An experimental investigation on the effect of viscosity on bubbles moving in horizontal and slightly inclined pipes

Accepted Manuscript An experimental investigation on the effect of viscosity on bubbles moving in horizontal and slightly inclined pipes Gianluca Losi, Pietro Poesio PII: DOI: Reference:

S0894-1777(16)00011-X http://dx.doi.org/10.1016/j.expthermflusci.2016.01.010 ETF 8681

To appear in:

Experimental Thermal and Fluid Science

Received Date: Revised Date: Accepted Date:

8 May 2015 30 October 2015 24 January 2016

Please cite this article as: G. Losi, P. Poesio, An experimental investigation on the effect of viscosity on bubbles moving in horizontal and slightly inclined pipes, Experimental Thermal and Fluid Science (2016), doi: http:// dx.doi.org/10.1016/j.expthermflusci.2016.01.010

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An experimental investigation on the effect of viscosity on bubbles moving in horizontal and slightly inclined pipes Gianluca Losia , Pietro Poesioa,∗ a Universit` a

degli Studi di Brescia, Dipartimento di Ingegneria Meccanica ed Industriale, Via Branze, 38, 25123, Brescia, Italy

Abstract The motion of long bubbles in horizontal and slightly inclined pipes is a thoroughly investigated phenomenon. However, few results are available in literature for the drift velocity of a gas bubble in viscous liquids. The aim of this study is to experimentally analyse the effect of viscosity on the drift velocity of an air bubble in horizontal and inclined pipes. Two different measurement techniques are used: five capacitance probes are placed along the pipe to monitor the velocity evolution and image analysis is used to measure the bubble front velocity at two different distances from the outlet. The results of the measurements show that for horizontal case, as opposite to Benjamin (1968), the bubble drift velocity is not constant while the front displaces along the pipe and, in general, the viscosity slows down the propagation rate of the bubble. Finally, two drift velocity closure equations are tested with collected data. Keywords: Drift velocity, high viscosity oils, Benjamin bubble

1. Introduction Few different approaches to slug flow modelling can be found in literature and one of the most used is the slug-unit model. In this formulation, a gas bubble in a slug unit moves with a velocity modelled as the sum of the drift velocity, Ud , and of the mixture velocity, Um , multiplied by the distribution parameter, C0 . Some Authors, Gregory and Scott (1969), Dukler and Hubbard (1975), and Heywood and Richardson (1979) considered a null drift velocity for the slug flow in horizontal or nearly horizontal pipes. However, it has been demonstrated theoretically, Benjamin (1968) and Weber (1981), and experimentally, Zukoski (1966), Bendiksen (1984), Weber et al. (1986), and Alves et al. (1993), that ∗ Corresponding

author, Tel. +39 030 3715646 Email addresses: [email protected] (Gianluca Losi), [email protected] (Pietro Poesio )

Preprint submitted to Elsevier

January 30, 2016

the drift velocity is not zero, even in horizontal flow. Nowadays it is widely accepted, Bendiksen (1984), Taitel and Barnea (1990), Fabre and Line (1992), Andreussi et al. (1993), and Hanratty (2013), that it should be accounted for in the calculation of the bubble translational velocity. The pioneering work of Benjamin (1968) explores the possibilities of applying the inviscid-fluid theory to steady gravity currents, analysing the problem of an empty cavity advancing along a horizontal pipe filled with liquid. In that analysis, the effect of viscosity and surface tension are neglected. After an initial transient stage, whose analysis is not carried out in the model, the air-filled cavity replacing the out-flowing liquid moves steadily along the tube. Benjamin (1968) provided a value for the propagation velocity of the front of the cavity, which depends on the pipe diameter and on the bubble shape. The well-known √ Benjamin velocity is Ud = 0.54 gD with g and D being the gravitational acceleration and the pipe diameter, respectively. Employing this value in slug flow models for low viscosity and nearly horizontal systems at low mixture velocities is a well established practice, see Taitel and Barnea (1990) and Orell (2005). However, when viscous effects become important, inviscid hypothesis may lose its validity and may not be sufficient to properly describe the motion of the bubble. The first who experimentally investigated the effects of viscosity and surface tension on the motion of long bubbles in horizontal and inclined pipes is Zukoski (1966). In his work, he explained the combined effect of inclination and surface tension. Wallis (1969) indicated three independent dimensionless groups that influence the rising motion of long bubbles in vertical pipes, the Froude number (F r), the E¨ otv¨ os number (E¨ o) and the Reynonlds number (Re); at the same time, Wallis (1969) defined three regions of influence, the inertia dominant region, the viscosity dominant region, and the surface tension dominant region. Although he formulated these considerations only for vertical flows, in many works, Gokcal et al. (2009), Khaledi et al. (2014), Moreiras et al. (2014), his approach is followed also to describe horizontal drift velocity, with good agreement with experimental results. A thorough experimental study on long bubbles in inclined pipes was performed by Bendiksen (1984). In order to take into account the effects of inclination, he presented a correlation for the drift velocity, in terms of F r, at all inclination angles. The correlation combines the Froude number for the two limit cases, horizontal flow (F rH ) and vertical flow (F rV ), by means of the cosine and the sine of the inclination angles. As he showed, the correlation is in good agreement with Zukoski’s data and his own when the E¨ otv¨ os number is above 100. Weber et al. (1986) proposed an extension of the correlation proposed in Bendiksen (1984) to improve its behaviour for E¨ o > 100. Weber et al. (1986) reduced the error in the drift velocity by a correction term that depends on the difference between F rH and F rV and on the sine of the inclination angle. In all experimental studies, the drift velocity is reported to increase as the inclination angle increases, reaching a maximum between 30◦ and 60◦ . An interesting study, both theoretical and experimental, is presented by 2

Baines et al. (1985). They investigated the motion of bubbles in a closed, horizontal, square section tube. Removing a moving wall, they released a fixed volume of air into the duct and observed the evolution of its motion. As they reported, the bubble passes through three different phases: initially, its front displaces with a constant speed, then this speed decreases monotonically and it finally goes toward a series of starts and stops until it comes to rest. Moreover, they classified the shapes of the bubbles into three categories according as the pipe end is completely or partially opened. A smooth steady cavity exists when the pipe end is completely opened while, as it is throttled, an hydraulic jump appears. Recently, several experimental campaigns have been performed by Gokcal et al. (2009), Jeyachandra (2011), and Moreiras et al. (2014) to investigate the effect of viscosity. In particular, all these works suggested that the viscosity has a deep impact on the value of drift velocity. In Gokcal et al. (2009), a procedure similar to that presented in Alves et al. (1993) is adapted to compute the total head loss for the horizontal flow and the drift velocity for this inclination. Furthermore, a correlation for inclined flow is presented in a form similar to the one by Bendiksen (1984). Moreiras et al. (2014) developed a new approach to model the horizontal drift velocity and, in general, the drift velocity in inclined pipes and it will be discussed in Section 4. Most of the experimental results regarding drift velocities of viscous fluids available in literature, Zukoski (1966) and Weber et al. (1986), is obtained in very short pipes and, as already pointed out by Zukoski (1966), for these tubes the distance from the bubble front to the exit was not enough for viscous effects to be important. There are some studies that reported experiments carried out with longer pipes, Alves et al. (1993), Gokcal et al. (2009), and Moreiras et al. (2014), however measurements are performed only in one point and there is no evidence of the velocity reduction along the pipe. Probably the first work to move towards this direction could be found moving to numerical simulations. Andreussi et al. (2009) showed that, for large liquid viscosity, the drift velocity of a bubble penetrating in a horizontal draining pipe decreases along the pipe. In this work, different simulations were carried out using the same pipe geometry and different liquid viscosities, showing that the reduction is always present, even for water, and it is more evident as viscosity increases. No further considerations about the influence of surface tension are reported. In line with Andreussi et al. (2009), Ramdin and Henkes (2012) and Kroes and Henkes (2014) reported the results of 2D and 3D simulations performed with a commercial CFD code. In their work they showed that a viscous Benjamin bubble decelerates while it expands along the pipe. The final value of the velocity depends on the viscosity set in the calculation. Moreover, Kroes and Henkes (2014) proposed a correlation for the nose displacement as a function of time. The functional dependence is a power law where the power depends the Reynolds number. Moving from these evidences, the aim of the present work is to experimentally investigate the evolution of the motion of long bubbles intruding in a draining pipe with the use of more than one measuring station. In this manner, 3

the reduction of the velocity of the bubble could be tracked and reported. Few inclinations, including the horizontal condition, will be analysed. Finally, two recently developed drift velocity closure equations will be tested against our results.

2. Experimental set-up The experimental facility, especially designed to study multiphase flows, consists of a L = 9 m long glass pipe with an inner diameter of D = 0.022 m. The pipe is formed by six 1.5 m long sections. A sketch of the set-up is given in Fig. 1. The pipe inclination during this work is set between 0◦ and 5◦ . Monitoring the pipe inclination is extremely important, in particular for the horizontal case, since the drift velocities we expect to measure are extremely low and, therefore, even a slight deviation from the set inclination can significantly affect the data. The inclination angle is periodically checked with an electronic level sensor,

Figure 1: Experimental multiphase flow set-up. C: capacitance probe; OW: observation window.

which offers an uncertainty of ±0.1◦ . Since the facility is composed of six pipes jointed together, some deviations from a perfect alignment of each single section with respect to the previous and the following ones may be present; they are of the same order of magnitude of the level accuracy, i.e. ±0.1◦ . Three different paraffin oils are used in these experiments. Their densities are measured by a hydrometer, their viscosities are measured with a rotational rheometer (Ultra Programmable Rheometer LV-DV III+, Brookfield, Middleboro, USA) while their surface tensions are measured by a Wilhelmy plate tensiometer; the values are all reported in Table 1. According to Kroes and Henkes (2014), using dimensionless groups to analyse of the motion of long bubbles in stagnant liquids is helpful for a better understanding of the phenomenon. In particular, a characteristic Reynolds number is √ defined as Re = ρl Dμl gD and it is used to indicate the effect of viscosity on the bubble motion; note that

√ ρl D gD μl

can be seen as the square root of the Galileo 4

ρ [kg/m3 ] μ [mP a · s] σ [N/m]

Water 998 1 0.0717 ± 7.9 10−3

OSO22 860 37.5 ± 0.3 0.0263 ± 8.4 10−3

OSO100 875 195.5 ± 1.7 0.0267 ± 5.3 10−3

TURBO320 886 804.7 ± 4.1 0.0151 ± 5.9 10−3

Table 1: Values of density (ρ), viscosity (μ) and surface tension (σ) measured at 24◦ C.

number. The importance of surface tension and pipe diameter is underlined by 2 the E¨ otv¨ os number defined as E¨ o = ρl gD σ . Using the values from Table 1, the map shown in Fig. 2 is built. As can be seen from Fig. 2, the conditions investigated in this work are far from those represented by water. The Reynolds numbers are far below the ones for water, showing the predominant influence of viscosity over density. Moreover, the surface tension is the other controlling parameter, since all the oils show an E¨ o > 100. In Fig.2, we report the working conditions by Weber et al. (1986), Moreiras et 900 800 700 1 mPas - This work 37.5 mPas 195.5 mPas 804 mPas 0.957 mPas - Weber et al. (1986) 51.1 mPas 195 mPas 518 mPas 0.957 mPas 39 mPas - Moreiras et al.(2014) 200 mPas - Gockal et al.(2009) 692 mPas

E¨ o

600 500 Water 400 300 200 100 0 1 10

2

10

3

10 Re

4

10

5

10

Figure 2: Flow map obtained plotting the E¨ otv¨ os number versus the characteristic Reynolds number. In blue – this work; in red – Weber et al. (1986); in green – Moreiras et al. (2014) and Gokcal et al. (2009).

al. (2014), and Gokcal et al. (2009). Weber et al. (1986) performed experiments in a D = 0.0221 m pipe and they are very close, both in terms of Reynolds number and E¨ otv¨ os number, to the conditions of our experiments. Data from Moreiras et al. (2014) are characterised by a very high E¨ otv¨ os number, due to the different pipe diameter (D = 0.0508 m). The experimental procedure has two steps: firstly the open pipe is filled up and, only when all the residual bubbles of air are removed, the outlet section is capped. Once the pipe is completely filled up, the other end is sealed with particular attention to eliminate all possible disturbances. The end cap (see 5

Fig. 1) is quickly removed in order to let air flow in. The bubble drift velocity is measured. Two different methods are used to investigate the evolution of the bubble drift velocity along the pipe. Five capacitance probes are mounted with their mid-point at 2, 4, 5.14, 6.5 and 7.9 m from the outlet section. Two observation windows are obtained enclosing the pipe into two transparent glass boxes; the two glass boxes have a length of 0.48 m and are placed with their mid-point at 0.6 and 3.5 m from the outlet section. The glass boxes are filled with water in order to eliminate optical distortion due to the cylindrical shape of the pipe. The flow is captured with two cameras with their lenses perpendicular to the box surface. During the tests, the out-flowing mass of oil is collected in a pitcher, which is placed over the plate of a precision balance. The balance reading is acquired by a PC and the mass discharge rate is measured. The balance has a capacity of 4100 g and a readability of 0.01 g. 2.1. Capacitance measurements

Output signal [−]

Output signal [−]

Capacitance probes take advantage of the difference in dielectric permittivity between oil and air; when a bubble passage occurs, it is detected by capacitance variations between two copper electrodes flush-mounted on the external surface of the non-conductive pipe Strazza et al. (2011). 2.42

x 10

4

2.4 2.38 2.36 4 2.41

x 10

4.5

5

5.5 Time [10−2 s]

6

5.5 −2 Time [10 s]

6

6.5

7 4

x 10

4

2.4

2.39 4

4.5

5

6.5

7 4

x 10

Figure 3: Top: original signal from a capacitance probe. Down: de-noised signal.

Each probe is made up of two electrodes, three guarding sections, and one polarization electrode, for more information on the sensors refer to Strazza et al. (2011). The mean distance ΔX between the two electrodes is 197.8±0.5 mm; this distance is used to compute the bubble velocity during data post-processing. Each electrode, which is the sensible area of the probe, has an opening angle of 179◦ , to ensure a full coverage of the test section and an axial extension of 10 mm. 6

a)

b) 0.1

Signal 1 Signal 2 Upper edges Lower edges

80 Output signal [−]

Derivative of the output signal [−]

100

60

τ 40

20

0 0

1000 2000 Time [10−2 s]

0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 0

3000

1000 2000 Time [10−2 s]

3000

Figure 4: a) de-noised cut signals; b) signal derivative - negative minimum are identified by circles.

The electric outputs of the probes are acquired at 100 Hz sampling frequency and passed on to a PC where they are recorded for further post-processing. The electrical noise is removed from each signal using the Fast Fourier Transform and setting to zero all frequencies above a chosen threshold. A comparison between an original signal and the same de-noised signal is given in Fig. 3. The signal is normalized between 0 and 100, see Fig. 4(a). 350 300 μ = 804 mPas

Output signal [−]

250 200

μ = 195.5 mPas

150 100

μ = 37.5 mPas

50 0 0

1

2 3 Time [10−2 s]

4

5 4

x 10

Figure 5: Capacitance probes output signals for the three different oils. The boxes indicate the part of the signal influenced by the drainage of the oil on pipe walls.

As can be noticed, after the passage of a bubble, which can be located in Figs. 3 - 5 as the discontinuity, the signal continues to decrease. This effect is due to the reduction of the liquid level in the pipe and to the persistence of an oil film at pipe walls, which slowly drains with a circumferential motion. The time required to the signal to stabilize depends on the liquid viscosity and increases as 7

2.09

x 10

4

Output signal [−]

2.088 2.086 2.084 2.082 δτ

2.08 2.078 0

0.5

1

1.5 Time [10−2 s]

2

2.5

3 x 10

4

Figure 6: The passage from the maximum to the minimum value requires a certain time (δτ ).

oil viscosity increases, see Fig. 5 where normalized signals are shifted vertically for clarity. Boxes highlight the change of signal behaviour between the time instant right after the bubble passage and when the effect of drainage decays. At this stage, the time derivative of each couple of signals is computed. The derivative has a negative peak, see Fig. 4(b), where the signal shows the discontinuity; the negative peak is individuated and its position is used to locate the upper and the lower edges of the discontinuity zone. The time τ , which is an estimate of the time spent by the front to pass from the first electrode to the second one, is the time distance between the upper edges of the discontinuities. The upper edges, which are the points in the two signals that correspond to the beginning of the step, are taken as reference points in the signal because the bubble passage is not detected by the probe as a discontinuous step. As can be noticed in Fig. 4(a) and even better in Fig. 6, the signal passes from the upper to the lower edge of the discontinuity in a certain time, which can be indicated as δτ . This behaviour of the instrumentation is due to several reasons and among them there are the shape of the bubble, a small electric conductivity of the oil, the shape, and imperfections of the sensors. In any case, this effect is negligible for this measurements since its characteristic time (≈ 0.1 s) is much smaller than τ (from ≈ 1.25 s to ≈ 60 s depending on the test case). Since the distance between the sensible areas, ΔX, is known, it is possible to compute the bubble nose velocity as the ratio between this distance (ΔX) and the time lag (τ ): ΔX . (1) Ud = τ For each inclination and viscosity, several different tests are conducted and error bars are built on the standard deviation of results. Also horizontal error bars are reported when experimental data are shown to take into account the axial extension of the capacitance sensors.

8

2.2. Image processing The identification of bubble characteristics is performed using back-lighted images. This method relies on the acquisition of images with an intense and uniform background illumination. Depending on the nature and composition of the oil, the light diffuses differently, but, in any case, the air bubble is characterized by a color, which is different from the surrounding oil. This evidence makes possible to adopt a background subtraction technique to track the bubble position. Videos are captured placing a color high-speed camera in front of the observation window number 1 (OW1, Fig. 1). During all tests, the frame rate is set to 33 fps to avoid flickering. This is a sufficient frame-rate to capture the motion and the shape of the bubble and, at the same time, to maintain an acceptable dimension of video files for further post-processing. Furthermore, for the horizontal case, photos of the bubble have been taken with a 10.1 MPixel digital single-lens reflex camera located in front of the observation window 2 (OW2, Fig. 1). From the analysis of video recordings it is possible to extract the front position and, subsequently, its average velocity in the observation window. The value is shown with a horizontal error bar, to account for the axial extension of the observation window, see Fig. 1. The position of the nose of the bubble is tracked frame by frame using a background subtraction technique. Background subtraction is a crucial step because it manages to detect moving objects within a video stream, without any a priori knowledge about them. In a first stage, each frame is converted to black and white by an open-source library of background subtraction algorithms, BGSLibrary - Sobral (2013). In Original colour image

BGS result

Figure 7: An example of the background subtraction operation using the DPEigenbackgroundBGS algorithm.

Sobral and Vacavant (2013) all available algorithms in Sobral (2013) are tested against synthetic and real cases. We tested different algorithms and we found the one that works best with our images is the DPEigenbackgroundBGS, Oliver et al. (2000). As can be seen in Fig. 7, the DPEigenbackgroundBGS algorithm returns a precise description of the entire bubble; on the other hand, it requires a high 9

computational effort. The black and white images are used to compute the displacement of the front. An algorithm for edge detection, see Canny (1986), is applied to black and white images. The average velocity of the front in observation windows is calculated as the slope of the linear regression of the successive points. 2.3. Calibration experiments To validate the setup and the experimental procedure, several tests have been carried out using water. To take advantage of the image analysis techniques considered in Section 2.2, a dye is added to water. The addition of the colourant does not affect water properties. 0.35

Drift velocity [m/s]

0.3 0.25 0.2 0.15 0.1 μ = 1 mPas Benjamin (1968) Weber (1986)

0.05 0 0

2

4 6 Position along the pipe [m]

8

Figure 8: Calibration test with water for the horizontal condition - drift velocity.

Since the electronic devices that control the capacitance probes are not able to pilot the sensors so to allow capacitance measurements with water, an optical probe is used to measure the bubble velocity 7 m upstream the outlet section. The probe has two columns of LEDs 19.5 cm apart. On the other side of it, opposite the LEDs, two columns of photoresistors capture the variation of the light intensity due to the bubble passage. The recorded signals are identical to those obtained by the capacitance probes, thus the same algorithm is used to analyse those data. The result of the tests are shown in Fig. 8. The measured velocity, Udexp = 0.162 ± 0.01 m/s, is constant√along the pipe, but its value is lower than the theoretical value of Ud = 0.54 gD = 0.252 m/s provided by Benjamin (1968). As reported in Fig. 8, this value is in good agreement with the measurements by Weber et al. (1986) for the same conditions (Udexp = 0.16 m/s). The only parameter that can affect this result is the E¨ otv¨ os number as was already noticed by Weber et al. (1986). At the same time, the mass outflow has been measured over time, see Fig. 9. The experimental value is compared against the one

10

obtained by of Eq. 2, see Benjamin (1968),  M˙ = 0.1355 ρl gD πD2

(2)

where ρl is the liquid density. 0.14

√ M˙ /(ρl gDπD2 )

0.12 μ = 1 mPas Benjamin (1968)

0.1 0.08 0.06 0.04 0.02 0

10

20

30 40 Time [s]

50

60

70

Figure 9: Dimensionless mass outflow for the calibration test with water for the horizontal condition.

Despite slightly different values, the flow rate is constant over time, as expected from Benjamin (1968).

11

3. Experimental results Experiments confirmed that the drift velocity is strongly influenced by the inclination of the pipe, as already reported in literature, see Zukoski (1966), Bendiksen (1984), and Weber et al. (1986). 0.09

0.035

μ = 37.5 mPas

μ = 195.5 mPas

0.08 0.03 Drift velocity [m/s]

Drift velocity [m/s]

0.07 0.06 0.05 0.04 0.03

0.025 0.02 0.015

0.02 0.01

0.01 0 0

1

2

3 4 5 6 Position along the pipe [m]

7

8

Figure 10: Drift velocities for μ = 37.5 mPas for the horizontal condition.

8

x 10

0.005 0

9

1

2

3 4 5 6 Position along the pipe [m]

7

8

9

Figure 11: Drift velocities for μ = 195.5 mPas for the horizontal condition.

−3

μ = 804 mPas

Drift velocity [m/s]

7 6 5 4 3 2 0

1

2

3 4 5 6 Position along the pipe [m]

7

8

9

Figure 12: Drift velocities for μ = 804 mPas for the horizontal condition.

As can be seen in Figs. 10–12, velocities for the horizontal flow decrease to a very low value, but they do not vanish to zero while the bubble intrudes into the stagnant liquid in the pipe. These measurements justify the results by Farsetti et al. (2014) and Foletti et al. (2011), which showed that, in horizontal slug flow with high viscosity oils, the drift velocity can be assumed to be negligible. The reduction of velocity along the pipe is more noticeable for the horizontal condition while it gradually disappears increasing the tilt angle of the pipe. Vertical error bars are obtained as explained in Section 3. Where they are not visible, their dimension is smaller than the symbol. Horizontal error bars

12

express the fact that both observation windows and capacitance probes have an extension and we are not reporting a punctual value, but a mean value over that portion of the pipe. Besides the overall pipe inclination, it is important to underline that even the misalignments of each pipe section affects the measurements since velocities are extremely low. These effects are more important for higher viscosities and disturb the motion, as can be seen in Fig. 12. μ = 37.5 mPas Water Moreiras (2014) - μ = 39 mPas Weber (1986) - μ = 51.1 mPas Andreussi (2009) - μ = 40 mPas

0.5 0.4 0.3 0.2 0.1 0 0

50

100

150 200 250 300 Position along the pipe [D]

350

0.5 0.4 0.3 0.2 0.1 0 0

400

Figure 13: Comparison between experimental results for μ = 37.5 mPas and available data from literature.

50

100

150 200 250 300 Position along the pipe [D]

350

400

Figure 14: Comparison between experimental results for μ = 195.5 mPas and available data from literature.

μ = 804 mPas Water Gokcal (2009) - μ = 692 mPas Weber (1986) - μ = 518 mPas Andreussi (2009) - μ = 692 mPas

0.6 Dimensionless drift velocity (Fr)

μ = 195.5 mPas Water Gokcal (2009) - μ = 200 mPas Weber (1986) - μ = 194 mPas

0.6 Dimensionless drift velocity (Fr)

Dimensionless drift velocity (Fr)

0.6

0.5 0.4 0.3 0.2 0.1 0 0

50

100

150 200 250 300 Position along the pipe [D]

350

400

Figure 15: Comparison between experimental results for μ = 804 mPas and available data from literature.

Figs. 13–15 show the comparison between our data and literature data. The drift velocities are presented in a dimensionless form as Froude numbers, F r = √Ud . Experimental results from Moreiras et al. (2014), Gokcal et al. (2009), gD and Weber et al. (1986), FLUENT simulation results from Andreussi et al. (2009) and our results with water are also reported for comparison. Since in previous experimental works the drift velocity is measured in a single point or 13

it is considered constant, a unique value is provided. Fig. 13 and Fig. 15 show a quite good agreement between Andreussi et al. (2009) and experimental evidences – Moreiras et al. (2014) and Gokcal et al. (2009) – although, the comparison can be made only in a qualitative sense, since only one experimental value is available. The drift velocity obtained from Andreussi et al. (2009) simulations decreases, as shown in Fig. 15, coherently with our experimental results, see Figs. 10-12; although values are different, the shape of the profile along the pipe shows a certain similarity. Despite the drift velocity has been made dimensionless (and plotted as Froude number), a dependence from the diameter is still present and suggests an influence of the surface tension. As can be seen in Fig. 14, even if our oil and the liquid used in Weber et al. (1986) exhibit a very similar viscosity and the pipe diameter is the same, the lower surface tension of our oil results in a lower drift velocity. Even more noticeable is the case shown in Fig. 13: the viscosity of the liquid used in Weber et al. (1986) is higher than that of our oil. In spite of that, its higher surface tension entails a higher dimensionless drift velocity. 0.6 0.5 D = 0.0508 m

Fr

0.4 0.3 D = 0.022 m

0.2 0.1 0 −4 10

10

−3

−2

10

10

−1

0

10

Z

Figure 16: Dimensionless drift velocity (Fr) as a function of Ohnesorge number. Refer to Fig. 2 for the legend.

The residual dependence of dimensionless drift velocity (e.g. Fr) from the pipe diameter is even clearer if we plot it as a function of the Ohnesorge number, defined as the ratio between the viscous forces and the square root of the μ . Its choice is an attempt of conproduct of inertia and surface forces, Z = √ρDσ centrating the physical parameters that influence the bubble motion in a unique dimensionless group, see Fig. 16. Data from Moreiras et al. (2014) and Gokcal et al. (2009) are obtained with a pipe diameter of 0.0508 m while Weber et al. (1986) and our data correspond to a diameter of 0.022 m and changing the pipe diameter affects the drift velocity as discussed in Weber (1981). Another way to analyse the reduction of velocity to directly plot it as a  is g , see Kroes and Henkes function of the dimensionless time defined as t˜ = t D (2014), who also proposed a power law approximation for the bubble velocity with the form F r = C t˜p . The coefficient C and the power p are functions of the

14

Reynolds number and they are set so to fit the simulation results. In Figs. 17– 19, experimental results from the two observation windows (OW1 and OW2, see Fig. 1) are plotted alongside the power function with the coefficients calculated to fit our cases. 0.7

0.5

Re = 234 F r = 0.58t˜−0.17

0.6

Re = 45 F r = 0.58t˜−0.34

0.4 0.5 0.3 Fr

Fr

0.4 0.3

0.2

0.2 0.1 0.1 0 0

500

1000

1500

2000

0 0

2500

1000

t*

Figure 17: Comparison between experimental results for Re = 234 and correlation by Kroes and Henkes (2014).

2000 t*

3000

4000

Figure 18: Comparison between experimental results for Re = 45 and correlation by Kroes and Henkes (2014).

0.12

Re = 11 F r = 0.22t˜−0.31

0.1

Fr

0.08 0.06 0.04 0.02 0 0

0.5

1

1.5 t*

2

2.5 4 x 10

Figure 19: Comparison between experimental results for Re = 11 and correlation by Kroes and Henkes (2014).

The correlation shows a good agreement with data. However, it can be probably enhanced introducing the dependence from surface tension. The reduction of the drift velocity along the pipe can be explained considering that, while the bubble propagates into the stagnant liquid, the liquid carpet left behind increases its length. Thus, frictional losses at pipe wall increase, slowing down the motion of the liquid layer under the bubble. Dissipation by the wall shear stresses drain an increasing amount of the gravitational potential energy made available in the bubble nose zone by the difference of height in the liquid level, see Alves et al. (1993).

15

Figure 20: A sketch of the physical system, as also reported in Alves et al. (1993).

Referring to Fig. 20, let consider a reference frame in which the bubble is at rest. In this reference frame, which moves with the bubble velocity, Ud = v1 , the system is in a steady state. The mass conservation for the liquid can be expressed as follows (3) Av1 = A2 v2 . The momentum balance for the liquid contained in the control volume 1 − 0 − 2, see Fig. 20, can be written as    h 1 2 2 ρl v1 A − ρl v2 A2 + P1 + ρl gD cos β A − P2 A2 − cos β ρl gc(h − z)dz+ 2 0    b

+ aA + 0

A2 dx ρl g sin β − Fv = 0, (4)

where Fv are the viscous forces, the terms that depend on the cos β are the hydrostatic contributes to pressure and the terms that depend from the sin β are the body forces. When the angle β = 0◦ , these last terms vanish and the gravity acts only by transverse pressure gradients proportional to the elevation difference along the bubble nose (the terms that depend on the cos β). These actions do not overcome the viscous forces and the bubble velocity slows down. In a tilted pipe, β > 0◦ and both contributes of hydrostatic pressure gradients and buoyancy body forces act on the system. They are balanced by wall viscous forces, which are of the same order of magnitude, thus, the system can reach a steady condition and it is possible to define a unique value for the drift velocity. 16

Along with viscous effects, the influence of surface tension clearly appears when bubble shapes are compared. A higher surface tension (water-air systems) results in a bubble with a higher radius at the tip, see Fig. 21–top. When the surface tension decreases, the bubble radius decreases and so does the liquid level difference along the bubble nose and the transverse pressure gradient, which is proportional to this difference.

Water σ = 0.717 N/m

OSO22 σ = 0.263 N/m

OSO100 σ = 0.267 N/m

TURBO320 σ = 0.151 N/m

Figure 21: Different shapes corresponding to different surface tensions influence the drift velocity.

In Figs. 22-26 the effect of inclination is shown. A slight reduction is still present for the inclination of 1◦ , but it is negligible for the other inclinations. As discussed earlier, when the pipe is inclined, the hydrostatic pressure gradients and the buoyancy forces are balanced by viscous forces and the motion of the bubble becomes steady after an initial transient stage, during which the bubble accelerates. This effect is visible for inclinations above 1◦ (see Figs. 23-26) and it is more evident increasing the tilting angle. The viscosity damps down this behaviour, which is more evident for the low viscosity oil.

17

0.14

0.18 0.16

0.12 μ = 37.5 mPas μ = 195.5 mPas μ = 804 mPas

0.08

Drift velocity [m/s]

Drift velocity [m/s]

0.14 0.1

0.06 0.04

μ = 37.5 mPas μ = 195.5 mPas μ = 804 mPas

0.12 0.1 0.08 0.06 0.04

0.02 0 0

0.02 1

2

6 5 4 3 Position along the pipe [m]

0 0

9

8

7

Figure 22: Drift velocities along the pipe inclined of 1◦ .

Figure 23: Drift velocities along the pipe inclined of 2◦ .

0.18

0.18

0.16

0.16 0.14 μ = 37.5 mPas μ = 195.5 mPas μ = 804 mPas

0.12 0.1

Drift velocity [m/s]

Drift velocity [m/s]

0.14

0.08 0.06

0.1 0.08 0.06 0.04

0.02

0.02 1

2

3 4 5 6 Position along the pipe [m]

7

8

0 0

9

Figure 24: Drift velocities along the pipe inclined of 3◦ .

μ = 37.5 mPas μ = 195.5 mPas μ = 804 mPas

0.12

0.04

0 0

9

8

7

6 5 4 3 Position along the pipe [m]

2

1

1

2

3 4 5 6 Position along the pipe [m]

0.16

Drift velocity [m/s]

0.14 μ = 37.5 mPas μ = 195.5 mPas μ = 804 mPas

0.12 0.1 0.08 0.06 0.04 1

2

8

9

Figure 25: Drift velocities along the pipe inclined of 4◦ .

0.18

0.02 0

7

3 4 5 6 Position along the pipe [m]

7

8

9

Figure 26: Drift velocities along the pipe inclined of 5◦ .

18

1800

5°

1600

4

°

1200 3°

2°

1000 °

1

1 Out−flown mass [g]

Out−flown mass [g]

1400

5° 4° ° 3 ° 2

1200 1000

°

0

800 600

°

800 0

°

600 400

400 200 200 0 0

50

100

0 0

150

100

200

300

Time [s]

Figure 27: Drained mass as a function of time for different tilt angles and μ = 37.5 mP as.

1200 5

°

°

4

Out−flown mass [g]

1000

400 500 Time [s]

600

700

800

Figure 28: Drained mass as a function of time for different tilt angles and μ = 195.5 mP as.

3° 2°

800

1° 0°

600 400 200 0 0

500

1000 1500 Time [s]

2000

2500

Figure 29: Drained mass as a function of time for different tilt angles and μ = 804 mP as.

These results are confirmed by the trends of the drained mass as a function of time. In Figs. 27-29 we report the mass of oil drained from the pipe as a function of time until the bubble reaches the last measuring station. It can be noticed that the amount of drained mass depends on the inclination, since the bubble height increases as the inclination angle increases and the amount of liquid left in the pipe decreases. Moreover, the slope of the 0◦ curve is not constant while the value of the slope increases and tends to be constant when the pipe is tilted. This effect is more evident at high viscosities. In Fig. 30 the derivative of the measured mass outflow for 0◦ is plotted after it was put in a dimensionless form (see Eq. 2) where we can clearly see a difference with water.

19

14

x 10

12

−3

μ = 37.5 mPas

√ M˙ /(ρl gDπD2 )

10 8 6 4

μ = 195.5 mPas μ = 804 mPas

2 0 −2 0

500

1000

1500 Time [s]

2000

2500

3000

Figure 30: Measured flow rates for μ = 37.5 mP as, μ = 195.5 mP as and μ = 804 mP as.

4. Model validation Two closure relationships for drift velocity available in literature are tested against our experimental data. In other experimental works, Weber et al. (1986), Jeyachandra (2011), and Moreiras et al. (2014), the drift velocity is measured near the outlet section. For this reason and since its value sensibly varies along the pipe for horizontal and 1◦ inclinations, only the values obtained from the first observation window (OW1, see Fig. 1) are considered to compare our data with the predictions from the two correlations. For inclinations above 1◦ , a mean value of the drift velocities measured along the pipe is considered. The equations of both the correlations are reported in Table 2. They are built following the correlation first introduced by Bendiksen (1984) and then extended by Weber et al. (1986). According to this last correlation, the Froude number (i.e. the dimensionless drift velocity) for any inclination can be computed combining the Froude number for the horizontal flow and that for the vertical flow in the form: F r = F rH (cosθ)c + F rV (sinθ)d + Q.

(5)

Moreiras et al. (2014) adapted the correlation presented in Weber et al. (1986) to take into account the effects of viscosity. Based on a large set of experimental data collected among several previous studies, Zukoski (1966), Weber et al. (1986), Gokcal et al. (2009) and Jeyachandra (2011), they proposed a new correlation to determine the horizontal Froude number (F rH ), see Table 2. 20

Moreiras et al. (2014) Jeyachandra et al. (2012) μ = 0.0375 mPas μ = 0.1955 mPas μ = 0.804 mPas

Drift velocity [m/s]

0.15

0.1

0.05

0

0

1

3 2 Inclination angle [°]

4

5

Figure 31: Drift velocity as a function of the inclination angle.

The correlation is valid for diameters larger than 0.03 m. According to Wallis (1969), the equation type holds for E¨ o > 100, since surface tension effects can be neglected and only viscosity acts to slow the motion of bubbles. For F rV , Moreiras et al. (2014) modified the rise velocity of a bubble in a vertical tube according to Joseph (2003). In particular, they corrected the rise velocity as suggested by Davies and Taylor (1950). Starting from the viscous potential result (the first two terms in the expression for F rV in Table 2) Moreiras et al. (2014) subtracted the difference between potential results for the cap shaped bubble and long bubble, the last two terms in the same expression, see Table 2. Once the Froude number is computed by Eq. 5, the drift velocity can be obtained using the follow relationship:  gD(ρl − ρg ) (6) Ud = F r ρl ρl − ρg . ρl When tested against our data (see Figs. 31-32), this correlation proves to be very good for medium viscosities and for inclination angles above 1◦ ; however, it lacks in accuracy both when the inclination angle decreases and when viscosity increases. In Jeyachandra et al. (2012), F rH depends on viscosity and also on surface tension. It is probably for this reason that this correlation shows better agreement with the behaviour of the most viscous oil. However, it underestimates velocities for less viscous oils. The other difference with the previous correlation where g  = g

21

d

U calculated [m/s]

0.25

Moreiras et al. (2014) Jeyachandra et al. (2012)

+35%

0.2

0.15

0.1

−35%

0.05

0 0

0.05

0.1 0.15 U experimental [m/s] d

0.2

Figure 32: Comparison between the calculated drift velocity and the experimental drift velocity.

is the manner the drift velocity Ud is calculated:  Ud = F r gD,

(7)

without introducing the densities of the two phases. The two ways of expressing the Froude number, by means of the reduced gravity g  and of the standard gravity acceleration g, do not produce a noticeable effect in results, since the difference between the phase densities is close to the liquid density. In Fig. 32 the agreement of the correlation to experimental data is shown. As can be seen, the overall accuracy of the correlation developed by Jeyachandra et al. (2012) is higher but the minimum of relative error is achieved by the correlation from Moreiras et al. (2014). For this comparison, a mean value of the drift velocity along the pipe is computed using our results for inclinations above 1◦ , to which correspond a quite constant value along the pipe. On the contrary, for 0◦ and 1◦ only the first and highest value of drift velocity is considered.

22

23

1 1 0

0.351

0.53e−13.7Nvis

0.46

Eo−0.1

Jeyachandra et al. (2012)

μ Table 2: Parameters a, b, e and f are obtained from data fitting; the viscosity number Nvis is defined as Nvis =  ; the E¨ otv¨ os gD3 (ρl − ρg )ρl ρl gD2 number E o¨ is defined as E o¨ = . σ

c d Q

F rV

F rH

Moreiras et al. (2014) Nvis 0.54 − a + bN vis√  ρl 64 2 2 ρl 8 + Nvis − − 0.35 − Nvis + 3 9 (ρl − ρg ) 9 3 (ρl − ρg ) data fitting data fitting 0 if F rV − F rH < 0 or if θ = 0 e(F rV − F rH )f sinθ(1 − sinθ) if F rV − F rH ≥ 0

5. Conclusions Liquid viscosity effect on drift velocity for horizontal and inclined flow is investigated in this study. Experiments are performed with three different viscosity paraffin oils. Two different measurement techniques, capacitance measurements, and image analysis, are used to determine the bubble velocity. Five capacitance probes are placed along the pipe to track the bubble evolution while a high-speed camera and a digital single-lens reflex camera are used to capture the bubble motion at two different distances from the outlet section, increasing the number of measuring points. The observed behaviour for horizontal flow agrees with simulations performed by Andreussi et al. (2009) and Ramdin and Henkes (2012) and, in particular, the bubble velocity decrease along the pipe while the liquid layer under and beyond it elongates. The reduction of the velocity is ascribable to the subsequent growth in wall shear stresses in the liquid phase, which drains an increasingly amount of the gravitational potential energy available in the difference of level in the nose zone of the bubble and needed for its motion. The situation is slightly different when the pipe is inclined; in those cases, buoyancy force helps the motion of the bubble, which does not significantly change its speed along the pipe. In the last section, two recently developed closure equations for drift velocity are tested against our data. They showed a different capacity of predicting the drift velocity as the viscosity varies. The correlation presented by Moreiras et al. (2014) is the best choice for low to medium viscosity, while the approach of Jeyachandra et al. (2012) in modelling the drift velocity is more convenient in dealing with high viscosities.

Aknowledgements We are grateful to T.E.A. S.p.A. that supported this work. A special thank goes to Davide Arnone for his precious help during all the experiments. We would also like to thank the reviewers for their suggestions that helped improving the paper. References References Alves, I.N., Shoham, O., Taitel, Y., 1993. Drift velocity of elongated bubbles in inclined pipes, Chem. Eng. Science, pp. 3063-3070. Andreussi, P., Minervini, A., Paglianti, A., 1993. Mechanistic model of slug flow in near-horizontal pipes, AIChE J., Vol. 39, No.8, pp. 1281-1291. Andreussi, P., Bonizzi, M., Vignali, A., 2009. Motion of elongated gas bubbles over a horizontal liquid layer, Proceedings of 14th International Conference on Multiphase Production Technology, Cannes, pp. 309-318.

24

Baines, W.D., Rottman, J.W., Simpson, J.E., 1985. The motion of constantvolume air cavities in long horizontal tubes, J. Fluid Mech., Vol.161, pp. 313327. Bendiksen, K.H., 1984. An experimental investigation of the motion of long bubbles in inclined tubes, Int. J. Multiphase Flow, Vol. 10, No.4, pp. 467-483. Benjamin, T.B., 1968. Gravity currents and related phenomena, J. Fluid Mech., Vol. 31, pp. 209-248. Canny, J.F., 1986. A computational approach to edge detection, IEEE Trans. Pattern Analysis and Machine Intelligence, pp. 679-698. Davies, R.M., Taylor, G.I., 1950. The mechanics of large bubbles rising through liquids in tubes, Proc. R. Soc. Lond. A 200, pp. 375-390. Dukler, A.E., Hubbard M.G., 1975. A model for gas-liquid slug flow in horizontal and near horizontal tubes, Ind. Eng. Chem. Fundam., Vol. 14, pp. 337-347. Fabre, J., Line, A., 1992. Modeling of Two-Phase Slug Flow, Ann. Rev. of Fluid Mech., Vol. 24, pp. 21-46. Farsetti, S., Faris`e, S., Poesio, P., 2014. Experimental investigation of high viscosity oilair intermittent flow, Exp. Th. Fl. Science, Vol. 57, pp. 285-292. Foletti, C., Faris`e, S., Grassi, B., Strazza, D., Lancini, M., Poesio, P., 2011. Experimental investigation on two-phase air/high-viscosity-oil flow in a horizontal pipe, Chem. Eng. Science, Vol. 66, pp. 5968-5975. Gregory, G.A., Scott, D.S., 1969. Correlation of liquid slug velocity and frequency in horizontal cocurrent gas-liquid slug flow, AIChE J., Vol. 15, No. 6, pp. 933935. Gokcal, B., Al-Sarkhi, A.S., Sarica, C., 2009. Effects of high oil viscosity on drift velocity for horizontal and upward inclined pipes, SPE Projects, Facilities and Construction SPE 115342. Hanratty, T.J., 2013. Physics of Gas-Liquid Flows, Cambridge University Press. Heywood, N.I., Richardson, J.F., 1979. Slug flow of airwater mixtures in a horizontal pipe: Determination of liquid holdup by γ-ray absorption Chem. Eng. Sci. 34, pp. 17-30. Jeyachandra, B.C.G.B., 2011. Effect of high oil viscosity on two-phase oil-gas flow behaviour in horizontal pipes, M.S. Thesis, University of Tulsa, Tulsa. Jeyachandra, B.C.G.B., Al-Sarkhi, A., Al-Sarica, C., Sharma, A., 2012. Driftvelocity closure relationships for slug two-phase high-viscosity oil flow in pipes. SPE J. 17, pp. 593601.

25

Joseph, D.D., 2003. Rise velocity of a spherical cap bubble, J. Fluid Mech., Vol. 488, pp.213-223. Khaledi, H.A., Smith, E.I., Unander, T.E., Nossen, J., 2014. Investigation of two-phase flow pattern, liquid holdup and pressure drop in viscous oil-gas flow, Int. J. Multiphase Flow Vol. 67, pp. 37-51. Kroes, R.F., Henkes, R.A.W.M., 2014. CFD for the motion of elongated gas bubbles in viscous liquid, Proceedings of the 9th North American Conference on Multiphase Technology 2014, pp. 283-298. Moreiras, J., Pereyra, E., Sarica, C.,Torres, C.F., 2014. Unified drift velocity closure relationship for large bubbles rising in stagnant viscous fluids in pipes, J. Petr. Sc. Eng., Vol. 124, pp. 359-366. Oliver, N.M., Rosario, B., Pentland, A.P., 2000. A Bayesian computer vision system for modeling human interactions, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 22 (8), pp. 831-843. Orell, A., 2005. Experimental validation of a simple model for gas-liquid slug flow in horizontal pipes, Chem. Eng. Science, Vol. 60, pp. 1371-1381. Ramdin, M., Henkes, R., 2012. Computational fluid dynamics modelling of Benjamin and Taylor bubbles in two-phase flow in pipes, J. Fluid Eng., Vol. 134, pp. 1-8. Strazza, D., Demori, M., Ferrari, V., Poesio, P., 2011. Capacitance sensor for hold-up measurement in high-viscous-oil/conductive-water core-annular flows. Flow Meas. Instr., pp. 360-369. Sobral, A., 2013. BGSLibrary: An OpenCV C++ Background Subtraction Library. In: IX Workshop de Vis˜ ao Computacional (WVC’2013), Rio de Janeiro, Brazil. Sobral, A., Vacavant, A., 2013. A comprehensive review of background subtraction algorithms evaluated with synthetic and real videos. Comp. Vision Im. Underst., pp. 4-21. Taitel, Y., Barnea, D., 1990. A consistent approach for calculating pressure drop in inclined slug flow, Chem. Eng. Sc., Vol. 45, No. 5, pp. 1199-1206. Wallis, G.B., 1969. One-dimensional two-phase flow, McGraw-Hill, New York. Weber, M.E., 1981. Drift in intermittent two-phase flow in horizontal pipes, The Can. J. of Ch. Eng., Vol. 59, pp. 398-399. Weber, M.E., Alarie, A., Ryan, M.E., 1986. Velocities of extended bubbles in inclined tubes Ch. Eng. Science, pp. 2235-2240. Zukoski, E.E., 1966. Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes, J. Fluid Mech, pp. 821-837. 26

List of Figures 1 2

3 4 5

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Experimental multiphase flow set-up. C: capacitance probe; OW: observation window. . . . . . . . . . . . . . . . . . . . . . . . . . Flow map obtained plotting the E¨ otv¨ os number versus the characteristic Reynolds number. In blue – this work; in red – Weber et al. (1986); in green – Moreiras et al. (2014) and Gokcal et al. (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Top: original signal from a capacitance probe. Down: de-noised signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a) de-noised cut signals; b) signal derivative - negative minimum are identified by circles. . . . . . . . . . . . . . . . . . . . . . . . Capacitance probes output signals for the three different oils. The boxes indicate the part of the signal influenced by the drainage of the oil on pipe walls. . . . . . . . . . . . . . . . . . . . . . . . The passage from the maximum to the minimum value requires a certain time (δτ ). . . . . . . . . . . . . . . . . . . . . . . . . . . An example of the background subtraction operation using the DPEigenbackgroundBGS algorithm. . . . . . . . . . . . . . . . . Calibration test with water for the horizontal condition - drift velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless mass outflow for the calibration test with water for the horizontal condition. . . . . . . . . . . . . . . . . . . . . . Drift velocities for μ = 37.5 mPas for the horizontal condition. . Drift velocities for μ = 195.5 mPas for the horizontal condition. . Drift velocities for μ = 804 mPas for the horizontal condition. . . Comparison between experimental results for μ = 37.5 mPas and available data from literature. . . . . . . . . . . . . . . . . . . . . Comparison between experimental results for μ = 195.5 mPas and available data from literature. . . . . . . . . . . . . . . . . . Comparison between experimental results for μ = 804 mPas and available data from literature. . . . . . . . . . . . . . . . . . . . . Dimensionless drift velocity (Fr) as a function of Ohnesorge number. Refer to Fig. 2 for the legend. . . . . . . . . . . . . . . . . . Comparison between experimental results for Re = 234 and correlation by Kroes and Henkes (2014). . . . . . . . . . . . . . . . . Comparison between experimental results for Re = 45 and correlation by Kroes and Henkes (2014). . . . . . . . . . . . . . . . . Comparison between experimental results for Re = 11 and correlation by Kroes and Henkes (2014). . . . . . . . . . . . . . . . . A sketch of the physical system, as also reported in Alves et al. (1993). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different shapes corresponding to different surface tensions influence the drift velocity. . . . . . . . . . . . . . . . . . . . . . . . . Drift velocities along the pipe inclined of 1◦ . . . . . . . . . . . . . Drift velocities along the pipe inclined of 2◦ . . . . . . . . . . . . . 27

4

5 6 7

7 8 9 10 11 12 12 12 13 13 13 14 15 15 15 16 17 18 18

24 25 26 27 28 29 30 31 32

Drift velocities along the pipe inclined of 3◦ . . . . . . . . . . . . . Drift velocities along the pipe inclined of 4◦ . . . . . . . . . . . . . Drift velocities along the pipe inclined of 5◦ . . . . . . . . . . . . . Drained mass as a function of time for different tilt angles and μ = 37.5 mP as. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drained mass as a function of time for different tilt angles and μ = 195.5 mP as. . . . . . . . . . . . . . . . . . . . . . . . . . . . Drained mass as a function of time for different tilt angles and μ = 804 mP as. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured flow rates for μ = 37.5 mP as, μ = 195.5 mP as and μ = 804 mP as. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drift velocity as a function of the inclination angle. . . . . . . . . Comparison between the calculated drift velocity and the experimental drift velocity. . . . . . . . . . . . . . . . . . . . . . . . . .

18 18 18 19 19 19 20 21 22

List of Tables 1 2

Values of density (ρ), viscosity (μ) and surface tension (σ) measured at 24◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters a, b, e and f are obtained from data fitting; the μ viscosity number Nvis is defined as Nvis =  ; 3 gD (ρl − ρg )ρl ρl gD2 .. . . . . . . . . the E¨ otv¨ os number E o¨ is defined as E o¨ = σ

28

5

23

Highlights ◮ The effect of viscosity on drift velocity is investigated. ◮ The evolution of the bubble drift velocity is monitored along the pipe. ◮ We show that a relevant reduction of the velocity occurs along the pipe. ◮ We tested two existing correlation for drift velocity with collected data.

1