A new technique for in-situ measurements of bubble characteristics in bubble columns operated in the heterogeneous regime

Author’s Accepted Manuscript A new technique for in-situ measurements of bubble characteristics in bubble columns operated in the heterogeneous regime P. Maximiano Raimundo, A. Cartellier, D. Beneventi, A. Forret, F. Augier www.elsevier.com/locate/ces

PII: DOI: Reference:

S0009-2509(16)30479-1 http://dx.doi.org/10.1016/j.ces.2016.08.041 CES13138

To appear in: Chemical Engineering Science Received date: 10 March 2016 Revised date: 25 August 2016 Accepted date: 31 August 2016 Cite this article as: P. Maximiano Raimundo, A. Cartellier, D. Beneventi, A. Forret and F. Augier, A new technique for in-situ measurements of bubble characteristics in bubble columns operated in the heterogeneous regime, Chemical Engineering Science, http://dx.doi.org/10.1016/j.ces.2016.08.041 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A new technique for in-situ measurements of bubble characteristics in bubble columns operated in the heterogeneous regime.

Authors P. Maximiano Raimundoa,c,e, A. Cartelliera,b, D. Beneventic,d, A. Forrete, F. Augiere a b

Université Grenoble Alpes, LEGI, F-38000 Grenoble, France CNRS, LEGI, F-38000 Grenoble, France

c

Université Grenoble Alpes, LGP2, F-38000 Grenoble, France

d e

CNRS, LGP2, F-38000 Grenoble, France

IFP Energies Nouvelles, Rond-Point de l’échangeur de Solaize, BP3, 69360, Solaize, France

Highlights  A new method is proposed to measure the Sauter mean horizontal diameter of bubbles,  That technique exploits the spatial correlation coefficient of phase indicator functions,  Optimum sensor design is defined depending on bubble sizes and concentration,  Comparisons with alternate techniques are achieved in industrially relevant conditions,  The technique proves efficient and reliable in dense heterogeneous bubbly flows.

Keywords: Bubbly flow; Optical probe; Bubble size measurement; Industrial bubble column; Heterogeneous regime; high gas holdup

Abstract: In order to characterize bubbles in dense, heterogeneous bubbly flows such as those encountered in industrial bubble columns, a new measuring technique based on the spatial correlation of phase indicator functions is proposed. By analyzing its principle of operation, it is shown that the correlation coefficient decreases with the ratio of the distance between two optical probes to the bubble size. Hence for a known distance, the bubble size, and more precisely the Sauter mean horizontal diameter of bubbles is accessible. A dedicated sensor has been designed using conical monofiber optical probes. It has been compared to alternate measuring techniques in the complex bubbly flows produced in an I.D. 400mm bubble column with gas superficial velocities from 3 to 35 cm/s. The performances of the correlation sensor happen to be quite satisfactory, with typical uncertainties about 15-20%, and even less (<10%) in absence of flow reversal. Similar performances are also obtained for conical mono-fiber optical probes when measuring the Sauter mean vertical diameter of bubbles.

1. Context and objectives Bubbly flows are widely used in industry, in particular in chemical and bio-chemical engineering, 1

oil industry, water treatment, and in various separation operation units. Quite common contacting devices consist of aerated stirred reactors, bubble columns, or flotation cells… In all these systems, the gas phase characteristics such as bubble shape and size, gas velocity and volume fraction have a strong influence on hydrodynamics and mass/heat transfer between phases. Most of the time, such quantities need to be characterized in-situ as they depend on the reactor geometry including the injection device, as well as on operating conditions. Bubble characteristics at injection are notably needed to feed numerical simulations (see for example Leonard et al., 2015, Ziegenhein et al., 2015, Alméras et al., 2016). Moreover, spatially resolved measurements are required as bubble size and/or shape distributions can evolve within the system because of segregation, coalescence and/or break-up, processes that are still quite complex to accurately predict (Buffo et al., 2013). Globally, information regarding gas phase characteristics are required to better understand the coupling between phases by way of interfacial momentum, mass/heat transfers, and to derive appropriate closure laws or to acquire reliable data bases for two-fluids models (see for example Vikas et. 2015, McClure et al. 2015, Renze et al. 2014). In this respect, multiphase flows occurring in bubble columns provide an emblematic example of current challenges. Indeed, industrial columns have a diameter D typically ranging from 2 to 8 m with heights between 2 and 5 times D, and superficial gas velocities (defined as the ratio of the injected gas flow rate to the column cross-section) are typically in the range of [5-30]cm/s, leading to void fractions from 10% to 35%. Most industrial systems are therefore operated in the heterogeneous regime characterized by high gas volume fraction and complex, unsteady motions of both phases (Forret et al. 2006). Despite active research since the 80’s, the scale-up of bubble columns is not yet mastered and the engineering practice heavily relies on empiricism (Forret et al. 2006; Rollbusch et al., 2015; Leonard et al., 2015). This situation is partly due to the lack of reliable in-situ measurements of the gas characteristics in industrial conditions, as most of the results reported in the literature have been obtained either in 2D or 3D bubble columns of small size (typ. ≤ 20 cm) and at low to moderate void fractions (typ.<20%) Clearly, it is of primary importance to develop robust and reliable measuring techniques to determine the gas phase characteristics in operating conditions that are relevant for industry. The available options are few. Among common techniques, imaging gas inclusions provides a direct way to measure shape and size (Deckwer, 1992) either using standard (e.g. Wilkinson et al., 1994; Lage and Esposito, 1999) or high-speed cameras (e.g. Kazakis et al., 2006; Ferreira et al., 2012). The relative uncertainty on size is rather good (typically 10%). Beside, the technique is also exploited for velocity measurements. However, it requires transparent walls and low optical density of the twophase medium: external imaging is thus restricted to dilute conditions or to the investigation of wall regions in dense bubbly flows. To get rid of these limitations, endoscopic imaging has also been exploited (Maas et al., 2011), making then the technique somewhat intrusive. It usually leads to weaker contrasts because of magnification and illumination constraints, and the image processing becomes much more delicate and time consuming compared to external imaging techniques. The chemical method has been used in early studies to determine the interfacial area in bubble columns using reactions such as sulphite oxidation, hydrazine oxidation, glucose oxidation or hydrogen peroxide degradation (Pinelli et al., 2010). Usually, the chemical specie present in the liquid phase reacts with the oxygen provided by the gas phase, and for fast enough reactions, the mean interfacial area controls the transformation rate (Shultz and Gaden, 1956; Akita and Yoshida, 1974). Thus, if the oxygen concentration is known at air inlet, at air outlet and in the liquid, the flux of oxygen from the gas phase to the liquid phase can be evaluated. This method, which requires very well controlled 2

experiments, only delivers a single global mean bubble size, the Sauter diameter, and no information on the size distribution (Popovic and Robinson, 1987). Although intrusive in nature, phase detection probes (either optical, electrical, thermal or electrochemical) have been extensively used since the 60’s in a variety of gas-liquid flows (see Cartellier and Achard 1991 for a review of early works). Basically these sensors provide the k phase indicator function Xk at a given location from which one can infer the local concentration using time averaging. In addition, various probes configurations have been proposed to access velocity, making thus extra variables available such as chord and then size distributions, volume flux, number density flux… (see Cartellier, 1999). Among these extensions, the double probe, consisting in two sensors whose extremities are set some distance apart along the main flow, is quite common. The post-processing is aimed at identifying the signatures of the same bubble on the signals delivered by the two probes and at measuring the corresponding transit time. Knowing the typical distance between the probe tips (typically of the order of 1mm), a displacement velocity can be estimated those significance depends on the relative distance between the probes compared with the size of gas inclusions (Cartellier, 2006). As one should ideally avoid to pair different bubbles or interfaces arriving on the probes, the posttreatment is crucial in the implementation and in the performance of the technique. The correct pairing becomes tricky at high void fractions, especially in presence of recirculating, unsteady flows. In such flow conditions, Chaumat et al. (2005, 2007) demonstrated that the extent of the velocity distribution becomes very sensitive to the selected association criteria. In addition, Chaumat et al. (2005) clearly show that bi-probes detect short waiting times out of a single bubble when the latter experiences strong lateral motions: when so, the measured bubble velocity distribution, and hence the chord distribution, exhibit very high unphysical values, an outcome already predicted by Xue et al. (2003). Therefore, for complex bubbly flows, Chaumat et al. (2007) recommend the exploitation of an “average” velocity Vmax based on the most probable transit time. For the size, they proposed to use the relationship D32 = 3 Vmax g / (2f) based on average quantities(1), namely the gas concentration g, the most probable velocity Vmax and the bubble detection frequency f, which are believed to be less sensitive to signal processing criteria. The above proposals were tested on bubbles in the range 3 to 9mm in diameter. In bubble plumes, typical deviations from direct imaging measurements were found to be less than 20% for the velocity as well as for the size, although size deviations can increase up to 40% in more dense situations (Chaumat et al., 2005). The performance of bi-probes in bubble columns at superficial velocities from 3 to 10cm/s were tested by comparing integrated local values to global ones (Chaumat et al., 2007): the gas hold up was recovered within 20% in cyclohexane while the difference reached 30% in water. The gas flow rate was overestimated by 20% in water, but was found almost twice the injected flow rate in cyclohexane. The origin of such differences between the two liquids is unclear. Note also that part of these overestimations arise from the fact that only upward

1

Note that the relation D32 = 3 Vmax g / (2f) arises from the comparison of the classical expression of the interfacial area density = 6 g / D32 and of the expression = 4 f <1/Vi> established for convex inclusions experiencing 1D motion, where Vi is the ith interface velocity and the brackets denote the average over all events. Combining these equations leads to D32 = 3g/(2f< vi>). As the average of the inverse of interfacial velocities (<1/ Vi >) is not the inverse of the average (1/), the expression used by Chaumat et al. (2005, 2007) is an approximation. A second approximation is to identify the average with the most probable velocity Vmax. 3

velocities were detected by the bi-probe so that the downward flux was not accounted for when integrating radial profiles. To better select meaningful events, and in particular to detect and eliminate strong lateral bubble motions, a configuration with four independent fibers has been developed. That arrangement consists in a central, longer probe with three others shorter probes organized as a triangle (e.g. Saito and Mudde, 2001; Guet et al., 2003; Luther et al. 2004; Xue et al., 2003, 2008 A, 2008 B). The typically distances between probes are of the order of 500µm along the radial direction and about 1mm along the axial direction. By analyzing the sequence of detection by the four probes, one can identify, or at least classify, bubble trajectories (Guet et al., 2003), and thus collect reliable velocity and size information on selected events such as for examples bubbles almost centered on the central probe. Such detection routines are thus adapted to the characterization of bubbles larger than the radial interprobe distance. On isolated, large (deformed) bubbles, the uncertainty on velocity and on eccentricity has been estimated to be less than 10% (Guet et al., 2003) and at most 20% (Luther et al., 2004). These four-probes sensors were also tested versus imaging techniques on bubbles with an equivalent diameter in the range 2-6mm organized as bubble trains, or confined Taylor bubbles (Xue et al., 2008 A) with similar performances. These four–point optical probes have also been exploited by Xue et al. (2008 B) in a 16 cm diameter column, at high gas superficial velocities - up to 60 cm/s. The bubble sizes were evolving from 2 to 6mm, corresponding to Sauter mean diameters in the range 3-9mm. In such conditions, statistics on chord and on velocity were collected in the column. In particular, the statistics on velocity were reconstructed using a probe successively oriented upward and downward in order to better account for the mean flow recirculation. Although the experimental results provided on the flow structure seem consistent (Xue et al., 2008 B), no comparison with alternate techniques was achieved. Besides, the integrated profiles were not tested versus global parameters such as global void fraction and gas flow rate. It is therefore difficult to appreciate the performances of four–point optical probes in complex bubbly flows. In particular, their capability in terms of the minimum detectable bubble size is not determined. An alternate way to measure velocity exploits the fact that the de-wetting time of the probe tip evolves as the inverse of the interface velocity (Cartellier, 1990, 1992). That technique only requires one probe. Compared with multiple tip probes, the bubble trajectories are less perturbed and there is no longer any issue concerning the association of signals from different probes. Consequently, all gas inclusions detected are taken into account in the statistics. That technique provides the bubble velocity and thus the pierced chord along the direction of the probe axis. Several tip geometries have been successfully tested (Cartellier 1992, Cartellier and Barrau, 1998), in various conditions including bubbly and slug flows in ducts (Barrau et al., 1999) and sprays (Hong et al., 2004). Global performances in duct flows at superficial velocities from 0.2 to 1m/s amount for a relative error -15/+0% on the global void fraction and on the gas flow rate. On isolated ellipsoidal bubbles with a 2mm diameter along the horizontal and 1.4mm along the vertical, Barrau et al. (1999) report a 10% error in the detected size when piercing the bubble along its axis of symmetry. Vejražka et al. (2010) achieved an in-depth investigation of measurement uncertainties of conical mono-fiber optical probes when applied on bubbles ranging from 1 to 3 mm in diameter. These authors demonstrated that the deceleration of the bubble at impact and its subsequent deformation and rolling motion are the main mechanisms affecting the probe performances. Solid guidelines were provided for an appropriate use of single fiber probes in bubbly flows. In particular, the uncertainties on chord, concentration and velocity measurements were shown to be controlled by a modified Weber number, namely M = 4

[Db2ub2/DbDprobe) where ub is the bubble approaching velocity with respect to the probe, Dprobe the probe diameter, Db the bubble diameter, the fluid density and  the surface tension. Typically, these uncertainties are less than 10% for M above about 50. Such a condition is fulfilled in industrial bubble columns, so that the single probe technique is potentially relevant for these systems. The present work addresses the issue of local measurements of bubble characteristics in dense, unsteady flows, such as those encountered in industrial bubble columns. Among the above-mentioned techniques, the single probe sensor seems quite promising. However, all the information available so far is relative to quasi-unidirectional flows. Larger uncertainties are to be expected in threedimensional, unsteady two-phase flows. In particular, severe drawbacks may arise when flow reversal occurs as found in bubble columns. It is thus worth investigating the performances of the single probe technique in such complex bubbly flows. Yet, that technique provides a single bubble dimension (namely, along the direction of the probe axis), while bubble columns are usually operated with deformed bubbles. In order to quantify the lateral dimension of bubbles, we propose a new method based on the spatial cross-correlation of the signals delivered by two side-by-side optical probes. The paper is organized as follows. The principle of operation of that new method is presented in section 2 together with a sensitivity analysis to various parameters such as bubble shape and orientation, size distribution and gas concentration. Uncertainties are also quantified. The method is then compared with alternate measuring techniques in bubbly flows generated in a 40 cm I.D. column over a wide range of superficial gas velocities (up to 35 cm/s). The experimental set-up is presented in section 3 as well as the endoscopic imaging technique that provides bubble shape, size and orientation. Then, the performances of single optical probes in bubble columns are scrutinized, with emphasis put on the heterogeneous regime. Finally, in Section 4, bubble size measurements obtained from the correlation technique are compared with endoscopic imaging and with single optical probe results and the performance of the correlation technique is discussed.

2. Bubble size derived from spatial correlation To illustrate the measuring principle, let us consider two phase detection probes located at the same elevation but separated by some horizontal distance d. The idea is to examine the spatial correlation between the signals collected by each probe. The correlation is maximum for probes located at the same point in space i.e. for a distance d=0mm. For a distance d smaller than the horizontal bubble size, the probes are quite often detecting the same bubble so that the correlation value remains significant. As the distance further increases, the probability to hit the same bubble decreases up to the point where it goes to zero. That latter situation is expected to occur when d becomes larger than the maximum bubble horizontal diameter present in the system. The spatial correlation is therefore expected to be non-zero for distances d smaller than the horizontal dimension of bubbles, and some information related to the lateral extent of bubbles could possibly be extracted from the shape of the correlation as a function of the distance between probes. Quantitatively, it is convenient to handle with the Eulerian spatial correlation coefficient between k-phase indicator functions at each probe positions, namely and . is defined as: 〈 〈

〉

〈

〉 〉

(1) 5

where brackets denote averaging. Since are telegraphic functions evolving between 0 and 1, then 〉 〈 〉 so that 〈 which is the k phase fraction at location x. Assuming that the concentrations at locations x and x+d are nearly the same for d small enough (alternately concentrations gradients at scale d are neglected), is rewritten: 〈

〉

(2)

In practice, we consider the cross-correlation between the two raw signals SignalA and SignalB simultaneously delivered by probe A and by probe B. These signals, normalized between 0 for the liquid phase and 1 for the gas phase, are a good approximation of the corresponding local gas phase indicator functions (How raw signals deviate from an ideal response is a key issue for the design of reliable phase detection probes and for the development of the optimal signal processing. These questions will not be addressed here, refer for example to (Cartellier, 2006). Using time averaging, the correlation coefficient for the gas is therefore evaluated as: ∫

∫

(3)

where the measurement duration is selected large enough to ensure convergence. also represents the conditional probability to detect some gas at x while gas is also present at x+d. In a gasliquid flow medium, widely separated probes detect different bubbles at the same time, so that the correlation does not drop to zero at large distances but instead it should asymptote to the mean void fraction. Indeed, in most flow conditions, the phase indicator functions at x and at x+d become uncorrelated for large distances d and if the concentration is nearly uniform in the system (= ), then: 〈

〉〈

〉

(4)

Fig.1 provides some measurements of correlation coefficients performed in a bubble column (the latter is presented in Section 3) using a pair of conical optical probes. Clearly and as expected, the correlation always monotonically decreases with the distance between probes. In addition, it never drops to zero, even for d=12mm a distance that significantly exceeds the largest bubble horizontal diameter (in these experimental conditions, the bubble size ranged from 4 mm to 9 mm). Indeed, and following the above discussion, the correlation at large distances reaches a value close to the global gas hold-up in the column. Hence, the information related the lateral extent of the bubbles should preferably be extracted from the behavior of the correlation coefficient at small gap between probes. The examples of Fig.1 correspond to nearly the same bubble size distributions: the correlations for different superficial velocities overlap up to a distance d about 1.5mm. This indicates that this is the zone to explore in order to obtain some information on the bubble size. In order to determine how the correlation coefficient evolves with the distance between probes and with the bubble size, we derived theoretical models considering situations of increasing complexity.

6

1 0.8 0.6 0.4

0.2 0 0

20 40 60 80 Distance between probes (mm)

3 cm/s

9 cm/s

16 cm/s

25 cm/s

100

35 cm/s

Fig.1 - Experimental correlation coefficients measured for various superficial gas velocities in a 400 mm I.D bubble column using conical optical probes. The dashed lines represent the corresponding global gas hold-up.

2.1.

Models of the correlation coefficient

Let us start with a 3D analytical model for an idealized bubbly flow. Two ideal (i.e. non perturbing) probes A and B are located side-by-side and separated by the distance d. Their axis is vertical (along z) so that their extremities define the horizontal plane (x,y), the direction x being set by the probe extremities (Fig.2). As seen above, the correlation coefficient is the conditional probability of detecting gas at probe A while there is some gas detected at B, i.e. . To evaluate these probabilities, let us consider mono-dispersed oblate ellipsoidal bubbles with a horizontal radius equal to Rh and a vertical radius equal to Ecc.Rh, Ecc being the eccentricity. All bubbles are assumed to travel along the vertical direction z, with a uniform probability P(x,y)=P0 for a bubble center to cross the horizontal plane at a location (x,y) during bubble motion. Moreover, we consider bubbles as isolated in the sense that only one bubble interacts with the probes at a given time; this rules out the void fraction effect illustrated above. With such assumptions, the two circles in Fig.2 top view represent the extreme positions of bubble centers such that the bubble is detected by probe A (respectively by probe B) when travelling along the vertical.

Top view

Lateral view y

z

d

Probe A

Probe B

x LA Bubble mo on

LB

2 Ecc Rh

Rh

A Rh Limit of bubble center posi ons for probe A

2 Rh

d

B Limit of bubble center posi ons for probe B

Intersec on = posi ons of bubble centers simultaneously detected by both probes.

Fig.2 – Schematic view of bubble - probes interaction accounted for in the analytical model.

7

x

The vertical chord L detected by a given probe is a function of the bubble size Rh and of the distance of the bubble center located at (x,y) to the tip of that probe (Fig.2 side view). For probe A √

located at x=y=0, √

with r2=x2+y2 and for probe B located at x=d, y=0,

. The probability P(A) to detect gas on probe A is given by the integral of chords

for bubble centers at a distance r ≤ Rh from the probe A weighted by the probability P(x,y) that the bubble center crosses the horizontal plane at (x,y), i.e.: ∫

(5)

Eq.(5) shows that, as expected, P(A) is given by the volume of the ellipsoid times P0. The probability P(A,B) to simultaneously detect gas on both probes corresponds to bubble positions such that the distance of their centers to probe A and to probe B are less than Rh: these events are only found at the intersection of the circles represented in the top view of Fig.2. In addition, to simultaneously detect gas on A and on B, one has to count only the minimum chord, min(LA, LB). For bubbles centers with an abscissa higher than d/2, the smallest chord is LA. Equivalently, for bubble centers with an abscissa less than d/2, the smallest chord is LB. Hence, one has to consider two times the integral involving the chord LA for x evolving between d/2 and Rh and for y between -ymax and ymax with ymax2+x2= Rh 2. Accounting also for the symmetry relative to the x-axis, one has: ∫

∫ [

(

∫

) ]

(6)

From eq.(5) and (6), the conditional probability, or equivalently the correlation coefficient is therefore: ( )

(7)

Eq.(7) demonstrates that the correlation coefficient is a unique function of the inter-probe distance divided by the horizontal bubble size Rh. In particular, for oblate ellipsoids, the prediction does not depend on the eccentricity. As expected, the correlation coefficient drops to zero at d=2Rh, i.e. at the maximum horizontal extent of bubbles (Fig.3). Therefore, the bubble size can be recovered by correlating the signals of two probes separated by a known distance of the order of the bubble lateral dimension. Besides, if one considers only the linear term in Eq.(7), the correlation coefficient is well predicted up to d ≈ Rh. All these features are quite encouraging in the perspective of devising a measuring technique for characterizing a mean lateral bubble size.

8

Correlation coefficient

1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

2.5

d/R Fig.3 – Analytical correlation coefficient according to Eq.(7); full equation (continuous line) and linear term only (dashed line).

Yet, a number of issues need to be addressed to ascertain the capability of the proposed method especially in complex two-phase flows. Among these, what is the sensitivity to the bubble shape, to the bubble orientation? What is the meaning of the size detected in case of an extended bubble size distribution? What is the typical uncertainty especially when accounting for the presence of neighbor bubbles? To investigate more complex situations, we relied on Monte Carlo simulations in which a bubble is held fixed while the two probes are moved over space to mimic the random position of bubbles interacting with the probes. The bubble was discretized using a mesh fine enough to properly resolve its entire volume. The typical spatial resolution was 0.01 mm (a more refined resolution does not change the results). The probe A was successively positioned at all nodes of the mesh, while the position of probe B relative to probe A was set at a fixed distance in a given selected direction. For each probe position, one has to check whether both probes are inside the bubble or not, and the correlation coefficient is calculated by cumulating such events divided by the number of events “probe A is in the gas”. This procedure was repeated for different inter probe distances. Some results for spherical bubbles are exemplified Fig.4-left. The key features identified from the analytical approach are recovered, namely curves for different bubble sizes collapse on a master curve when plotting the correlation coefficient versus the ratio d/Dh (Fig.4-right). The initial slope has been evaluated using a linear regression: at small inter-probes distances d, the horizontal diameter Dh happens to obey: (

(8)

)

1

1

Cross-correlation (-)

Cross-correlation (-)

The -1.472 simulated slope is almost equal to the -1.5 analytical prediction of Eq.(7). 0.8

0.8

0.6

0.6

0.4

0.4 0.2

0.2

0 0

5 10 15 Distance between probes (mm) R=6 mm R=4 mm R=2 mm

0 0 0.5 1 1.5 Distance between probes/Horizontal diameter Dh=4 mm

9

Dh=8 mm

Dh=12 mm

Fig.4 – Spatial correlations computed from Monte Carlo simulations for spherical bubbles: Left) correlation coefficient versus the distance between probes for different bubble sizes; Right) correlation coefficient plotted versus the ratio ‘distance between probes/bubble horizontal diameter’.

2.2.

Sensitivity to bubble shape and to bubble orientation.

In industrial reactors, bubbles are most of the time large enough to be significantly deformed. Let us recall that bubbles larger than 1 mm are no longer spherical in tap water/air systems. Although large bubbles can take a variety of shapes (Clift et al., 1978), only ellipsoids have been considered for the simulations. Even more, we restricted our analysis to spheroids, i.e. ellipsoids with two equal diameters, namely prolate and oblate as illustrated Fig.5. The distortion of such bubbles is thus described by the eccentricity Ecc defined as the ratio of the vertical (vertical refers here to the probe axis) diameter to the horizontal diameter. In two-phase flows, the inclination of such deformed bubbles, characterized by a tilt angle (Fig.5), is evolving in time. To simulate this, the tilt angle was randomly varied in the range +/-30 degrees: these values are in agreement with the observations of Xue et al. (2003) as well as with our own observations in bubble columns using endoscope imaging. To correctly reproduce the bubble-probe interactions, it was also necessary to consider that the bubble randomly rotates around the probe axis over a 0-360° range (angle α in Fig.5).

Fig.5 – The two spheroids considered in the simulations (angles accounted for are also shown).

As before, the probe A was fixed at a position inside the bubble. The second probe should be located at the same elevation, but in order to simulate the inclination and the rotation of bubbles, the probe B was located at the following coordinates: (9) For each probe A position, both and α were randomly varied over their respective intervals with typically a 1/1000 resolution. The probe A position scanned the entire volume of the bubble, with a spatial resolution of 0.01mm. The correlation was again calculated by accumulating positive events (gas at A and at B, gas at A), and the procedure was repeated for different distances d. Fig.6-left presents the computed correlation for an oblate spheroid with a fixed horizontal diameter Dh of 4 mm and for different eccentricities equal to 1, 0.7 and 0.5 respectively. Clearly, the predictions are insensitive to the eccentricity, and the slope in the limit of small distances is the same as 10

the one found for spheres. For a prolate spheroid, the results, shown Fig.6-right, are slightly different. The correlation becomes now somewhat sensitive to the eccentricity. In particular, the initial slope of the correlation neatly increases as the eccentricity decreases. A fit of the numerical results provides: (10) so that for prolate spheroids the horizontal diameter takes the following form: (

(11)

)

1

1

0.8 0.6 0.4 0.2 0 0

0.5 Ecc=1

1

d/Dh (-) Ecc=0.7

1.5

Cross-correlation (-)

Cross-correlation (-)

As the eccentricity now enters the relation, the method used so far to derive the bubble horizontal diameter cannot be applied as such for prolate spheroids. For moderate eccentricities, the difference compared with the coefficient for oblate spheroids is moderate: 8% at Ecc=0.8, 16% at Ecc=0.7 and 25% at Ecc=0.6. In general, Eq.(11) can be exploited if the eccentricity is known beforehand from others means. Alternately, the above measurements can be combined with some extra information (such as for example a vertical mean size estimated from a single probe) to simultaneously solve for Ecc and Dh.

0.8 0.6 0.4 0.2 0

0

0.5

Ecc=0.5

Ecc = 1

d/Dh Ecc=0.7

1

1.5

Ecc=0.5

Fig.6 – Simulated correlations as a function of the distance between probes divided by the horizontal bubble diameter for various eccentricities: Left) oblate spheroids Right) prolate spheroids. Results for Dh =4mm.

Yet, for most bubbly flows encountered in bubble columns or vertical ducts, the bubbles experience a vertical pressure gradient and are therefore compressed along that direction: they thus assume oblate instead of prolate shapes. Although gas inclusions are often deformed into distorted oblate spheroids, the shape sensitivity analysis has not been pursued further. Instead, we considered more realistic conditions occurring in a bubble column by accounting for actual distributions in eccentricities and in tilt angles. For that, various quantities were measured using endoscopic imaging in the bubble column described in Section 3. The distribution of horizontal eccentricities (ratio of horizontal diameters measured along two perpendicular directions) shown Fig.7-left demonstrates that the oblate assumption is valid over a large range of superficial gas velocities. In addition, Fig.7-right indicates that this conclusion holds whatever the bubble size class. 4

3

0.4 PDF

Frequency (-)

0.5 0.3 0.2

2 1

0.1

0

0

0.1

0.8 0.9 1 Horizontal eccentricity (-) 3 cm/s 9 cm/s 16 cm/s

0.2

0.7

<=6 mm

11

0.3

0.4 0.5 0.6 0.7 Eccentricity (-)

6-7 mm

7-8 mm

0.8

8-9 mm

0.9

1

>=9 mm

Fig.7 – Distribution of horizontal eccentricities in the I.D. 400mm bubble column measured on the column axis and at an elevation 3.75 D. Left) for various superficial gas velocities irrespective of the bubble size, Right) for various size classes at a given superficial gas velocities (Vsg=16cm/s).

Concerning now the vertical eccentricity Ecc, its distribution is shown Fig.8-left. Except in the homogeneous regime observed at 3cm/s, the distributions are the same whatever the gas superficial velocity. Besides, Fig.8-right provides tilt angle distributions for various vertical eccentricities. Clearly, there is no significant correlation between eccentricity and tilt angle. A second observation is that the tilt angle is not random and that its range widely exceeds the ±30° amplitude previously assumed. All the above results are valid over a wide range of superficial gas velocities. According to these observations, the Monte Carlo simulations were run as follows. For a given bubble size, the bubble eccentricity and the inclination were randomly chosen from their respective distributions while the probe was randomly located in space. This process was repeated enough times in order to ensure a correct convergence of the correlation coefficient. The process was then repeated for various distances between the two probes. 0.6 Frequency (-)

10

PDF

8 6 4

0.5 0.4 0.3 0.2 0.1

2

0

0 0.1 3 cm/s

0.2

0.3 9 cm/s

0.4 0.5 0.6 0.7 Eccentricity (-) 16 cm/s

0.8

25 cm/s

0.9

10 20 30 40 50 60 70 80 90 Angle θ(°) Ecc=0.5 Ecc=0.6 Ecc=0.7 Ecc=0.8 Ecc=0.9 Ecc=1

1

35 cm/s

Fig.8 – Distribution of vertical eccentricities for various superficial gas velocities (left) and of bubble tilt angle for a superficial gas velocity= 16 cm/s (right). Measurements performed in the I.D. 400mm column, on the column axis and at an elevation 3.75 D.

Fig.9 shows the predicted correlation coefficient as a function of the ratio ‘distance between probes/bubble horizontal diameter’ when accounting for the measured eccentricity and inclination distributions (the so called 2nd method). For sake of comparison, the prediction using the approach already presented where the eccentricity was fixed and the bubble inclination randomly selected between ±30° (the so called 1st method) is also shown Fig.9. Accounting for the actual eccentricity distribution slightly decreases the correlation at a given distance d between probes, but the difference is slim. Indeed, at small distances d, the horizontal diameter Dh is found to evolve as: ( ) with a prefactor 1.5978 to be compared to the value 1.472 found when using the 1st method.

12

(12)

Cross-correlation (-)

1

0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 Distance between probes /Horizontal diameter 1st method 2nd method

Fig.9 – Simulated correlation versus the ratio inter-probes distance/bubble horizontal diameter for oblate st nd spheroids: 1 method= fixed bubble eccentricity and tilt angle between ±30°. 2 method= bubble eccentricity and inclination obeying the probability distributions detected by endoscopic imaging in a I.D. 400mm bubble column. (Simulations for a fixed bubble horizontal diameter =4 mm).

Summarizing the analysis presented in this section, it happens that, regardless of the bubble shape, the relationship between the bubble horizontal diameter, the correlation coefficient and the inter-probes distance always expresses as follows: (

(13)

)

The impact of the bubble shape on the measured diameter is quantified by the variation of the prefactor a: available values are collected in Table 1. Spheres and oblate ellipsoids provide very similar prefactors, with a maximum difference about 8% when accounting for actual eccentricity and inclination distributions. For prolate shaped bubbles, the coefficient becomes a function of eccentricity. For the flow conditions considered here, the average eccentricity is equal to 0.7 and it leads to a 15% deviation compared with spheres. Such a reasonable difference demonstrates the limited sensitivity of the prefactor to the bubble shape, and thus the robustness of the proposed method. Simulation results Prefactor a in Eq.(13) Deviation from the sphere case (simulated)

nd

st

Sphere

Oblate (1 method)

-1.472

-1.472

-

0%

Oblate (2 method) -1.5978 8.54%

Prolate with Ecc=0,7 -1.7 15.5%

Table 1 – Prefactors for various bubble shapes, and their deviation compared with the prefactor for spheres.

2.3.

Effect of the void fraction

The approaches developed so far were based on probes interacting with isolated bubbles while, in actual flows, bubbles are present all over the space surrounding the probes. To evaluate the influence of the void fraction, two simulations were performed. In a first simulation, spherical bubbles of the same size were randomly placed within a control volume. Overlapping between bubbles and with the box boundaries was avoided (Fig.10 left). Different spatial arrangements were tested for a 23% void fraction: all computations lead to the same result, showing no sensitivity to the bubble positioning procedure. In a second simulation, the bubbles were manually positioned in space (Fig.10-left) so that the gas concentration has been increased up to 41%. The simulated correlations are presented in Fig.10-right. First, as expected from Eq.(4) and as seen in measurements (Fig.1), the correlation 13

coefficient tends to the void fraction for large distances between probes. Second, a finite void fraction affects the shape of the correlation as the probability that the two probes hit different bubbles increases with the gas hold up. Such events are becoming more and more rare as the distance between probes decreases, so that the correlation remains unaffected at very short distances. Hence, the proposed method remains valid even at high void fraction provided that the probes are close enough to each other. The practical question is then to determine, for a given distance between probes, what are the ranges of bubble diameters and void fraction on which the method can be used with confidence. According to Fig.10-right, in order to maintain an uncertainty on the estimation of the diameter below 10%, the ratio d/Dh should be less than 0.23 at a void fraction of 41%, and less than 0.57 for a void fraction of 23%. Alternately, for a given probe, that is for a given distance d, the above objectives restrict the range of sizes that can be measured with a given uncertainty. For example, if one sets d=0.8mm, the correlation method applied to a void fraction up to 23% correctly measures bubbles whose horizontal diameter is larger than 1.4mm. That limit increases to 3.5mm at a void fraction equal to 41%. Smaller bubbles will be detected but the technique will provide mean diameter with an increased uncertainty. For example, for Dh=2mm bubbles at a void fraction 41% and with a sensor such that d=0.8mm, the bubble size would be underestimated by 20% compared with its actual value. Note also that, whatever the flows conditions, and since the same bubble must be perceived on both probes for the technique to work, there is a minimum diameter equals to d/2 below which the correlation technique would be inoperative. The finite void fraction effect discussed above could in principle be accounted for in the processing. Indeed, the dependency of the correlation curve and/or of the prefactor a involved in Eq.(13) on the void fraction can be established using Monte Carlo simulations with varying void fractions, bubble shapes and sizes. Since the local void fraction is already available from probes, one could then select the proper prefactor a(g) (or the proper correlation curve to get rid of the linearization approximation) to estimate the bubble dimension. This procedure has not been attempted for lack of time since Monte Carlo simulations are somewhat lengthy, but it may be worth the try especially for applications at large gas hold up. g=23%

1

Cross-correlation (-)

g=41%

0.8 0.6

0.4

g=41%

0.2

g=23%

0

0 d/Dh single bubble

0.5 void fraction= 23 %

1

1.5 void fraction =41%

Fig.10 - Spatial arrangements of spherical bubbles considered in order to achieve a given void fraction (left) and simulated correlation coefficient versus the ratio ‘distance between probes /bubble horizontal diameter’ (right). (Simulations for a fixed bubble horizontal diameter = 4 mm).

2.4.

Size measurement uncertainty

In these first developments of the method, we relied on the linear approximation and thus on Eq.(13). Therefore, the uncertainty when performing bubble diameter measurements using the correlation technique originates from three quantities: the value of the correlation coefficient, the slope of the correlation as determined by a linear regression scheme and the distance between the two 14

fibers. To evaluate the uncertainty in the determination of the correlation, measurements were repeated several times for the same flow conditions, with the same sensor and for a total of 10000 bubbles detected. The values exhibit a variation of 0.44 % that affects the diameter estimate by 2.5 %. Second, the deviation between the linear approximation used for the measurements and the exact value of the correlation coefficient increases with the ratio d/Dh. To quantify this, let us consider a sensor such that d=0.8mm and bubble sizes ranging from 2 mm up to 14 mm which is representative of the conditions found in bubble columns. The ratio d/Dh varies then from 0.06 up to 0.4. If the linear regression is established up to d/Dh = 0.06 or 0.4, the slope varies by ±2 %. When transposed into a diameter prediction, that difference corresponds to a ±6% variation. Last, the sensor used for the experiments presented in Section 4 consists in two parallel optical probes separated by a distance of 0.8 mm. That distance was measured within +/- 6 µm using a microscope. Such an uncertainty affects the diameter estimation by 0.075%, and it will be disregarded. Therefore, the typical uncertainty on horizontal diameter measurements by the correlation technique is about ±6%. It is at most 10% when all the factors biasing the measurements are acting in the same way. That 10% magnitude also corresponds to the sensitivity of the prefactor a to the bubble shape (see Table 1). Besides, according to the discussion at the end of section 2.3, 10% is also a correct estimate of the influence of the void fraction when the distance between probes is small enough. Overall, the typical uncertainty on the horizontal diameter measured by the proposed technique is about 10%.

2.5.

Nature of the averaging

In actual two-phase flows, the measured correlation will result from bubbles of different sizes and shapes, based on their contribution to the local void fraction. It is therefore important to understand what kind of averaging is performed by the correlation technique. Two approaches are presented below to characterize the averaging process induced by the cross correlation; both lead to the same conclusion. In a first approach, we split the bubble size distribution into N classes, class i corresponding to bubbles with a horizontal diameter , a vertical diameter and a void fraction . Each of the signals involved in Eq.(3) can be decomposed as the sum of the bubble pulses relative to each bubble class: (14) ∑ ∑ Assuming that only one bubble can be detected by both probes at the same time, then: (15) and Eq. (14) becomes: ∑

(16)

Using Eq. (16) in the expression of the correlation coefficient (see Eq. (3)): ∫

[∑

]

∑ [∫

] (17)

∑

15

The correlation coefficient for the class i is now replaced by its linearized expression Eq.(13) versus the inter-probe distance d and the bubble size for that class. The volume of ellipsoidal bubbles is also introduced. Hence: ∑ [

∑ [

]

∑

∑ ∑

]

∑ [

(18)

]

∑

∑

∑ ∑

In Eq.(18), the factor a is negative. The diameter appearing in Eq.(18), namely ∑

is the Sauter mean

value of horizontal bubble diameters present in the flow: it is noted Dh32 in the sequel. A similar conclusion arises when revisiting the analytical model of Section 2.1 established for oblate spheroids. Indeed, accounting for the size distribution P(Rh) for bubbles with the same eccentricity, the probability to detect a bubble on probe A becomes: ∫

〈

∫

〉

(19)

and the joint probability to simultaneously detect gas at probes A and B writes: ∫

∫ (

[

∫

) ]

(20)

Therefore, 〈 〈

〉 〉

(〈

)

〉

(21)

This equation show that the conditional probability to find gas at A while gas is at B is a complex function of the two moments and of the true size distribution in the system. Neglecting the third term in Eq.(21), the measured diameter is then the Sauter mean diameter / = 2 / of the horizontal dimension of the bubbles. We thus recover the same conclusion as before for a distribution of oblate spheroids with the same eccentricity. In particular, the prefactor coefficient for a poly-dispersed size distribution remains the same as in the mono-dispersed case. Note that Eq.(21) is valid provided that all the bubbles contribute to the correlation coefficient. In practice, bubbles smaller than d/2 do not contribute to the conditional probability so that the size distribution perceived by the probe (and hence the one used when estimating the Sauter average) is the fraction of the actual distribution for horizontal diameters above d/2. According to the analysis presented in this section, the proposed correlation technique seems promising. However, its actual performances need to be tested in actual bubbly flows in particular when bubbles of different shapes are present and also in strongly inhomogeneous conditions involving recirculation and transient motions such as those encountered in bubble columns. The next section will present the experimental set-up, two alternate measuring techniques, namely endoscopic imaging and single optical probes as well as the sensor developed to exploit the spatial correlation technique. In section 4, a set of flow conditions will be first detailed. The performances of conical probes in 16

heterogeneous regime will be analyzed. Then the measurements performed with the correlation technique will be compared with those collected from endoscopic imaging and from mono-fiber optical probes, and the reliability of the correlation technique in complex bubbly flows will be discussed.

3. Experimental conditions and alternate measuring techniques 3.1.

Experimental set-up

Experiments have been carried out in a 0.4 m I.D., 3.25 m high bubble column made of Plexiglas as schematized Fig.11-left. Gas injection was ensured using a 10 mm thick perforated plate with 391 holes of diameter 1 mm, arranged with a 15 mm triangular pitch, and yielding a plate porosity of 0.2%. Tap water and dried air were used as liquid and gas phases. The surface tension at 20°C was measured to be 67mN/m, indicating that a small amount of surfactants was indeed present in the tap water. The column was operated with no liquid flow rate, for a static height to diameter ratio (H0/D) of 4 and at atmospheric pressure and room temperature. Air flow rate was measured using three Brooks mass flow controllers spanning the range 0-60 Nm3/h with a 0.1 Nm3/h resolution. a

b Column walls

Column body

Air compressor

Air chamber

Mass flow controlers

Illumination bar

Spot light

Endoscope

CCD camera

5 cm

Perforated plate

Fig.11 - Schematic view of the bubble column (left); Schematic view of the endoscope and of the illumination bar (right).

3.2.

Endoscopic imaging

To ensure in situ flow visualization even at high void fraction, endoscopic imaging was implemented in the column. A Sirius endoscope from Foretec Company was inserted in a 0.7 m long and 8mm O.D tube. The focal plane is located 26mm away from the endoscope extremity with a depth of field of 3 mm. Since the endoscope is equipped with a front lens ensuring a ±45° viewing angle, the lateral extent of the field of view is 25 mm. Images were recorded with a Baumer HXC20 camera (2048 x 1088 pixels, 10 bits resolution) using a 50µs exposure duration. The frequency was set to 60 fps to collect uncorrelated events. Consequently, the minimum detectable size is about 300µm and the dimensions of an object in the focal plane is measured with a ±7% uncertainty. The endoscope was radially inserted in the column. To provide the necessary light to capture high-speed videos at high void fraction, an illumination bar was facing the collecting lens of the endoscope. The latter consisted high intensity LED located at one extremity of a 50mm O.D. Plexiglas bar. The bar was aligned with the endoscope optical axis, and it was placed 5 cm away from the endoscope (Fig.11-right). Note that for horizontal eccentricity measurements, the endoscope was vertically immerged in the column, and an external light source was used. For each detected bubble, the horizontal and vertical diameters, the eccentricity and tilt angle relative to the horizontal axis were measured. Due to the limited contrast of images, and a frequent 17

overlap between bubbles (Fig.12-left), it was not possible to develop an automatic measurement routine. Hence, all measurements were hand-made, a difficult and time-consuming process. That selection was based on somewhat subjective criteria based on sharpness. In addition, overlapping bubbles were discarded to avoid erroneous measurements. The final count rate was quite low with typically 1 out of 20 images providing reliable size measurements. Typical histograms thus collected are presented Fig.12-right.. As shown in Fig.13, 100 to 200 measurements are sufficient to ensure a ±5% convergence on the average for the three quantities of interest namely Dh, Dv and Ecc. A similar conclusion holds for the distributions. These statistics are subject to some bias but the latter is difficult to quantity. In terms of dimensions, the measured bubble sizes were always both well above the minimum resolution and well below the extent of the field of view. Thus, there was no artificial filtering of the size information. We also tested whether or not the endoscope and its illumination bar induce significant flow distortions. To check that, measurements were performed with optical probes located in the interval separating the illumination bars and the endoscope tube. The measurements were repeated without the imaging equipment. The differences observed between these two arrangements, both on the bubble size and on the void fraction happen to be negligible (at most about 1.8 %), and that conclusion holds over a large range of superficial velocities (from 3 to 35 cm/s). Therefore, flow disturbances when using the imaging technique and any potential bias associated to it are not significant. For the high void fractions considered, another effect is to be expected. Indeed, the frequent overlapping of bubbles in the images tends to minimize the number of small bubbles detected. The proportion of such missed bubbles, which is expected to be a function of the size, is difficult to quantify. Thus, there is no easy way to correct the size pdfs. Yet, the largest sizes detected using the imaging technique do represent the largest bubbles present in the flow. 0.4

Pdf (-)

0.3 0.2 0.1 0

0

2

3 cm/s

4

6 8 10 12 14 16 18 20 Bubble diameter (mm) 35 cm/s

3 cm/s

35 cm/s

Fig.12 - Left) typical images collected with the endoscope Right) PDF of bubble diameters (horizontal diameter filled symbols, vertical diameter – void symbols) detected by endoscopic imaging at Vsg = 3 cm/s (rhombus) and 35 cm/s (squares) (h/D=3.75, column center).

18

1

8

0.9

6

0.8

4

0.7

2

Eccentricity (-)

Bubble diameter (mm)

10

0

0.6 50 100 150 200 250 Distance between probes (m) Vertical diameter Horizontal diameter Eccentricity

0

Fig.13 - Convergence of mean horizontal and vertical diameters (left hand side scale) and of mean vertical eccentricity (right hand side scale) using endoscopic imaging. Dashed lines represent ±5% deviations from the corresponding average.

3.3.

Mono-fiber conical optical probe

A conical (1C) mono-fiber optical probe has been exploited to measure the vertical dimension of bubbles. The signal processing provides for each gas inclusion the arrival time, the de-wetting duration Tm (which is the rise time between liquid and gas) and the gas residence time TG (Barrau et al., 1999). The interface velocity Vi along the probe axis is deduced from the de-wetting duration using Vi = Ls / Tm, where Ls is the sensitive length of the probe (Cartellier 1990, 1992, Cartellier and Barrau, 1998). In the present work, we used probes from A2 Photonic Sensors company: their sensitive lengths were typically about 22-23µm (for thresholds set at 10% and 90% of the full signal amplitude). The dewetting duration Tm is not measured on distorted signals corresponding to partial probe drying. In such cases, a routine is used to interpolate the velocity from the nearest bubbles along the time line. In fine, each inclusion i is characterized by a velocity Vi and the chord c = Vi TG intersected by the probe during the motion of the gas inclusion. From these raw data, a number of parameters can be inferred (Cartellier 1999). In particular, in unidirectional bubbly flows, the chords distribution can be transformed into a diameter distribution provided that the bubble shape is fixed (Clark and Turton, 1988; Liu and Clark, 1995). Alternately, some moments of the diameter distribution are directly accessible. In particular, for oblate ellipsoids of fixed eccentricity, the arithmetic mean chord ̅ is related with the Sauter mean vertical diameter of the bubbles present in the system (i.e. the Sauter mean is computed using the actual size distribution in the system), namely: ̅

(22)

The mean chord ̅ is also related with the arithmetic mean vertical diameter according to ̅ =(2/3) Dv10 but in that case the average Dv10 must be computed using the detected size distribution (i.e. as detected by the probe) and not the actual size distribution in the system. In the sequel, we will rely on Eq.(22) to compare single probe measurements with other techniques. In practice, the optical probe is connected to a detection analog module, an A/D converter and then to a real time signal processing (Software SO6 from A2 Photonic Sensors company). For the flow conditions considered, the sampling frequency was set to 400 kHz to resolve all physical time scales present in the signal. The discretization was achieved over 12 bits, a resolution well adapted to the amplitude and to the signal to noise ratio. Convergence was ensured with measuring durations comprised between 70 seconds (at Vsg=35cm/s) and 300 seconds (at Vsg=3cm/s), leading to 10000 detected bubbles by run whatever the flow conditions. 19

As evoked in Section 1, the response of optical probes in unsteady, dense bubbly flows is not well understood. As a first step, the raw signals have been examined. For the flow conditions considered, bubble size ranges from 3 to 12 mm and are thus always quite large compared with the probe latency length. Consequently, most signals should correspond to a full de-wetting of the probe and should exhibit a clear plateau at the gas level. This is indeed what is observed: signatures with a clear plateau, referred to as T1 signals, are common while the so-called T2 signatures corresponding to a partial dewetting and thus to signals of limited amplitude are rare (Fig.14-left). The percentage of T1 signal has been measured along a column radius (Fig.14-right): even in the recirculating region of the bubble column (i.e. for dimensionless radial positions above 0.7), the occurrence of T1 signals is higher than 94%. This result holds whatever the superficial velocity from 3 to 35 cm/s.

T1 bubbles (%)

100 98 96

94 92 90 0

0.2

0.4 0.6 Radial position (-)

0.8

1

Fig.14 – Left) Raw signal collected optical probe signal at a void fraction of 46%. Right) Radial distribution of the percentage of T1 signatures collected in the 0.4 m diameter bubble column for a superficial gas velocity of 16 cm/s at h/D of 2.5.

The above features have strong consequences on velocity measurements. Indeed, the bubble velocity is evaluated whenever the ratio of the gas residence time to the de-wetting time exceeds a few units (Barrau et al., 1999) that is for all T1 signatures. That criterion ensures that the impact angle between the probe axis and the normal to the interface is within an acceptable range: that condition was quite efficient for bubbles up to 2-3mm in size for which T1 amounts to 70 to 80% at most (Cartellier 1992, Barrau et al., 1999). In the present flow conditions involving large bubbles, almost all signatures are of the T1 type. Therefore, and owing to the strong velocity fluctuations (up to 30%) and the variable flow orientation with respect to the probe axis (see Section 4), the transformation de-wetting time to velocity can be unduly applied to large impact angles. This has three consequences. First, the percentage of T1 bubbles is no longer a meaningful criterion to ascertain the reliability of velocity measurements. Second, as velocities are detected also for strongly inclined bubble trajectories, the interpretation of the data as the velocity component projected along the probe axis is not ascertained. Third, incorrect velocity information are collected in such situations. On one hand, de-wetting times larger than expected occur for impact angles in the range say 40°- 80° (Cartellier 1992), leading then to underestimated interface velocities. On another hand, de-wetting times shorter than expected are also recorded notably for bubbles approaching almost normal to the probe axis: these events induce overestimated velocities and thus overestimated chords as we will see in Section 4. This drawback due to bubbles moving radially is similar to the one observed by Chaumat et al. (2005, 2007) on bi-probes. In addition, as some bubbles trajectories are normal to the probe axis, the chords detected are no longer uniquely related with the bubble size along the direction of the probe axis but are also related with the transverse dimensions of bubbles. The validity of velocity and chord measurements in such conditions will be discussed in Section 4. 20

In dense bubbly flows, another specific situation occurs. When two bubbles separated by a distance smaller than the probe sensitive length hit the probe, the latter is not completely wetted by the liquid bridge and the signal does not drop down to the liquid level. An example of such a situation is shown Fig.15. For such signatures, the signal processing used here detects a single bubble, with a velocity given by the de-wetting time on the first bubble and with a gas residence time equal to the sum of both contributions. Such events generate very large chords. However, they happen to be quite rare, less than 0.2% of bubble signatures, even at the largest void fraction considered (≈40%). Their impact on the mean chord is negligible (2.5% at most) so that no additional processing criteria were implemented to detect and correct such events.

Fig.15 – Signature recorded from two bubbles very close to each other. Signal acquired at superficial gas velocity of 35 cm/s in the column center.

3.4.

Sensor developed for the correlation technique

Size measurements using the correlation technique were achieved using two optical probes located side-by-side as shown Fig.16. That sensor was manufactured by A2 Photonic Sensors company. Each probe is a conical mono-fiber probe with a latency length comprised between 30 and 40µm. According to the discussion in Section 2, the lateral distance between the probes extremities was set to 0.8mm. The two fibers were held in a support tube (O.D. 0.9mm) as shown in Fig.16. To minimize flow disturbances, the length of the glass fibers located outside the holding stainless steel tube was at least 6 mm. The circuits used for signal detection and digitalization were the same as for single conical probes. A post-processing program was developed to compute the correlation coefficient from the two raw signals according to Eq.(3). In the experiments, the holder tube was vertical so that the probes extremities were in a horizontal plane. The probes were pointing downward. For each measurement, at least 10000 bubbles were detected, which correspond to measurement durations comprised between 70 and 300 seconds depending on the gas superficial velocity. Such conditions ensure both a meaningful scan of the bubbly flow and the correct convergence of the correlation coefficient measurements (Fig.17).

21

Fig.16 – Left) Image of the correlation sensor. Right) Magnified view of the extremity of a conical probe. The optical fiber outer diameter is 125µm.

Correlaiton coefficients

100 98 96 94

92 90 0

50

100

150

Time (s) Fig. 17 – Convergence of the correlation coefficient. The dashed lines correspond to ±2.5 % deviations (data collected on the axis of the column, at Vsg=35 cm/s).

As an accurate measurement of the correlation coefficient requires the detection of all bubbles passing at the probe position, the bubble detection capability was checked considering void fraction measurements. It happens that the void fraction detected by any of the two probes of the correlation sensor is the same 2% absolute difference as the void fraction detected by a single conical probe. This agreement holds whatever the flow conditions (See Section 4.1). Therefore, the disturbances induced by the correlation sensor are of the same order as those due to a single probe. The performances of the single probe in terms of bubble detection are discussed in Section 4 here below.

4. Results and discussion 4.1.

Flow conditions

Before discussing the performances of the measuring techniques, it is worth providing some information regarding the type of bubbly flows considered here. The comparisons will be achieved for superficial velocities ranging from 3 to 35 cm/s, the corresponding global void fractions evolving from 8% up to 35%. The regime becomes heterogeneous at Vsg above 4-5cm/s: almost all the flow conditions exploited will concern thus pertain to this regime. A few data have been collected at Vsg=3cm/s, in a homogeneous regime. In terms of bubble size, endoscopic observations were 22

0

2

4

6 8 10 12 14 16 18 20 Bubble diameter (mm)

3 cm/s

35 cm/s

12 10 8 6 4 2 0

1 0.8 0.6

0.4 0.2

0

Eccentricity (-)

0.3 0.25 0.2 0.15 0.1 0.05 0

Bubble diameter (mm)

Pdf (-)

performed at 1.5 m (h/D =3.75) above the gas distributor. Horizontal diameter distributions for two extreme values of Vsg are provided Fig.18-left. No significant evolutions of the mean horizontal diameter and of the vertical eccentricity were perceived when varying the superficial gas velocity as shown Fig.18-right. In this figure, the error bars represent the standard deviation of the corresponding distributions. The mean horizontal diameter is about 7mm with a standard deviation about 2mm while the mean eccentricity is 0.7 with a standard deviation about 0.12 Along the radial direction, the eccentricity remains the same but the bubble size slightly diminishes when moving away from the axis (Fig.19). In particular, the size distributions (Fig.19-left) show that bubbles larger than 10 mm are detected on the axis but not at x/R=0.8. Similarly, bubbles between 2 and 4 mm are not perceived on the axis but are present near walls. This radial evolution is due to the segregation induced by the mean flow recirculation: in the present conditions, that segregation effect does not drastically alter the mean size while the shape of bubbles remains the same (see also Fig.7).

0 10 20 30 40 Superficial gas velocity (cm/s) Horizontal diameter Eccentricity

0

2

4

6 8 10 12 14 16 18 20 Bubble diameter (mm)

x/R=0

x/R=0.8

12 10 8 6 4 2 0

1

0.8 0.6 0.4

0.2 0

Eccentricity (-)

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Bubble diameter (mm)

Pdf (-)

Fig.18 – Left) PDF of bubble horizontal diameter detected by endoscopic measurements at a Vsg of 3 cm/s and 35 cm/s; Right) Evolution of the mean horizontal diameter and of the vertical eccentricity with Vsg (h/D=3.75, on the column axis).

0 0.2 0.4 0.6 0.8 1 Radial position (x/R) Horizontal diameter Eccentricity

Fig.19 – Left) PDF of bubble horizontal diameter detected by endoscopic measurements at two radial positions x/R=0 and x/R=0.8, Right) Evolution of the mean horizontal diameter and mean eccentricity along the column radius (data for Vsg=16cm/s at h/D=3.75).

Liquid velocities were measured with Pavlov tubes (see Forret et al. 2006 for a description of the technique) with an 8Hz resolution. On the axis, the mean vertical liquid velocity increases with Vsg: it covers the range 0.5 to 0.9m/s. Besides and as shown Fig.20-left, the mean vertical velocity follows rather well the form of the radial profile identified by Schweitzer et al. (2001) which is recalled here: ( ⁄ )

[

( ⁄ )

]

(23)

where is the local mean vertical liquid velocity, and where A=2.976, B=0.943 and C=1.848. In particular, the mean flow reversal occurs at x/R=0.7. The RMS of liquid velocity fluctuations plotted 23

Fig.20-left happen to be quite large: the turbulent intensity amounts to 25 to 30% on the axis and it increases at larger radial distances. This behavior is in agreement with the observations of Forret et al. (2006) and of Menzel et al. (1990). Such strong fluctuations lead to flow reversal in the frame of the laboratory. This is clearly seen on the liquid velocity distributions. On the axis, Fig.21-left indicate that for superficial velocities above 9cm/s, the flow is always directed upward but strong downward motions do happen at lower superficial gas velocities. Notably, at Vsg=3cm/s, the downward liquid velocity can be as high as 0.3m/s, which is enough to entrain bubbles. Similarly, at x/r=0.8 the mean flow is downward directed but the distributions indicate that strong upward and downward motions do exist whatever the superficial liquid velocity. Such flow reversals are also quite significant in the region where the mean flow approaches zero. To quantify their importance, the fraction of time the velocity was directed downward (in the frame of the laboratory) is plotted Fig.22 versus the radial position for three superficial velocities. For the lowest superficial velocity upward and downward motions are present over the entire column radius. At higher Vsg, the flow is upward in a zone close to the axis, up to x/R≈ 0.2. At larger radial distances, up and down flows are always present whatever the superficial velocity, and their magnitude is sufficient to entrain even large bubbles (in our flow conditions, the terminal velocity is about 0.3m/s over the entire bubble population). Note that the 50% probability occurs at x/R=0.7 which is indeed the position of mean flow reversal reported in the literature. Liquid velocity fluctuation (rms)

0.8

0.6 0.4 0.2

0 0

0.5

1

Radial position (x/R)

9 cms-1

16 cms-1

25 cms-1

Fig.20 – Left) Radial distributions of the mean vertical liquid velocity (left) and the RMS of velocity fluctuation in m/s (right) for various Vsg (h/D=2.5). 4

-1 3 cm/s

0 1 Liquid velocity (m/s) 9 cm/s 16 cm/s

3

PDF(-)

PDF(-)

3 2.5 2 1.5 1 0.5 0

2 1

2 25 cm/s

-2 3cm/s

0 -1 0 1 Liquid velocity (m/s) 9 m c/s 16cm/s

2 25 cm/s

Fig.21 – Liquid velocity distributions on the axis (left) and at x/R=0.8 (right) for several gas superficial velocities (h/D=2.5).

24

Downflow time fraction

1 0.8

0.6 0.4 0.2 0 0

0.2

0.4 0.6 Radial position (x/R)

3 cm/s

16 cm/s

0.8

1

25 cm/s

Fig.22 – Fraction of time the liquid flows downward as a function of the radial position and for various gas superficial velocities (h/D=2.5).

These features have strong consequences on the measuring techniques. First, the impact velocity of a bubble on the probe could be much less than the mean velocity value, so that the detection error due to bubble slow down and rolling motion discussed in Section 1 can neatly increase (see Vejražka et al., 2010). But the worst effect is that, for a significant fraction of the measuring duration, bubbles are arriving on the probe from the rear. That result also means that bubbles trajectories with respect to the probe axis can take any inclination from 0° to 360°. As seen above, such situations arise for almost all conditions except in the vicinity of the axis and at large Vsg. Hence, the bubbly flows considered here are really quite challenging for intrusive phase detection probes.

4.2.

Performance of conical mono-fiber optical probes in heterogeneous bubbly flows

We took benefit of the present investigation to analyze the response of conical mono-fiber optical probes in such complex bubbly flows. To check first the detection capability of mono-fiber probes, local void fraction profiles were gathered for different gas superficial velocities (Fig.23). These measurements are compared with the radial distribution proposed by Schweitzer et al. (2001). The latter expresses as: ( ⁄ )

̅{

[( ⁄ )

]

[( ⁄ )

]

[( ⁄ )

]} (24)

where x/R represents the dimensionless radial position in the column and ̅ is the global gas hold up. The empirical proposition of Schweitzer et al. (2001) is reliable: it was notably validated by Forret et al. (2006) in bubble columns of diameters 0.15, 0.4 and 1m and at several superficial gas velocities corresponding to the heterogeneous regime. For the comparison, ̅ was estimated using: ̅

(25)

where represents the static liquid height in the column (i.e. with no gas injection) and the dynamic height of the bed. The estimation of is not very accurate since it relies on the visual measurement of the bed’s free surface position, which can be quite challenging to determine in turbulent flows: the uncertainty is typically ±10%. The comparisons shown Fig.23 indicates a good agreement between the measured local void fractions and the predictions using Eq.(24) and Eq.(25): the deviations are typically within ±3% which is not much owing to the uncertainty on the global gas hold up. It is also noticeable that the agreement remains good even at large radial distances where the turbulent intensity drastically increases and also where the mean flow becomes directed downward. 25

Void fraction (-)

0.5

0.4 0.3 0.2 0.1 0 0

0.2

0.4 0.6 0.8 Radial position (-) Schweitzer et al. (2001) 9 cm/s 16 cm/s

1 35 cm/s

Fig.23 - Void fraction radial profiles at h/D=3.75 for various Vsg compared with the predictions from Eq.(24) combined with global gas fraction measurements using Eq.(25).

Volume gas hold-up (-)

As a second test, the radial void fraction profiles were integrated assuming that the void fraction is linear between the last measured point at x/R=0.8 and the wall position where is set to zero. As shown Fig.24, that integral is in good agreement with the global gas hold up ̅ , with deviations typically within ±15%. Note that the comparison between surface and volume averages is valid for fully developed bubbly flows: let us recall that this condition in fulfilled in bubble columns only over a fraction of their height (Forret et al., 2006). 0.4 0.3 0.2

0.1 0 0

0.1 0.2 0.3 Surface gas hold-up (-)

0.4

Phi 400 Fig.24 - Comparison between the volume gas hold-up and the surface gas hold-up deduced from optical probes measurements. (Dashed lines ±15 %).

Hence, the bubble detection capability of conical mono-fiber optical probes in dense, unsteady bubbly flows is validated for gas fractions of more than 30%. Such a good result is not too surprising as void fraction measurements with single optical probes are known to be weakly sensitive to the probe orientation with respect to the main flow (Barrau et al., 1999; Zun et al., 1995). The situation becomes much more delicate when considering velocity measurements. We have seen in Section 3.2 that the absence of discrimination on bubble trajectory can induce incorrect velocity data. In section 4.1, it was demonstrated that bubbles do approach the probe at any angle. Inconsistencies in velocity measurements are therefore to be expected on velocity statistics. To evaluate their importance, bubble velocity distributions as detected by conical probes are compared with liquid velocity distributions measured with a Pavlov tube in Fig.25. Even if we do not expect identical results for the two phases, the magnitudes should be comparable. This is not so as the tails of bubble velocity distributions exceed the maximum liquid velocity by a large factor (about 2), which is unphysical. In addition, for Vsg ≥ 9cm/s, some bubble velocities happen to be much lower than the smallest liquid velocity. Clearly, the bubble velocity distributions measured with a single probe in such 26

complex bubbly flows are strongly distorted and cannot be considered as reliable. This feature is further confirmed by examining the local volumetric gas flux detected by the mono-fiber probe given by ∑ . The local flux integrated over a column cross-section has been compared with the injected gas flow rate. Here, the probe was always directed downward and the changes in flow direction were not accounted for. The two quantities are in reasonable agreement (within 20%) at low gas superficial velocities (say for Vsg≤9cm/s) for which the recirculation is weak, but the underestimation increases with Vsg and reaches 50% at the highest superficial gas velocity i.e. 35cm/s. 3,0

PdF

Gas, Vsg = 3 cm/s

2,5

Gas, Vsg = 16 cm/s Gas, Vsg = 25 cm/s

2,0

Liquid, Vsg=9 cm/s Liquid, Vsg=16 cm/s

1,5

Liquid, Vsg=25 cm/s 1,0 0,5 0,0 -0,5

0

0,5

1

1,5

2

2,5

3

3,5

4

Velocity (m/s)

Fig.25 - Comparison of gas and liquid velocity distributions for various superficial velocities (h/d=2.5, on the column axis).

Incorrect bubble velocities lead to erroneous chords measurements. Beside, and as already mentioned, the presence of lateral motions implies that the chords detected are related to both the vertical and the horizontal dimensions of the bubbles. To evaluate the resulting distortions on the statistics, we reconstructed the vertical chords that should have been detected by an ideal probe from the size distributions measured with the endoscope technique and assuming that all bubbles are travelling vertically (that transformation is available in Clark and Turton, 1988 and in Cartellier, 1999). These predictions are compared with the chords detected by the optical probe in Fig.26. Vsg=3cm/s

1 P(C) op cal probe P(C) from endoscope

PdF 0,1

Max(Dv)

Max(Dh)

0,01 0,001 0,0001 0

2

4

6

8

10

12

14

Chord (mm)

27

1

Vsg=16 cm/s

Max(Dv)

PdF 0,1

P(C) op cal probe Max(Dh)

P(C) from endoscope

0,01 0,001 0,0001 0

10

20

30

40

50

60

Chord (mm)

PdF 1

Vsg=35 cm/s

Max(Dv)

P(C) op cal probe

0,1

Max(Dh)

P(C) from endoscope

0,01 0,001 0,0001 0

10

20

30

40

50 60 Chord (mm)

Fig.26 - Comparison between the chord distributions reconstructed from endoscopic size measurements and the chord distributions detected by the conical mono-fiber probe for three gas superficial velocities (h/D=3.75, x/R=0). The reconstruction process assumes vertical trajectories.

For all superficial velocities, the probabilities to detect small chords (below 2-3mm) as well as large chords with optical probes exceed the corresponding probabilities estimated by the reconstruction process. The difference observed for small chords may be due to a bias in the imaging technique. On another hand, the largest chords detected by the probe exceed both the maximum vertical diameter and the maximal horizontal diameter present in the flow. This feature is especially noticeable at the largest superficial velocities. Clearly, all chords above the maximum of Dh correspond to erroneous data. Such chords arise mainly from lateral impacts that lead to strongly overestimated bubble velocities. Another consequence of the above observation is that, contrary to more “regular” bubbly flows, the maximum chord detected by optical probes can no longer be considered as an estimate of the maximum bubble size. However, the mean chords of the distributions shown Fig.26 happen to be not so different at a given Vsg: it is thus worth examining the behavior of averaged quantities. For that, the vertical Sauter mean diameter deduced from the arithmetic mean of the chords using Eq.(22) has been compared with the one provided by the endoscope technique. In Fig.27, the comparison is achieved on the column axis for various gas superficial velocities. The measured from images is nearly constant, close to 6 mm, whatever the gas superficial velocity while the detected by the probe 28

increases for Vsg between 3cm/s and 15cm/s before remaining almost constant. Above 15cm/s, both techniques are in quite good agreement with a maximum absolute difference of 0.7mm (13% in relative value). The difference become much stronger at lower gas superficial velocities: it reaches 3mm for Vsg=3cm/s. To explain this feature, one has to go back to the analysis of the flow direction evoked in Section 4.1. According to Fig.21, at Vsg=3cm/s, the liquid flow assumes absolute negative velocities in the frame of the laboratory, that is with respect to the probe. Hence, bubbles are approaching the probe from behind and this happens 40% of the time (Fig.22). Above 15cm/s, such negative velocities no longer exist and bubbles are approaching the probe in the right direction even though their trajectories can be strongly inclined owing to the large velocity fluctuations. Therefore, the good agreement between the two techniques corresponds also to flow conditions ensuring a correct functioning of optical probes. Note that the situation is not so clear at the intermediate condition of 9cm/s since Fig.21 indicates that negative velocities do not exist but, at the same time, the agreement between the two techniques is poor. The same comparison has been achieved at Vsg=25cm/s along a column radius (Fig.28). Maximum deviations arise at intermediate radial positions where they amount for 1.4 mm at most (≈28%). This effect is probably due to strong unsteady structures present in that region: as shown Fig.22, flow reversal indeed occurs in that zone even at large gas superficial velocities. Oddly, the agreement between the two techniques is rather good within the recirculation zone (x/R=0.8) even though the probe was still pointing downward. Globally, and despite erroneous data on velocity and chords distributions, the mean chord detected by conical mono-fiber probes (which is related with the Sauter mean vertical diameter) is in reasonable agreement with direct image analysis provided that the superficial velocity is not too low. This is an encouraging result in the perspective of characterizing bubbly flows in the heterogeneous regime.

Dv32 (mm)

7 5

3 1 0

10

20 vsg (cm/s)

Optical probe

30

40

Endoscope

Fig.27 – Evolution of the Sauter mean vertical diameter with the superficial gas velocity. (Measurements at h/D=3.75, on the column axis). The arrow indicates the range of conditions for which the liquid flow remains upward directed at all times.

29

Dv32 (mm)

7

5 3

1 0.0

0.2

0.4 0.6 0.8 Radial position (x/R) Optical probe Endoscope

1.0

Fig.28 – Evolution of the Sauter mean vertical diameter along the column radius (measurements at h/D=3.75 for a superficial gas velocity of 16cm/s).

4.3.

Performance of the correlation measurements

technique

for

Dh32

Bubble size measurements were performed at an elevation of 3.75 column diameters from injection and for Vsg of 3, 9, 16, 25 and 35 cm/s. The Sauter mean horizontal diameter was deduced from the three measuring techniques as follows: - for the correlation technique, was calculated using Eq.(13) with the prefactor a set to 1.47 (see Table 1). was directly computed from the size distributions measured from endoscopic imaging. - for the conical mono-fiber probe, the Sauter mean vertical diameter was deduced from the mean chord using Eq.(22) and it was then transformed into a Sauter mean horizontal diameter using the mean eccentricity of 0.7 valid in the present flow conditions (see Fig.7 and 8). It should first be underlined that the minimum bubble size present in the system (see Section 4.1) exceeds the d/2 detection limit of the correlation sensor used here. Therefore, the comparison of the correlation technique with the others two is fully relevant. The data are presented Fig.29 as a function of the gas superficial velocity for a sensor located on the column axis. The results from the correlation technique assuming prolate spheroids with an eccentricity equal to 0.7 (i.e. for a prefactor a=-1.7, see Table 1) are also plotted on the same figure: as expected from the weak difference in prefactors, the results are barely distinguishable (maximum difference 0.3mm) from those obtained assuming oblate ellipsoids, and they will not be discussed further. Above Vsg=10cm/s, the agreement between the three techniques is quite good with differences less than 0.6mm (i.e. ≤7% in relative value). The agreement between the correlation technique and direct imaging is excellent, the difference being at most 0.3mm (i.e. ≤ 4%). Discrepancies between these two techniques appear at lower gas superficial velocities: they amount for 1.1mm at Vsg=9cm/s (15%), and for 2.3mm at Vsg=3cm/s (≈33%). Although less than those discussed in the previous section between the conical mono-fiber probe and the imaging technique, these discrepancies increase as Vsg decreases. As already discussed, this behavior is connected with the sign of the approaching velocity of the bubbles with respect to the sensor. In that respect, the correlation technique, that only requires the presence of gas at two positions, is much less sensitive to the flow orientation than the single probe technique for which a velocity estimate is mandatory. From the above discussion, one can therefore conclude that the performances of the correlation technique are quite satisfactory in the inhomogeneous flow regime provided that no backflow occurs.

30

Dh32 (mm)

10 8 6 4 0 Oblate

10 Prolate

20

vsg (cm/s) Endoscope

30

40

Optical probe

Fig.29 – Sauter mean horizontal diameters Dh32 measured by the correlation technique, by endoscopic imaging and with conical optical probes. Evolution with the gas superficial velocity (measurements on the column axis, h/d=2.5).

To pursue the analysis, let us consider now different radial positions. The profiles established for Vsg=16cm/s are provided Fig.30: again, the agreement between the correlation technique and direct imaging is quite good with differences in Dh32 about 0.5mm (≤ 8% in relative value) except at x/R≈0.2 where the difference reaches 1mm (≈13%). At that position, the difference between the correlation and the single probe techniques is also maximum, equal to 1.3mm (≈18%). Let us emphasis that these intermediate radial positions are quite challenging for measurements with probes due to the high level of fluctuations compared with the mean velocity (see Fig.20) and also due to the presence of flow reversal (see Fig.22). Further away from the axis, at x/R=0.8, a reasonable agreement is also obtained even though the correlation sensor was pointing downward, i.e. in a direction opposite to the mean flow direction. In that zone too, velocity fluctuations are strong enough to lead to flow reversals and thus to produce meaningful bubble signatures on the sensor. Overall, the agreement between the three techniques is quite reasonable despite the challenging flow conditions considered.

Fig.30 - Sauter mean horizontal diameters Dh32 measured by the correlation technique, by endoscopic imaging and with a conical mono-fiber optical probe. Radial evolution for Vsg=16cm/s (measurements at h/D=2.5).

As the correlation technique has been successfully applied to bubble size measurements on the axis and for several gas superficial velocities as well as along the column radius for a given Vsg, it was 31

then employed to systematically characterize the bubble size in the column. So doing, extra comparisons were performed between the correlation and the single probe techniques at various heights and gas superficial velocities. The data collected at h/D=0.5, i.e. closer to the injector, are provided Fig.31. The maximum differences in terms of size are 1.2mm at 35cm/s (16%), 1.3mm at 16cm/s (20%) and they reach 2.5mm at 3cm/s (≈50%). The data collected at h/D=3.75 are provided Fig.32: the maximum differences are 1.8mm at 35cm/s (20%), 1.4mm at 16cm/s (20%) and 1.8mm at 3cm/s (≈40%). For all conditions, the maximum differences occur at intermediate radial positions where velocity fluctuations are large compared with the mean and where flow reversals occur. Again, the performances of the correlation technique are quite satisfactory with deviations in heterogeneous conditions of 20% when compared with the single probe technique.

Dh32 (mm)

10 8 6

4 2 0 0

0.5 Radial position (x/R)

16cm/s

3cm/s

1

35cm/s

Cross-correlation Optical probe Fig.31 - Radial distributions of the Sauter mean horizontal diameters Dh32 measured at h/D=0.5 by the correlation technique (filled symbols) and with conical optical probes (shallow symbols) for various gas superficial velocities.

Dh32 (mm)

10 8

6 4 2

0 0

0.5 Radial position (x/R)

16cm/s

3cm/s

1

35cm/s

Cross-correlation Optical probe Fig.32 – Radial distributions of the Sauter mean horizontal diameters Dh32 measured at h/D=3.75 by the correlation technique (filled symbols) and with conical optical probes (shallow symbols) for various gas superficial velocities.

32

5. Conclusion Gas phase characteristics are quite difficult to measure in dense bubbly flows and this issue is especially critical in bubble columns operating in industrial conditions. To progress in that area, a new technique dedicated to bubble size measurements has been proposed which is based on the spatial correlation of phase indicator function delivered by two optical probes located side by side. From an analytical model and from Monte Carlo simulations, the correlation coefficient has been shown to depend on the distance between probes divided by the bubble dimension along the direction defined by the two probe extremities. In the limit of small distances between probes, that relationship becomes linear so that the bubble size can be estimated from the correlation coefficient for a given, known distance between probes. Rules are provided for an optimum choice of that distance as a function of the bubble size distribution and the void fraction. Beside, we have shown that the size measured by the spatial correlation technique corresponds to a Sauter mean diameter. In a second step, a sensor adapted to correlation measurement in dense conditions has been manufactured and its performances have been tested in a bubble column by comparison with direct endoscopic imaging and with data gathered from a conical mono-fiber optical probe. The Sauter mean horizontal diameters detected with the correlation technique happen to be in good agreement with those provided by these alternate measuring techniques. Observed discrepancies typically amount for 15-20%, and they are much less in absence of any flow reversal. These results hold over a large range of gas superficial velocities, corresponding to global void fractions up to 35%. Thus, the proposed technique, which is easy to implement, appears reliable for the investigation of heterogeneous bubbly flows. More precisely, the present method has been tested in poly-dispersed bubbly flows with sizes from 1mm up to 10mm. The size distributions considered evolved from narrow to large (the standard deviations of chords ranged from 0.8 up to 1.86 times the mean) but they always consisted of a single peak. In presence of multi-peaked size distributions, for instance when large Taylor bubbles or slugs are also present in the flow, the detection of a characteristic size may become difficult or even not relevant as the correlation will be smoothed out by the contributions of bubble populations of different sizes. We also took advantage of this study to investigate the response of conical mono-fiber optical probes in these challenging bubbly flows. Bubble detection happens to be efficient and void fractions measurements quite reliable. However, we have shown that lateral motions of bubbles induce erroneous velocity measurements, so that the chord pdfs exhibit strongly overestimated bubble sizes. Therefore, in strongly agitated bubbly flows, the maximum chord is not representative of the size of the largest inclusions. However, the mean chord and thus the Sauter mean vertical diameter happen to be reasonably well captured, provided that flow reversal with respect to the probe is avoided or is weak enough. Although complementary analysis of the sensitivity of the above techniques to the flow orientation would be welcome, the proposed improvements already open the way to reliable investigations of dense, unsteady bubbly flows such as those occurring in bubble columns operated in industrial conditions. These techniques will be exploited in a future paper devoted to the up-scaling of bubble columns.

Notation A, B, C Coefficients, a prefactor, 33

c D d Db Dh Dprobe Dv Ecc f H0 HD jg LA, LB Ls

Chord, m Column diameter, m Distance between optical probes, m Bubble diameter Bubble horizontal diameter (perpendicular to probe axis), m Optical fiber diameter, m Bubble vertical diameter (along probe axis), m Bubble eccentricity (-) Bubble detection frequency, Hz Static height of the liquid, m Dynamic height of the liquid, m local gas flux, m/s Chords, m Sensitive length of optical probe, m Modified Weber number

P() P0 r R Rh Rk Signali t Tg Tm ub texp Vi Vl Vmax Vsg Xk x, y, z

Probability, Probability, distance, m Column radius, m Bubble horizontal radius, m spatial correlation coefficient, Signal of the optical probe n°i time, s Gas residence time, s De-wetting time, s Bubble velocity , m/s Acquisition time, s Interface velocity, m/s Liquid phase velocity, m/s Most probable velocity, m/s Gas superficial velocity, m/s k-phase indicator function, spatial coordinates, m

Greek letters      

Density of interfacial area, m-1 Phase fraction, Liquid density, kg/m3 Tilt angle, ° Angle of rotation, ° Liquid-air surface tension, N/m

34

Indices 32 g k

Sauter mean Gas phase k phase

Acknowledgment This research was supported by « IFP Energies Nouvelles » under Grant N°242 404. The LEGI and LGP2 laboratories are part of the LabEx Tec 21 (Investissements d’Avenir - grant agreement n° ANR-11-LABX-0030).

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