- Email: [email protected]
Examination of disturbed pipe flow and its effects on flow measurement using orifice plates
Flow Measurement and Instrumentation 10 (1999) 223–240 www.elsevier.com/locate/flowmeasinst
Examination of disturbed pipe flow and its effects on flow measurement using orifice plates H. Zimmermann
*
Institute of Applied Thermodynamics and Fluid Dynamics, University of Applied Sciences Mannheim, Windeckstrasse 110, D-68163 Mannheim, Germany Received 17 February 1999; received in revised form 4 May 1999; accepted 8 June 1999
Abstract In a great number of measurements the influence of a disturbed flow on the flow coefficient of a standard orifice plate was investigated. Single bends and double bends out of plane with and without spacer tubes were used as typical disturbances. Experiments were also performed using a combination with a star-shaped flow straightener. The necessary correction factors of the flow coefficient were determined for upstream straight length shorter than detailed in ISO 5167. The flow velocity profiles produced by the disturbances were examined and on this basis profile numbers were calculated. The examinations presented here show that the existing standard should be revised as regards the definition of the fully developed turbulent flow profile and the selection of the required upstream straight lengths. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Orifice plate; Disturbed pipe flow; Flow measurement
Symbols and units Symbol Units b 1 ba, bk, 1 br, bLe 1 c1 d m D D bLe
m ⫺ 1
fa L Le
% m m
Le1
m
Le2
m
Lz
m
Meaning profile constant correction factors for flow coefficient profile constant internal diameter of orifice plate acc. to standard pipe diameter measurement plane correction factor for α coefficient deviation of α coefficient pipe length length between disturbance and orifice plate length between disturbance and flow straightener length between flow straightener and orifice plate length of the spacer tube of the double bend out of plane
* Tel.: +49-0621-2926-201; fax: +49-0621-2926-240. E-mail address: [email protected] (H. Zimmermann)
pG PG
Pa 1
r
m
R R Re s
m m 1 m
Sa Sd u,v v
1 1 m m/s
vm vax⫽vZ(r=0) vr=f(r) vu=f(r) vz=f(r) vmax v* x y y+
m/s m/s m/s m/s m/s m/s m/s m m 1
0955-5986/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 5 - 5 9 8 6 ( 9 9 ) 0 0 0 1 5 - 1
pressure upstream of gas meter measurement plane at the gas meter measurement radius within coordinate system bend radius pipe radius Reynolds number plate thickness of flow straightener circumference number radial number measurement coordinate vector velocity of swirl, velocity in general average pipe velocity velocity within the pipe axis radial velocity of swirl circumference velocity of swirl axial velocity of swirl maximum flow velocity wall shear stress velocity measurement coordinate measurement coordinate dimensionless wall distance
224
z a a
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
m measurement coordinate 1 flow coefficient degree swirl angle of the circumference component 1 flow coefficient without correction factors 1 flow coefficient for upstream disturbance Le 1 flow coefficient for inlet ⬇100·D 1 diameter ratio of orifice plate degree swirl angle degree swirl angle of radial component % profile deviation % maximum profile deviation % total profile deviation 1 profile constant 1 von-Ka´rma´n constant 1 pipe friction factor m2/s kinematic viscosity kg/m3 fluid density, general Pa wall shear stress
The technical and commercial significance of this publication can be described as follows:
1. Introduction
앫 The installation lengths specified in ISO 5167 have neither been verified by experiment nor by theory. They are based on studies dating back to the 1930s. The straight lengths considered necessary at that time to achieve a turbulent profile have simply been doubled for ‘safety reasons’ since a sound analytical relationship does not exist. The original lengths are included in brackets in ISO standards and are defined with a 0.5% uncertainty with respect to the flow coefficient a. The different upstream straight lengths specified in AGA 3 and ISO 5167 can be explained in the same way. 앫 In many technical facilities, it is impossible to conduct mass flow measurements which conform to the standards since the upstream straight length requirements for orifice plate measurements cannot be met. This is regrettable as exact measurement data are crucial in various modes of operation, particularly during start-up and shutdown, when malfunctions occur or when the facility is operated in preparation for handover to the client or acceptance testing.
As part of this study, experiments were conducted to examine the influence of a disturbed flow on the flow coefficient a of a standard orifice plate. The objective was to show a connection between velocity distribution and swirl as described by profile numbers and the change in the flow coefficient. Apart from its scientific importance, this study has a great many practical benefits. On the one hand, the necessary upstream straight lengths can be shortened. On the other hand, the results help to bring the international standard ISO 5167 and US standard AGA 3 into line with each other.
The first series of tests were experiments to determine the change in the flow coefficient a with shortened upstream straight lengths. Single bends and double bends out of plane installed at different cylindrical distances were used as typical disturbances. Tests were also performed using a combination of a star-shaped flow straightener and greatly shortened upstream straight lengths. The necessary correction factors (cf VDI 2040, sheet 1) of the flow coefficient were determined for a disturbed upstream straight length shorter than detailed in ISO 5167. It is thus possible to quantify the measure-
a0 ae a⬁(akal) b b d ⌬A ⌬Amax ⌬Atotal η1=11 =0.4 l n r tw
Fig. 1.
Developed and disturbed flow profile upstream of an orifice plate.
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Table 1 Upstream straight length for measurements with flow straightener Le1 Le2
7·D1 12·D1
4·D1 7·D1
2·D1 12·D1
4·D1 12·D1
ment errors for a large number of practical cases. Moreover, a meter run which does not conform to standard can be modified by installing a simple star-shaped flow straightener so that the measurements are as accurate as those for a installation which conforms to standard. In the second series of tests, the flow profiles produced by the disturbances described above were examined on the different straight upstream lengths themselves. For these tests, a spherical pressure tube (5-hole tube) was used. This part of the tests is designed to verify the general conditions in section 7.4 and their connection with Table (3) in ISO 5167.
225
ever, if turbulent flow has not developed because the straight lengths downstream of disturbances are too short, the contraction at the orifice and hence the differential pressure for the same mass flow will change. This relationship is shown in the Fig. 1. According to ISO 5167, undisturbed straight lengths upstream and downstream of the orifice plate are required depending on the type of disturbance and the diameter ratio b. Shorter upstream lengths increase systematic measurement uncertainties. The required undisturbed straight lengths for orifice plates are defined in Table (3) of ISO 5167 for different types of disturbances. The general evaluation criteria to ISO 5167, section 6.4, evaluate velocity profiles and swirl directly:
2.1. Fluid flow requirements
Acceptable velocity profile conditions can be presumed to prevail when at each point across the pipe cross-section the ratio of the local axial velocity to the maximum axial velocity at the crosssection agrees to within ±5% with that which would be achieved in swirl-free flow at the same radial position at a crosssection located at the end of a very long straight length (over 100 D), of similar pipe.
Before fluid flow can be measured as prescribed by the standards, turbulent flow must have developed. How-
Swirl-free conditions can be taken to exist when the swirl angle over the pipe is less than 2°.
2. Basic principles of orifice plate measurement
Fig. 2. Main dimensions for measurements with star-shaped flow straightener. Table 2 Re numbers for measurements with flow straightener b Re
0.80 200 000
0.70 200 000
0.60 200 000
0.50 160 000
0.40 105 000
0.30 60 000
226
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 3.
Test stand to investigate the swirl influences on the alpha coefficient. Config: Two bends out of plane with a spacer.
Table 3 Used standard orifice plates d b
128 0.797
120 0.747
112 0.697
In experiments [1,2] it was found that the requirements specified in Table (3) and those of the general evaluation criteria do not match. Depending on the type of disturbance, the one constitutes a stricter or more lenient requirement than the other. Thus, the objective of these extensive studies was to narrow down these differences using the data obtained. 2.2. Deviations from the standards If the calculation principles discussed above are to be used for measuring fluid flow, the measuring facility has to conform to the standards. If, however, the orifice plate, pipe lengths or state of flow do not meet standards, the flow coefficient changes. The change in the flow coefficient a0 is allowed for by the correction factors ba, bk, br…
95 0.591
80 0.498
64 0.398
48 0.299
a⫽a0·ba·bk·br·…·bLe where Le=length of straight pipe. If ba, bk, br…=1, then: a⫽a0. For disturbances not mentioned in ISO 51 67, the respective correction factor cannot generally be given. Where necessary, it must be defined e.g. in a model experiment. Correction factor tendencies are listed in Table (5) of VDI/VDE 2040, part 1. A correction factor greater than 1 means that the fluid flow calculated using the standard flow coefficients is too small, and vice versa. At present, a quantitative correction of the flow coefficient can only be made in a small number of cases. The experimental results presented here for different
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
227
Table 4 Coordinates for spherical pressure tube measurements ±x, y, u, v in mm ±r/R
Fig. 4.
70 0.875
65 0.813
60 0.750
50 0.625
40 0.500
30 0.375
20 0.250
10 0.125
0 0.000
Coordinates for spherical pressure tube measurements.
Fig. 6. Sign convention for swirl measurements.
a⫽a0·bLe
3. Experiments 3.1. Experimental programme The experimental programme conducted can be divided into two parts. Fig. 5. ments.
Velocity components for spherical pressure tube measure-
installations upstream of standard orifice plates and for shortened upstream straight lengths Le for different diameter ratios b now permit correction. The correction factor bLe for the flow coefficient α results from:
冉
bLe⫽ 1⫹
冊
fa 100
fa in % so that the corrected flow coefficient a is:
3.1.1. Part 1 Influence of disturbed flow on the flow coefficient a of a standard orifice plate as a function of the diameter ratio b 1st test section: Measurements at orifice plates with disturbed inlet profiles caused by (a)shortened upstream straight lengths Le=5; 10; 15; 20; 25; 30; 35; 42.5; 52.5; 62.5; and 72.5D, and (b)the disturbances single 90° bend, R/D=1.5 double bend out of plane, 2*90°, R/D=1.5 without spacer tube Lz=0
228
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 7.
Schematic explanation to the general evaluation criteria in ISO 5167.
double bend out of plane, 2*90°, R/D=1.5 with spacer tubes Lz=1D and 2D. The distance between the curved portions of the bends is zero because plain flanges were used. 2nd test section: Measurements at orifice plates with star-shaped flow straightener combined with a single bend and double bend out of plane (without spacer tube Lz=0D) (Table 1). For test configuration with main dimensions (D=D1) see Fig. 2. The Re numbers for Part 1 are listed in Table 2.
Fig. 8. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a single bend.
3.1.2. Part 2 Development of disturbed pipe flow downstream of single bends, double bends out of plane etc. Measurements of swirl for the variants described in Part 1 were carried out using a spherical pressure tube (5-hole tube) in the measurement plane 2, Fig. 3. The orifice plate was removed for the measurements, the Re number was always Re=2×105. Velocity measurements for an undisturbed inlet flow in the measurement planes 1 and 2 were carried out using a Prandtl tube. The measurements were performed
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
229
Fig. 9. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a double bend.
in two and four coordinate planes, respectively (see Fig. 4 for definition). 3.2. Test configuration and preparations Fig. 3 shows the test configuration. The suction side of the pipe system was connected to a rotational frequencycontrolled radial fan. The entire pipe system consisted of longitudinally welded stainless steel pipes. The DN 160 standard orifice plates (Table 3) with corner taps and carrier rings with annular slots were used. In a first step, these orifice plates were calibrated to reduce the manufacturer’s guaranteed uncertainty t0 of the flow coefficient a to ISO 5167. The carrier rings were fixed in place to ensure that the orifice plate would not move in the pipe. A 91D upstream straight length with a combined flow straightener to ISO 5167, Type A (68.5D upstream of the orifice) was installed for the calibrations. The velocity profiles measured in measurement plane 2 in two co-ordinates using a Prandtl tube revealed deviations of less than 1.0% with respect to the reference profile described in Section 3.5. Thus, a turbulent uniform flow could be assumed for the calibration of the orifice plates. The normal mass flow was meas-
Fig. 10. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a double bend with a cylindrical spacer tube Lz=1D.
ured by a turbine flow meter calibrated by the manufacturer (Elster) at low pressure, the reproducibility of the error curve being better than 0.2%. With of all relevant factors taken into account, the accuracy of the calibrated a coefficient (a⬁) was calculated to be ga⬁=±0.25%. When the orifices were later re-calibrated, the deviations for the flow coefficients were a maximum 0.18%. Fig. 3 shows a ‘double bend out of plane with spacer tube’ configuration as an example. The straight length Le upstream of the disturbances examined (bend, double bend out of plane etc.) was normally 22.5D. At the entrance to the upstream straight length, there was a perforated plate with a free surface area of 45%, which ensured an undisturbed flow development in the upstream straight length. Additional experiments were carried out with an upstream straight length sized L0=42.5D together with the double bend out of plane to show that the standard upstream straight length sized 22.5D was long enough to prevent an impact on the development of fluid flow downstream of the disturbances.
230
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 11. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a double bend with a cylindrical spacer tube Lz=2D.
Fig. 12. Measured deviation fa=0.5% compared to 0.0% and 0.5% uncertainty of ISO 5167.
3.3. Measuring instruments Prandtl tubes with a 3 mm diameter were used to measure the velocity without the swirl (axial component). The swirl flow was measured using a spherical pressure tube (5-hole tube) with a sphere diameter of 8 mm. A support for accurately aligning and traversing the tubes within the pipe was developed, which permitted the pressure tube to be aligned at an angle of ±0.30° with respect to the pipe axis. This was the value determined for swirl-free flow using the above configuration for orifice plate calibration. The calibration curves provided by the tube manufacturer (Schildknecht) are based on parallel flow. It was therefore necessary to do up-front experiments to define the impact of a velocity distribution with an arbitrary profile on pressure measurement at the spherical head in order to calculate the correction angles needed for an exact definition of the swirl angles. Moreover, the restriction of the free cross-sectional area in the tube was also taken into account. The normal volumetric flow was established with a
turbine flow meter (Elster) calibrated at low pressure. This involved the use of a HE pulse generator to determine the flow at the gas meter using the pulse counting method. With this method, incoming pulses are totalised over 5 s and shown as a sliding window average. This sliding window average was repeatedly read. In order to calculate the normal mass flow qmG, the air density rG at the gas meter was determined by measuring the static pressure and the temperature in the measuring plane pG. Prior to the measurements it was shown that the turbine meter is not affected by the swirl. Velocity profiles were measured 4.5 D2 upstream of the turbine meter for the worst case installation (i.e. double bend out of plane, Le=5 D1 and b=0,8) using the spherical pressure tube. The maximum swirl angle determined was 2.5° and the average value was 1.1° respectively 1.6° over the traverses in x and y directions. It is well known that turbine meters for fiscal purposes can be installed 5D downstream of disturbances for example caused by double bends out of plane without being more affected than ±0.3% compared to ideal flow conditions. At that pos-
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 13.
231
Schematic of swirl center position downstream of double bend out of plane.
ition the swirl angle can be as large as 10–20°, see Fig. 16(a) and (b). Thus the inlet length of 20 D2 is large enough to allow for an unaffected flow rate measurement with the turbine meter. Measurement of the volumetric flow at the standard orifice plate was carried out according to ISO 5167. The differential pressure ⌬pD and the upstream pressure ⌬p+ were measured by a capacitive pressure transducer. The manufacturer (Setra) guarantees a total accuracy of ±0.14% with respect to the final value for these instruments (RSS method). This total accuracy takes account of a linearity of ±0.1%, a hysteresis of ±0.05% and a repeatability of ±0.02%. The pressure transducers were repeatedly calibrated during the experiments, with projection manometers used as pressure standards. Two specially developed computer programs along with a DAS 16G card (Keithly) were used for recording the measured values obtained during the spherical pressure tube swirl measurements and the orifice plate flow measurement. Calibrated instruments were available for measuring the atmospheric pressure and the temperature. 3.4. Definitions for the profile and swirl measurements In order to determine the velocity profile and the swirl angles a and b, a number of measuring points were defined in the measuring plane on the x and y-axes. Moreover, two additional directions u and v were defined for the swirl measurements. The different measuring instruments, i.e. 앫 the Prandtl tube for measuring velocity in undisturbed flow; and 앫 the spherical presure tube for measuring swirl and velocity in disturbed flow,
as well as prevailing test conditions lead to the following definitions: 3.4.1. Profile measurement with Prandtl tube Results are positive for the direction of the x and yaxes and flow into the paper plane (z-axis) (Fig. 4). Measuring points on the x and y-axes: ⫺75; ⫺70; ⫺65; ⫺50; ⫺30; 0; 30; 50; 65; 70; 75 mm 3.4.2. Swirl and profile measurement with spherical pressure tube The swirl angles a and b are best determined at points with the same radius r. Based on the definition of the x and y-axes for the profile measurement, an xy coordinate system is laid into the pipe axis, with pipe flow being in the direction of the paper plane (z-axis). Moreover, two additional axes u and v are used to determine the development of swirl downstream of the double bend out of plane with and without spacer (Table 4). 3.4.3. Definitions for swirl and profile measurements using spherical pressure tube Fig. 5 shows the coordinate system using spherical pressure tube and Fig. 6 shows sign convention for swirl measurement. The signs of the velocity components vx, vy and the swirl angles a and d refer to the relative position of the tube and to its direction of insertion, respectively. In order to transform the velocity components into a global coordinate system, the sign convention as shown in Fig. 6 is used. The main direction of flow z is into the paper plane. 3.5. The turbulent reference profile (Fig. 7) According to the requirements detailed in Section 2.1, the deviation is calculated as follows:
232
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 14. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a double bend, main tests with L0=22,5D and separate tests with L0=42,4D.
(v/vmax)measured−(v/vmax)100·D ⌬A⫽ (v/vmax)100·D The reference profile is calculated according to [3]. The velocity distribution, given here in dimensionless form, is calculated as follows:
冉
冊 冉
v 1 1.5·(1+r/R) ⫽ ·ln (1⫹·y+)· ⫹c1· 1⫺e−y+/η1 v∗ 1+2·(r/R)2
冊
y+ ⫺ ·e−b·y+ h1 where
wall shear stress velocity v∗⫽
冪 冪 tw ⫽ r
l·v¯2m 8
l friction coefficient; v¯m volumetrically averaged velocity. dimensionless wall distance v∗·(R−r) y+= n n kinematic viscosity. von-Ka´rma´n constant: =0.4i profile constants: c1=7.8; h1=11 and b=0.33. The velocity distribution according to [3] was chosen because it applies to the entire domain all the way into
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
the viscous layer and on the pipe wall, respectively. Moreover, the weaknesses of other distribution laws in the pipe centre do not occur here. 3.6. Definition of the a coefficient deviation fa Deviation of the a coefficient: fa⫽
ae−a⬁ ·100 a⬁
fa in % ae is the actual value for the disturbed flow, a⬁ is the target value for undisturbed conditions downstream of a 100D straight length.
4. Results 4.1. Part 1
233
are the single bend, double bend out of plane and double bend out of plane with cylindrical 2D spacer tube disturbances. The points for the ±0.5% deviations are taken from Figs. 2, 3 and 5, with the original data levelled out by best fit curves. Particularly in the case of the double bend out of plane, there are no clear intersections for the fixed deviations fa=+0.5% and fa=⫺0.5%, respectively, owing to the strong variations in the results for fa=f(Le/D,b). The points plotted in Fig. 12 show strong scatter in parts. Yet for the single bend and for the double bend out of plane with the spacer tube Lz=2D, there are clear intersections. The negative and positive deviations of the a coefficient as determined by the diameter ratio b, found for the double bend out of plane without a spacer tube, correspond to the findings in [4]. However, it must be assumed that the 0.5% uncertainty curve in Fig. 12, taken from ISO 5167, is a best fit curve which compensates the characteristic features: fa⫽⫺0.5% for b⬎0.65 fa⫽⫹0.5% for b⬍0.65 The uncertainty of the ISO results will be approximately
4.1.1. Experiments without star-shaped flow straightener The influence of a disturbed, swirl-type flow on the flow coefficient a is shown in Figs. 8–11 for the disturbances described above, with the a flow coefficient deviation as defined in section 2.6 plotted as a function of the relative upstream straight length ratio Le/D. The parameter is the diameter ratio b of the orifice plates. For the single bend, there are only negative deviations fa, i.e. the flow coefficients ae become smaller than in the case of a sufficiently long straight pipe. The influence of the swirl-type flow caused by the single bend (double vortex, see also Section 4.2) ceases downstream of a pipe length of approximately 45D. For the double bend out of plane without the cylindrical spacer tube, there are positive and negative deviations fa, depending on the diameter ratio b. Deviations are positive for b=0.30 to 0.70 and negative if b=0.75 or more. The influence of the swirl-type flow caused by the double bend out of plane (single vortex with alternating centre, see also Section 4.2) ceases downstream of a pipe length of approximately 80D. For the double bend out of plane with the cylindrical spacer tube Lz=1D, the positive deviations are already significantly smaller (less than 0.5%). The double bend out of plane with the cylindrical spacer tube Lz=2D already produces only negative deviations, comparable with the results for the single bend. However, the influence of the straight pipe length ceases only at approximately 65D. Fig. 12 shows the diameter ratio b over the upstream straight length Le/D for the ±0.5% deviation. Parameters
Fig. 15. Deviation fa of a coefficient as a function of the diameter ratio b and the relative upstream straight length Le/D with star- shaped flow straightener for a single and double bend.
234
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 16. (a) Max. swirl angles a and d for single bend and double bend. (b) Max. swirl angles a and d for double bend with spacer Lz=1D and Lz=2D.
the same as in the case of the experiments presented here. The minimum, i.e. Le/D=0, is also confirmed for these experiments at b=0.67. As the b curves in Fig. 12 approach the deviation fa=0% asymptotically and a clear upstream straight length Le cannot thus be defined, the resulting upstream straight lengths for the 0.5% deviation have probably been doubled. These lengths are then valid for the 0.0% uncertainty (arbitrary fixing). It is interesting that, according to these tests, the double bend out of plane with the spacer tube Lz=2·D behaves more favourably than the single bend. In the case of the double bend out of plane, the fa deviations show extremely strong variations for b=0.75 and 0.80. This effect was condoned by repeating the experiments. The experiments in the second part showed that the rotation centre of the swirl alternates, centring only downstream of Le=70D. This means that the orifice plates influence the contraction of the jet depending on their position with respect to the disturbance. This influence is determined by the position of the rotation centre. In Fig. 13 this effect is first explained schematically. The influence of the incompletely developed flow downstream of a straight length sized L0=22.5D on the
development of swirl downstream of the disturbances and hence on the actual flow coefficient was examined in a separate test. These measurements were repeated for all diameter ratios b with a 42.5D upstream straight length and a downstream “double bend out of plane” disturbance and recalibration of the orifice plates. The results are shown in Fig. 14 and compared with the measurements for 22.5D. There is no significant influence of the velocity distribution downstream of the 22.5D straight length of the disturbance on the flow coefficient deviation fa. 4.1.2. Experiments with star-shaped flow straightener Experiments with a star-shaped flow straightener were carried out based on the assumption that the velocity distribution does not influence the jet contraction at the orifice as strongly as the swirl. The ⌬Amax=±5% requirement in the standard seems too strong. Fig. 15 gives a summary of the test results. Deviations are negative for the ‘double bend out of plane’ and the ‘single bend’ disturbances. For the combinations examined they increase with the diameter ratio b. For b=0.3 to 0.5, all combinations examined result in fa values below the 0.5%
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
deviation. For the combinations 7D/12D and 4D/12D, the results for all diameter ratios b are always in the faⱕ⫺0.5% range. 4.2. Part 2 In this part of the tests [6], swirl measurements with the spherical pressure tube described above were conducted on the x, y, u and v-axes as explained in Section 3.4. The Re number was always 2×105. The first and foremost objective of these measurements was to find the maximum swirl angle. Fig. 16 shows the maximum swirl angles a and d for the single bend and the double bend out of plane. The single bend satisfies the general evaluation criteria according to ISO 5167 of a, d⬍2° at approximately 15D, the double bend out of plane at 80D (based on extrapolation) and the double bend out of plane with spacer tube Lz=2D at approximately 50D. Fig. 17(a) and (b) show the comparison between the measured velocity profiles and the reference profile. The actual length from the disturbance to measuring plane,
235
i.e. L⬘e=Le⫺2·D, is plotted on the abscissa. The graph shows both the average value and the maximum value over the traverses in the x and y directions. The disturbances tested thus satisfy the general evaluation criteria according to ISO 5167 of ⌬Amaxⱕ5% for the following upstream straight lengths: Single bend Double bend out of plane Double bend out of plane with Lz=1D Double bend out of plane with Lz=2D
approx. approx. approx. approx.
48D 83D 65D 40D
The main test results in combination with the star-shaped flow straightener are summarised in Fig. 18(a) and (b). As already outlined in Section 4.1.2, the deviation fa of the a coefficient for the combinations 7D/12D and 4D/12D is always smaller than 0.5% for all diameter ratios b. The associated maximum profile deviations are listed in Table 5. Therefore, measurements in line with the standard can be carried out with a total range of the profile deviations
Fig. 17. (a) Max. and average deviation between local axial velocity and reference profile of ISO 5167 as a function of the relative upstream straight length Le/D for single and double bend. (b) Max. and average deviation between local axial velocity and reference profile of ISO 5167 as a function of the relative upstream straight length Le/D for double bend with spacer Lz=1D and Lz=2D.
236
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 18. (a) Min. and max. deviation of axial velocity as a function of the relative upstream straight length Le/D with star-shaped flow straightener for single and double bend. (b) Min. and max. swirl angles a and d as a function of the relative upstream straight length Le/D with star-shaped flow straightener for single and double bend. Table 5 Maximum profile deviations ⌬A in %
⌬Atotal in %
Disturbance
Combination
+14.2 ⫺4.9 +16.5 ⫺8.1 +11.3 ⫺5.8 +16.1 ⫺3.9
19.1
Single bend
4D/12D
24.6
Double bend out of plane Single bend
7D/12D
17.1 20.0
Double bend out of plane
of up to approx. 25%, especially since the swirl angles d and a are less than 2°! For the combinations 4·D/7D and 2D/12D, the total profile deviations are between 26 and 31%. The maximum of ⫺3° for the swirl angle a is reached for the 4D/7D combination. The requirement in the standard calling for a maximum allowable profile deviation of ±5% is far too stringent and seems to be based on the results for the double bend out of plane shown in Fig. 12 and 17(a). For
the 0% uncertainty and b=0.80, this gives a minimum upstream straight length of Le=80D. The maximum profile deviation determined here is ⌬Amax⬵7%. The requirement of a swirl angle of 2° maximum has been confirmed by these experiments [Fig. 16(a)]. In order to represent the complete swirl development, especially for the double bend out of plane with and without the spacer tube, two more axes u and v were introduced. In qualitative terms the circumference component of the velocity for the single bend and the double bend out of plane can be described as follows: Single bend: Double vortex with the two stable rotation centres on the radius r=R/2 (i.e. no dependency on the upstream straight length Le) (Fig. 19). Double bend out of plane: Rotational swirl with its centre not on the pipe axis (Fig. 20). Fig. 21 shows an example for the swirl profile downstream of the double bend out of plane for a downstream pipe length of 3D. The transportation velocity vz was approximately 20 m/s. The figure also indicates the respective rotation centre which is defined according to a self-developed evaluation method. A particular characteristic of the double bend out of plane is the eccentricity
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
237
Fig. 19. Double vortex downstream of a single bend.
Fig. 21. Velocity components perpendicular to main flow direction after 3D straight length behind a double bend, main flow direction into paper plane.
Fig. 20. Single vortex out of center line downstream of a double bend out of plane.
of the swirl and the circumferential velocities vu increasing towards the pipe wall. Fig. 22 shows the position of the rotation centres on the x/y plane. One can recognise the ‘corkscrew’ movement of the swirl. For the double bend out of plane with spacer tube, there was no comparable ‘corkscrew’ approximation of the swirl towards the pipe axis. Apparently, the short spacer tube of 1D already causes the transition to a double vortex as in the case of the single bend. For the description of the swirl intensity, dimensionless numbers (swirl numbers) were defined, with the kinetic energy of the circumference and radial velocities, respectively examined in relation to the kinetic energy of the transportation velocity. Circumference number: Radial number:
Fig. 22. Swirl center in x-y-plane for a double bend, main flow direction into paper plane.
冕 冕
v2r·vz·dA
A S a⫽
A
v3z·dA
238
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
The integration was approximated via a summation over sub-domains. In qualitative terms this process can be described for the swirl numbers as shown in Fig. 23. Fig. 24 shows the swirl numbers defined above for the double bend out of plane with and without the spacer tube. They represent the damping curves of the vortex over the pipe length referred to the diameter. The double bend out of plane without the spacer tube produces the strongest swirl, starting with some 3% of kinetic energy with respect to the transportation energy along the zaxis. This energy component decreases with an increase in spacer tube length. If the latter is sufficiently long, the form of the swirl will correspond to that of the single bend with a double vortex. If the length is 2·D. this is not yet the case. The swirl profiles [5] clearly show a single vortex which is, however, increasingly distorted by the influence of the spacer tube. This can also be seen in the development of the radial number. For Le=3D, it is greater than for the double bend out of plane with a ID spacer tube, although the latter generates a stronger swirl intensity. Presumably, this is due to the tendency to form a double vortex, which increases with the length of the spacer tube. The radial numbers in Fig. 24 are smaller than the circumference numbers by approximately one order of magnitude. It is striking that the double bend out of plane without a spacer tube has a slightly different characteristic than the double bends out of plane with a spacer tube. If there is a spacer tube, the radial number decreases much more quickly than for a double bend out of plane without spacer tube. The radial number could be regarded as an indicator for the eccentricity of the swirl. This would mean that in the case of the double
Fig. 23.
Fig. 24. Circumference number Sa and radial number Sd for double bends with and without spacer.
Sub-domains for circumference- and radial number calculation.
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
bend out of plane with spacer tube, the swirl centres far more quickly than without spacer tube, which is in keeping with the position of the rotational centres described above. In the case of the double bend out of plane with spacer tube, one can see that the radial numbers do not continuously decrease. Instead, they remain almost constant downstream of around 20D. The reason for this behaviour are likely to be arbitrary measurement errors which add up to this result. The swirl numbers were checked for the double bend out of plane for three different Reynolds numbers 100 000; 215 000; 300 000 and three upstream straight lengths 3D; 18D; 40.5D on a random basis. A comparison of the results does not show any significant differences. Thus, it seems as though the relative swirl intensity is largely independent of the Reynolds number in the domain examined.
5. Summary A general solution which is characterised by a single profile number and describes the effect of disturbed flows on the flow coefficient a unmistakably by way of a correction factor cannot be derived where diameter ratios b differ. Several profile numbers are always necessary. From the first part of the tests a correction factor for the flow coefficient a can be calculated for the disturbances examined with shortened upstream straight lengths from the following formula:
冉
bLe⫽ 1⫹
fa 100
冊
Table (5) in VDI 2040, Sheet 1, is thus extended in quantitative terms. Further examinations show that the upstream straight lengths specified in Table (3) of ISO 5167 are the result of the doubling of the upstream straight lengths, for which the deviation fa is only just ±0.5%. The tests with the star-shaped flow straightener in combination with a double bend out of plane and a single bend always show negative deviations. For the combinations examined, these deviations increase with the diameter ratio b; for b=0.3 to 0.5, fa values below 0.5% deviation can always be obtained. For the combinations 7D/12D and 4D/12D, the value is always within the faⱕ⫺0.5% range. These results were obtained although the profile deviations found were well above the general evaluation criteria in section 7.4 of ISO 5167. According to the results of the second part of the tests, measurements in keeping with the standard can be achieved with a total profile deviation range of up to some 25%, especially since the swirl angles a and b are below 2°. For the combinations 4·D/7·D and 2·D/12·D, the total
239
profile deviations are between 26 and 31%. The maximum of ⫺3° for the swirl angle a is obtained for the 4·D/7·D combination. This result indicates that the general evaluation criteria regarding the ±5% profile deviation are too stringent a requirement. This finding is corroborated by the following examinations. Putting a tolerance band of ±0.5% on either side of the zero line in Figs. 8–11 gives the relevant upstream straight lengths for the disturbances examined. If the orifice plate with the largest diameter ratio is selected as b=0.75 and if the values obtained are plotted in Fig. 17(a) and (b) respectively, the corresponding profile deviations are obtained. The striking thing is that for these upstream straight lengths the maximum values over the traverses of the x and y directions is several times larger then the average values over these traverses. When the upstream straight lengths obtained are doubled, the maximum values for the profile deviation at the bend are approx. 5%, for the double bend out of plane Lz=2D approx. 4% and for the double bend out of plane Lz=ID approx. 7.5%. The double bend out of plane without the spacer is more difficult to include in these considerations because the curve of the deviation fa (Fig. 9) already varies around the 0.5% tolerance band for Le/D=10. However, fa⬍0.5% is maintained at Le/D=60, the maximum profile deviation being approx. 12% and the average being approx. 2% according to Fig. 17(a). By the same token, the swirl strength of less than 2° defined in the general evaluation criteria seems too stringent. It seems to have been derived from the results of the double bend out of plane measurements. Fig. 16(a) shows an 80D upstream straight length for 2°. For the double bends out of plane Lz=1D and 2D, the maximum swirl angles for the upstream straight lengths of 28D and 23D determined above are approx. 6° and 5°, respectively. Doubling the upstream straight lengths gives approx. 3° in each case.
6. Conclusion The examinations presented here show that the existing ISO 5167 standard should be revised as regards the definition of the fully formed turbulent flow profile and the selection of the required upstream straight lengths. This study has provided the necessary data for this to be done.
References [1] B. Harbrink, W. Zirig (Essen), H.-U. Hassenpflug, W. Kerber, H. Zimmermann (Mannheim), The Disturbance of Flow through an Orifice Plate Meter run by the Upstream Header. VDI reports no. 768, 1989. [2] H.-U. Hassenpflug, W. Kerber, H. Zimmermann (Mannheim),
240
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Messungen am Modell einer Gasmeßlstation im Maßstab 1:4; Reports A and B of Ruhrgas AG Essen 1988/1989. [3] H. Reichardt, Vollsta¨ndige Darstellung der turbulenten Geschwindigkeitsverteilung in glatten Leitungen. Max-Planck Institut Go¨ttingen, 1950. [4] H. Calame, A. Schro¨der, Problematik der Einlaufstrecken von Durchflußmessem—Versuch einer Definition und eines Vergleichs. VDI report Nr. 375, 1980.
[5] H. Zimmemann, Versuche u¨ber den Einfluß einer drallbehafteten Stro¨mung auf die Durchflußzahl a einer Normblende in Abha¨ngigkeit vom Durchmesserverha¨ltnis b. Zwischenbericht Dezember 1994. Abschlußbericht Mai 1996. Karl-Vo¨lker Stiftung an der Fachhochschule fu¨r Technik und Gestaltung, Mannheim. [6] P. Mu¨ller, Untersuchung von Drallstro¨mungen nach einem Raumkru¨mmer. Institut fu¨r Angewandte Thermo- und Fluiddynamik. Fachhochschule fu¨r Technik und Gestaltung, Mannheim.
Examination of disturbed pipe flow and its effects on flow measurement using orifice plates H. Zimmermann
*
Institute of Applied Thermodynamics and Fluid Dynamics, University of Applied Sciences Mannheim, Windeckstrasse 110, D-68163 Mannheim, Germany Received 17 February 1999; received in revised form 4 May 1999; accepted 8 June 1999
Abstract In a great number of measurements the influence of a disturbed flow on the flow coefficient of a standard orifice plate was investigated. Single bends and double bends out of plane with and without spacer tubes were used as typical disturbances. Experiments were also performed using a combination with a star-shaped flow straightener. The necessary correction factors of the flow coefficient were determined for upstream straight length shorter than detailed in ISO 5167. The flow velocity profiles produced by the disturbances were examined and on this basis profile numbers were calculated. The examinations presented here show that the existing standard should be revised as regards the definition of the fully developed turbulent flow profile and the selection of the required upstream straight lengths. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Orifice plate; Disturbed pipe flow; Flow measurement
Symbols and units Symbol Units b 1 ba, bk, 1 br, bLe 1 c1 d m D D bLe
m ⫺ 1
fa L Le
% m m
Le1
m
Le2
m
Lz
m
Meaning profile constant correction factors for flow coefficient profile constant internal diameter of orifice plate acc. to standard pipe diameter measurement plane correction factor for α coefficient deviation of α coefficient pipe length length between disturbance and orifice plate length between disturbance and flow straightener length between flow straightener and orifice plate length of the spacer tube of the double bend out of plane
* Tel.: +49-0621-2926-201; fax: +49-0621-2926-240. E-mail address: [email protected] (H. Zimmermann)
pG PG
Pa 1
r
m
R R Re s
m m 1 m
Sa Sd u,v v
1 1 m m/s
vm vax⫽vZ(r=0) vr=f(r) vu=f(r) vz=f(r) vmax v* x y y+
m/s m/s m/s m/s m/s m/s m/s m m 1
0955-5986/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 5 - 5 9 8 6 ( 9 9 ) 0 0 0 1 5 - 1
pressure upstream of gas meter measurement plane at the gas meter measurement radius within coordinate system bend radius pipe radius Reynolds number plate thickness of flow straightener circumference number radial number measurement coordinate vector velocity of swirl, velocity in general average pipe velocity velocity within the pipe axis radial velocity of swirl circumference velocity of swirl axial velocity of swirl maximum flow velocity wall shear stress velocity measurement coordinate measurement coordinate dimensionless wall distance
224
z a a
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
m measurement coordinate 1 flow coefficient degree swirl angle of the circumference component 1 flow coefficient without correction factors 1 flow coefficient for upstream disturbance Le 1 flow coefficient for inlet ⬇100·D 1 diameter ratio of orifice plate degree swirl angle degree swirl angle of radial component % profile deviation % maximum profile deviation % total profile deviation 1 profile constant 1 von-Ka´rma´n constant 1 pipe friction factor m2/s kinematic viscosity kg/m3 fluid density, general Pa wall shear stress
The technical and commercial significance of this publication can be described as follows:
1. Introduction
앫 The installation lengths specified in ISO 5167 have neither been verified by experiment nor by theory. They are based on studies dating back to the 1930s. The straight lengths considered necessary at that time to achieve a turbulent profile have simply been doubled for ‘safety reasons’ since a sound analytical relationship does not exist. The original lengths are included in brackets in ISO standards and are defined with a 0.5% uncertainty with respect to the flow coefficient a. The different upstream straight lengths specified in AGA 3 and ISO 5167 can be explained in the same way. 앫 In many technical facilities, it is impossible to conduct mass flow measurements which conform to the standards since the upstream straight length requirements for orifice plate measurements cannot be met. This is regrettable as exact measurement data are crucial in various modes of operation, particularly during start-up and shutdown, when malfunctions occur or when the facility is operated in preparation for handover to the client or acceptance testing.
As part of this study, experiments were conducted to examine the influence of a disturbed flow on the flow coefficient a of a standard orifice plate. The objective was to show a connection between velocity distribution and swirl as described by profile numbers and the change in the flow coefficient. Apart from its scientific importance, this study has a great many practical benefits. On the one hand, the necessary upstream straight lengths can be shortened. On the other hand, the results help to bring the international standard ISO 5167 and US standard AGA 3 into line with each other.
The first series of tests were experiments to determine the change in the flow coefficient a with shortened upstream straight lengths. Single bends and double bends out of plane installed at different cylindrical distances were used as typical disturbances. Tests were also performed using a combination of a star-shaped flow straightener and greatly shortened upstream straight lengths. The necessary correction factors (cf VDI 2040, sheet 1) of the flow coefficient were determined for a disturbed upstream straight length shorter than detailed in ISO 5167. It is thus possible to quantify the measure-
a0 ae a⬁(akal) b b d ⌬A ⌬Amax ⌬Atotal η1=11 =0.4 l n r tw
Fig. 1.
Developed and disturbed flow profile upstream of an orifice plate.
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Table 1 Upstream straight length for measurements with flow straightener Le1 Le2
7·D1 12·D1
4·D1 7·D1
2·D1 12·D1
4·D1 12·D1
ment errors for a large number of practical cases. Moreover, a meter run which does not conform to standard can be modified by installing a simple star-shaped flow straightener so that the measurements are as accurate as those for a installation which conforms to standard. In the second series of tests, the flow profiles produced by the disturbances described above were examined on the different straight upstream lengths themselves. For these tests, a spherical pressure tube (5-hole tube) was used. This part of the tests is designed to verify the general conditions in section 7.4 and their connection with Table (3) in ISO 5167.
225
ever, if turbulent flow has not developed because the straight lengths downstream of disturbances are too short, the contraction at the orifice and hence the differential pressure for the same mass flow will change. This relationship is shown in the Fig. 1. According to ISO 5167, undisturbed straight lengths upstream and downstream of the orifice plate are required depending on the type of disturbance and the diameter ratio b. Shorter upstream lengths increase systematic measurement uncertainties. The required undisturbed straight lengths for orifice plates are defined in Table (3) of ISO 5167 for different types of disturbances. The general evaluation criteria to ISO 5167, section 6.4, evaluate velocity profiles and swirl directly:
2.1. Fluid flow requirements
Acceptable velocity profile conditions can be presumed to prevail when at each point across the pipe cross-section the ratio of the local axial velocity to the maximum axial velocity at the crosssection agrees to within ±5% with that which would be achieved in swirl-free flow at the same radial position at a crosssection located at the end of a very long straight length (over 100 D), of similar pipe.
Before fluid flow can be measured as prescribed by the standards, turbulent flow must have developed. How-
Swirl-free conditions can be taken to exist when the swirl angle over the pipe is less than 2°.
2. Basic principles of orifice plate measurement
Fig. 2. Main dimensions for measurements with star-shaped flow straightener. Table 2 Re numbers for measurements with flow straightener b Re
0.80 200 000
0.70 200 000
0.60 200 000
0.50 160 000
0.40 105 000
0.30 60 000
226
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 3.
Test stand to investigate the swirl influences on the alpha coefficient. Config: Two bends out of plane with a spacer.
Table 3 Used standard orifice plates d b
128 0.797
120 0.747
112 0.697
In experiments [1,2] it was found that the requirements specified in Table (3) and those of the general evaluation criteria do not match. Depending on the type of disturbance, the one constitutes a stricter or more lenient requirement than the other. Thus, the objective of these extensive studies was to narrow down these differences using the data obtained. 2.2. Deviations from the standards If the calculation principles discussed above are to be used for measuring fluid flow, the measuring facility has to conform to the standards. If, however, the orifice plate, pipe lengths or state of flow do not meet standards, the flow coefficient changes. The change in the flow coefficient a0 is allowed for by the correction factors ba, bk, br…
95 0.591
80 0.498
64 0.398
48 0.299
a⫽a0·ba·bk·br·…·bLe where Le=length of straight pipe. If ba, bk, br…=1, then: a⫽a0. For disturbances not mentioned in ISO 51 67, the respective correction factor cannot generally be given. Where necessary, it must be defined e.g. in a model experiment. Correction factor tendencies are listed in Table (5) of VDI/VDE 2040, part 1. A correction factor greater than 1 means that the fluid flow calculated using the standard flow coefficients is too small, and vice versa. At present, a quantitative correction of the flow coefficient can only be made in a small number of cases. The experimental results presented here for different
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
227
Table 4 Coordinates for spherical pressure tube measurements ±x, y, u, v in mm ±r/R
Fig. 4.
70 0.875
65 0.813
60 0.750
50 0.625
40 0.500
30 0.375
20 0.250
10 0.125
0 0.000
Coordinates for spherical pressure tube measurements.
Fig. 6. Sign convention for swirl measurements.
a⫽a0·bLe
3. Experiments 3.1. Experimental programme The experimental programme conducted can be divided into two parts. Fig. 5. ments.
Velocity components for spherical pressure tube measure-
installations upstream of standard orifice plates and for shortened upstream straight lengths Le for different diameter ratios b now permit correction. The correction factor bLe for the flow coefficient α results from:
冉
bLe⫽ 1⫹
冊
fa 100
fa in % so that the corrected flow coefficient a is:
3.1.1. Part 1 Influence of disturbed flow on the flow coefficient a of a standard orifice plate as a function of the diameter ratio b 1st test section: Measurements at orifice plates with disturbed inlet profiles caused by (a)shortened upstream straight lengths Le=5; 10; 15; 20; 25; 30; 35; 42.5; 52.5; 62.5; and 72.5D, and (b)the disturbances single 90° bend, R/D=1.5 double bend out of plane, 2*90°, R/D=1.5 without spacer tube Lz=0
228
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 7.
Schematic explanation to the general evaluation criteria in ISO 5167.
double bend out of plane, 2*90°, R/D=1.5 with spacer tubes Lz=1D and 2D. The distance between the curved portions of the bends is zero because plain flanges were used. 2nd test section: Measurements at orifice plates with star-shaped flow straightener combined with a single bend and double bend out of plane (without spacer tube Lz=0D) (Table 1). For test configuration with main dimensions (D=D1) see Fig. 2. The Re numbers for Part 1 are listed in Table 2.
Fig. 8. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a single bend.
3.1.2. Part 2 Development of disturbed pipe flow downstream of single bends, double bends out of plane etc. Measurements of swirl for the variants described in Part 1 were carried out using a spherical pressure tube (5-hole tube) in the measurement plane 2, Fig. 3. The orifice plate was removed for the measurements, the Re number was always Re=2×105. Velocity measurements for an undisturbed inlet flow in the measurement planes 1 and 2 were carried out using a Prandtl tube. The measurements were performed
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
229
Fig. 9. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a double bend.
in two and four coordinate planes, respectively (see Fig. 4 for definition). 3.2. Test configuration and preparations Fig. 3 shows the test configuration. The suction side of the pipe system was connected to a rotational frequencycontrolled radial fan. The entire pipe system consisted of longitudinally welded stainless steel pipes. The DN 160 standard orifice plates (Table 3) with corner taps and carrier rings with annular slots were used. In a first step, these orifice plates were calibrated to reduce the manufacturer’s guaranteed uncertainty t0 of the flow coefficient a to ISO 5167. The carrier rings were fixed in place to ensure that the orifice plate would not move in the pipe. A 91D upstream straight length with a combined flow straightener to ISO 5167, Type A (68.5D upstream of the orifice) was installed for the calibrations. The velocity profiles measured in measurement plane 2 in two co-ordinates using a Prandtl tube revealed deviations of less than 1.0% with respect to the reference profile described in Section 3.5. Thus, a turbulent uniform flow could be assumed for the calibration of the orifice plates. The normal mass flow was meas-
Fig. 10. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a double bend with a cylindrical spacer tube Lz=1D.
ured by a turbine flow meter calibrated by the manufacturer (Elster) at low pressure, the reproducibility of the error curve being better than 0.2%. With of all relevant factors taken into account, the accuracy of the calibrated a coefficient (a⬁) was calculated to be ga⬁=±0.25%. When the orifices were later re-calibrated, the deviations for the flow coefficients were a maximum 0.18%. Fig. 3 shows a ‘double bend out of plane with spacer tube’ configuration as an example. The straight length Le upstream of the disturbances examined (bend, double bend out of plane etc.) was normally 22.5D. At the entrance to the upstream straight length, there was a perforated plate with a free surface area of 45%, which ensured an undisturbed flow development in the upstream straight length. Additional experiments were carried out with an upstream straight length sized L0=42.5D together with the double bend out of plane to show that the standard upstream straight length sized 22.5D was long enough to prevent an impact on the development of fluid flow downstream of the disturbances.
230
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 11. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a double bend with a cylindrical spacer tube Lz=2D.
Fig. 12. Measured deviation fa=0.5% compared to 0.0% and 0.5% uncertainty of ISO 5167.
3.3. Measuring instruments Prandtl tubes with a 3 mm diameter were used to measure the velocity without the swirl (axial component). The swirl flow was measured using a spherical pressure tube (5-hole tube) with a sphere diameter of 8 mm. A support for accurately aligning and traversing the tubes within the pipe was developed, which permitted the pressure tube to be aligned at an angle of ±0.30° with respect to the pipe axis. This was the value determined for swirl-free flow using the above configuration for orifice plate calibration. The calibration curves provided by the tube manufacturer (Schildknecht) are based on parallel flow. It was therefore necessary to do up-front experiments to define the impact of a velocity distribution with an arbitrary profile on pressure measurement at the spherical head in order to calculate the correction angles needed for an exact definition of the swirl angles. Moreover, the restriction of the free cross-sectional area in the tube was also taken into account. The normal volumetric flow was established with a
turbine flow meter (Elster) calibrated at low pressure. This involved the use of a HE pulse generator to determine the flow at the gas meter using the pulse counting method. With this method, incoming pulses are totalised over 5 s and shown as a sliding window average. This sliding window average was repeatedly read. In order to calculate the normal mass flow qmG, the air density rG at the gas meter was determined by measuring the static pressure and the temperature in the measuring plane pG. Prior to the measurements it was shown that the turbine meter is not affected by the swirl. Velocity profiles were measured 4.5 D2 upstream of the turbine meter for the worst case installation (i.e. double bend out of plane, Le=5 D1 and b=0,8) using the spherical pressure tube. The maximum swirl angle determined was 2.5° and the average value was 1.1° respectively 1.6° over the traverses in x and y directions. It is well known that turbine meters for fiscal purposes can be installed 5D downstream of disturbances for example caused by double bends out of plane without being more affected than ±0.3% compared to ideal flow conditions. At that pos-
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 13.
231
Schematic of swirl center position downstream of double bend out of plane.
ition the swirl angle can be as large as 10–20°, see Fig. 16(a) and (b). Thus the inlet length of 20 D2 is large enough to allow for an unaffected flow rate measurement with the turbine meter. Measurement of the volumetric flow at the standard orifice plate was carried out according to ISO 5167. The differential pressure ⌬pD and the upstream pressure ⌬p+ were measured by a capacitive pressure transducer. The manufacturer (Setra) guarantees a total accuracy of ±0.14% with respect to the final value for these instruments (RSS method). This total accuracy takes account of a linearity of ±0.1%, a hysteresis of ±0.05% and a repeatability of ±0.02%. The pressure transducers were repeatedly calibrated during the experiments, with projection manometers used as pressure standards. Two specially developed computer programs along with a DAS 16G card (Keithly) were used for recording the measured values obtained during the spherical pressure tube swirl measurements and the orifice plate flow measurement. Calibrated instruments were available for measuring the atmospheric pressure and the temperature. 3.4. Definitions for the profile and swirl measurements In order to determine the velocity profile and the swirl angles a and b, a number of measuring points were defined in the measuring plane on the x and y-axes. Moreover, two additional directions u and v were defined for the swirl measurements. The different measuring instruments, i.e. 앫 the Prandtl tube for measuring velocity in undisturbed flow; and 앫 the spherical presure tube for measuring swirl and velocity in disturbed flow,
as well as prevailing test conditions lead to the following definitions: 3.4.1. Profile measurement with Prandtl tube Results are positive for the direction of the x and yaxes and flow into the paper plane (z-axis) (Fig. 4). Measuring points on the x and y-axes: ⫺75; ⫺70; ⫺65; ⫺50; ⫺30; 0; 30; 50; 65; 70; 75 mm 3.4.2. Swirl and profile measurement with spherical pressure tube The swirl angles a and b are best determined at points with the same radius r. Based on the definition of the x and y-axes for the profile measurement, an xy coordinate system is laid into the pipe axis, with pipe flow being in the direction of the paper plane (z-axis). Moreover, two additional axes u and v are used to determine the development of swirl downstream of the double bend out of plane with and without spacer (Table 4). 3.4.3. Definitions for swirl and profile measurements using spherical pressure tube Fig. 5 shows the coordinate system using spherical pressure tube and Fig. 6 shows sign convention for swirl measurement. The signs of the velocity components vx, vy and the swirl angles a and d refer to the relative position of the tube and to its direction of insertion, respectively. In order to transform the velocity components into a global coordinate system, the sign convention as shown in Fig. 6 is used. The main direction of flow z is into the paper plane. 3.5. The turbulent reference profile (Fig. 7) According to the requirements detailed in Section 2.1, the deviation is calculated as follows:
232
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 14. Deviation fa of a coefficient as a function of the relative upstream straight length Le/D for a double bend, main tests with L0=22,5D and separate tests with L0=42,4D.
(v/vmax)measured−(v/vmax)100·D ⌬A⫽ (v/vmax)100·D The reference profile is calculated according to [3]. The velocity distribution, given here in dimensionless form, is calculated as follows:
冉
冊 冉
v 1 1.5·(1+r/R) ⫽ ·ln (1⫹·y+)· ⫹c1· 1⫺e−y+/η1 v∗ 1+2·(r/R)2
冊
y+ ⫺ ·e−b·y+ h1 where
wall shear stress velocity v∗⫽
冪 冪 tw ⫽ r
l·v¯2m 8
l friction coefficient; v¯m volumetrically averaged velocity. dimensionless wall distance v∗·(R−r) y+= n n kinematic viscosity. von-Ka´rma´n constant: =0.4i profile constants: c1=7.8; h1=11 and b=0.33. The velocity distribution according to [3] was chosen because it applies to the entire domain all the way into
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
the viscous layer and on the pipe wall, respectively. Moreover, the weaknesses of other distribution laws in the pipe centre do not occur here. 3.6. Definition of the a coefficient deviation fa Deviation of the a coefficient: fa⫽
ae−a⬁ ·100 a⬁
fa in % ae is the actual value for the disturbed flow, a⬁ is the target value for undisturbed conditions downstream of a 100D straight length.
4. Results 4.1. Part 1
233
are the single bend, double bend out of plane and double bend out of plane with cylindrical 2D spacer tube disturbances. The points for the ±0.5% deviations are taken from Figs. 2, 3 and 5, with the original data levelled out by best fit curves. Particularly in the case of the double bend out of plane, there are no clear intersections for the fixed deviations fa=+0.5% and fa=⫺0.5%, respectively, owing to the strong variations in the results for fa=f(Le/D,b). The points plotted in Fig. 12 show strong scatter in parts. Yet for the single bend and for the double bend out of plane with the spacer tube Lz=2D, there are clear intersections. The negative and positive deviations of the a coefficient as determined by the diameter ratio b, found for the double bend out of plane without a spacer tube, correspond to the findings in [4]. However, it must be assumed that the 0.5% uncertainty curve in Fig. 12, taken from ISO 5167, is a best fit curve which compensates the characteristic features: fa⫽⫺0.5% for b⬎0.65 fa⫽⫹0.5% for b⬍0.65 The uncertainty of the ISO results will be approximately
4.1.1. Experiments without star-shaped flow straightener The influence of a disturbed, swirl-type flow on the flow coefficient a is shown in Figs. 8–11 for the disturbances described above, with the a flow coefficient deviation as defined in section 2.6 plotted as a function of the relative upstream straight length ratio Le/D. The parameter is the diameter ratio b of the orifice plates. For the single bend, there are only negative deviations fa, i.e. the flow coefficients ae become smaller than in the case of a sufficiently long straight pipe. The influence of the swirl-type flow caused by the single bend (double vortex, see also Section 4.2) ceases downstream of a pipe length of approximately 45D. For the double bend out of plane without the cylindrical spacer tube, there are positive and negative deviations fa, depending on the diameter ratio b. Deviations are positive for b=0.30 to 0.70 and negative if b=0.75 or more. The influence of the swirl-type flow caused by the double bend out of plane (single vortex with alternating centre, see also Section 4.2) ceases downstream of a pipe length of approximately 80D. For the double bend out of plane with the cylindrical spacer tube Lz=1D, the positive deviations are already significantly smaller (less than 0.5%). The double bend out of plane with the cylindrical spacer tube Lz=2D already produces only negative deviations, comparable with the results for the single bend. However, the influence of the straight pipe length ceases only at approximately 65D. Fig. 12 shows the diameter ratio b over the upstream straight length Le/D for the ±0.5% deviation. Parameters
Fig. 15. Deviation fa of a coefficient as a function of the diameter ratio b and the relative upstream straight length Le/D with star- shaped flow straightener for a single and double bend.
234
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 16. (a) Max. swirl angles a and d for single bend and double bend. (b) Max. swirl angles a and d for double bend with spacer Lz=1D and Lz=2D.
the same as in the case of the experiments presented here. The minimum, i.e. Le/D=0, is also confirmed for these experiments at b=0.67. As the b curves in Fig. 12 approach the deviation fa=0% asymptotically and a clear upstream straight length Le cannot thus be defined, the resulting upstream straight lengths for the 0.5% deviation have probably been doubled. These lengths are then valid for the 0.0% uncertainty (arbitrary fixing). It is interesting that, according to these tests, the double bend out of plane with the spacer tube Lz=2·D behaves more favourably than the single bend. In the case of the double bend out of plane, the fa deviations show extremely strong variations for b=0.75 and 0.80. This effect was condoned by repeating the experiments. The experiments in the second part showed that the rotation centre of the swirl alternates, centring only downstream of Le=70D. This means that the orifice plates influence the contraction of the jet depending on their position with respect to the disturbance. This influence is determined by the position of the rotation centre. In Fig. 13 this effect is first explained schematically. The influence of the incompletely developed flow downstream of a straight length sized L0=22.5D on the
development of swirl downstream of the disturbances and hence on the actual flow coefficient was examined in a separate test. These measurements were repeated for all diameter ratios b with a 42.5D upstream straight length and a downstream “double bend out of plane” disturbance and recalibration of the orifice plates. The results are shown in Fig. 14 and compared with the measurements for 22.5D. There is no significant influence of the velocity distribution downstream of the 22.5D straight length of the disturbance on the flow coefficient deviation fa. 4.1.2. Experiments with star-shaped flow straightener Experiments with a star-shaped flow straightener were carried out based on the assumption that the velocity distribution does not influence the jet contraction at the orifice as strongly as the swirl. The ⌬Amax=±5% requirement in the standard seems too strong. Fig. 15 gives a summary of the test results. Deviations are negative for the ‘double bend out of plane’ and the ‘single bend’ disturbances. For the combinations examined they increase with the diameter ratio b. For b=0.3 to 0.5, all combinations examined result in fa values below the 0.5%
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
deviation. For the combinations 7D/12D and 4D/12D, the results for all diameter ratios b are always in the faⱕ⫺0.5% range. 4.2. Part 2 In this part of the tests [6], swirl measurements with the spherical pressure tube described above were conducted on the x, y, u and v-axes as explained in Section 3.4. The Re number was always 2×105. The first and foremost objective of these measurements was to find the maximum swirl angle. Fig. 16 shows the maximum swirl angles a and d for the single bend and the double bend out of plane. The single bend satisfies the general evaluation criteria according to ISO 5167 of a, d⬍2° at approximately 15D, the double bend out of plane at 80D (based on extrapolation) and the double bend out of plane with spacer tube Lz=2D at approximately 50D. Fig. 17(a) and (b) show the comparison between the measured velocity profiles and the reference profile. The actual length from the disturbance to measuring plane,
235
i.e. L⬘e=Le⫺2·D, is plotted on the abscissa. The graph shows both the average value and the maximum value over the traverses in the x and y directions. The disturbances tested thus satisfy the general evaluation criteria according to ISO 5167 of ⌬Amaxⱕ5% for the following upstream straight lengths: Single bend Double bend out of plane Double bend out of plane with Lz=1D Double bend out of plane with Lz=2D
approx. approx. approx. approx.
48D 83D 65D 40D
The main test results in combination with the star-shaped flow straightener are summarised in Fig. 18(a) and (b). As already outlined in Section 4.1.2, the deviation fa of the a coefficient for the combinations 7D/12D and 4D/12D is always smaller than 0.5% for all diameter ratios b. The associated maximum profile deviations are listed in Table 5. Therefore, measurements in line with the standard can be carried out with a total range of the profile deviations
Fig. 17. (a) Max. and average deviation between local axial velocity and reference profile of ISO 5167 as a function of the relative upstream straight length Le/D for single and double bend. (b) Max. and average deviation between local axial velocity and reference profile of ISO 5167 as a function of the relative upstream straight length Le/D for double bend with spacer Lz=1D and Lz=2D.
236
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Fig. 18. (a) Min. and max. deviation of axial velocity as a function of the relative upstream straight length Le/D with star-shaped flow straightener for single and double bend. (b) Min. and max. swirl angles a and d as a function of the relative upstream straight length Le/D with star-shaped flow straightener for single and double bend. Table 5 Maximum profile deviations ⌬A in %
⌬Atotal in %
Disturbance
Combination
+14.2 ⫺4.9 +16.5 ⫺8.1 +11.3 ⫺5.8 +16.1 ⫺3.9
19.1
Single bend
4D/12D
24.6
Double bend out of plane Single bend
7D/12D
17.1 20.0
Double bend out of plane
of up to approx. 25%, especially since the swirl angles d and a are less than 2°! For the combinations 4·D/7D and 2D/12D, the total profile deviations are between 26 and 31%. The maximum of ⫺3° for the swirl angle a is reached for the 4D/7D combination. The requirement in the standard calling for a maximum allowable profile deviation of ±5% is far too stringent and seems to be based on the results for the double bend out of plane shown in Fig. 12 and 17(a). For
the 0% uncertainty and b=0.80, this gives a minimum upstream straight length of Le=80D. The maximum profile deviation determined here is ⌬Amax⬵7%. The requirement of a swirl angle of 2° maximum has been confirmed by these experiments [Fig. 16(a)]. In order to represent the complete swirl development, especially for the double bend out of plane with and without the spacer tube, two more axes u and v were introduced. In qualitative terms the circumference component of the velocity for the single bend and the double bend out of plane can be described as follows: Single bend: Double vortex with the two stable rotation centres on the radius r=R/2 (i.e. no dependency on the upstream straight length Le) (Fig. 19). Double bend out of plane: Rotational swirl with its centre not on the pipe axis (Fig. 20). Fig. 21 shows an example for the swirl profile downstream of the double bend out of plane for a downstream pipe length of 3D. The transportation velocity vz was approximately 20 m/s. The figure also indicates the respective rotation centre which is defined according to a self-developed evaluation method. A particular characteristic of the double bend out of plane is the eccentricity
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
237
Fig. 19. Double vortex downstream of a single bend.
Fig. 21. Velocity components perpendicular to main flow direction after 3D straight length behind a double bend, main flow direction into paper plane.
Fig. 20. Single vortex out of center line downstream of a double bend out of plane.
of the swirl and the circumferential velocities vu increasing towards the pipe wall. Fig. 22 shows the position of the rotation centres on the x/y plane. One can recognise the ‘corkscrew’ movement of the swirl. For the double bend out of plane with spacer tube, there was no comparable ‘corkscrew’ approximation of the swirl towards the pipe axis. Apparently, the short spacer tube of 1D already causes the transition to a double vortex as in the case of the single bend. For the description of the swirl intensity, dimensionless numbers (swirl numbers) were defined, with the kinetic energy of the circumference and radial velocities, respectively examined in relation to the kinetic energy of the transportation velocity. Circumference number: Radial number:
Fig. 22. Swirl center in x-y-plane for a double bend, main flow direction into paper plane.
冕 冕
v2r·vz·dA
A S a⫽
A
v3z·dA
238
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
The integration was approximated via a summation over sub-domains. In qualitative terms this process can be described for the swirl numbers as shown in Fig. 23. Fig. 24 shows the swirl numbers defined above for the double bend out of plane with and without the spacer tube. They represent the damping curves of the vortex over the pipe length referred to the diameter. The double bend out of plane without the spacer tube produces the strongest swirl, starting with some 3% of kinetic energy with respect to the transportation energy along the zaxis. This energy component decreases with an increase in spacer tube length. If the latter is sufficiently long, the form of the swirl will correspond to that of the single bend with a double vortex. If the length is 2·D. this is not yet the case. The swirl profiles [5] clearly show a single vortex which is, however, increasingly distorted by the influence of the spacer tube. This can also be seen in the development of the radial number. For Le=3D, it is greater than for the double bend out of plane with a ID spacer tube, although the latter generates a stronger swirl intensity. Presumably, this is due to the tendency to form a double vortex, which increases with the length of the spacer tube. The radial numbers in Fig. 24 are smaller than the circumference numbers by approximately one order of magnitude. It is striking that the double bend out of plane without a spacer tube has a slightly different characteristic than the double bends out of plane with a spacer tube. If there is a spacer tube, the radial number decreases much more quickly than for a double bend out of plane without spacer tube. The radial number could be regarded as an indicator for the eccentricity of the swirl. This would mean that in the case of the double
Fig. 23.
Fig. 24. Circumference number Sa and radial number Sd for double bends with and without spacer.
Sub-domains for circumference- and radial number calculation.
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
bend out of plane with spacer tube, the swirl centres far more quickly than without spacer tube, which is in keeping with the position of the rotational centres described above. In the case of the double bend out of plane with spacer tube, one can see that the radial numbers do not continuously decrease. Instead, they remain almost constant downstream of around 20D. The reason for this behaviour are likely to be arbitrary measurement errors which add up to this result. The swirl numbers were checked for the double bend out of plane for three different Reynolds numbers 100 000; 215 000; 300 000 and three upstream straight lengths 3D; 18D; 40.5D on a random basis. A comparison of the results does not show any significant differences. Thus, it seems as though the relative swirl intensity is largely independent of the Reynolds number in the domain examined.
5. Summary A general solution which is characterised by a single profile number and describes the effect of disturbed flows on the flow coefficient a unmistakably by way of a correction factor cannot be derived where diameter ratios b differ. Several profile numbers are always necessary. From the first part of the tests a correction factor for the flow coefficient a can be calculated for the disturbances examined with shortened upstream straight lengths from the following formula:
冉
bLe⫽ 1⫹
fa 100
冊
Table (5) in VDI 2040, Sheet 1, is thus extended in quantitative terms. Further examinations show that the upstream straight lengths specified in Table (3) of ISO 5167 are the result of the doubling of the upstream straight lengths, for which the deviation fa is only just ±0.5%. The tests with the star-shaped flow straightener in combination with a double bend out of plane and a single bend always show negative deviations. For the combinations examined, these deviations increase with the diameter ratio b; for b=0.3 to 0.5, fa values below 0.5% deviation can always be obtained. For the combinations 7D/12D and 4D/12D, the value is always within the faⱕ⫺0.5% range. These results were obtained although the profile deviations found were well above the general evaluation criteria in section 7.4 of ISO 5167. According to the results of the second part of the tests, measurements in keeping with the standard can be achieved with a total profile deviation range of up to some 25%, especially since the swirl angles a and b are below 2°. For the combinations 4·D/7·D and 2·D/12·D, the total
239
profile deviations are between 26 and 31%. The maximum of ⫺3° for the swirl angle a is obtained for the 4·D/7·D combination. This result indicates that the general evaluation criteria regarding the ±5% profile deviation are too stringent a requirement. This finding is corroborated by the following examinations. Putting a tolerance band of ±0.5% on either side of the zero line in Figs. 8–11 gives the relevant upstream straight lengths for the disturbances examined. If the orifice plate with the largest diameter ratio is selected as b=0.75 and if the values obtained are plotted in Fig. 17(a) and (b) respectively, the corresponding profile deviations are obtained. The striking thing is that for these upstream straight lengths the maximum values over the traverses of the x and y directions is several times larger then the average values over these traverses. When the upstream straight lengths obtained are doubled, the maximum values for the profile deviation at the bend are approx. 5%, for the double bend out of plane Lz=2D approx. 4% and for the double bend out of plane Lz=ID approx. 7.5%. The double bend out of plane without the spacer is more difficult to include in these considerations because the curve of the deviation fa (Fig. 9) already varies around the 0.5% tolerance band for Le/D=10. However, fa⬍0.5% is maintained at Le/D=60, the maximum profile deviation being approx. 12% and the average being approx. 2% according to Fig. 17(a). By the same token, the swirl strength of less than 2° defined in the general evaluation criteria seems too stringent. It seems to have been derived from the results of the double bend out of plane measurements. Fig. 16(a) shows an 80D upstream straight length for 2°. For the double bends out of plane Lz=1D and 2D, the maximum swirl angles for the upstream straight lengths of 28D and 23D determined above are approx. 6° and 5°, respectively. Doubling the upstream straight lengths gives approx. 3° in each case.
6. Conclusion The examinations presented here show that the existing ISO 5167 standard should be revised as regards the definition of the fully formed turbulent flow profile and the selection of the required upstream straight lengths. This study has provided the necessary data for this to be done.
References [1] B. Harbrink, W. Zirig (Essen), H.-U. Hassenpflug, W. Kerber, H. Zimmermann (Mannheim), The Disturbance of Flow through an Orifice Plate Meter run by the Upstream Header. VDI reports no. 768, 1989. [2] H.-U. Hassenpflug, W. Kerber, H. Zimmermann (Mannheim),
240
H. Zimmermann / Flow Measurement and Instrumentation 10 (1999) 223–240
Messungen am Modell einer Gasmeßlstation im Maßstab 1:4; Reports A and B of Ruhrgas AG Essen 1988/1989. [3] H. Reichardt, Vollsta¨ndige Darstellung der turbulenten Geschwindigkeitsverteilung in glatten Leitungen. Max-Planck Institut Go¨ttingen, 1950. [4] H. Calame, A. Schro¨der, Problematik der Einlaufstrecken von Durchflußmessem—Versuch einer Definition und eines Vergleichs. VDI report Nr. 375, 1980.
[5] H. Zimmemann, Versuche u¨ber den Einfluß einer drallbehafteten Stro¨mung auf die Durchflußzahl a einer Normblende in Abha¨ngigkeit vom Durchmesserverha¨ltnis b. Zwischenbericht Dezember 1994. Abschlußbericht Mai 1996. Karl-Vo¨lker Stiftung an der Fachhochschule fu¨r Technik und Gestaltung, Mannheim. [6] P. Mu¨ller, Untersuchung von Drallstro¨mungen nach einem Raumkru¨mmer. Institut fu¨r Angewandte Thermo- und Fluiddynamik. Fachhochschule fu¨r Technik und Gestaltung, Mannheim.