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Power law approximations to gas volume fraction and velocity profiles in low void fraction vertical gas–liquid flows
Flow Measurement and Instrumentation 15 (2004) 271–283 www.elsevier.com/locate/flowmeasinst
Power law approximations to gas volume fraction and velocity profiles in low void fraction vertical gas–liquid flows G.P. Lucas , R. Mishra, N. Panayotopoulos School of Computing and Engineering, University of Huddersfield, Queensgate, Huddersfield HD1 3DH, UK Received 30 March 2004; received in revised form 11 May 2004; accepted 14 June 2004
Abstract A dual sensor conductance probe was used to measure the distributions of the local gas volume fraction and the local gas axial velocity in vertical upward, bubby air–water flows in which the mean gas volume fraction was less than 0.1. Very limited data are available in the literature for such low volume fraction flows. The measured local gas volume fraction and velocity distributions were approximated by power law functions. The power law exponents associated with the measured local gas volume fraction profiles were found to be up to 30% higher than values predicted in the literature. The power law exponents associated with the measured local gas velocity profiles were also found to be somewhat higher than values predicted in the literature. The power law exponents for the measured local gas volume fraction and local axial gas velocity distributions at a given flow condition were combined to obtain an estimate of the ‘Zuber–Findlay’ distribution parameter C0 at that flow condition. The mean value of C0 for all of the flow conditions investigated was 1.09. This value of C0 was found to give good agreement with the gradient of a plot of the mean gas velocity ug versus the homogeneous velocity uh, where ug and uh were obtained from reference measurements. This agreement is evidence for the good accuracy of the measured volume fraction and velocity profiles. Finally, the paper casts doubt upon previously published criteria regarding the optimum axial sensor separation in dual sensor probes. # 2004 Elsevier Ltd. All rights reserved. Keywords: Gas velocity; Volume fraction profiles
1. Introduction This paper describes the use of a local, intrusive, dual sensor conductance probe to obtain profiles of the local gas volume fraction and the local gas velocity in low volume fraction, vertical upward, bubbly air–water flows. The main motivation for this work was to produce power law approximations to the local gas velocity and volume fraction profiles in the authors’ multiphase flow loop. These power law approximations would then be available for quantitative comparison with profiles obtained using novel measurement techniques such as electrical resistance tomography. It was also required to investigate whether correlations for the power law exponents for the gas velocity and volume fraction profiles obtained by van der Welle [1] would Corresponding author. Tel.: +44-1484-472266; fax: +44-1484451883. E-mail address: [email protected] (G.P. Lucas).
0955-5986/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.flowmeasinst.2004.06.004
be valid for flows relevant to the present study, namely flows in an 80 mm internal diameter pipe for which the mean volume fraction is less than 0.1. van der Welle [1] based his correlations on measurements taken in a 100 mm internal diameter pipe and at values of mean gas volume fraction in the range 0.28–0.75. A further purpose of the research described in this paper was to determine the accuracy with which the Zuber–Findlay [2] distribution parameter C0 can be estimated from measured gas volume fraction and velocity profiles. C0 is an important parameter in enabling the mean velocity of the dispersed phase in two phase flows to be determined from measurements of the mixture superficial velocity (also called the homogeneous velocity) using ‘drift velocity’ models [2]. Knowledge of the Zuber–Findlay [2] distribution parameter C0 is therefore of importance, for example, to oil companies making downhole velocity estimates in two phase gas–liquid wells.
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Nomenclature A Aj C0 d D i j jl n p q Qg Qw r R Rref Rs s t1f t2f t1r t2r ug ug;est ugl ugl,max ugs uh ug;ref ut0 uws Va Vf Vin Vout Vr a aest al amax aref dt1,i dt2,i dtmax dtmin
flow cross-sectional area (m2) cross-sectional area of jth region (m2) distribution parameter mean bubble diameter (m) internal pipe diameter (m) subscript referring to bubble number subscript referring to region number local homogeneous velocity (m s1) exponent used to calculate slip velocity power law exponent for velocity profile power law exponent for volume fraction profile gas volumetric flow rate (m3 s1) liquid volumetric flow rate (m3 s1) radial position (m) internal pipe radius (m) reference resistance (X) sensor resistance (X) sensor separation (m) time when bubble first contacts front sensor (s) time when bubble last contacts front sensor (s) time when bubble first contacts rear sensor (s) time when bubble last contacts rear sensor (s) mean gas velocity (m s1) estimated mean gas velocity (m s1) local gas velocity (m s1) maximum local gas velocity (m s1) superficial gas velocity (m s1) homogeneous velocity (m s1) reference mean gas velocity (m s1) single bubble terminal velocity (m s1) liquid superficial velocity (m s1) output voltage from op-amp (V) front sensor threshold voltage (V) input voltage (V) output voltage (V) rear sensor threshold voltage (V) mean gas volume fraction estimated mean gas volume fraction local gas volume fraction maximum gas volume fraction reference mean gas volume fraction (from dp cell) time interval defined in Eq. (1) (s) time interval defined in Eq. (2) (s) maximum time interval (s) minimum time interval (s)
Optical, non-intrusive methods such as particle image velocimetry can potentially be used to make local volume fraction and velocity measurements in two phase flows but often cannot achieve sufficient depth of investigation due to the scattering of light from multiple bubble surfaces. Intrusive local probes
have successfully been used to make measurements in two phase flows in a number of previous investigations. For example, Toral [3] used a hot wire anemometer, Hamad et al. [4] used an intrusive, local optical probe. Many others including Serizawa et al. [5], Herringe and Davis [6], Steinemann and Buchholz [7], van der Welle
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[1], Wu et al. [8] and Liu [9] have all successfully used dual sensor probes which measure variations in the local electrical impedance of the multiphase mixture to determine volume fraction or velocity profiles of the dispersed phase. The principle of operation of a local, dual sensor conductance probe can be explained as follows. Consider an idealized situation where the two sensors of the probe are separated by an axial distance s in a vertical upward bubbly air–water flow in which the bubbles are assumed to travel in the axial direction only. Let us assume that the surface of a bubble makes first contact with the upstream (front) sensor at time t1f. At this time, the measured conductance at the front sensor will fall sharply (see Fig. 1) as it is immersed in air instead of water. Let us further assume that the front sensor makes last contact with the surface of the bubble at time t2f (this is the time at which the bubble leaves the front sensor). At time t2f, the measured conductance at the front sensor will rise sharply as the sensor is again surrounded by water. The times at which the rear sensor makes first and last contact with the surface of the bubble are t1r and t2r, respectively, as shown in Fig. 1. Suppose N bubbles hit both the front and rear sensors during a sampling period T. For the ith bubble, two time intervals dt1,i and dt2,i may be defined as follows dt1;i ¼ t1f;i t1r;i
of curvature of bubbles that hit the probe ‘off centre’, as demonstrated by Steinemann and Buchholz [7]. The mean local volume fraction al of the bubbles at the position of the probe can be estimated from the conductance signal from either the front or the rear sensor (see Serizawa et al. [5]). For the front sensor, al is given by al ¼
N 1X ðt2f;i t1f;i Þ T i¼1
ð4Þ
In real, as opposed to ideal, vertical upward air– water flows the velocities of the air bubbles are not purely axial but have small lateral components as well. Wu et al. [8] showed that these lateral velocity components can cause the surface of a bubble to strike both the front and rear sensors nearly simultaneously, giving rise to very small values of dt1,i and dt2,i. With reference to Eq. (3), this in turn can give rise to gross overestimates of the mean local axial bubble velocity ugl. To minimize the effects of this problem in typical vertical upward, bubbly air–water flows Wu et al. [8] suggested that the axial separation s of the sensors should be in the range 0:5d s 2d
ð5Þ
where d is the mean bubble diameter.
ð1Þ 2. Construction of the dual sensor probe and associated circuitry
and dt2;i ¼ t2f;i t2r;i
ð2Þ
The mean local axial bubble velocity ugl at the position of the probe is then given by ugl ¼
273
N 2s X 1 N i¼1 ðdt1;i þ dt2;i Þ
ð3Þ
If the air bubbles have a plane of symmetry normal to their direction of motion then use of Eq. (3) minimizes the errors in the calculated value of ugl due to the effects
Fig. 1. Ideal signals obtained from a gas bubble striking a local, dual sensor conductance probe.
In the present investigation, a number of dual sensor probes were constructed. Each probe was manufactured from two stainless steel acupuncture needles which were 0.3 mm in diameter and which offered a relatively high level of rigidity. The acupuncture needles were mounted inside a stainless steel tube with an outer diameter of 4 mm as shown in Fig. 2. Each acupuncture needle was coated with waterproof paint and insulating varnish but this was removed from the very tip of the needle using fine emery paper. Thus, the front and rear sensors of each dual sensor probe were located at the very tips of the acupuncture needles. The probe dimensions were accurately measured using a magnifying shadowgraph system. For the probes used in the experiments described in this paper, the axial sensor separation s was typically 5 mm whilst the lateral sensor separation was typically 1 mm. Since the mean bubble diameter in the experiments described below was about 5 mm, the value for s which was used was comfortably within the range suggested by Wu et al. [8]. The distance from the tip of the rear sensor to the support tube was typically about 10 mm. The stainless steel tube forming the probe body was used as a common earth electrode for the front and rear sensors. The conductance at each sensor was
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transition times (e.g. t1f and t2f for the front sensor) much more difficult. The output voltage Va from the operational amplifier in the circuit shown in Fig. 3 is given by Rref Va ¼ Vin 1 þ ð6Þ Rs When the tip of a given acupuncture needle is immersed in water Rs is relatively small compared to Rref (which has a typical value of 1.5 MX) and so Va saturates at the positive rail voltage used to supply the operational amplifier (+15 V). When the tip of the acupuncture needle is immersed in the air bubble, the quantity Rref =Rs approaches zero and so Va approaches Vin (+5 V). Thus, as each sensor is alternately immersed in water and air, output signals similar to those shown in Fig. 1 are obtained. Note that the potentiometer shown in Fig. 3 was used to scale the voltage Va from each sensor to provide an output voltage Vout which covered the full range of the analogue to digital converters of the data acquisition system that was used.
3. Signal processing
Fig. 2. Construction and geometry of the dual sensor conductance probe (drawing not to scale).
measured using a circuit based on the design given in Fig. 3, in which the sensor resistance Rs is the resistance between the tip of the relevant needle and the stainless steel tube forming the probe body. Each circuit is dc because an ac circuit would have required a demodulator, the low pass filter of which would have removed high frequency components from the sensor output signal making accurate identification of the
Fig. 3. Outline diagram of circuit used to measure conductance changes at each sensor. Rref is typically 1.5 MX.
The dual sensor probe described in the previous section was used to make measurements in vertical upward, bubbly air–water flows for a range of values of water superficial velocity uws and air superficial velocity ugs in an 80 mm internal diameter pipe as described in Section 4. At each flow condition, the dual sensor probe was successively moved to each of 81 distinct spatial locations in the flow cross-section using an automatic traversing mechanism. At each spatial location, the probe was used to obtain signals from the flowing air bubbles for a period of 30 s and these signals were then stored. After data were taken at each of the 81 distinct spatial locations in the flow cross-section, it was processed off-line in the following manner. Real (as opposed to ideal) signals from the two sensors are shown in Fig. 4. It was found that the best estimates for the mean local gas volume fraction al and the mean local axial bubble velocity ugl were obtained if threshold voltages Vf and Vr were defined for the front and rear sensors, respectively, as shown in Fig. 4. This need to use threshold voltages arises from the fact that there is a small reduction in the measured conductance from a given sensor just before the air bubble actually touches the sensor. This initial drop in conductance is probably due to the air bubble partially blocking the flow of electrical current through the water from the sensor tip to the earthed probe body. For a similar reason, the measured sensor conductance does not return to its maximum value until a short time after the air bubble has ceased to be in contact with the
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Fig. 4. The solid and dashed lines are portions of real output signals obtained from the front and rear sensors, respectively. Threshold voltage levels Vf and Vr are illustrative only.
sensor. Thus, the times t1f,i and t2f,i at which the surface of the ith bubble comes into contact with the front sensor occur when the output voltage from the front sensor is at a value Vf. Similarly, the times t1r,i and t2r,i at which the surface of the ith bubble comes into contact with the rear sensor occur when the output voltage from the rear sensor is at a value Vr. For a given flow condition, the values of Vf and Vr were calculated as follows by making use of a reference measurement aref of the area averaged gas volume fraction, obtained using a differential pressure measurement technique (see Section 4). If we consider the front sensor, an initial value of Vf is guessed and then the local gas volume fraction is obtained for each of the 81 spatial locations of the probe using Eq. (4). If, at the jth spatial location, the local gas volume fraction is al,j then an estimate aest of the area average gas volume fraction can be made using the expression 81 P
aest ¼
al;j Aj
j¼1
A
ð7Þ
where A is the cross-sectional area of the flow and where Aj is one of 81 discrete areas into which the flow cross-section is divided, the jth discrete area corresponding to the jth spatial location of the probe. If aest is not equal to aref , then the threshold voltage Vf is adjusted and the process repeated. When aest and aref are equal, the relevant value of Vf is used to determine the bubble contact times t1f,i and t2f,i which in turn are used, in conjunction with Eq. (4), to produce plots of the local gas volume fraction distribution (see for example Figs. 5 and 6). A similar procedure to that outlined above is used to determine the threshold voltage Vr for the rear sensor, from which the bubble contact times t1r,i and t2r,i are determined. (Note that this method for setting the threshold voltages is very straightforward because the reference measurement aref
is very simple to make. The technique described herein therefore represents a significant improvement over the complex and often subjective signal processing techniques described elsewhere, see for example Liu [9].) Once t1f,i, t2f,i, t1r,i and t2r,i have been found for a given probe location at a particular flow condition, Eq. (3) can be used to determine the local axial velocity ugl at that spatial location of the probe. However, two additional signal processing features were introduced, as described below, to make the local axial bubble velocity estimates as accurate as possible. As described in Section 1, bubbles with significant lateral velocities can strike both the front and rear sensors virtually simultaneously giving rise to very small values for dt1,i and dt2,i and hence (with reference to Eq. (3)) causing gross overestimates in the value of ugl. In an attempt to counteract this phenomenon, a minimum time interval dtmin was introduced such that If dt1;i < dtmin or dt2;i < dtmin then ith bubble is ignored when calculating ugl
ð8Þ
For a given flow condition, the value of dtmin was found in the following way. The global mean gas velocity ug;ref was obtained from reference measurements of the gas volumetric flow rate Qg and the area average gas volume fraction aref (see Section 4) using the relationship Qg ð9Þ Aaref An estimate ug;est of the mean gas velocity can then be obtained using the expression ug;ref ¼
81 P
ug;est ¼
ugl;j al;j Aj
j¼1 81 P
ð10Þ al;j Aj
j¼1
where for the jth probe location ugl,j is calculated from Eq. (3) and al,j is calculated from Eq. (4). (Note that
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Fig. 5. Local gas volume fraction distributions with the water superficial velocity uws constant at 0.64 m s1 and the gas superficial velocity ugs at 0.017, 0.035 and 0.073 m s1 in a–c, respectively.
when using Eq. (10) it should be understood that the criterion given in Eq. (8) is employed.) With dtmin set to zero (in the criterion given in Eq. (8)) ug;est is gener ally greater than ug;ref . However, for any particular flow condition, a value of dtmin can be found for which ug;ref . For the given flow condition, this ug;est is equal to value of dtmin was subsequently used in calculating values of the local axial velocity ugl from Eq. (3), in conjunction with the criterion given in Eq. (8). A typical value for dtmin was 2:5 104 s.
Fig. 6. Local gas volume fraction distributions with the gas superficial velocity ugs held constant at 0.033 m s1 and with the water superficial velocity uws at 0.1, 0.38 and 0.91 m s1 in a–c, respectively.
In calculating the local axial bubble velocity, the following additional criterion was used to eliminate a further potential source of error. The slip velocity of a 5 mm air bubble in water is approximately equal to 0.25 m s1 and so in a vertical upward air–water flow, the maximum time taken for a gas bubble to travel the 5 mm axial distance between the two sensors of the probe would be expected to be 0.02 s (close to the pipe walls it could take slightly longer than this because, as noted by Steinemann and Buchholz [7], there may be a
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moderate down flow of water at the walls). A time interval dtmax can be defined (which is greater than 0.02 s) and which represents the maximum expected transit time of a gas bubble between the two sensors of the dual sensor probe. If, for what is assumed to be the ith bubble, either dt1,i or dt2,i is greater than dtmax then it is highly unlikely that a single bubble is producing the signals at the two sensors. A more likely explanation is that the bubble striking the front sensor misses the rear sensor and that, slightly later, a different bubble strikes the rear sensor having missed the front sensor (the phenomenon of missing bubbles is discussed in detail by Liu [9]). To prevent inaccuracies in the estimate of ugl the following criterion was introduced. If dt1;i > dtmax or dt2;i > dtmax then what is assumed to be the ith bubble is ignored when calculating ugl ð11Þ The criterion given in Eq. (11) was particularly helpful in ensuring that values of the local axial bubble velocity ugl were not underestimated. In the present investigation, a conservative value of dtmax equal to 0.05 s was employed. The local axial bubble velocity profiles shown in Fig. 9 were obtained using Eq. (3) in conjunction with the criteria given in Eqs. (8) and (11).
4. The flow loop and its associated measurement instrumentation Measurements of the local axial bubble velocity distribution and the local gas volume fraction distribution in vertical upward bubbly air–water flows were made using the dual sensor probe in a flow loop with a 2.5 m long, 80 mm internal diameter, transparent, vertical working section. Water was pumped into the base of the working section via a turbine meter which enabled the water volumetric flow rate Qw to be measured. The water superficial velocity uws in the working section is then given by uws
Qw ¼ A
ð12Þ
where A is the flow cross-sectional area. Air was pumped into the working section via a series of 1 mm diameter holes equispaced around the circumference of the base of the working section. The mass flow rate of the air was measured before it entered the working section using a thermal mass flow meter. Measurements of the pressure and temperature in the working section enabled the mean air volumetric flow rate Qg in the working section to be determined. The air superficial
277
velocity ugs was then calculated using Qg ð13Þ A The mean air volume fraction aref in the working section at a given flow condition was obtained using a differential pressure measurement, compensated for the effects of frictional pressure loss. This technique is widely described in the literature (see for example Lucas and Jin [10]) and so no further description will be given here. The mean air velocity ug;ref was then found using Eq. (9). The dual sensor probe described in Section 2 was mounted in a fully automated, two-axis traversing mechanism which enabled the probe to be moved to any spatial location in a plane at a distance of 2 m from the inlet to the working section. In the experiments described in Section 5, the probe was traversed across eight equispaced diameters in the flow cross-section. Data were obtained at 11 equispaced locations on each diameter (at the pipe centre and at distances from the pipe centre of 7.6, 15.2, 22.8, 30.4 and 38 mm) giving a total of 81 distinct measurement locations. (NB multiple measurements at the pipe centre were useful in establishing that the flow conditions inside the working section were not changing greatly with time. In some of the data presented in this paper (see for example the local axial bubble velocity distribution plots in Fig. 9) all eight measurements taken at the pipe centre are shown. However, it should be noted that for calculations such as those given in Eqs. (7) and (10), the average of the eight values of local velocity or local volume fraction taken at the pipe centre was used.) At each measurement location, and for each flow condition, data were acquired from the dual sensor probe for a period of 30 s. Due to the thickness of the probe holder, it was not possible to obtain data at distances of less than 2 mm from the internal pipe wall.
ugs ¼
5. The local gas volume fraction distribution in vertical upward, bubbly air–water flows A series of experiments was carried out to measure the local gas volume fraction distribution in vertical upward air water flows using the dual sensor probe described previously in this paper. These experiments were carried out at water superficial velocities uws in the range 0.1–1.15 m s1 and with appropriate values of the gas superficial velocity ugs to ensure that the flow was always in the bubbly regime. In the present investigation, the mean gas volume fraction aref was never more than 0.1, thus ensuring that the two phase flow was always bubbly. Examples of the local gas volume fraction distributions that were obtained from the front sensor are
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shown in Figs. 5a–c and 6a–c. In Fig. 5a–c, the water superficial velocity uws is constant at a value of 0.64 m s1 whilst the gas superficial velocity ugs takes values of 0.017, 0.035 and 0.073 m s1 in Fig. 5a–c, respectively. The mean gas volume fraction aref (obtained using the differential pressure measurement technique mentioned in Section 4) is 0.013, 0.033 and 0.062 in Fig. 5a–c, respectively. It is apparent from Fig. 5a–c that as aref increases, as a result of ugs being increased, the local gas volume fraction distribution changes from a ‘peaky’ profile, where the gas is relatively highly concentrated at the pipe centre, to a smoother profile where the gas is more uniformly distributed in the flow cross-section. This phenomenon can also be seen in Fig. 6a–c where the gas superficial velocity ugs is held constant at 0.033 m s1 whilst the water superficial velocity uws takes values of 0.1, 0.38 and 0.91 m s1. The mean gas volume fraction aref has values of 0.069, 0.038 and 0.026 in Fig. 6a–c, respectively. It is again apparent that as aref decreases (as uws is increased with ugs being held constant) the local gas volume fraction distribution changes from a relatively smooth distribution in Fig. 6a to a relatively ‘peaky’ shape in Fig. 6c. (Note that in Figs. 5a–c and 6a–d the x and y co-ordinates represent the spatial location of the probe, non-dimensionalised with respect to the internal pipe diameter D.) van der Welle [1] suggested that the local gas volume fraction in vertical upward gas–liquid flows could be represented by a power law expression of the form
increasing aref , higher values of q representing more ‘peaky’ local gas volume fraction distributions. A correlation between q and a for the data taken in the present study, and which is valid for 0:01 a 0:08, is q ¼ 0:9014e4:823a
ð16Þ
This correlation is shown in Fig. 7 as a solid dark line. On the basis of experiments performed in gas–liquid flows in which the volume fraction ranged from 0.28 to 0.75, van der Welle [1] suggested the following correlation between q and a q¼
1 a 2
0:7 ð17Þ
ð15Þ
Van der Welle’s correlation is shown in Fig. 7 as a solid light line. It is clear from Fig. 7 that van der Welle’s correlation agrees well with the data taken in the present study for values of a around 0.08 (a not unremarkable result given the differences in his experimental conditions mentioned in Section 1). However, as a is reduced below 0.08 van der Welle’s correlation increasingly under predicts q. For a ¼ 0:02, this under prediction is 30%. For a ¼ 0:02, Eq. (15) can be used to show the gas volume fraction distributions that are obtained using the correlations for q given in Eqs. (16) and (17). These distributions are shown in Fig. 8 where it is clear that van der Welle’s correlation predicts a somewhat smoother profile than that which is actually observed. (Note that Hibiki and Ishii [11] have stated that the local gas volume fraction distribution is dependent upon many factors including pipe diameter, bubble size and method of bubble injection. The correlation between q and a given in Eq. (16) should only be regarded as valid for experimental conditions similar to those described in this paper.)
For the local volume fraction distributions obtained in the present investigation, Eq. (15) was used to determine the exponent q for each of the flow conditions investigated. This was done by finding the value of q which gave the best fit of the measured values of al at a given flow condition to Eq. (15). (NB in Eq. (15) a is the value of aest that is obtained from Eq. (7) when the correct threshold voltage Vf is applied. From the discussion given in Section 3, it should be apparent to the reader that this is virtually identical to the value of aref obtained using the differential pressure measurement technique mentioned in Section 4.) Fig. 7 shows the exponent q plotted against aref for all of the flow conditions investigated in the present study. It is apparent that there is a tendency for q to decrease slowly with
Fig. 7. Exponent q versus aref . The dark line is defined by Eq. (16). The light line represents the van der Welle correlation for q.
al ¼ amax ð1 r=RÞq
ð14Þ
where amax is the maximum value of the local gas volume fraction, at a given flow condition, which occurs at the pipe centre, r is radial position within the pipe, R is the pipe radius and q is an exponent. It is fairly straightforward to show that Eq. (14) can also be written in the form að1 r=RÞq ðq þ 1Þðq þ 2Þ al ¼ 0:5
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279
Fig. 8. Local gas volume fraction distributions for a ¼ 0:02 from present study (dark line) and from van der Welle correlation (light line).
exponent p
6. The local gas velocity distribution in vertical upward, bubbly air–water flows
p¼ The dual sensor probe was also used to obtain profiles of the local axial bubble velocity distribution for the same flow conditions (given in Section 5) at which the local gas volume fraction profiles were obtained. Examples of the gas velocity profiles obtained are shown in Fig. 9 in which the water superficial velocity uws was held constant at 0.38 m s1 and the gas superficial velocity took values of 0.017, 0.032 and 0.072 m s1 in Fig. 9a–c, respectively. The mean gas volume fraction aref is 0.02, 0.038 and 0.079 in Fig. 9a–c, respectively. It is apparent from Fig. 9a–c that the measured velocity profiles are much flatter than the volume fraction profiles shown in Figs. 5 and 6. It also apparent that, as expected, the velocity profiles are essentially axisymmetric, any minor deviations from axisymmetry almost certainly being due to either the blockage effect of the probe and its holder, or asymmetry in the bubble injection conditions. van der Welle [1] proposed that the local axial gas velocity distribution in vertical air–water flows could be represented by a power law approximation of the form ugl ¼ ugl;max ð1 r=RÞp
ð18Þ
where ugl,max is the maximum value of the local gas velocity at the pipe centre. By finding the best fit of the experimental data to the relationship given in Eq. (18) (using techniques similar to those described in the previous section), the value of the exponent p was found for each of the flow conditions investigated. In Fig. 10, p is plotted against the mean gas volume fraction aref and it is apparent that there is some scatter in the value of p about a mean value of 0.173. On the basis of experiments which he performed in the volume fraction range 0.28–0.75 van der Welle [1] suggested the following empirical correlation for the
1 10ð1 aÞ
ð19Þ
This correlation is shown in Fig. 10 and predicts slightly lower values for p than were observed in the present study, thus predicting slightly flatter gas velocity profiles. In Fig. 11, the non-dimensional velocity profile ugl =ugl;max is plotted for p ¼ 0:173, the mean value observed in the present study. Also, shown in Fig. 11 is the non-dimensional velocity profile for p ¼ 0:105, the mean value predicted by the van der Welle correlation for 0 < a 0:1. It is apparent that although the profile predicted by van der Welle is slightly flatter, the difference between the two profiles is marginal. It should be noted that some of the values of p shown in Fig. 10 may be inaccurate for the following reason. As described in Section 3, if a gas bubble strikes the front sensor but not the rear sensor and shortly afterwards another gas bubble strikes the rear sensor but not the front sensor, then if the relevant sensor signals are wrongly assumed to be from the same bubble, an incorrect bubble velocity will be obtained. This incorrect bubble velocity will almost certainly be lower than the true mean local gas velocity at the probe position. It was found that if dtmax was reduced below 0.05 s there was a tendency for the values of p to reduce slightly, indicating that the velocity profiles had been made slightly flatter. This could be explained by the phenomenon of air bubbles being increasingly likely to strike only one of the two sensors as the probe was moved closer to the pipe wall (such a phenomenon could perhaps be due to the fact that velocity gradients within the water are greater closer to the pipe wall and hence lateral forces on the bubbles, causing them to deviate from purely axial motion, are greater closer to the pipe wall). By reducing dtmax, the effect of this phenomenon on the velocity profile may have been
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that the values of p which were obtained in the present study are probably quite reasonable. It is much less likely that a bubble which strikes the first sensor will miss the second sensor if the axial sensor separation s is relatively small. In the present study, the axial sensor separation s of 5 mm was approximately equal to the mean bubble diameter and yet there was still some uncertainty as to whether particular signals came from the same bubble. It is therefore possible that the criterion of Wu et al. [8] given in Eq. (5), which recommends that s can be up to twice the mean bubble diameter, may lead to the use of axial sensor separations which can give rise to ambiguity when processing the sensor output signals.
7. Estimates of the Zuber–Findlay distribution parameter from the local gas volume fraction and velocity distributions With reference to Zuber and Findlay [2], it can be shown that for a two phase gas–liquid flow, the mean gas velocity ug is given by an expression of the form ug ¼ C0 uh þ ut0 ð1 aÞn
ð20Þ
where C0 is a distribution parameter, ut0 is the rise velocity of a single gas bubble through the stationary liquid (note that the bubble size for which ut0 is calculated must be a size which is representative of the two phase flow under consideration), n is an appropriate exponent and uh is the homogeneous velocity (or mixture superficial velocity) which is given by uh ¼ uws þ ugs
Fig. 9. Local gas velocity profiles with the water superficial velocity uws held constant at 0.38 m s1 and with the gas superficial velocity at 0.017, 0.032 and 0.072 m s1 in a–c, respectively (diamond shapes indicate measured velocities at relevant probe locations).
reduced. If a value for dtmax of 0.05 s is too high, then (from the reasoning given above) this could lead to an overestimation of the corresponding values of the exponent p. Dependency of dtmax on such factors as uh or on the probe position was not investigated further but could well be worthy of future investigation. However, it should also be noted that in Section 7 it is shown that the distribution parameter C0, calculated using the values of p obtained in the present study, agrees quite well with the expected value. This suggests
ð21Þ
As stated in Section 1, relationships of the form shown in Eq. (20) are important when calculating the mean dispersed phase velocity from measurements of the homogeneous velocity uh. Lucas and Jin (see for example [10]) have described various practical techniques by which uh can be measured for two phase flows. In the present study, since only relatively low values of the mean gas volume fraction a have been investigated, ut0 ð1 aÞn ut0
ð22Þ
Furthermore, with reference to Govier and Aziz [12], the value of ut0 for 5 mm diameter air bubbles (a typical size in the present study) rising through stationary water is ut0 ¼ 0:25 m s1
ð23Þ
This value of ut0 has also been found experimentally to be valid for low volume fraction vertical air–water flows by, for example, Johnson and White [13]. With reference to Zuber and Findlay [2], the distribution
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281
Fig. 10. Exponent p versus aref . Solid line represents van der Welle correlation for p.
parameter C0 can be obtained from the expression C0 ¼
al j l auh
ð24Þ
where the overbar in the numerator represents averaging in the flow cross-section, and where jl is the local homogeneous velocity (i.e. it is the sum of the local flux densities of the air and water). If the assumption is now made that the local homogeneous velocity jl has a power law distribution in the flow cross-section, and that the relevant exponent p associated with this power law distribution is the same as for the local gas velocity distribution, then it is not difficult to show from Eqs. (14), (18) and (24) that C0 ¼
ðq þ 1Þðq þ 2Þðp þ 1Þðp þ 2Þ 2ðp þ q þ 1Þðp þ q þ 2Þ
ð25Þ
where q is the power law exponent associated with the local gas volume fraction distribution. For the flow conditions investigated, Eq. (25) was used to calculate
C0. Fig. 12 shows C0 plotted against the mean gas volume fraction aref . The mean value of C0 for all of the flow conditions investigated was 1.09 and so, with reference to Eqs. (20), (22) and (23) an expression for the mean gas velocity ug derived (in part) from the measured gas volume fraction and velocity distributions, is given by ug ¼ 1:09uh þ 0:25 m s1
ð26Þ
In an attempt to test the accuracy of Eq. (26), the mean gas velocity ug;ref obtained from global reference measurements (see Eq. (9) and Section 4) was plotted against uh, also obtained from global reference measurements (see Eqs. (12), (13) and (21)), for a number of flow conditions in the range relevant to the present study. This plot of ug;ref versus uh is shown in Fig. 13. It is apparent from Fig. 13, that there is considerable scatter in the data, but this is typical for such plots obtained at low values of the mean gas volume fraction a (for example at a flow condition where the true value of a is 0.02 but where a small error of 0.01 is made in
Fig. 11. Mean non-dimensional velocity profiles for 0 < a < 0:1 from present study (dark line) and from van der Welle correlation (light line).
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Fig. 12. Distribution parameter C0 versus mean gas volume fraction aref . Solid line is best fit to data and has the equation C0 ¼ 0:5053 aref þ 1:11.
the measurement of a then, with reference to Eq. (9), this would lead to the estimated value of ug being twice its true value). Also, shown in Fig. 13 is the straight line for the predicted relationship between ug and uh given by Eq. (26). It can be seen that there is reasonable agreement between the predicted relationship and the observed values of these variables. The distribution parameter C0 of 1.09 obtained in the present study is also in reasonable agreement with the slightly lower value of 1.05 obtained by Johnson and White [13] for bubbly air–water flows in a 200 mm diameter pipe. The difference in values may be explained by the work of Hibiki et al. [14] who suggest that, for a given mean bubble diameter, the value of C0 will increase as the pipe diameter is reduced. Fig. 12 also hints that the value of C0 reduces very slightly as aref is increased. If this trend is real, it would agree with observations made by Lucas and Jin [15] for vertical, bubbly oil-in-water flows, where it was found that C0 reduced with increasing oil volume fraction for values of oil volume fraction in the range 0–0.3.
Fig. 13. Mean gas velocity ug;ref versus the homogeneous velocity uh (diamond shapes). The solid line represents the predicted relationship between ug and uh given by Eq. (26).
8. Conclusions A dual sensor conductance probe has been designed and built and used, with appropriate signal processing techniques, to obtain distributions of the local gas volume fraction and the local gas velocity in vertical upward, bubbly air–water flows in which the mean gas volume fraction was less than 0.1. Both the gas volume fraction and the gas velocity distributions were approximated by power law functions. The power law exponents q for the local gas volume fraction distributions were found to be higher than the values predicted by van der Welle [1]. For a ¼ 0:02, the exponent q was 30% higher than predicted by van der Welle indicating that, for this low value of a, the local gas volume fraction distribution is much less smooth than previously thought, with the gas preferentially clustering at the pipe centre. The power law exponents p for the gas velocity distributions were on average slightly higher than the values predicted by van der Welle. This may be due to the fact that van der Welle’s predictions were based on data taken at much higher values of a. However, it is also possible that this discrepancy may be due to signal processing errors arising from an axial sensor separation s which was inappropriately large. This in turn would suggest that the criterion for axial sensor separation recommended by Wu et al. [8], and which was adopted in the present study, may cause ambiguity in deciding whether signals from the two sensors are from the same bubble or from different bubbles (as described in Section 6). For each flow condition investigated, a Zuber– Findlay distribution parameter C0 was calculated from the power law exponents for the gas velocity and volume fraction distributions. The mean value of C0 for all of the flow conditions investigated was 1.09. When this value of C0 was combined with an appropriate slip velocity (for gas bubbles in water) a relationship
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g and the homogeneous between the mean gas velocity u velocity uh was obtained which agreed closely with the relationship between ug and uh obtained from reference measurements. This result suggests that the power law exponents obtained for the local gas volume fraction and velocity distributions are probably correct.
Acknowledgements The authors wish to acknowledge the financial support of the UK Engineering and Physical Sciences Research Council (EPSRC). The work presented here represents part of the work undertaken via EPSRC grant GR/N28610.
References [1] R. van der Welle, Void fraction, bubble velocity and bubble size in two phase flow, Int. J. Multiphase Flow 11 (1985) 317–345. [2] N. Zuber, J.A. Findlay, Average volumetric concentration in two phase flow systems, ASME J. Heat Transfer 87 (1965) 453–468. [3] H. Toral, A study of the hot wire anemometer for measuring void fraction in two phase flow, J. Phys. E: Sci. Instrum. 14 (1981). [4] F.A. Hamad, B.K. Pierscionek, H.H. Bruun, A dual optical probe for volume fraction, drop velocity and drop size measurements in liquid–liquid two phase flow, Meas. Sci. Technol. 11 (2000) 1307–1318.
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[5] A. Serizawa, I. Kataoko, I. Michiyoshi, Turbulence structure of air–water bubbly flow—I. Measuring techniques, Int. J. Multiphase Flow 2 (1975) 221–233. [6] R.A. Herringe, M.R. Davis, Structural development of gas– liquid mixture flows, J. Fluid Mech. 73 (part 1) (1976) 97–123. [7] J. Steinemann, R. Buchholz, Application of an electrical conductivity microprobe for the characterization of bubble behavior in gas–liquid bubble flow, Part. Charact. 1 (1984) 102–107. [8] Q. Wu, K. Welter, D. McCreary, J.N. Reyes, Theoretical studies on the design criteria of double-sensor probe for the measurement of bubble velocity, J. Flow Meas. Instrum. 12 (issue 1) (2001) 43–51. [9] T.-J. Liu, An effective signal processing method for resistivity probe measurements in a two phase bubbly flow, Meas. Sci. Technol. 13 (2) (2002) 206–217. [10] G.P. Lucas, N.D. Jin, Measurement of the homogeneous velocity of inclined oil-in-water flows using a resistance cross correlation flow meter, Meas. Sci. Technol. 12 (2001) 1529–1537. [11] T. Hibiki, M. Ishii, Experimental study on interfacial area transport in bubbly two-phase flows, Int. J. Heat Mass Transfer 42 (1999) 3019–3035. [12] G.W. Govier, K. Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold Company, 1972. [13] A.B. Johnson, D.B. White, Gas rise velocities during kicks, SPE 20431, 65th Annual Technical Conference of Society of Petroleum Engineers, New Orleans, LA, September 23–26, 1990. [14] T. Hibiki, R. Situ, Y. Mi, M. Ishii, Local flow measurements of vertical upward bubbly flow in an annulus, Int. J. Heat Mass Transfer 46 (2003) 1479–1496. [15] G.P. Lucas, N.D. Jin, A new kinematic wave model for interpreting cross correlation velocity measurements in vertically upward bubbly oil-in-water flows, Meas. Sci. Technol. 12 (2001) 1538–1545.
Power law approximations to gas volume fraction and velocity profiles in low void fraction vertical gas–liquid flows G.P. Lucas , R. Mishra, N. Panayotopoulos School of Computing and Engineering, University of Huddersfield, Queensgate, Huddersfield HD1 3DH, UK Received 30 March 2004; received in revised form 11 May 2004; accepted 14 June 2004
Abstract A dual sensor conductance probe was used to measure the distributions of the local gas volume fraction and the local gas axial velocity in vertical upward, bubby air–water flows in which the mean gas volume fraction was less than 0.1. Very limited data are available in the literature for such low volume fraction flows. The measured local gas volume fraction and velocity distributions were approximated by power law functions. The power law exponents associated with the measured local gas volume fraction profiles were found to be up to 30% higher than values predicted in the literature. The power law exponents associated with the measured local gas velocity profiles were also found to be somewhat higher than values predicted in the literature. The power law exponents for the measured local gas volume fraction and local axial gas velocity distributions at a given flow condition were combined to obtain an estimate of the ‘Zuber–Findlay’ distribution parameter C0 at that flow condition. The mean value of C0 for all of the flow conditions investigated was 1.09. This value of C0 was found to give good agreement with the gradient of a plot of the mean gas velocity ug versus the homogeneous velocity uh, where ug and uh were obtained from reference measurements. This agreement is evidence for the good accuracy of the measured volume fraction and velocity profiles. Finally, the paper casts doubt upon previously published criteria regarding the optimum axial sensor separation in dual sensor probes. # 2004 Elsevier Ltd. All rights reserved. Keywords: Gas velocity; Volume fraction profiles
1. Introduction This paper describes the use of a local, intrusive, dual sensor conductance probe to obtain profiles of the local gas volume fraction and the local gas velocity in low volume fraction, vertical upward, bubbly air–water flows. The main motivation for this work was to produce power law approximations to the local gas velocity and volume fraction profiles in the authors’ multiphase flow loop. These power law approximations would then be available for quantitative comparison with profiles obtained using novel measurement techniques such as electrical resistance tomography. It was also required to investigate whether correlations for the power law exponents for the gas velocity and volume fraction profiles obtained by van der Welle [1] would Corresponding author. Tel.: +44-1484-472266; fax: +44-1484451883. E-mail address: [email protected] (G.P. Lucas).
0955-5986/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.flowmeasinst.2004.06.004
be valid for flows relevant to the present study, namely flows in an 80 mm internal diameter pipe for which the mean volume fraction is less than 0.1. van der Welle [1] based his correlations on measurements taken in a 100 mm internal diameter pipe and at values of mean gas volume fraction in the range 0.28–0.75. A further purpose of the research described in this paper was to determine the accuracy with which the Zuber–Findlay [2] distribution parameter C0 can be estimated from measured gas volume fraction and velocity profiles. C0 is an important parameter in enabling the mean velocity of the dispersed phase in two phase flows to be determined from measurements of the mixture superficial velocity (also called the homogeneous velocity) using ‘drift velocity’ models [2]. Knowledge of the Zuber–Findlay [2] distribution parameter C0 is therefore of importance, for example, to oil companies making downhole velocity estimates in two phase gas–liquid wells.
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Nomenclature A Aj C0 d D i j jl n p q Qg Qw r R Rref Rs s t1f t2f t1r t2r ug ug;est ugl ugl,max ugs uh ug;ref ut0 uws Va Vf Vin Vout Vr a aest al amax aref dt1,i dt2,i dtmax dtmin
flow cross-sectional area (m2) cross-sectional area of jth region (m2) distribution parameter mean bubble diameter (m) internal pipe diameter (m) subscript referring to bubble number subscript referring to region number local homogeneous velocity (m s1) exponent used to calculate slip velocity power law exponent for velocity profile power law exponent for volume fraction profile gas volumetric flow rate (m3 s1) liquid volumetric flow rate (m3 s1) radial position (m) internal pipe radius (m) reference resistance (X) sensor resistance (X) sensor separation (m) time when bubble first contacts front sensor (s) time when bubble last contacts front sensor (s) time when bubble first contacts rear sensor (s) time when bubble last contacts rear sensor (s) mean gas velocity (m s1) estimated mean gas velocity (m s1) local gas velocity (m s1) maximum local gas velocity (m s1) superficial gas velocity (m s1) homogeneous velocity (m s1) reference mean gas velocity (m s1) single bubble terminal velocity (m s1) liquid superficial velocity (m s1) output voltage from op-amp (V) front sensor threshold voltage (V) input voltage (V) output voltage (V) rear sensor threshold voltage (V) mean gas volume fraction estimated mean gas volume fraction local gas volume fraction maximum gas volume fraction reference mean gas volume fraction (from dp cell) time interval defined in Eq. (1) (s) time interval defined in Eq. (2) (s) maximum time interval (s) minimum time interval (s)
Optical, non-intrusive methods such as particle image velocimetry can potentially be used to make local volume fraction and velocity measurements in two phase flows but often cannot achieve sufficient depth of investigation due to the scattering of light from multiple bubble surfaces. Intrusive local probes
have successfully been used to make measurements in two phase flows in a number of previous investigations. For example, Toral [3] used a hot wire anemometer, Hamad et al. [4] used an intrusive, local optical probe. Many others including Serizawa et al. [5], Herringe and Davis [6], Steinemann and Buchholz [7], van der Welle
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[1], Wu et al. [8] and Liu [9] have all successfully used dual sensor probes which measure variations in the local electrical impedance of the multiphase mixture to determine volume fraction or velocity profiles of the dispersed phase. The principle of operation of a local, dual sensor conductance probe can be explained as follows. Consider an idealized situation where the two sensors of the probe are separated by an axial distance s in a vertical upward bubbly air–water flow in which the bubbles are assumed to travel in the axial direction only. Let us assume that the surface of a bubble makes first contact with the upstream (front) sensor at time t1f. At this time, the measured conductance at the front sensor will fall sharply (see Fig. 1) as it is immersed in air instead of water. Let us further assume that the front sensor makes last contact with the surface of the bubble at time t2f (this is the time at which the bubble leaves the front sensor). At time t2f, the measured conductance at the front sensor will rise sharply as the sensor is again surrounded by water. The times at which the rear sensor makes first and last contact with the surface of the bubble are t1r and t2r, respectively, as shown in Fig. 1. Suppose N bubbles hit both the front and rear sensors during a sampling period T. For the ith bubble, two time intervals dt1,i and dt2,i may be defined as follows dt1;i ¼ t1f;i t1r;i
of curvature of bubbles that hit the probe ‘off centre’, as demonstrated by Steinemann and Buchholz [7]. The mean local volume fraction al of the bubbles at the position of the probe can be estimated from the conductance signal from either the front or the rear sensor (see Serizawa et al. [5]). For the front sensor, al is given by al ¼
N 1X ðt2f;i t1f;i Þ T i¼1
ð4Þ
In real, as opposed to ideal, vertical upward air– water flows the velocities of the air bubbles are not purely axial but have small lateral components as well. Wu et al. [8] showed that these lateral velocity components can cause the surface of a bubble to strike both the front and rear sensors nearly simultaneously, giving rise to very small values of dt1,i and dt2,i. With reference to Eq. (3), this in turn can give rise to gross overestimates of the mean local axial bubble velocity ugl. To minimize the effects of this problem in typical vertical upward, bubbly air–water flows Wu et al. [8] suggested that the axial separation s of the sensors should be in the range 0:5d s 2d
ð5Þ
where d is the mean bubble diameter.
ð1Þ 2. Construction of the dual sensor probe and associated circuitry
and dt2;i ¼ t2f;i t2r;i
ð2Þ
The mean local axial bubble velocity ugl at the position of the probe is then given by ugl ¼
273
N 2s X 1 N i¼1 ðdt1;i þ dt2;i Þ
ð3Þ
If the air bubbles have a plane of symmetry normal to their direction of motion then use of Eq. (3) minimizes the errors in the calculated value of ugl due to the effects
Fig. 1. Ideal signals obtained from a gas bubble striking a local, dual sensor conductance probe.
In the present investigation, a number of dual sensor probes were constructed. Each probe was manufactured from two stainless steel acupuncture needles which were 0.3 mm in diameter and which offered a relatively high level of rigidity. The acupuncture needles were mounted inside a stainless steel tube with an outer diameter of 4 mm as shown in Fig. 2. Each acupuncture needle was coated with waterproof paint and insulating varnish but this was removed from the very tip of the needle using fine emery paper. Thus, the front and rear sensors of each dual sensor probe were located at the very tips of the acupuncture needles. The probe dimensions were accurately measured using a magnifying shadowgraph system. For the probes used in the experiments described in this paper, the axial sensor separation s was typically 5 mm whilst the lateral sensor separation was typically 1 mm. Since the mean bubble diameter in the experiments described below was about 5 mm, the value for s which was used was comfortably within the range suggested by Wu et al. [8]. The distance from the tip of the rear sensor to the support tube was typically about 10 mm. The stainless steel tube forming the probe body was used as a common earth electrode for the front and rear sensors. The conductance at each sensor was
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transition times (e.g. t1f and t2f for the front sensor) much more difficult. The output voltage Va from the operational amplifier in the circuit shown in Fig. 3 is given by Rref Va ¼ Vin 1 þ ð6Þ Rs When the tip of a given acupuncture needle is immersed in water Rs is relatively small compared to Rref (which has a typical value of 1.5 MX) and so Va saturates at the positive rail voltage used to supply the operational amplifier (+15 V). When the tip of the acupuncture needle is immersed in the air bubble, the quantity Rref =Rs approaches zero and so Va approaches Vin (+5 V). Thus, as each sensor is alternately immersed in water and air, output signals similar to those shown in Fig. 1 are obtained. Note that the potentiometer shown in Fig. 3 was used to scale the voltage Va from each sensor to provide an output voltage Vout which covered the full range of the analogue to digital converters of the data acquisition system that was used.
3. Signal processing
Fig. 2. Construction and geometry of the dual sensor conductance probe (drawing not to scale).
measured using a circuit based on the design given in Fig. 3, in which the sensor resistance Rs is the resistance between the tip of the relevant needle and the stainless steel tube forming the probe body. Each circuit is dc because an ac circuit would have required a demodulator, the low pass filter of which would have removed high frequency components from the sensor output signal making accurate identification of the
Fig. 3. Outline diagram of circuit used to measure conductance changes at each sensor. Rref is typically 1.5 MX.
The dual sensor probe described in the previous section was used to make measurements in vertical upward, bubbly air–water flows for a range of values of water superficial velocity uws and air superficial velocity ugs in an 80 mm internal diameter pipe as described in Section 4. At each flow condition, the dual sensor probe was successively moved to each of 81 distinct spatial locations in the flow cross-section using an automatic traversing mechanism. At each spatial location, the probe was used to obtain signals from the flowing air bubbles for a period of 30 s and these signals were then stored. After data were taken at each of the 81 distinct spatial locations in the flow cross-section, it was processed off-line in the following manner. Real (as opposed to ideal) signals from the two sensors are shown in Fig. 4. It was found that the best estimates for the mean local gas volume fraction al and the mean local axial bubble velocity ugl were obtained if threshold voltages Vf and Vr were defined for the front and rear sensors, respectively, as shown in Fig. 4. This need to use threshold voltages arises from the fact that there is a small reduction in the measured conductance from a given sensor just before the air bubble actually touches the sensor. This initial drop in conductance is probably due to the air bubble partially blocking the flow of electrical current through the water from the sensor tip to the earthed probe body. For a similar reason, the measured sensor conductance does not return to its maximum value until a short time after the air bubble has ceased to be in contact with the
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Fig. 4. The solid and dashed lines are portions of real output signals obtained from the front and rear sensors, respectively. Threshold voltage levels Vf and Vr are illustrative only.
sensor. Thus, the times t1f,i and t2f,i at which the surface of the ith bubble comes into contact with the front sensor occur when the output voltage from the front sensor is at a value Vf. Similarly, the times t1r,i and t2r,i at which the surface of the ith bubble comes into contact with the rear sensor occur when the output voltage from the rear sensor is at a value Vr. For a given flow condition, the values of Vf and Vr were calculated as follows by making use of a reference measurement aref of the area averaged gas volume fraction, obtained using a differential pressure measurement technique (see Section 4). If we consider the front sensor, an initial value of Vf is guessed and then the local gas volume fraction is obtained for each of the 81 spatial locations of the probe using Eq. (4). If, at the jth spatial location, the local gas volume fraction is al,j then an estimate aest of the area average gas volume fraction can be made using the expression 81 P
aest ¼
al;j Aj
j¼1
A
ð7Þ
where A is the cross-sectional area of the flow and where Aj is one of 81 discrete areas into which the flow cross-section is divided, the jth discrete area corresponding to the jth spatial location of the probe. If aest is not equal to aref , then the threshold voltage Vf is adjusted and the process repeated. When aest and aref are equal, the relevant value of Vf is used to determine the bubble contact times t1f,i and t2f,i which in turn are used, in conjunction with Eq. (4), to produce plots of the local gas volume fraction distribution (see for example Figs. 5 and 6). A similar procedure to that outlined above is used to determine the threshold voltage Vr for the rear sensor, from which the bubble contact times t1r,i and t2r,i are determined. (Note that this method for setting the threshold voltages is very straightforward because the reference measurement aref
is very simple to make. The technique described herein therefore represents a significant improvement over the complex and often subjective signal processing techniques described elsewhere, see for example Liu [9].) Once t1f,i, t2f,i, t1r,i and t2r,i have been found for a given probe location at a particular flow condition, Eq. (3) can be used to determine the local axial velocity ugl at that spatial location of the probe. However, two additional signal processing features were introduced, as described below, to make the local axial bubble velocity estimates as accurate as possible. As described in Section 1, bubbles with significant lateral velocities can strike both the front and rear sensors virtually simultaneously giving rise to very small values for dt1,i and dt2,i and hence (with reference to Eq. (3)) causing gross overestimates in the value of ugl. In an attempt to counteract this phenomenon, a minimum time interval dtmin was introduced such that If dt1;i < dtmin or dt2;i < dtmin then ith bubble is ignored when calculating ugl
ð8Þ
For a given flow condition, the value of dtmin was found in the following way. The global mean gas velocity ug;ref was obtained from reference measurements of the gas volumetric flow rate Qg and the area average gas volume fraction aref (see Section 4) using the relationship Qg ð9Þ Aaref An estimate ug;est of the mean gas velocity can then be obtained using the expression ug;ref ¼
81 P
ug;est ¼
ugl;j al;j Aj
j¼1 81 P
ð10Þ al;j Aj
j¼1
where for the jth probe location ugl,j is calculated from Eq. (3) and al,j is calculated from Eq. (4). (Note that
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Fig. 5. Local gas volume fraction distributions with the water superficial velocity uws constant at 0.64 m s1 and the gas superficial velocity ugs at 0.017, 0.035 and 0.073 m s1 in a–c, respectively.
when using Eq. (10) it should be understood that the criterion given in Eq. (8) is employed.) With dtmin set to zero (in the criterion given in Eq. (8)) ug;est is gener ally greater than ug;ref . However, for any particular flow condition, a value of dtmin can be found for which ug;ref . For the given flow condition, this ug;est is equal to value of dtmin was subsequently used in calculating values of the local axial velocity ugl from Eq. (3), in conjunction with the criterion given in Eq. (8). A typical value for dtmin was 2:5 104 s.
Fig. 6. Local gas volume fraction distributions with the gas superficial velocity ugs held constant at 0.033 m s1 and with the water superficial velocity uws at 0.1, 0.38 and 0.91 m s1 in a–c, respectively.
In calculating the local axial bubble velocity, the following additional criterion was used to eliminate a further potential source of error. The slip velocity of a 5 mm air bubble in water is approximately equal to 0.25 m s1 and so in a vertical upward air–water flow, the maximum time taken for a gas bubble to travel the 5 mm axial distance between the two sensors of the probe would be expected to be 0.02 s (close to the pipe walls it could take slightly longer than this because, as noted by Steinemann and Buchholz [7], there may be a
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moderate down flow of water at the walls). A time interval dtmax can be defined (which is greater than 0.02 s) and which represents the maximum expected transit time of a gas bubble between the two sensors of the dual sensor probe. If, for what is assumed to be the ith bubble, either dt1,i or dt2,i is greater than dtmax then it is highly unlikely that a single bubble is producing the signals at the two sensors. A more likely explanation is that the bubble striking the front sensor misses the rear sensor and that, slightly later, a different bubble strikes the rear sensor having missed the front sensor (the phenomenon of missing bubbles is discussed in detail by Liu [9]). To prevent inaccuracies in the estimate of ugl the following criterion was introduced. If dt1;i > dtmax or dt2;i > dtmax then what is assumed to be the ith bubble is ignored when calculating ugl ð11Þ The criterion given in Eq. (11) was particularly helpful in ensuring that values of the local axial bubble velocity ugl were not underestimated. In the present investigation, a conservative value of dtmax equal to 0.05 s was employed. The local axial bubble velocity profiles shown in Fig. 9 were obtained using Eq. (3) in conjunction with the criteria given in Eqs. (8) and (11).
4. The flow loop and its associated measurement instrumentation Measurements of the local axial bubble velocity distribution and the local gas volume fraction distribution in vertical upward bubbly air–water flows were made using the dual sensor probe in a flow loop with a 2.5 m long, 80 mm internal diameter, transparent, vertical working section. Water was pumped into the base of the working section via a turbine meter which enabled the water volumetric flow rate Qw to be measured. The water superficial velocity uws in the working section is then given by uws
Qw ¼ A
ð12Þ
where A is the flow cross-sectional area. Air was pumped into the working section via a series of 1 mm diameter holes equispaced around the circumference of the base of the working section. The mass flow rate of the air was measured before it entered the working section using a thermal mass flow meter. Measurements of the pressure and temperature in the working section enabled the mean air volumetric flow rate Qg in the working section to be determined. The air superficial
277
velocity ugs was then calculated using Qg ð13Þ A The mean air volume fraction aref in the working section at a given flow condition was obtained using a differential pressure measurement, compensated for the effects of frictional pressure loss. This technique is widely described in the literature (see for example Lucas and Jin [10]) and so no further description will be given here. The mean air velocity ug;ref was then found using Eq. (9). The dual sensor probe described in Section 2 was mounted in a fully automated, two-axis traversing mechanism which enabled the probe to be moved to any spatial location in a plane at a distance of 2 m from the inlet to the working section. In the experiments described in Section 5, the probe was traversed across eight equispaced diameters in the flow cross-section. Data were obtained at 11 equispaced locations on each diameter (at the pipe centre and at distances from the pipe centre of 7.6, 15.2, 22.8, 30.4 and 38 mm) giving a total of 81 distinct measurement locations. (NB multiple measurements at the pipe centre were useful in establishing that the flow conditions inside the working section were not changing greatly with time. In some of the data presented in this paper (see for example the local axial bubble velocity distribution plots in Fig. 9) all eight measurements taken at the pipe centre are shown. However, it should be noted that for calculations such as those given in Eqs. (7) and (10), the average of the eight values of local velocity or local volume fraction taken at the pipe centre was used.) At each measurement location, and for each flow condition, data were acquired from the dual sensor probe for a period of 30 s. Due to the thickness of the probe holder, it was not possible to obtain data at distances of less than 2 mm from the internal pipe wall.
ugs ¼
5. The local gas volume fraction distribution in vertical upward, bubbly air–water flows A series of experiments was carried out to measure the local gas volume fraction distribution in vertical upward air water flows using the dual sensor probe described previously in this paper. These experiments were carried out at water superficial velocities uws in the range 0.1–1.15 m s1 and with appropriate values of the gas superficial velocity ugs to ensure that the flow was always in the bubbly regime. In the present investigation, the mean gas volume fraction aref was never more than 0.1, thus ensuring that the two phase flow was always bubbly. Examples of the local gas volume fraction distributions that were obtained from the front sensor are
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shown in Figs. 5a–c and 6a–c. In Fig. 5a–c, the water superficial velocity uws is constant at a value of 0.64 m s1 whilst the gas superficial velocity ugs takes values of 0.017, 0.035 and 0.073 m s1 in Fig. 5a–c, respectively. The mean gas volume fraction aref (obtained using the differential pressure measurement technique mentioned in Section 4) is 0.013, 0.033 and 0.062 in Fig. 5a–c, respectively. It is apparent from Fig. 5a–c that as aref increases, as a result of ugs being increased, the local gas volume fraction distribution changes from a ‘peaky’ profile, where the gas is relatively highly concentrated at the pipe centre, to a smoother profile where the gas is more uniformly distributed in the flow cross-section. This phenomenon can also be seen in Fig. 6a–c where the gas superficial velocity ugs is held constant at 0.033 m s1 whilst the water superficial velocity uws takes values of 0.1, 0.38 and 0.91 m s1. The mean gas volume fraction aref has values of 0.069, 0.038 and 0.026 in Fig. 6a–c, respectively. It is again apparent that as aref decreases (as uws is increased with ugs being held constant) the local gas volume fraction distribution changes from a relatively smooth distribution in Fig. 6a to a relatively ‘peaky’ shape in Fig. 6c. (Note that in Figs. 5a–c and 6a–d the x and y co-ordinates represent the spatial location of the probe, non-dimensionalised with respect to the internal pipe diameter D.) van der Welle [1] suggested that the local gas volume fraction in vertical upward gas–liquid flows could be represented by a power law expression of the form
increasing aref , higher values of q representing more ‘peaky’ local gas volume fraction distributions. A correlation between q and a for the data taken in the present study, and which is valid for 0:01 a 0:08, is q ¼ 0:9014e4:823a
ð16Þ
This correlation is shown in Fig. 7 as a solid dark line. On the basis of experiments performed in gas–liquid flows in which the volume fraction ranged from 0.28 to 0.75, van der Welle [1] suggested the following correlation between q and a q¼
1 a 2
0:7 ð17Þ
ð15Þ
Van der Welle’s correlation is shown in Fig. 7 as a solid light line. It is clear from Fig. 7 that van der Welle’s correlation agrees well with the data taken in the present study for values of a around 0.08 (a not unremarkable result given the differences in his experimental conditions mentioned in Section 1). However, as a is reduced below 0.08 van der Welle’s correlation increasingly under predicts q. For a ¼ 0:02, this under prediction is 30%. For a ¼ 0:02, Eq. (15) can be used to show the gas volume fraction distributions that are obtained using the correlations for q given in Eqs. (16) and (17). These distributions are shown in Fig. 8 where it is clear that van der Welle’s correlation predicts a somewhat smoother profile than that which is actually observed. (Note that Hibiki and Ishii [11] have stated that the local gas volume fraction distribution is dependent upon many factors including pipe diameter, bubble size and method of bubble injection. The correlation between q and a given in Eq. (16) should only be regarded as valid for experimental conditions similar to those described in this paper.)
For the local volume fraction distributions obtained in the present investigation, Eq. (15) was used to determine the exponent q for each of the flow conditions investigated. This was done by finding the value of q which gave the best fit of the measured values of al at a given flow condition to Eq. (15). (NB in Eq. (15) a is the value of aest that is obtained from Eq. (7) when the correct threshold voltage Vf is applied. From the discussion given in Section 3, it should be apparent to the reader that this is virtually identical to the value of aref obtained using the differential pressure measurement technique mentioned in Section 4.) Fig. 7 shows the exponent q plotted against aref for all of the flow conditions investigated in the present study. It is apparent that there is a tendency for q to decrease slowly with
Fig. 7. Exponent q versus aref . The dark line is defined by Eq. (16). The light line represents the van der Welle correlation for q.
al ¼ amax ð1 r=RÞq
ð14Þ
where amax is the maximum value of the local gas volume fraction, at a given flow condition, which occurs at the pipe centre, r is radial position within the pipe, R is the pipe radius and q is an exponent. It is fairly straightforward to show that Eq. (14) can also be written in the form að1 r=RÞq ðq þ 1Þðq þ 2Þ al ¼ 0:5
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Fig. 8. Local gas volume fraction distributions for a ¼ 0:02 from present study (dark line) and from van der Welle correlation (light line).
exponent p
6. The local gas velocity distribution in vertical upward, bubbly air–water flows
p¼ The dual sensor probe was also used to obtain profiles of the local axial bubble velocity distribution for the same flow conditions (given in Section 5) at which the local gas volume fraction profiles were obtained. Examples of the gas velocity profiles obtained are shown in Fig. 9 in which the water superficial velocity uws was held constant at 0.38 m s1 and the gas superficial velocity took values of 0.017, 0.032 and 0.072 m s1 in Fig. 9a–c, respectively. The mean gas volume fraction aref is 0.02, 0.038 and 0.079 in Fig. 9a–c, respectively. It is apparent from Fig. 9a–c that the measured velocity profiles are much flatter than the volume fraction profiles shown in Figs. 5 and 6. It also apparent that, as expected, the velocity profiles are essentially axisymmetric, any minor deviations from axisymmetry almost certainly being due to either the blockage effect of the probe and its holder, or asymmetry in the bubble injection conditions. van der Welle [1] proposed that the local axial gas velocity distribution in vertical air–water flows could be represented by a power law approximation of the form ugl ¼ ugl;max ð1 r=RÞp
ð18Þ
where ugl,max is the maximum value of the local gas velocity at the pipe centre. By finding the best fit of the experimental data to the relationship given in Eq. (18) (using techniques similar to those described in the previous section), the value of the exponent p was found for each of the flow conditions investigated. In Fig. 10, p is plotted against the mean gas volume fraction aref and it is apparent that there is some scatter in the value of p about a mean value of 0.173. On the basis of experiments which he performed in the volume fraction range 0.28–0.75 van der Welle [1] suggested the following empirical correlation for the
1 10ð1 aÞ
ð19Þ
This correlation is shown in Fig. 10 and predicts slightly lower values for p than were observed in the present study, thus predicting slightly flatter gas velocity profiles. In Fig. 11, the non-dimensional velocity profile ugl =ugl;max is plotted for p ¼ 0:173, the mean value observed in the present study. Also, shown in Fig. 11 is the non-dimensional velocity profile for p ¼ 0:105, the mean value predicted by the van der Welle correlation for 0 < a 0:1. It is apparent that although the profile predicted by van der Welle is slightly flatter, the difference between the two profiles is marginal. It should be noted that some of the values of p shown in Fig. 10 may be inaccurate for the following reason. As described in Section 3, if a gas bubble strikes the front sensor but not the rear sensor and shortly afterwards another gas bubble strikes the rear sensor but not the front sensor, then if the relevant sensor signals are wrongly assumed to be from the same bubble, an incorrect bubble velocity will be obtained. This incorrect bubble velocity will almost certainly be lower than the true mean local gas velocity at the probe position. It was found that if dtmax was reduced below 0.05 s there was a tendency for the values of p to reduce slightly, indicating that the velocity profiles had been made slightly flatter. This could be explained by the phenomenon of air bubbles being increasingly likely to strike only one of the two sensors as the probe was moved closer to the pipe wall (such a phenomenon could perhaps be due to the fact that velocity gradients within the water are greater closer to the pipe wall and hence lateral forces on the bubbles, causing them to deviate from purely axial motion, are greater closer to the pipe wall). By reducing dtmax, the effect of this phenomenon on the velocity profile may have been
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that the values of p which were obtained in the present study are probably quite reasonable. It is much less likely that a bubble which strikes the first sensor will miss the second sensor if the axial sensor separation s is relatively small. In the present study, the axial sensor separation s of 5 mm was approximately equal to the mean bubble diameter and yet there was still some uncertainty as to whether particular signals came from the same bubble. It is therefore possible that the criterion of Wu et al. [8] given in Eq. (5), which recommends that s can be up to twice the mean bubble diameter, may lead to the use of axial sensor separations which can give rise to ambiguity when processing the sensor output signals.
7. Estimates of the Zuber–Findlay distribution parameter from the local gas volume fraction and velocity distributions With reference to Zuber and Findlay [2], it can be shown that for a two phase gas–liquid flow, the mean gas velocity ug is given by an expression of the form ug ¼ C0 uh þ ut0 ð1 aÞn
ð20Þ
where C0 is a distribution parameter, ut0 is the rise velocity of a single gas bubble through the stationary liquid (note that the bubble size for which ut0 is calculated must be a size which is representative of the two phase flow under consideration), n is an appropriate exponent and uh is the homogeneous velocity (or mixture superficial velocity) which is given by uh ¼ uws þ ugs
Fig. 9. Local gas velocity profiles with the water superficial velocity uws held constant at 0.38 m s1 and with the gas superficial velocity at 0.017, 0.032 and 0.072 m s1 in a–c, respectively (diamond shapes indicate measured velocities at relevant probe locations).
reduced. If a value for dtmax of 0.05 s is too high, then (from the reasoning given above) this could lead to an overestimation of the corresponding values of the exponent p. Dependency of dtmax on such factors as uh or on the probe position was not investigated further but could well be worthy of future investigation. However, it should also be noted that in Section 7 it is shown that the distribution parameter C0, calculated using the values of p obtained in the present study, agrees quite well with the expected value. This suggests
ð21Þ
As stated in Section 1, relationships of the form shown in Eq. (20) are important when calculating the mean dispersed phase velocity from measurements of the homogeneous velocity uh. Lucas and Jin (see for example [10]) have described various practical techniques by which uh can be measured for two phase flows. In the present study, since only relatively low values of the mean gas volume fraction a have been investigated, ut0 ð1 aÞn ut0
ð22Þ
Furthermore, with reference to Govier and Aziz [12], the value of ut0 for 5 mm diameter air bubbles (a typical size in the present study) rising through stationary water is ut0 ¼ 0:25 m s1
ð23Þ
This value of ut0 has also been found experimentally to be valid for low volume fraction vertical air–water flows by, for example, Johnson and White [13]. With reference to Zuber and Findlay [2], the distribution
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Fig. 10. Exponent p versus aref . Solid line represents van der Welle correlation for p.
parameter C0 can be obtained from the expression C0 ¼
al j l auh
ð24Þ
where the overbar in the numerator represents averaging in the flow cross-section, and where jl is the local homogeneous velocity (i.e. it is the sum of the local flux densities of the air and water). If the assumption is now made that the local homogeneous velocity jl has a power law distribution in the flow cross-section, and that the relevant exponent p associated with this power law distribution is the same as for the local gas velocity distribution, then it is not difficult to show from Eqs. (14), (18) and (24) that C0 ¼
ðq þ 1Þðq þ 2Þðp þ 1Þðp þ 2Þ 2ðp þ q þ 1Þðp þ q þ 2Þ
ð25Þ
where q is the power law exponent associated with the local gas volume fraction distribution. For the flow conditions investigated, Eq. (25) was used to calculate
C0. Fig. 12 shows C0 plotted against the mean gas volume fraction aref . The mean value of C0 for all of the flow conditions investigated was 1.09 and so, with reference to Eqs. (20), (22) and (23) an expression for the mean gas velocity ug derived (in part) from the measured gas volume fraction and velocity distributions, is given by ug ¼ 1:09uh þ 0:25 m s1
ð26Þ
In an attempt to test the accuracy of Eq. (26), the mean gas velocity ug;ref obtained from global reference measurements (see Eq. (9) and Section 4) was plotted against uh, also obtained from global reference measurements (see Eqs. (12), (13) and (21)), for a number of flow conditions in the range relevant to the present study. This plot of ug;ref versus uh is shown in Fig. 13. It is apparent from Fig. 13, that there is considerable scatter in the data, but this is typical for such plots obtained at low values of the mean gas volume fraction a (for example at a flow condition where the true value of a is 0.02 but where a small error of 0.01 is made in
Fig. 11. Mean non-dimensional velocity profiles for 0 < a < 0:1 from present study (dark line) and from van der Welle correlation (light line).
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Fig. 12. Distribution parameter C0 versus mean gas volume fraction aref . Solid line is best fit to data and has the equation C0 ¼ 0:5053 aref þ 1:11.
the measurement of a then, with reference to Eq. (9), this would lead to the estimated value of ug being twice its true value). Also, shown in Fig. 13 is the straight line for the predicted relationship between ug and uh given by Eq. (26). It can be seen that there is reasonable agreement between the predicted relationship and the observed values of these variables. The distribution parameter C0 of 1.09 obtained in the present study is also in reasonable agreement with the slightly lower value of 1.05 obtained by Johnson and White [13] for bubbly air–water flows in a 200 mm diameter pipe. The difference in values may be explained by the work of Hibiki et al. [14] who suggest that, for a given mean bubble diameter, the value of C0 will increase as the pipe diameter is reduced. Fig. 12 also hints that the value of C0 reduces very slightly as aref is increased. If this trend is real, it would agree with observations made by Lucas and Jin [15] for vertical, bubbly oil-in-water flows, where it was found that C0 reduced with increasing oil volume fraction for values of oil volume fraction in the range 0–0.3.
Fig. 13. Mean gas velocity ug;ref versus the homogeneous velocity uh (diamond shapes). The solid line represents the predicted relationship between ug and uh given by Eq. (26).
8. Conclusions A dual sensor conductance probe has been designed and built and used, with appropriate signal processing techniques, to obtain distributions of the local gas volume fraction and the local gas velocity in vertical upward, bubbly air–water flows in which the mean gas volume fraction was less than 0.1. Both the gas volume fraction and the gas velocity distributions were approximated by power law functions. The power law exponents q for the local gas volume fraction distributions were found to be higher than the values predicted by van der Welle [1]. For a ¼ 0:02, the exponent q was 30% higher than predicted by van der Welle indicating that, for this low value of a, the local gas volume fraction distribution is much less smooth than previously thought, with the gas preferentially clustering at the pipe centre. The power law exponents p for the gas velocity distributions were on average slightly higher than the values predicted by van der Welle. This may be due to the fact that van der Welle’s predictions were based on data taken at much higher values of a. However, it is also possible that this discrepancy may be due to signal processing errors arising from an axial sensor separation s which was inappropriately large. This in turn would suggest that the criterion for axial sensor separation recommended by Wu et al. [8], and which was adopted in the present study, may cause ambiguity in deciding whether signals from the two sensors are from the same bubble or from different bubbles (as described in Section 6). For each flow condition investigated, a Zuber– Findlay distribution parameter C0 was calculated from the power law exponents for the gas velocity and volume fraction distributions. The mean value of C0 for all of the flow conditions investigated was 1.09. When this value of C0 was combined with an appropriate slip velocity (for gas bubbles in water) a relationship
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g and the homogeneous between the mean gas velocity u velocity uh was obtained which agreed closely with the relationship between ug and uh obtained from reference measurements. This result suggests that the power law exponents obtained for the local gas volume fraction and velocity distributions are probably correct.
Acknowledgements The authors wish to acknowledge the financial support of the UK Engineering and Physical Sciences Research Council (EPSRC). The work presented here represents part of the work undertaken via EPSRC grant GR/N28610.
References [1] R. van der Welle, Void fraction, bubble velocity and bubble size in two phase flow, Int. J. Multiphase Flow 11 (1985) 317–345. [2] N. Zuber, J.A. Findlay, Average volumetric concentration in two phase flow systems, ASME J. Heat Transfer 87 (1965) 453–468. [3] H. Toral, A study of the hot wire anemometer for measuring void fraction in two phase flow, J. Phys. E: Sci. Instrum. 14 (1981). [4] F.A. Hamad, B.K. Pierscionek, H.H. Bruun, A dual optical probe for volume fraction, drop velocity and drop size measurements in liquid–liquid two phase flow, Meas. Sci. Technol. 11 (2000) 1307–1318.
283
[5] A. Serizawa, I. Kataoko, I. Michiyoshi, Turbulence structure of air–water bubbly flow—I. Measuring techniques, Int. J. Multiphase Flow 2 (1975) 221–233. [6] R.A. Herringe, M.R. Davis, Structural development of gas– liquid mixture flows, J. Fluid Mech. 73 (part 1) (1976) 97–123. [7] J. Steinemann, R. Buchholz, Application of an electrical conductivity microprobe for the characterization of bubble behavior in gas–liquid bubble flow, Part. Charact. 1 (1984) 102–107. [8] Q. Wu, K. Welter, D. McCreary, J.N. Reyes, Theoretical studies on the design criteria of double-sensor probe for the measurement of bubble velocity, J. Flow Meas. Instrum. 12 (issue 1) (2001) 43–51. [9] T.-J. Liu, An effective signal processing method for resistivity probe measurements in a two phase bubbly flow, Meas. Sci. Technol. 13 (2) (2002) 206–217. [10] G.P. Lucas, N.D. Jin, Measurement of the homogeneous velocity of inclined oil-in-water flows using a resistance cross correlation flow meter, Meas. Sci. Technol. 12 (2001) 1529–1537. [11] T. Hibiki, M. Ishii, Experimental study on interfacial area transport in bubbly two-phase flows, Int. J. Heat Mass Transfer 42 (1999) 3019–3035. [12] G.W. Govier, K. Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold Company, 1972. [13] A.B. Johnson, D.B. White, Gas rise velocities during kicks, SPE 20431, 65th Annual Technical Conference of Society of Petroleum Engineers, New Orleans, LA, September 23–26, 1990. [14] T. Hibiki, R. Situ, Y. Mi, M. Ishii, Local flow measurements of vertical upward bubbly flow in an annulus, Int. J. Heat Mass Transfer 46 (2003) 1479–1496. [15] G.P. Lucas, N.D. Jin, A new kinematic wave model for interpreting cross correlation velocity measurements in vertically upward bubbly oil-in-water flows, Meas. Sci. Technol. 12 (2001) 1538–1545.