Implementation of the quadrature method of moments in CFD codes for aggregation–breakage problems
Introduction
Aggregation and breakage play an important role in a number of important chemical processes such as precipitation, crystallization, separation processes, and reaction in multiphase systems. Modeling and simulation of these processes is complicated due to the difficulties inherent in describing the evolution of a distribution of particle sizes and because of the incomplete understanding of the mechanisms by which aggregation and breakage occur, including the role of hydrodynamics. This latter problem is often neglected, despite considerable evidence that aggregation is strongly influenced by mixing. For example, Brown and Glatz (1987) investigated the effect of the operating conditions on particle size established during breakage of protein particles prepared under isoelectric precipitation in an agitated vessel. They found that the aggregation rate increases with both particle concentration and shear rate. In another study, the influence of the type of flow on the aggregation rate was investigated using a Taylor–Couette reactor, a pipe-flow reactor, and a flat-bottomed tank reactor (Krutzer, van Diemen, & Stein, 1995). These experiments showed that at equal energy dissipation rates, the aggregation rate is higher for isotropic turbulent flows than for non-isotropic flows.
Raphael and Rohani (1996) investigated the effect of aggregation on the particle size distribution (PSD) during sunflower protein precipitation and found that the maximum size of the aggregates is determined by the hydrodynamics of the reactor and the mean residence time of the particles in the reactor. Serra, Colomer, and Casamitjana (1997) investigated aggregation and breakup of particles in a Taylor–Couette reactor. In their experimental work they analyzed the effect of particle concentration, shear rate, and particle initial diameter. Their results showed the existence of three regions determined by particle concentration and type of flow established (laminar or turbulent). Moreover, they found that the final aggregate diameter in the turbulent regime is independent of monomer size and is instead controlled by the Kolmogorov micro-scale, whereas in laminar flow the final mean particle size decreases as the diameter of the primary particles is increased.
One means for characterizing the morphology of clusters of particles formed by aggregation–breakage processes is the fractal dimension, which provides an indication of the compactness of aggregates. The relationship between fractal dimension and physical properties has been studied from both the experimental (Serra & Casamitjana, 1998a), and theoretical viewpoints (Filippov, Zurita, & Rosner, 2000; Jlang & Logan, 1991). Hansen and co-workers used a Taylor–Couette reactor to study orthokinetic aggregation for monodisperse and bidisperse colloidal systems (Hansen, Malmsten, Bergenstahl, & Bergstrom, 1999). Particle aggregation was investigated by direct observation using a CCD camera, and the observed aggregates were characterized by a high fractal dimension, suggesting that clusters are rearranged and densified by the shear.
Aggregation and breakage are often the last steps of a complex sequence of phenomena, such as nucleation, fast reactions, combustion, and molecular growth (Marchisio, Barresi, & Garbero, 2002; Rosner & Pyykonen, 2002; Baldyga & Orciuch, 2001). Such systems inevitably lead to non-negligible spatial heterogeneities, and therefore a method of modeling these processes that accounts for hydrodynamics is crucial for predicting reactor performance.
One approach to account for non-ideal mixing is to use CFD methods. In such an approach the reactor is represented by a computational grid and the continuity and Navier–Stokes equations are solved over the computational domain. When dealing with turbulent flows, the set of equations is unclosed and turbulence models are used to solve the closure problem. In addition to these equations, the population balance equation for the solid phase has to be solved.
The population balance is a continuity statement written in terms of the PSD and has consistently received attention since Smoluchowski (1917) introduced the mathematical formalism nearly a century ago. A comprehensive overview of the mathematical issues involved, the numerical methods available, and possible developments for the future have been given by Ramkrishna 1985, Ramkrishna 2000 and Ramkrishna and Mahoney (2002).
A general form of the mean-field population balance for a spatially extended system can be written as follows (repeated indices implies summation):where is the property vector that specifies the state of the particle, is the number density function, 〈ui〉 is the Reynolds-average velocity in the ith direction, xi is the spatial coordinate in the ith direction, Γ is the molecular diffusivity and Γt is the turbulent diffusivity. For turbulent flows Γt≫Γ and thus it is commonly assumed that Γt+Γ≈Γt. The “flux in -space” is denoted byand represents the net rate of introduction of new particles into the system (Hulburt & Katz, 1964).
The main problem in solving the above equation is the presence of the extra variables ξi, which define the particle size, shape, etc. In most CFD codes it is possible to introduce user-defined scalars by using user-defined subroutines, but these scalars must only be functions of time and space. Hence, in order to reduce the dimension of the problem, several methods have been developed.
The discretized population balance (DPB) approach is based on the discretization of the internal coordinates (i.e., the components of the property vector). A detailed comparison of the performance of the most popular DPB methods has been carried out by Vanni (2000a). The principle advantage of the DPB method is that the PSD is calculated directly. However, in order to maintain reasonable accuracy, a large number of scalars (i.e., particle classes) are required. As a consequence, the DPB approach is computationally intractable for spatially heterogeneous systems and therefore not suitable for CFD applications.
An alternative to PBE approaches is to implement a stochastic analog via a Monte Carlo algorithm (Smith & Matsoukas, 1998; Lee & Matsoukas, 2000; Rosner & Yu, 2001). These methods have the advantage of satisfying mass conservation as well as correctly accounting for fluctuations that arise as the system mass accumulates in a small number of large aggregates. However, incorporation of these methods into CFD codes is also not computationally tractable because of the large number of scalars required.
In contrast to the DPB or stochastic approaches, the moment method (MM) is suitable for use with CFD codes because the internal coordinates are integrated out such that solution only requires a small number of scalars (i.e., 4–6 moments of the PSD) at each grid point. Of course the vast reduction in the number of scalars, which makes implementation in CFD codes feasible, comes at the cost of a less-detailed description of the PSD. However, provided that the PSD function is sufficiently simple (e.g., monomodal or bimodal), a low-order moment description may be sufficient. The method was first proposed many years ago by Hulburt and Katz (1964), but it has not found wide applicability due to the difficulty of expressing the transport equations for the moments of the PSD in terms of the moments themselves. More recently, several approaches for contending with this “closure” problem have been developed, and a discussion of these can be found in Diemer and Olson (2002).
When the population balance is written in terms of one internal coordinate (e.g., particle length or particle volume) the closure problem has been successfully solved with the use of a quadrature approximation (McGraw, 1997), where weights and abscissas of the quadrature approximation can be found by using the product-difference (PD) algorithm described by Gordon (1968). The so-called quadrature method of moments (QMOM) has been validated for several problems (e.g., molecular growth, aggregation, breakage) and by using different internal coordinates (Marchisio, Pikturna, Fox, Vigil, & Barresi, 2003a; Barret & Webb, 1998). Moreover, the method has been extended to the study of aerosol dynamics by using population balances with two internal coordinates: particle volume and surface area (Wright, McGraw, & Rosner, 2002).
The main purpose of this work is to implement the QMOM approach to solving the population balance equation in a commercial CFD code and to verify the feasibility of these calculations for practical applications. The specific problem considered is aggregation and breakage in turbulent Taylor–Couette flow.
The CFD-based predictions for the particle size distributions are generated by using several different combinations of conventional expressions for the aggregation and breakup kernels and are compared with experimental data from Serra et al. (1997) and Serra and Casamitjana 1998a, Serra and Casamitjana 1998b. The paper is organized as follows: governing equations are discussed, aggregation and breakage models are presented, and the results of the case study considered, particularly with respect to the number of scalars involved in the calculations, the quality of the predictions, and the CPU time in comparison with alternative approaches.
Section snippets
Governing equations
In this section the QMOM is presented and explained. Particular attention will be devoted to its implementation in FLUENT.
Computational details and case study
The experimental data used in this study are taken from Serra et al. (1997) and Serra and Casamitjana 1998a, Serra and Casamitjana 1998b. The reactor used in their work is a Taylor–Couette device, which consists of a fluid confined to the annular region between two concentric cylinders (outer cylinder stationary, inner cylinder rotating). The wetted diameters of the inner and outer cylinders are and , respectively, thereby giving an annular gap . The reactor
Results and discussion
In Fig. 3, the mean velocity vectors in a meridian section of the Taylor–Couette reactor are shown for 75, 125, 165, and . For the four rotational speeds reported above we find that Re/Rec falls in the range between 50 and 160, which corresponds to the Turbulent Vortex Flow regime (see Marchisio, 2002).
In all cases, the contour plots show the expected counter-rotating vortical structure. Although no experimental velocity data are available for these specific cases, the results are
Conclusions
Simultaneous aggregation and breakage of particles in a Taylor–Couette reactor was simulated by implementing the QMOM in a commercial CFD code (FLUENT). Experimental data taken from the literature was compared with the CFD predictions for a range of operating conditions and for several combinations of aggregation and breakage kernels.
The implementation of the QMOM in FLUENT was found to be very convenient. In fact, by tracking only six scalars and using the quadrature approximation the moments
Notation
breakage kernel A Hamaker constant daughter distribution function kth moment of the daughter distribution function for birth term due to aggregation and breakage kth moment transform of the birth term due to aggregation kth moment transform of the birth term due to breakage d dimensionless mean particle size diameter of the inner cylinder of the Taylor–Couette reactor diameter of the outer cylinder of the Taylor–Couette reactor fractal dimension D annular
Acknowledgements
This work has been financially supported the US Department of Energy (Project award number DE-FC07-01ID14087).
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