between the flow disturber and the orifice plate showed a deviation of

To find an accepted extrapolation model for the discharge coefficient of an orifice plate, measurements were performed at SP. It is well-known that the discharge coefficient depends on the velocity profile. On the other hand the velocity profile is affected by temperature effects due to density and viscosity changes and installation effects. In practice, the two effects cannot be separated from each other. A particular focus in this work was placed on the characterization of the temperature dependence of the orifice plate. To separate the temperature effects from the installation effects, the evidence of a fully developed undisturbed velocity profile must be provided. A fully developed velocity profile does not change with the axial inlet section and depends therefore in the axial direction only on the flow parameter and no longer on the installation effects. As a result, the evidence for a fully developed turbulent velocity profile on SP’s test and calibration rig was provided by comparison with a theoretical standard profile and the evaluation with flow indicators with the aid of LDV measurements. Based on experimental data by measuring at different flow rates and temperatures, an extrapolation model for the temperature dependence was developed and validated. There is overall confidence that the model allows the prediction of the discharge coefficient for the higher Reynolds numbers under power plant process conditions with an uncertainty less than 0.5%. In the first phase the experimental data measured at SP enable a prediction for the discharge coefficient for the higher Reynolds numbers at the high temperature test rig (HTPA) at the PTB— Institute Berlin. In the second phase the model should finally be improved for power plant applications by using the measured values at the HTPA available in 2013. Additionally, by the use of a swirl generating and an asymmetry generating disturber the installation effects were investigated.

Acknowledgements This research was undertaken in the scope of the EU-funded EMRP research project ENG06: ‘‘Metrology for improved power plant efficiency’’. The authors would like to express their thanks to Dr. Ulrich ¨ Muller (OPTOLUTION GmbH, Switzerland) and Dr. Michael Dues

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(ILA GmbH, Germany) for their long-time support, especially in the area of the LDV measuring technique and their significant contributions to the evaluation of flow profiles in the framework of the working group Laseroptische Str¨ omungsdiagnose. In addition, the authors thank Department 7.5 of PTB, which contributed the measuring chamber and the staff for the LDV measurements. References [1] Powerplants – metrology for improved power plant efficiency. /http://www. power-plant-efficiency.de/S; 2010. [online; accessed 2012-10-15]. [2] Takamoto M. Traceability and calibration – what is real measurement uncertainty in feedwater flow metering. 2008, Workshop on flow measurement in nuclear power plants, Tokio, Japan. [3] DIN EN 1434. Heat meters – part 4: pattern approval tests; 2007. ¨ ¨ ¨ [4] Muller U, Dues M, Baumann H. Vollflachige Erfassung von ungestorten und ¨ gestorten Geschwindigkeitsverteilungen in Rohrleitungen mittels der LaserDoppler-Velocimetrie. Technisches Messen 2007;74:342–352. [5] Yeh TT, Mattingly GE. Pipeflow downstream of a reducer and its effects on flowmeters. Flow Measurement and Instrumentation 1994;5(3):181–187. [6] Yeh TT, Mattingly GE. Laser doppler velocimeter studies of the pipeflow produced by a generic header. NIST Technical Note 1995;1409:1–78. ¨ ¨ [7] Gersten K, Herwig G. Stromungsmechanik. Grundlagen der Impuls-, Warme¨ und Stoffubertragung aus asymptotischer Sicht. Vieweg-Verlag Braunschweig/ Wiesbaden, 1st ed.; 1992. [8] Gersten K. Fully developed turbulent pipe flow. In: Merzkirch W, editor. Fluid mechanics of flow metering, 1 ed. Springer-Verlag, Berlin, Heidelberg, New York; 2005. p. 1–22. [9] McKeon BJ, Li J, Jiang W, Morrison JF, Smits AJ. Further observations on the mean velocity in fully developed pipe flow. Journal of Fluid Mechanics 2004;501:135–147.

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