Population balance modeling of aggregation and breakage in turbulent Taylor–Couette flow

https://doi.org/10.1016/j.jcis.2006.12.016Get rights and content

Abstract

An experimental and computational study of aggregation and breakage processes for fully destabilized polystyrene latex particles under turbulent-flow conditions in a Taylor–Couette apparatus is presented. To monitor the aggregation and breakage processes, an in situ optical imaging technique was used. Consequently, a computational study using a population balance model was carried out to test the various parameters in the aggregation and breakage models. Very good agreement was found between the time evolution of the cluster size distribution (CSD) calculated with the model and that obtained from experiment. In order to correctly model the left-hand side of the CSD (small clusters), it was necessary to use a highly unsymmetric fragment-distribution function for breakage. As another test of the model, measurements with different solid volume fractions were performed. Within the range of the solid volume fractions considered here, the steady-state CSD was not significantly affected. In order to correctly capture the right-hand side of the CSD (large aggregates) at the higher solid volume fraction, a modified aggregation rate prefactor was used in the population balance model.

Graphical abstract

An experimental and computational study of aggregation and breakage processes for fully destabilized polystyrene latex particles under turbulent-flow conditions in a Taylor–Couette apparatus is presented.

  1. Download : Download full-size image

Introduction

Processing of colloidal dispersions, such as polymer latexes, inorganic nanoparticle suspensions or biological systems is an important step in the production of particulate materials in numerous industrial applications. Properties of the final particulate product can be controlled through the aggregation processing step, where primary particles, with typical sizes between nanometers and micrometers, are aggregated into clusters with sizes from micrometers to millimeters. The resulting size, structure and shape of the aggregates significantly impact the final quality of the product. To relate required product characteristics to process conditions, one needs to take into account many process parameters, such as the flow field, the solid volume fraction, amount of destabilizer, which need to be monitored and properly manipulated in order to control the size and structure of the aggregates as they evolve in time during the aggregation process.

To understand, predict, and control the aggregate formation processes, it is necessary to develop quantitative models that are able to describe aggregation and breakage under a wide range of processing conditions. The common modeling approach for such systems is based on the solution of a population balance equation (PBE) [1]. For particulate systems, it is convenient to take mass as the internal coordinate, because mass is the primary conserved quantity in aggregation and breakage events. The necessary inputs into the PBE are the values of aggregation and breakage rates (i.e., the kernels), which include information about operating conditions and the relevant physicochemical characteristics of the primary particles.

Unfortunately, the physical understanding of breakage and structure evolution is not yet satisfactory, and therefore precise experimental measurements of relevant quantities and their proper interpretation is necessary in order to understand and quantify key mechanisms during aggregation, breakage, and restructuring. Nowadays there are several measuring techniques available to measure the evolution of the relevant quantities, in terms of certain particle sizes and numbers, in time. These can be classified into two groups: single particle measurements and ensemble measurements. In the first group are techniques such as the Coulter counter (measures particle volume distribution), focused beam reflectance (measures chord length distribution) and microscopy combined with image analysis (determines 2D projections of aggregates). As was presented by several authors [2], [3], [4], [5], [6], [7], microscopy is a very valuable technique, but it has significant limitations due to the minimum size of primary particles visible with optical microscopy (around 1 micron), and highly demanding collection and sampling procedures. In the second group are techniques such as light scattering and sound spectroscopy that, in principle, allow in situ measurement of population-averaged quantities in a non-invasive way.

Nevertheless, not every technique is appropriate for the measurement of the fragile aggregates obtained in colloidal systems, which can be destroyed relatively easily. Moreover, often the interpretation and conversion of experimental data to the corresponding cluster mass distribution (CMD) or cluster size distribution is based on rather crude assumptions about the shape or structure of clusters. In addition, the resulting inverse problems are often ill-posed. Even more significantly, experimental data from kinetic studies of turbulent aggregation are often reported in terms of only one moment of the CMD (usually some sort of average size) [8], [9], [10], [11], [12], [13], [14]. There are only a few published data for the time evolution of two independent moments of the CMD [15], [16], [17]. The consequence is that the population balance models have been typically validated against just one moment of the CMD. As was clearly shown by [18], because of the complexity of aggregation and breakage kernels, using only one moment of the CMD is insufficient for discriminating between the various models. For these reasons, it is now clearly evident that information about the time evolution of more than one moment is necessary in order to distinguish between the various models for aggregation, breakage and restructuring.

Another very important issue that is directly related to modeling is the effect of shear-rate heterogeneity on the time evolution of the CMD. Up to now most of the presented comparisons between simulations and experiments were done with strong assumptions about the shear-rate distribution. Only a few previous studies addressed issues of flow heterogeneity by using detailed modeling approaches in either laminar [19] or turbulent conditions [17], [20], [21], [22], [23]. They clearly showed significant differences compared to simplified models assuming a homogeneous distribution of the shear rate. This is in agreement with the experimental results [24], [25], [26] obtained using various types of impellers and vessel volumes operated at the same volume-averaged shear rate. Based on the operating conditions, particularly the solid volume fraction, there are several approaches to include the spatial distribution of the shear rate in population balance models. In the case where the system is rather dilute and the timescales of aggregation and breakage are much longer than the ones for mixing, it is sufficient to use a properly averaged breakage kernel inside the population balance model instead of a full computational fluid dynamics (CFD) simulation without losing accuracy [23]. On the other hand, with increasing solid volume fraction the timescales for aggregation and breakage start to be comparable to the ones for mixing and a population balance model coupled with CFD must be used.

Reflecting the above-mentioned limitations, the objective of the present work is to show the effect of operating conditions (rotational speed and initial solid volume fraction) on the time evolution of the whole CSD, and related moments, for a dispersion of polystyrene microspheres with diameter 9.6 μm under fully destabilized conditions in a turbulent Taylor–Couette apparatus using an in situ particle imaging technique to monitor time evolution of the CSD. The obtained results are combined with a modeling study of different sub-models for the breakage and the fragment-distribution function. Results of the simulation show that based on the shape of the CSD at steady state, it is possible to specify the approximate shape of the fragment-distribution function. In addition, the region of validity of particle imaging technique is discussed.

Section snippets

Experimental methods

In the presented work, aggregation and breakage experiments were performed under turbulent-flow conditions in a Taylor–Couette (TC) flow [7], [17]. The Taylor–Couette apparatus was built from two concentric cylinders with the inner one rotating while the outer one was at rest. The radius of the inner cylinder was 3.49 cm and was constructed from polished stainless steel. The transparent outer cylinder was constructed from precision glass Pyrex tubing and had an inside radius of 4.86 cm. The

Population balance model

One of the main goals of this work is to validate, develop and test models for the cluster size distribution in the presence of aggregation and breakage. In this section, we give a brief overview of the population balance model used to describe the experimental data. As discussed elsewhere [7], [17], the aggregation and breakage kinetics in the experimental system considered here are slow relative to the timescales of the turbulent flow. It is therefore possible to treat the Taylor–Couette

Numerical methods

There are several methods available in the literature to solve Eq. (1) [1], e.g., the method of successive approximations, Laplace transforms, the method of moments, the method of weighted residuals, the classes method, and Monte-Carlo simulations. In our previous work [7], [17], the quadrature method of moments (QMOM) combined with CFD was successfully used to reproduce the time evolution of two moments of the equivalent diameter distribution. Nevertheless, for a detailed investigation of the

Results and discussion

The starting point of our quantitative analysis is to reproduce the time evolution of the two moments of the CSD (d10 and d43) obtained from experiments at two different average shear rates with a constant solid volume fraction. As mentioned in the description of the PBE model above in order to solve the set of ODE (Eq. (1)), except other model parameters such as shear rate, parameters appearing in the collision efficiency and breakage kernel, it is necessary to define the initial distribution

Conclusions

In this work the aggregation and breakage processes of polystyrene microspheres under turbulent flow conditions in a Taylor–Couette (TC) apparatus were monitored via a particle imaging technique. It was shown that the time evolution of two independent moments of the cluster size distribution (CSD) for two different rotation speeds of the TC apparatus can be well reproduced with standard population balance models combining aggregation and breakage. With such a model, we were able to reproduce

Acknowledgements

This work was financially supported by Swiss National Science Foundation (Grant No. 200020-101724) and the US National Science Foundation (CTS-0403864).

References (65)

  • S. Schuetz et al.

    Chem. Eng. Sci.

    (2002)
  • X.Y. Li et al.

    J. Colloid Interface Sci.

    (2003)
  • L. Wang et al.

    J. Colloid Interface Sci.

    (2005)
  • V. Oles

    J. Colloid Interface Sci.

    (1992)
  • K.A. Kusters et al.

    Chem. Eng. Sci.

    (1997)
  • T. Serra et al.

    J. Colloid Interface Sci.

    (1998)
  • P.T. Spicer et al.

    Powder Technol.

    (1998)
  • C. Selomulya et al.

    J. Colloid Interface Sci.

    (2001)
  • L. Wang et al.

    J. Colloid Interface Sci.

    (2005)
  • M. Soos et al.

    Chem. Eng. Sci.

    (2006)
  • T. Kramer et al.

    J. Colloid Interface Sci.

    (2000)
  • E.D. Hollander et al.

    Chem. Eng. Sci.

    (2001)
  • M. Vanni et al.
  • S. Blaser

    J. Colloid Interface Sci.

    (2000)
  • L.B. Brakalov

    Chem. Eng. Sci.

    (1987)
  • L.L.M. Krutzer et al.

    J. Colloid Interface Sci.

    (1995)
  • M.H. Waldner et al.

    Powder Technol.

    (2005)
  • C. Selomulya et al.

    Chem. Eng. Sci.

    (2003)
  • J.D. Pandya et al.

    Chem. Eng. Sci.

    (1983)
  • M. Lattuada et al.

    J. Colloid Interface Sci.

    (2003)
  • M. Lattuada et al.

    J. Colloid Interface Sci.

    (2003)
  • C.M. Sorensen et al.

    J. Colloid Interface Sci.

    (1997)
  • P.T. Spicer et al.

    J. Colloid Interface Sci.

    (1996)
  • C. Lee et al.

    Adv. Colloid Interface Sci.

    (2004)
  • S. Kumar et al.

    AIChE J.

    (1996)
  • P.T. Spicer et al.

    Water Res.

    (1996)
  • T. Serra et al.

    J. Colloid Interface Sci.

    (1997)
  • J.R. Kadambi et al.

    Powder Technol.

    (1998)
  • D. Ramkrishna

    Population Balances

    (2000)
  • M. Kobayashi et al.

    Langmuir

    (1999)
  • J.J. Zhang et al.

    AIChE J.

    (2003)
  • V.A. Tolpekin et al.

    Langmuir

    (2004)
  • Cited by (48)

    • Agglomeration of Li(NixMnyCoz)O2 particles in Couette–Taylor flow reactor

      2019, Journal of Industrial and Engineering Chemistry
    • Influence of COM-peptides/proteins on the properties of flocs formed at different shear rates

      2019, Journal of Environmental Sciences (China)
      Citation Excerpt :

      An important feature of self-similar fractals is that their mass and density decrease as they grow larger (Meakin, 1990). A number of studies have investigated the effect of hydrodynamics on floc properties, many of them employing artificial model particles, such as silica, kaolinite, latex and polystyrene (Jarvis et al., 2005b; Oles, 1992; Selomulya et al., 2001; Soos et al., 2007; Spicer et al., 1998). However, the effect of hydrodynamics on floc properties in natural water has largely been overlooked (Bubakova et al., 2013; Bubakova and Pivokonsky, 2012).

    View all citing articles on Scopus
    View full text