- Email: [email protected]
The crossed beam correlation technique for two-phase flow measurements
Progress in Nuclear Energy, Vol. 29, No. 3/4, pp. 337-346, 1995
1995 Elsevier Science Ltd Printed in Great Brit,fin 0149-1970/95 $29.00
Pergamon
0149-1970(95)00016-X THE CROSSED
BEAM CORRELATION
TWO-PHASE
TECHNIQUE
FOR
FLOW MEASUREMENTS
O. Thomson and I. P ~ s i t Department of Reactor Physics, Chalmers University of Technology, S-412 96 G6teborg, Sweden
ABSTRACT A number of experiments have been performed using the crossed beam correlation technique with laser beams. The purpose has been to establish the applicability of the technique to non-intrusive, local flow measurements in a two-phase or two-component flow. The measured parameters were: 1) perturbation velocity, 2) void fraction and 3) correlation length. In another series of experiments, a slightly different geometry with three beams was used, allowing determination of the correlation length in two dimensions instead of one.
KEYWORDS Two-phase flow, two-component flow, radiation correlation, local parameters, perturbation velocity, correlation length, mean density, laser measurements, crossed beam correlation technique.
INTRODUCTION It has long been known that the cross-correlation function (CCF) can be used for two-phase flow velocity determination in BWRs. Such measurements are made by cross-correlating the fluctuating part of the signals from two neutron detectors, situated at different heights (along the flow direction) in the core. If there is a pronounced peak in the resulting CCF, the position of this peak corresponds to the mean transit time of the flow perturbations between the detectors. The flow perturbations are associated with the transport of the minority component, i.e. bubbles in a sparse bubbly flow (low void fraction), or with the droplets in vapour in case of high void fraction. The situation becomes much more complicated for void fractions around 50%, in which case it is not possible to identify a minority component. Since the two flow components usually have different velocities, it is not clear whose velocity such a correlation measurement determines. Another complication arises when there is a distribution of the flow parameters, e.g. a velocity profile, present across the detector viewing area, in which case the detected value represents a weighted integral over this viewing area. Much effort has been made in order to interpret exactly which transit time is obtained from such measurements (Analytis & Ltlbbesmeyer, 1983). There are other two-phase flow parameters that can be determined by noise analysis techniques, for example qq7
338
O. Thomsonand I. Pfizsit
the void fraction. This latter can be estimated from the variance of a detector signal. Efforts have also been made to determine the flow regime, i.e. the topological structure of the flow, in which case also higher moments (skewness, curtosis) have been investigated (Jones & Zuber, 1976; Vince & Lahey, 1982). The type of measurements mentioned above have been conducted either in operating reactors, using in-core neutron detectors, or in model assemblies, using external radiation sources, such as X-rays. A common difficulty in most of these measurements, at least when non-intrusive detectors are used, is the rather large control volume (Hewitt, 1978) that is generally obtained. The control volume is defined as the volume over which the measured parameter is averaged, and is determined by both the detector viewing area and the structure of the flow. Within this volume there may exist an unknown distribution of the measured flow parameter.
THE CROSSED BEAM CORRELATION TECHNIQUE A solution to some of the mentioned difficulties can be found in the crossed beam correlation technique. This technique was originally developed for non-intrusive measurements of local parameters in turbulent flow by Fisher & Krause (1967). Later, it was suggested by P~sit (1986) and P~sit & G1Ockler (1985), that the assumption of a short correlation length of the density fluctuations would permit the use of the same technique for two-phase or two-component flow measurements. Experiments by Thomson & PEzsit 0994) have proven the technique to be capable of measuring three important flow parameters simultaneously: perturbation velocity, void fraction and correlation length. All three parameters are measured locally, at any position within the flow. In case the perturbations have a regular shape, the correlation length is closely related to their size. For example, in a bubbly flow with spherical bubbles, the correlation length is equal to the maximum diameter of the bubbles. A summary of the method, as applied in the above mentioned experiments, can be given as follows. For simplicity, we will study a water-vapour flow within a circular tube, where the vapour forms bubbles of various shapes and sizes. As information carder, we use visible light, which has different refractive indices in the water and the vapour. If we apply a collimated light beam (laser), entering the tube on one side and being monitored by a small detector on the other side, a binary fluctuating signal will be obtained. This is because due to light reflection on the bubble surface, the passing bubbles will screen the detector from the light. An example of such a signal is shown in Fig. 1.
.
.
.
.
L
_
_
.
.
.
.
!
_
_
_
2
.
.
.
.
~
.
.
.
.
.
.
m
E >
2
0
-2
.... 0
I . . . . . . . . 5 10
15
20
25
l i m e [ms]
Detector signal from two-component flow experiment, with laser beam as information carder.
If we assume a radially homogeneous, bubbly flow with isotropic (spherical) bubbles, it is in principle possi-
The crossed beam correlation technique
339
ble to determine the maximum bubble diameter (correlation length) and the void fraction from one such sig-
nal (i(t)). This is done by first forming the autocovariance: C(z) = (Si(t)Si(t+'c))
(1)
from the detected signal and determining the time at which this function becomes zero (the cut-off time, "% in Fig. 2.). This is also called the correlation time, which (for randomly distributed bubbles) is equal to the maximum passage time of the bubbles. The correlation length (maximum bubble diameter) is then obtained by multiplying "~c with the bubble velocity.
C(z~)
-~ CO (' y "[c
Fig. 2.
-co
%
Autocovariance from one beam and cross-covariance from two, parallel and axially shifted beams.
The value of C('c) for z = 0 is equal to the variance of the signal fluctuations and can, as already mentioned, be used for determination of the void fraction. If the bubble velocity is not known (which is normally the case), it has to be determined from cross-correlation measurements. To this end one uses a second beam, parallel to the first, but at a different axial level. The cross-covariance C('t) = (5i 1 (t) 5 i 2 ( t + ' 0 )
(2)
will then show a peak at a time, z 0 , which is equal to the average time it takes for the bubbles to pass the axial distance between the beams. A condition for this is that we have a transport of the flow components in the axial direction only, and that the flow properties do not change significantly between the beams. This is illustrated qualitatively in Fig. 2., where the covariances are shown with a triangular shape for simplicity. The velocity is calculated by dividing the axial inter-beam distance with the transit time. If the flow is not homogeneous, for instance if we have a radial distribution of the bubble sizes, void fraction and/or velocity over a cross-section of the tube, the above method might give results that are hard or impossible to interpret. The reason for this is that all parameters will be integrated along the beams. This is where the new method is introduced. One rotates one of the beams such that it becomes perpendicular to the other one, while the axial separation between the two beams is preserved (see Fig. 3.). When forming the cross-covariance from these two signals, we get the same information as before, but now integrated over a small control volume around the crossing point of the two beams only. The size of this control volume is determined by the local correlation length itself. The void fraction in this case is given by: a(r)
= rc C('%) 4 R (Zo)
(3)
O. Thomsonand I. P~zsit
340
@j''
E" Source
I
)
D~ector
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
V--1 Detector Fig. 3.
Geometries for axial correlation length determination.
i.e. a geometrical factor times the ratio between the peak values of the cross-covariance and the cross-correlation of the two detected signals. The cross-correlation is defined as R ('Q = (i 1 (t) i2 (t + "0 ). For further details on the derivation of this relation, see Thomson & P~sit (1994). All the above applies to cases where the bubbles can be treated as being isotropic (spherical) only. If we have a flow where the bubbles have different extensions in different directions, the situation becomes more complicated. Then, generalization of the underlying theory to non-isotropie flows and a method of measurement of the correlation length in different directions becomes necessary. This is given in (P~sit, 1994). In a case where cylindrical symmetry can be assumed, the method can be used to determine the correlation length in two dimensions, namely axially and radially, in the following way. We start again by assuming a radially homogeneous flow and two parallel beams. With these the axial correlation length can be determined as before. If we then gradually increase the radial distance between the beams (from zero), the eross-covarianee peak height will decrease. The radial correlation length is now given by the radial inter-beam distance for which the eross-covarianee becomes zero everywhere. In the case of a radially inhomogeneous flow we can combine the above with the idea of using crossed beams. In this case, however, we need three beams, where the first two are crossing in the same axial plane and the third beam is parallel to either one of the other two, but lying in another axial plane (downstream the flow).
~
Sour~
V
~-1 Detector Fig. 4.
Geometries for axial and radial correlation length determination.
The crossed beam correlation technique
341
Then we form the triple covariance from the three detected signals: C ('~) = (5i 1 (t) 5i 2 (t) 5i 3 (t + "~))
(4)
At first thought, one would expect this function to yield the same information as for two parallel beams, but now for a small control volume around the crossing point. Unfortunately, this is not true for all cases, and this is explained with the following. While the cross-covariance of two signals always takes a positive value when the signals are (directly) correlated, the triple cross-covariance can take both positive and negative values, as it is an odd order moment. This means that the symmetry properties of the three signals will also affect the result (compare with the third moment or skewness for one signal). For perfectly symmetrical signals, the triple cross-covariance will be zero everywhere, regardless whether the signals are correlated or not. By symmetrical it is meant that the signals have symmetrical amplitude probability distributions. Thus, it is only for correlated and asymmetrical signals, that a peak (positive or negative) in the triple cross-covariance can be expected. To date, no solution to this complication has been found, but the method has still been able to produce useful results, in cases where the signals are not symmetrical.
EXPERIMENTAL
Setuo In order to verify the practical applicability of the described methods, an experimental facility was set up at the Department of Reactor Physics, Chalmers University of Technology. To achieve better control of the parameters involved, a flow consisting of glass beads was used, instead of water-vapour flow. The facility consists of: i. Two different test sections: 1) A single, vertical plexiglass tube into which glass beads are dropped at a constant rate and 2) dual, concentric tubes where glass beads (of possibly different diameter) are dropped from different heights with different filling rates into the two tubes. 2. A measurement equipment with 3 laser beams (2 parallel and I crossing), together with aligned photodiode detectors. Each beam can be accurately positioned, using micrometer screws (see Fig. 5.), 3. A high capacity PC-based data acquisition system containing analog filters and a 16-bit, 100 kHz analogto-digital converter. 4. Data analysis and presentation software, including specially designed, very fast correlation programs. Flow direction
Laser beams
Fig. 5.
Geometrical arrangement of laser beams.
342
O. Thomson and I. Pkzsit
Results from one-dimensional measurements (axial correlation length only) A number of measurement series were carried out in order to verify the different features of the methods described. In the following, a short description of each measurement is given, together with the most important results. 1. Velocity and mean densitv measurements: single tube. parallel beams. As the glass beads are falling freely, their velocity will depend on which vertical position of the tube the measurements are made at. As the expected velocity can easily be calculated from the height of fall, measurements at different vertical positions provide a means of determining the accuracy of the measured velocity. A comparison of measured (triangles) and theoretical (solid line) velocities is shown in Fig. 6. As can be seen, the precision is very good. 10 ~,
8
v.
•~'
6
>~ •6
4
8
2 0 0
10
20
30
40
50
Height of fall [cm]
Fig. 6.
Square of measured velocity as a function of height of fall.
7
6i_
•
5
3
1 0 0
0.2
0.4
0.6
Inverse of velocity Is/m]
Fig. 7.
Mean flow density as a function of 1/v.
0.8
1
The crossed beam correlation technique
343
For a constant bead dropping rate, the mean density (amount of beads) is inversely proportional to the velocity. This fact can, again, be used to make a comparison of theoretical and measured values, as shown in Fig. 7. In this case the precision is less good - to achieve better precision, longer measurement times are necessary. However, for purposes of flow regime identification, the measurement precision is rather satisfactory. 2, Correlation lenglda measurements: single tube. parallel beams. Two measurement series were made, one with each of the two available bead diameters of 3 and 4 mm. The resulting cross-covariance is shown in Fig. 8. Taking the half-widths of these curves (the correlation time) and multiplying with the velocity, the resulting correlation lengths of 2.0 and 3.1 mm respectively are obtained. The deviation can be explained by the finite (0.5 mm) diameter of the laser beams. 2.5 2 1.5 > o o
1 0.5 0 -0.5 6
Fig. 8.
7
8
9 Time [ms]
10
11
12
Cross-covariances from measurements with two different bead diameters.
3. Dual tube measurements of velocity, mean density_ and correlation length: crossed beams. This was the first attempt to determine local flow parameters. For this purpose, a radially inhomogeneous flow was created by using two concentric tubes. Two different sets of parameters were created in the central and the peripheral tube regions, respectively, by using different bead sizes and dropping them from different heights into each region. Two measurements were made: the first with the two beams crossing in the central region, and the second with one beam shifted radially to the peripheral region. The resulting covariances are shown in Fig. 9. The resulting parameters are as follows.
Velocity. Each CCV in Fig. 9. contains only one peak, corresponding to the velocity in the dual tube region in which the crossing point lies. The velocities were determined to be 2.64 and 1.92 m/s in the central and peripheral region respectively, in very good agreement with the theoretical values (2.62 and 1.98 m/s).
Correlation length. The correlation lengths can be obtained from the CCVs of in the same way as above. In this case however, the resulting correlation lengths are slightly larger than before, which may depend on less perfect alignment of the detectors (misalignment actually counterbalances the effect of finite beam diameter, yielding more correct results). The measured correlation lengths were 2.6 mm in the central, and 4.0 mm in the peripheral region respectively, corresponding to the bead diameters of 3.0 and 4.0 mm.
Mean density. From the CCV peak values, the mean densities were calculated to be 0.023 and 0.09 in the central and peripheral region respectively. The "true" values were 0.020 and 0.08. The last value is a rather uncertain estimate, due to technical difficulties.
344
O. Thomson and I. P~sit 1.6
i
i
cen~e peripheryJ
~ ' 0.8 > (3 ¢o 0.4
_
_
~
.
.
.
.
, _
_
,/-\i
-0.4
'''
4
F i g . 9.
_
;''' 6
; ' 8
10 12 Time [ms]
14
16
18
CCVs in the dual tube experiment with different velocities and bead diameters in the central and the peripheral region.
Results from two-dimensional measurements (axial and radial correlationlengths] I. Two beams, single tube. In this case, the correlationlength should be the same everywhere, determined by the bead diameter. The geometry shown to the leftin Fig. 4. (parallelbeams) was used. The purpose was to show that there is a consistency between the correlation lengths measured in the radial and axial direction. TWo measurements were made, with 3 and 4 mm beads respectively.The results are shown in 3-D plots in Fig. 10. and Fig. 11. In these and the foUowing plots, the time delay of the cross-covariance has been convertcd to a spatialdisplacement, by multiplying with the velocity and subtracting the axial beam separation.
F2 0
Fig. 10.
Axial and radial cross-eovariance for 3 mm glass beads.
2. Three beams, dual tube. As before, dual, concentric tubes were used to verify that the measurements were truly local. Three laser beams were used, as shown to the right in Fig. 4. and in Fig. 5. Two measurements
The crossed beam correlation technique
Fig. 11.
345
Axial and radial cross-covariance for 4 mm glass beads.
were made, one in the central and one in the peripheral region respectively. The results are shown in Fig. 12. and Fig. 13.
Fig. 12.
Axial and radial cross-covariance in central region.
CONCLUSIONS AND DISCUSSION The experimental results above clearly show that the crossed beam correlation technique provides a very useful tool for determination of the local perturbation velocity in a two-phase or two-component flow, at least in cases where the mixing ratio is far from 1 (void fraction far from 50%). The local correlation length can also be determined with a reasonable accuracy, at least in the one-dimensional case, whereas the two-dimensional technique may need further development. Finally, an aceurate determination of the local void fraction generally requires rather long measurement times, although it should be noted that, for this parameter, we only need
346
O. Thomsonand I. P~sit
Fig. 13.
Axial and radial cross-covariance in peripheral region.
to calculate one point on the cross-covariance curve (or simply the variance, if the beams are crossing in the same axial plane).
REFERENCES Analytis, G. Th. and LUbbesmeyer, D. (1983) Studies of annular flows in an air-water loop by stochastic analysis techniques. Trans. Am. Nucl. Soc. 45, 845-846. Fisher, M. J. and Krause, F. R. (1967) The crossed-beam correlation technique. J. Fluid Mech. 28, 705-717. Hewitt, G. F. (1978) Measurement of Two-Phase Flow Parameters. Academic Press, London. Jones, O. C. and Zuber, N. (1976) Transient and statistical measurement techniques for two-phase flow - a critical review. Int. J. Multiphase Flow 3, 89-116. Ptizsit, I. (1986) Two-phase flow identification by correlation techniques. Ann. nucl. Energy 13, 37-41. Ptizsit, I. (1994) Density correlations in two-phase flow and fusion plasma transport. To appear in J. Phys. D. Ptizsit, I. and G1Ockler, O. (1985) Cross-sectional identification of two-phase flow by correlation techniques. Prog. nucl. Energy 15, 661-669. Rouhani, S. Z. and Sohal, M.S. (1983) Two-phase flow patterns: a review of research results. Prog. nucl. Energy 11,219-259. Thomson, O. and P~sit, I. (1994) Determination of local two-phase flow parameters and flow structure with radiation correlation. Submitted to Int. J. Multiphase Flow. V'mce, M. A. and Lahey Jr, R. T. (1982) On the development of an objective flow regime indicator. Int. J. Multiphase Flow 8, 93-124.
1995 Elsevier Science Ltd Printed in Great Brit,fin 0149-1970/95 $29.00
Pergamon
0149-1970(95)00016-X THE CROSSED
BEAM CORRELATION
TWO-PHASE
TECHNIQUE
FOR
FLOW MEASUREMENTS
O. Thomson and I. P ~ s i t Department of Reactor Physics, Chalmers University of Technology, S-412 96 G6teborg, Sweden
ABSTRACT A number of experiments have been performed using the crossed beam correlation technique with laser beams. The purpose has been to establish the applicability of the technique to non-intrusive, local flow measurements in a two-phase or two-component flow. The measured parameters were: 1) perturbation velocity, 2) void fraction and 3) correlation length. In another series of experiments, a slightly different geometry with three beams was used, allowing determination of the correlation length in two dimensions instead of one.
KEYWORDS Two-phase flow, two-component flow, radiation correlation, local parameters, perturbation velocity, correlation length, mean density, laser measurements, crossed beam correlation technique.
INTRODUCTION It has long been known that the cross-correlation function (CCF) can be used for two-phase flow velocity determination in BWRs. Such measurements are made by cross-correlating the fluctuating part of the signals from two neutron detectors, situated at different heights (along the flow direction) in the core. If there is a pronounced peak in the resulting CCF, the position of this peak corresponds to the mean transit time of the flow perturbations between the detectors. The flow perturbations are associated with the transport of the minority component, i.e. bubbles in a sparse bubbly flow (low void fraction), or with the droplets in vapour in case of high void fraction. The situation becomes much more complicated for void fractions around 50%, in which case it is not possible to identify a minority component. Since the two flow components usually have different velocities, it is not clear whose velocity such a correlation measurement determines. Another complication arises when there is a distribution of the flow parameters, e.g. a velocity profile, present across the detector viewing area, in which case the detected value represents a weighted integral over this viewing area. Much effort has been made in order to interpret exactly which transit time is obtained from such measurements (Analytis & Ltlbbesmeyer, 1983). There are other two-phase flow parameters that can be determined by noise analysis techniques, for example qq7
338
O. Thomsonand I. Pfizsit
the void fraction. This latter can be estimated from the variance of a detector signal. Efforts have also been made to determine the flow regime, i.e. the topological structure of the flow, in which case also higher moments (skewness, curtosis) have been investigated (Jones & Zuber, 1976; Vince & Lahey, 1982). The type of measurements mentioned above have been conducted either in operating reactors, using in-core neutron detectors, or in model assemblies, using external radiation sources, such as X-rays. A common difficulty in most of these measurements, at least when non-intrusive detectors are used, is the rather large control volume (Hewitt, 1978) that is generally obtained. The control volume is defined as the volume over which the measured parameter is averaged, and is determined by both the detector viewing area and the structure of the flow. Within this volume there may exist an unknown distribution of the measured flow parameter.
THE CROSSED BEAM CORRELATION TECHNIQUE A solution to some of the mentioned difficulties can be found in the crossed beam correlation technique. This technique was originally developed for non-intrusive measurements of local parameters in turbulent flow by Fisher & Krause (1967). Later, it was suggested by P~sit (1986) and P~sit & G1Ockler (1985), that the assumption of a short correlation length of the density fluctuations would permit the use of the same technique for two-phase or two-component flow measurements. Experiments by Thomson & PEzsit 0994) have proven the technique to be capable of measuring three important flow parameters simultaneously: perturbation velocity, void fraction and correlation length. All three parameters are measured locally, at any position within the flow. In case the perturbations have a regular shape, the correlation length is closely related to their size. For example, in a bubbly flow with spherical bubbles, the correlation length is equal to the maximum diameter of the bubbles. A summary of the method, as applied in the above mentioned experiments, can be given as follows. For simplicity, we will study a water-vapour flow within a circular tube, where the vapour forms bubbles of various shapes and sizes. As information carder, we use visible light, which has different refractive indices in the water and the vapour. If we apply a collimated light beam (laser), entering the tube on one side and being monitored by a small detector on the other side, a binary fluctuating signal will be obtained. This is because due to light reflection on the bubble surface, the passing bubbles will screen the detector from the light. An example of such a signal is shown in Fig. 1.
.
.
.
.
L
_
_
.
.
.
.
!
_
_
_
2
.
.
.
.
~
.
.
.
.
.
.
m
E >
2
0
-2
.... 0
I . . . . . . . . 5 10
15
20
25
l i m e [ms]
Detector signal from two-component flow experiment, with laser beam as information carder.
If we assume a radially homogeneous, bubbly flow with isotropic (spherical) bubbles, it is in principle possi-
The crossed beam correlation technique
339
ble to determine the maximum bubble diameter (correlation length) and the void fraction from one such sig-
nal (i(t)). This is done by first forming the autocovariance: C(z) = (Si(t)Si(t+'c))
(1)
from the detected signal and determining the time at which this function becomes zero (the cut-off time, "% in Fig. 2.). This is also called the correlation time, which (for randomly distributed bubbles) is equal to the maximum passage time of the bubbles. The correlation length (maximum bubble diameter) is then obtained by multiplying "~c with the bubble velocity.
C(z~)
-~ CO (' y "[c
Fig. 2.
-co
%
Autocovariance from one beam and cross-covariance from two, parallel and axially shifted beams.
The value of C('c) for z = 0 is equal to the variance of the signal fluctuations and can, as already mentioned, be used for determination of the void fraction. If the bubble velocity is not known (which is normally the case), it has to be determined from cross-correlation measurements. To this end one uses a second beam, parallel to the first, but at a different axial level. The cross-covariance C('t) = (5i 1 (t) 5 i 2 ( t + ' 0 )
(2)
will then show a peak at a time, z 0 , which is equal to the average time it takes for the bubbles to pass the axial distance between the beams. A condition for this is that we have a transport of the flow components in the axial direction only, and that the flow properties do not change significantly between the beams. This is illustrated qualitatively in Fig. 2., where the covariances are shown with a triangular shape for simplicity. The velocity is calculated by dividing the axial inter-beam distance with the transit time. If the flow is not homogeneous, for instance if we have a radial distribution of the bubble sizes, void fraction and/or velocity over a cross-section of the tube, the above method might give results that are hard or impossible to interpret. The reason for this is that all parameters will be integrated along the beams. This is where the new method is introduced. One rotates one of the beams such that it becomes perpendicular to the other one, while the axial separation between the two beams is preserved (see Fig. 3.). When forming the cross-covariance from these two signals, we get the same information as before, but now integrated over a small control volume around the crossing point of the two beams only. The size of this control volume is determined by the local correlation length itself. The void fraction in this case is given by: a(r)
= rc C('%) 4 R (Zo)
(3)
O. Thomsonand I. P~zsit
340
@j''
E" Source
I
)
D~ector
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
V--1 Detector Fig. 3.
Geometries for axial correlation length determination.
i.e. a geometrical factor times the ratio between the peak values of the cross-covariance and the cross-correlation of the two detected signals. The cross-correlation is defined as R ('Q = (i 1 (t) i2 (t + "0 ). For further details on the derivation of this relation, see Thomson & P~sit (1994). All the above applies to cases where the bubbles can be treated as being isotropic (spherical) only. If we have a flow where the bubbles have different extensions in different directions, the situation becomes more complicated. Then, generalization of the underlying theory to non-isotropie flows and a method of measurement of the correlation length in different directions becomes necessary. This is given in (P~sit, 1994). In a case where cylindrical symmetry can be assumed, the method can be used to determine the correlation length in two dimensions, namely axially and radially, in the following way. We start again by assuming a radially homogeneous flow and two parallel beams. With these the axial correlation length can be determined as before. If we then gradually increase the radial distance between the beams (from zero), the eross-covarianee peak height will decrease. The radial correlation length is now given by the radial inter-beam distance for which the eross-covarianee becomes zero everywhere. In the case of a radially inhomogeneous flow we can combine the above with the idea of using crossed beams. In this case, however, we need three beams, where the first two are crossing in the same axial plane and the third beam is parallel to either one of the other two, but lying in another axial plane (downstream the flow).
~
Sour~
V
~-1 Detector Fig. 4.
Geometries for axial and radial correlation length determination.
The crossed beam correlation technique
341
Then we form the triple covariance from the three detected signals: C ('~) = (5i 1 (t) 5i 2 (t) 5i 3 (t + "~))
(4)
At first thought, one would expect this function to yield the same information as for two parallel beams, but now for a small control volume around the crossing point. Unfortunately, this is not true for all cases, and this is explained with the following. While the cross-covariance of two signals always takes a positive value when the signals are (directly) correlated, the triple cross-covariance can take both positive and negative values, as it is an odd order moment. This means that the symmetry properties of the three signals will also affect the result (compare with the third moment or skewness for one signal). For perfectly symmetrical signals, the triple cross-covariance will be zero everywhere, regardless whether the signals are correlated or not. By symmetrical it is meant that the signals have symmetrical amplitude probability distributions. Thus, it is only for correlated and asymmetrical signals, that a peak (positive or negative) in the triple cross-covariance can be expected. To date, no solution to this complication has been found, but the method has still been able to produce useful results, in cases where the signals are not symmetrical.
EXPERIMENTAL
Setuo In order to verify the practical applicability of the described methods, an experimental facility was set up at the Department of Reactor Physics, Chalmers University of Technology. To achieve better control of the parameters involved, a flow consisting of glass beads was used, instead of water-vapour flow. The facility consists of: i. Two different test sections: 1) A single, vertical plexiglass tube into which glass beads are dropped at a constant rate and 2) dual, concentric tubes where glass beads (of possibly different diameter) are dropped from different heights with different filling rates into the two tubes. 2. A measurement equipment with 3 laser beams (2 parallel and I crossing), together with aligned photodiode detectors. Each beam can be accurately positioned, using micrometer screws (see Fig. 5.), 3. A high capacity PC-based data acquisition system containing analog filters and a 16-bit, 100 kHz analogto-digital converter. 4. Data analysis and presentation software, including specially designed, very fast correlation programs. Flow direction
Laser beams
Fig. 5.
Geometrical arrangement of laser beams.
342
O. Thomson and I. Pkzsit
Results from one-dimensional measurements (axial correlation length only) A number of measurement series were carried out in order to verify the different features of the methods described. In the following, a short description of each measurement is given, together with the most important results. 1. Velocity and mean densitv measurements: single tube. parallel beams. As the glass beads are falling freely, their velocity will depend on which vertical position of the tube the measurements are made at. As the expected velocity can easily be calculated from the height of fall, measurements at different vertical positions provide a means of determining the accuracy of the measured velocity. A comparison of measured (triangles) and theoretical (solid line) velocities is shown in Fig. 6. As can be seen, the precision is very good. 10 ~,
8
v.
•~'
6
>~ •6
4
8
2 0 0
10
20
30
40
50
Height of fall [cm]
Fig. 6.
Square of measured velocity as a function of height of fall.
7
6i_
•
5
3
1 0 0
0.2
0.4
0.6
Inverse of velocity Is/m]
Fig. 7.
Mean flow density as a function of 1/v.
0.8
1
The crossed beam correlation technique
343
For a constant bead dropping rate, the mean density (amount of beads) is inversely proportional to the velocity. This fact can, again, be used to make a comparison of theoretical and measured values, as shown in Fig. 7. In this case the precision is less good - to achieve better precision, longer measurement times are necessary. However, for purposes of flow regime identification, the measurement precision is rather satisfactory. 2, Correlation lenglda measurements: single tube. parallel beams. Two measurement series were made, one with each of the two available bead diameters of 3 and 4 mm. The resulting cross-covariance is shown in Fig. 8. Taking the half-widths of these curves (the correlation time) and multiplying with the velocity, the resulting correlation lengths of 2.0 and 3.1 mm respectively are obtained. The deviation can be explained by the finite (0.5 mm) diameter of the laser beams. 2.5 2 1.5 > o o
1 0.5 0 -0.5 6
Fig. 8.
7
8
9 Time [ms]
10
11
12
Cross-covariances from measurements with two different bead diameters.
3. Dual tube measurements of velocity, mean density_ and correlation length: crossed beams. This was the first attempt to determine local flow parameters. For this purpose, a radially inhomogeneous flow was created by using two concentric tubes. Two different sets of parameters were created in the central and the peripheral tube regions, respectively, by using different bead sizes and dropping them from different heights into each region. Two measurements were made: the first with the two beams crossing in the central region, and the second with one beam shifted radially to the peripheral region. The resulting covariances are shown in Fig. 9. The resulting parameters are as follows.
Velocity. Each CCV in Fig. 9. contains only one peak, corresponding to the velocity in the dual tube region in which the crossing point lies. The velocities were determined to be 2.64 and 1.92 m/s in the central and peripheral region respectively, in very good agreement with the theoretical values (2.62 and 1.98 m/s).
Correlation length. The correlation lengths can be obtained from the CCVs of in the same way as above. In this case however, the resulting correlation lengths are slightly larger than before, which may depend on less perfect alignment of the detectors (misalignment actually counterbalances the effect of finite beam diameter, yielding more correct results). The measured correlation lengths were 2.6 mm in the central, and 4.0 mm in the peripheral region respectively, corresponding to the bead diameters of 3.0 and 4.0 mm.
Mean density. From the CCV peak values, the mean densities were calculated to be 0.023 and 0.09 in the central and peripheral region respectively. The "true" values were 0.020 and 0.08. The last value is a rather uncertain estimate, due to technical difficulties.
344
O. Thomson and I. P~sit 1.6
i
i
cen~e peripheryJ
~ ' 0.8 > (3 ¢o 0.4
_
_
~
.
.
.
.
, _
_
,/-\i
-0.4
'''
4
F i g . 9.
_
;''' 6
; ' 8
10 12 Time [ms]
14
16
18
CCVs in the dual tube experiment with different velocities and bead diameters in the central and the peripheral region.
Results from two-dimensional measurements (axial and radial correlationlengths] I. Two beams, single tube. In this case, the correlationlength should be the same everywhere, determined by the bead diameter. The geometry shown to the leftin Fig. 4. (parallelbeams) was used. The purpose was to show that there is a consistency between the correlation lengths measured in the radial and axial direction. TWo measurements were made, with 3 and 4 mm beads respectively.The results are shown in 3-D plots in Fig. 10. and Fig. 11. In these and the foUowing plots, the time delay of the cross-covariance has been convertcd to a spatialdisplacement, by multiplying with the velocity and subtracting the axial beam separation.
F2 0
Fig. 10.
Axial and radial cross-eovariance for 3 mm glass beads.
2. Three beams, dual tube. As before, dual, concentric tubes were used to verify that the measurements were truly local. Three laser beams were used, as shown to the right in Fig. 4. and in Fig. 5. Two measurements
The crossed beam correlation technique
Fig. 11.
345
Axial and radial cross-covariance for 4 mm glass beads.
were made, one in the central and one in the peripheral region respectively. The results are shown in Fig. 12. and Fig. 13.
Fig. 12.
Axial and radial cross-covariance in central region.
CONCLUSIONS AND DISCUSSION The experimental results above clearly show that the crossed beam correlation technique provides a very useful tool for determination of the local perturbation velocity in a two-phase or two-component flow, at least in cases where the mixing ratio is far from 1 (void fraction far from 50%). The local correlation length can also be determined with a reasonable accuracy, at least in the one-dimensional case, whereas the two-dimensional technique may need further development. Finally, an aceurate determination of the local void fraction generally requires rather long measurement times, although it should be noted that, for this parameter, we only need
346
O. Thomsonand I. P~sit
Fig. 13.
Axial and radial cross-covariance in peripheral region.
to calculate one point on the cross-covariance curve (or simply the variance, if the beams are crossing in the same axial plane).
REFERENCES Analytis, G. Th. and LUbbesmeyer, D. (1983) Studies of annular flows in an air-water loop by stochastic analysis techniques. Trans. Am. Nucl. Soc. 45, 845-846. Fisher, M. J. and Krause, F. R. (1967) The crossed-beam correlation technique. J. Fluid Mech. 28, 705-717. Hewitt, G. F. (1978) Measurement of Two-Phase Flow Parameters. Academic Press, London. Jones, O. C. and Zuber, N. (1976) Transient and statistical measurement techniques for two-phase flow - a critical review. Int. J. Multiphase Flow 3, 89-116. Ptizsit, I. (1986) Two-phase flow identification by correlation techniques. Ann. nucl. Energy 13, 37-41. Ptizsit, I. (1994) Density correlations in two-phase flow and fusion plasma transport. To appear in J. Phys. D. Ptizsit, I. and G1Ockler, O. (1985) Cross-sectional identification of two-phase flow by correlation techniques. Prog. nucl. Energy 15, 661-669. Rouhani, S. Z. and Sohal, M.S. (1983) Two-phase flow patterns: a review of research results. Prog. nucl. Energy 11,219-259. Thomson, O. and P~sit, I. (1994) Determination of local two-phase flow parameters and flow structure with radiation correlation. Submitted to Int. J. Multiphase Flow. V'mce, M. A. and Lahey Jr, R. T. (1982) On the development of an objective flow regime indicator. Int. J. Multiphase Flow 8, 93-124.