Elsevier

Chemical Engineering Science

Volume 84, 24 December 2012, Pages 735-745
Chemical Engineering Science

Bubble size measurement using Bayesian magnetic resonance

https://doi.org/10.1016/j.ces.2012.08.024Get rights and content

Abstract

This paper describes a new magnetic resonance technique for measuring bubble size distributions in high voidage gas–liquid flows. The technique is based on Bayesian analysis of data acquired directly in the frequency domain, eliminating the need for complex imaging techniques. This means the technique can be applied on high- and low-field magnetic resonance systems, and therefore potentially enables in-plant measurements and online monitoring. The technique is demonstrated and validated experimentally. Bubble size distributions are then measured as a function of gas flow rate (voidage) and height in the column. The technique is capable of high temporal resolution measurements, which is demonstrated by monitoring the change in bubble size in a column as a result of introducing a pulse of surfactant.

Highlights

▸ We present a magnetic resonance technique for measuring bubble size distributions. ▸ Our technique is applicable to both high- and low-field magnetic resonance hardware. ▸ We use the technique to measure optically opaque flows in real-time. ▸ We validate the technique using optical measurements at low voidage. ▸ We compare measurements and correlations over a range of voidages.

Introduction

Gas–liquid contacting is of great importance to the process industries, being widely used in, e.g., oxidation, absorption, and froth flotation (Deckwer, 1992). Most reactors for contacting gas and liquid phases involve the formation of bubbles of gas within a continuous liquid phase. The efficiency of contacting is dominated by the size and size distribution of these gas bubbles as this determines the interfacial area available for mass transfer and the residence time distribution (Deckwer, 1992, Clift et al., 1978). Despite the significance of gas bubble sizes for determining the performance of chemical processes only limited techniques are available to obtain measurements of the bubble size distribution. The majority of studies presented to date have used optical imaging techniques to measure the bubble size distribution, however these are limited to observations in the near wall region of transparent systems and to low gas void fractions. In the present work a new method of measuring the bubble size distribution using magnetic resonance (MR) is developed and tested. This approach is applicable to both opaque systems and to high void fraction measurements.

The earliest reports of measurements of bubble size distribution were nearly 150 years ago (Tate, 1864), and since then there have been extensive attempts to measure bubble size distributions as reviewed in several publications (Saxena et al., 1988, Cartellier and Achard, 1991, Kulkarni and Joshi, 2005, Junker, 2006). Measurement techniques used to study gas–liquid flows include optical, electrical conductivity, electro-optical, and light scattering techniques (Junker, 2006), as well as, more recently, wire mesh sensors (Prasser et al., 2001). Many of these techniques can be difficult to apply to dense swarms of bubbles, due to the increased scattering of light by dense swarms or due to the interference or overlap of nearby bubbles, which has a tendency to bias results to larger bubble sizes (Mudde et al., 1998). Invasive techniques are applicable to dense bubble swarms, but can be unreliable as the presence of the probe may distort the local bubble size and shape (Julia et al., 2005).

Despite their limitations, several interesting results have been obtained using conventional optical or invasive bubble size measurement techniques. Bubble size distributions obtained using optical techniques have been typically well described by a log-normal bubble size distribution (Leibson et al., 1956, Camarasa et al., 1999, Kazakis et al., 2008, Cao et al., 2009). When using porous frits with a large pore size distribution, the bubble size distribution has been found to be bimodal (Kazakis et al., 2008, Otake et al., 1977). A number of empirical correlations are available to describe the bubble size (e.g., van Krevelen and Hoftijzer, 1950, Davidson and Schueler, 1960, Walters and Davidson, 1963, Kumar and Kuloor, 1970, Akita and Yoshida, 1974, Kumar et al., 1976, Gaddis and Vogelpohl, 1986, Jamialahmadi and Mueller-Steinhagen, 1993, Winterton, 1994, Jamialahmadi et al., 2001, Pohorecki et al., 2005, Kazakis et al., 2008). Individual bubbles are typically characterised by an ellipsoidal shape that is rotationally symmetric about the minor axis (Junker, 2006). For air bubbles in water, bubbles larger than 1mm are ellipsoidal, though the deviation from spherical may not be pronounced. Bubbles that are larger than 20mm in diameter tend to form spherical caps and are likely to break up (Clift et al., 1978).

MR presents a method of overcoming many of the limitations of conventional bubble size measurement techniques. MR can study optically opaque systems and does not suffer from light scattering effects. However, MR is sensitive to motion and is an inherently slow imaging technique, which could therefore lead to blurring or distortion of the bubbles (Callaghan, 1991). In fact, using standard spin-echo imaging protocols, the imaging time would be several minutes and it would be impossible to obtain an image of the bubble distribution. A variety of fast imaging techniques are available including EPI (Mansfield, 1977), RARE (Hennig et al., 1986) and FLASH (Haase et al., 1986). However, these imaging techniques are prone to artefacts arising from magnetic susceptibility differences, velocity attenuation, or poor signal-to-noise ratio. In related work the authors have overcome these problems through the use of fast spiral MR imaging (Tayler et al., 2012a, Tayler et al., 2012b). However, this technique requires a high signal-to-noise ratio and fast gradient switching, and therefore is not suitable for application to many low field instruments. The approach outlined in this work aims to incorporate the advantages of MR while overcoming the limitations of conventional MR imaging techniques by utilising a Bayesian approach to measure the bubble size distribution.

Bayesian analysis has been used in a variety of MR applications (Bretthorst et al., 1988, Bretthorst, 2005, Xing et al., 1995, Wise et al., 1996). It has been shown to improve the recovery of an MR spectrum from noisy data (Bretthorst et al., 1988) and to improve the accuracy of flow measurements by enabling a sparse sampling procedure to be used (Xing et al., 1995). In this work we exploit both these advantages of Bayesian analysis to enable measurements of the bubble size distribution in a dynamic system.

The approach used here is derived from texture analysis concepts in image processing (Gonzalez and Woods, 2008) and extends previous approaches for analysing MR data to characterise fibre bundles (Barrall et al., 1992) to provide quantitative measurements in dynamic systems. In our case, the “texture” refers to the characteristic shape of bubbles. The approach used exploits the fact that MR acquires data directly in the Fourier transform domain of an image, commonly referred to as the frequency domain in image processing and k-space in MR literature (Callaghan, 1991). In this paper we will use the term k-space to refer to the Fourier transform of an image, as this is the convention in MR imaging and is the domain in which the measurements are acquired. Fourier transforms are widely used throughout image processing as a method of characterising textures in images (Nixon and Aguado, 2002). For example, Fourier transforms have been used for texture analysis in topics as diverse as the detection of lung disease (Katsuragawa et al., 1988), defects in fabric (Chan and Pang, 2000), and changes in the surface colour of chocolate (Briones and Aguilera, 2005). The property that enables the use of the Fourier transform as a texture characterisation tool is the shift invariance. Shift invariance means that regardless of where an object appears in the image domain, the magnitude of its Fourier transform does not change. This means that the Fourier transform of the image domain is the ideal domain in which to characterise the size, shape and orientation of collections of objects in the image domain. MR inherently measures in k-space; images are obtained from MR after inverse Fourier transformation. Thus, MR measurements in k-space provide a method of directly accessing the desired information on the shape, size and orientation of objects such as bubbles.

This paper uses a Bayesian approach to obtain quantitative measurements of the bubble size in a bubble column. The Bayesian approach to bubble size measurement eliminates the requirement of conventional MR that every point in k-space is sampled with the same bubbles located at the same positions. Instead, it only requires that all samples are obtained for the same bubble size distribution. Therefore, the Bayesian approach described here avoids the need for complex ultra-fast imaging techniques, and so data are acquired using two simple imaging techniques. The paper builds on our recent publication that describes the concept of Bayesian MR (Holland et al., 2011) and extends the technique to enable the characterisation of elliptical bubbles and time-resolved measurements. The paper begins by describing the probability distribution that governs how the measured signal obtained from the MR experiment varies as a function of the bubble size distribution. This is the so-called likelihood function in Bayesian analysis and is used to provide a quantitative estimate of the bubble size in a gas–liquid flow. We present validation experiments illustrating that quantitative bubble size distribution measurements can be obtained. The technique is then demonstrated using a series of experiments: (i) measurements of the change in bubble size as a function of gas flow rate and height above a distributor, (ii) measurements of the bubble size and aspect ratio in pure water and solutions containing a surfactant, and (iii) time-resolved measurements of the change in bubble size and voidage resulting from the addition of a pulse of surfactant to a bubble column. A particular advantage of the proposed approach is that each point can be sampled independently, which can significantly improve the signal-to-noise ratio of the measurement. By maximising the signal-to-noise ratio of each data point, the technique can therefore be used with conventional high-field MR hardware and cheap, portable, low-field MR systems. Furthermore, although the approach outlined is developed for use with bubble size measurement, a similar approach would be applicable whenever it is useful to characterise the size of a regular feature of an image, e.g., in pore sizing in bead packs and rocks.

Section snippets

Theory

The measurement of bubble size is based on the characteristics of the Fourier transform of an image. In the Fourier domain of an image, the magnitude of the signal depends only on the size and shape of an object; i.e., changes in position will result in a change in phase but no change in the magnitude. This information can be used to formulate a method for rapidly and robustly estimating the size of an object, such as a bubble, using MR. The approach used has been described briefly (Holland et

Experimental

Experiments were performed on a Bruker AV400 MR spectrometer operating at a 1H resonance frequency of 400.23 MHz with a 38 mm imaging coil and a 3-axis shielded gradient set capable of producing a maximum gradient in each direction of 0.3 T m−1.

A schematic diagram of the bubble column used for these experiments, and how it fits in the MR system, is shown in Fig. 2. A perspex column was used of inner diameter 31 mm and length 2 m. Compressed air was supplied to the column using either a porous

Results

This section is divided into two parts. Firstly, a brief demonstration of the validity of the technique is presented. Secondly, measurements of the bubble size distribution in a bubble column are presented as a function of superficial gas velocity, height above the gas distributor and surfactant concentration. The second section concludes with a series of measurements of a dynamic system showing the evolution in bubble size over time resulting from a changing surfactant concentration.

Discussion

In this section, the results presented in Section 4.2 are discussed in the context of empirical correlations for the bubble size that have been derived using conventional sizing techniques. Table 1 provides a range of bubble size correlations taken from the literature. These correlations are derived from different sources and conditions. Correlations (a)–(g) are applicable to bubbles formed immediately above an orifice, whilst correlations (h)–(l) are derived far from the orifice and are

Conclusions

This paper presents a new technique for measuring the bubble size distribution in gas–liquid flows at high voidage. The approach is based on analysing magnetic resonance data directly in k-space and therefore overcomes many of the limitations associated with the long acquisition times of conventional magnetic resonance imaging techniques. The new MR approach was validated by comparison with optical measurements at low voidage, which showed excellent agreement. The technique was used to measure

Acknowledgements

The authors would like to acknowledge the support of the EPSRC under grants EP/F047991/1 and EP/G011397/1 and Microsoft Research Connections.

References (53)

  • R. Kumar et al.

    The formation of bubbles and drops

    Adv. Chem. Eng.

    (1970)
  • G. Lynch et al.

    Direct measurement of the void fraction of a two-phase fluid by nuclear magnetic resonance

    Int. J. Heat Mass Transfer

    (1977)
  • A.A. Mouza et al.

    Effect of liquid properties on the performance of bubble column reactors with fine pore spargers

    Chem. Eng. Sci.

    (2005)
  • R.F. Mudde et al.

    Application of LDA to bubbly flows

    Nucl. Eng. Des.

    (1998)
  • T. Otake et al.

    Coalescence and breakup of bubbles in liquids

    Chem. Eng. Sci.

    (1977)
  • R. Pohorecki et al.

    Diameter of bubbles in bubble column reactors operating with organic liquids

    Chem. Eng. Res. Des.

    (2005)
  • H.M. Prasser et al.

    Bubble size measurement using wire-mesh sensors

    Flow Meas. Instrum.

    (2001)
  • J.G. Ross et al.

    Extending the use of Earth's field NMR using Bayesian methodology: application to particle sizing

    J. Magn. Reson.

    (2012)
  • N.E. Tayali et al.

    Particle sizing techniques in multiphase flows: a review

    Flow Meas. Instrum.

    (1990)
  • A.B. Tayler et al.

    Applications of ultra-fast MRI to high voidage bubbly flow: measurement of bubble size distributions, interfacial area and hydrodynamics

    Chem. Eng. Sci.

    (2012)
  • A.B. Tayler et al.

    Time resolved velocity measurements of unsteady systems using spiral imaging

    J. Magn. Reson.

    (2011)
  • R.G. Wise et al.

    Measurement of pulsatile flow using MRI and a Bayesian technique of probability analysis

    Magn. Reson. Imaging

    (1996)
  • D. Xing et al.

    Bayesian analysis for quantitative NMR flow and diffusion imaging

    J. Magn. Reson.

    (1995)
  • K. Akita et al.

    Bubble size, interfacial area, and liquid-phase mass transfer coefficient in bubble columns

    Ind. Eng. Chem. Process Des. Dev.

    (1974)
  • G.A. Barrall et al.

    NMR diffraction and spatial statistics of stationary systems

    Science (New York, NY)

    (1992)
  • G.L. Bretthorst

    How accurately can parameters from exponential models be estimated? A Bayesian view

    Concepts Magn. Reson.

    (2005)
  • Cited by (21)

    • Applications of tomography in bubble column and fixed bed reactors

      2022, Industrial Tomography: Systems and Applications, Second Edition
    • Bubble growth obtained from pressure fluctuation in vibration separation fluidized bed using wavelet analysis

      2020, Advanced Powder Technology
      Citation Excerpt :

      Penn et al. [27] studied the effect of bed internals on the fluidization dynamics in three–dimensional (3D) cylindrical fluidized beds with real–time magnetic resonance imaging. Holland et al. [28] determined the bubble size evolution in a fluidized bed by magnetic resonance imaging technology. Though these experimental measurement techniques can all be employed to obtain the evolution characteristic and feature parameters of bubbles in three–dimensional and two–dimensional fluidized beds.

    • Flotation technique: Its mechanisms and design parameters

      2018, Chemical Engineering and Processing - Process Intensification
      Citation Excerpt :

      Dissolved air flotation (DAF) and electroflotation (EF) are some other important process, which is able to generate even small-sized bubble in the range of 10–100 μm diameter. Some of the methods like Mie scattering technique [134], Bayesian magnetic resonance technique [135], Coulter count method or pore electrical resistance method [136], dynamic gas disengagement technique [113] are utilized to estimate the bubble size. Image analysis technique is most widely used to estimate the bubble size but there are some drawbacks like it requires a transparent wall for image capture, low bubble concentration, complicated experimental setup, and time-consuming process.

    View all citing articles on Scopus
    View full text