Advances in Chemical Engineering, Volume 31: Computational Fluid Dynamics

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contribution begin. A. T. ANDREWS IV, Depa...

Author: Guy B. Marin (Editor)

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CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contribution begin.

A. T. ANDREWS IV, Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA (65) A. G. DIXON, Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USA (307) L.-S. FAN, Department of Chemical and Biomolecular Engineering, The Ohio State University, 140 West 19th Avenue, Columbus, OH 43210, USA (1) R. O. Fox, Herbert L. Stiles Professor of Chemical Engineering, Iowa State University, 3162 Sweeney Hall, Ames, IA 50011-2230, USA (231) Currently on sabbatical at: Swiss Federal Institute of Technology Zurich, ETHZ Institut fu¨r Chemieund Bioingenieurwissenschaften ETH-Ho¨nggerberg/HCI H 109 (Gruppe Morbidelli), CH-8093 Zurich, Switzerland Y. GE, Department of Chemical and Biomolecular Engineering, The Ohio State University, 140 West 19th Avenue, Columbus, OH 43210, USA (1) J. A. M. KUIPERS, University of Twente, Faculty of Science & Technology, PO Box 217, NL - 7500 AE Enschede, The Netherlands (65) M. NIJEMEISLAND, Johnson Matthey Catalysts, Billingham, UK (307) E. H. STITT, Johnson Matthey Catalysts, Billingham, UK (307) S. SUNDARESAN, Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA (65) H. E. A. VAN DEN AKKER, Delft University of Technology, Molenwindsingel 50, NL 4105 HK Culemborg, The Netherlands (151) M. A. VAN DER HOEF, Department of Science and Technology, University of Twente, PO 217, NL - 7500 AE Enschede, The Netherlands (65) M. VAN SINT ANNALAND, Department of Science and Technology, University of Twente, PO 217, NL - 7500 AE Enschede, The Netherlands (65) M. YE, Department of Science and Technology, University of Twente, PO 217, NL - 7500 AE Enschede, The Netherlands (65)

ix

PREFACE This issue attempts to give a feeling of the state-of-the-art of the application of computational ﬂuid dynamics (CFD) in chemical engineering. It is, however, not limited to a snap-shot but is aimed at providing a perspective: how did we arrive at the present status and where do we go from here? To do so, contributions from ﬁve complementary contributions are brought together. From the definition of CFD as the ensemble ‘‘of all computational approaches that solve for the spatial distribution of the velocity, concentration, and temperature ﬁelds’’ recalled by Fox, it is clear that a selection had to be made as to the topics covered. In the wake of volume 30 on ‘‘Multiscale Analysis’’ the present volume is organized from ‘‘small’’ to ‘‘large’’: from ‘‘bubbles and droplets’’ in the ﬁrst contribution, to a ‘‘ﬁxed catalyst bed’’ in the last one. The application of direct numerical simulations (DNS) clearly is still limited to the small scale. Today subgrid-scale (SGS) models are required to cover the full spectrum. The reader will be confronted with some redundancy but this allows each contribution to stand on its own. Also, a good balance is maintained between the style of a tutorial and that of a research paper. Those who will read the complete volume will realize that opinions can vary from looking at CFD as an alternative for experimentation to emphasizing the need of experimental validation. Some contributions are entirely limited to velocity and temperature ﬁelds. Others, on the contrary, emphasize the difﬁculties associated with the combination of transport and reaction. The latter can introduce stiffness even for laminar ﬂow. Averaging (e.g. Reynolds-averaged Navier–Stokes, RANS) or ﬁltering (e.g. large eddy simulations, LES), performed to model velocity ﬁelds, does not alleviate this difﬁculty. Clearly, this is still quite a challenge. The contribution from the Ohio State University by Ge and Fan is dealing with the simulation of gas–liquid bubble columns and gas–liquid–solid ﬂuidized beds. A scientist of a major engineering company told me a few years ago that when he wanted to know how serious an academic group was about CFD, he would ask whether they could simulate bubble columns. He would only engage into further conversation if the answer was negative. The group from Columbus is wise enough to focus on a single air bubble rising in water, and bubble formation from a single nozzle. In a second part the hydrodynamics and heat transfer phenomena of a liquid droplet in motion and during the impact process with a hot ﬂat surface, as well as with a particle are studied. The applied numerical techniques, such as the level set and immersed boundary method, are outlined and important contributions are highlighted. Next, detailed implementations for particular problems are presented. Finally, numerous simulation results are shown and compared with experimental data.

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PREFACE

The second contribution addresses the different levels of modeling that are required in order to cover the full spectrum of length scales that are important for industrial applications. It is a joint paper from Twente and Princeton University and claims to put ‘‘Emphasis on technical details.’’ The latter is a too modest description of what is really offered to the reader. The recent developments in two leading research groups on the modeling of gas-ﬂuidized beds are presented. The holy grail for those interested in the design of industrial units being the closure of the model equations in general and SGS modeling in particular. The latest developments of both the ‘‘ﬁltering’’ approach pursued at Princeton University by Sundaresan and coworkers and the ‘‘discrete bubble model’’ developed in Twente by the team of Kuipers are presented. The authors realize fully that there is still a long way to go, as evidenced by their last sentence: ‘‘Finally, the adapted model should be augmented with a thermal energy balance, and associated closures for the thermo-physical properties, to study heat transport in large scale ﬂuidized beds, such as FCC-regenerators and PE and PP gas-phase polymerization reactors.’’ This is even more so because inclusion of reaction kinetics remains beyond the scope of the contribution! Chemical reactions come into the picture in the context of stirred turbulent vessels in Chapter 3. Van den Akker from Delft strongly emphasizes the potential of LES and DNS for reproducing not only the hydrodynamics of turbulent stirred vessels but also for providing a basis for simulating a wide variety of physical and chemical processes in this equipment. The author advocates the use of the lattice–Boltzmann (LB) technique to this purpose. Van den Akker certainly belongs to those who believe that one can and should be much more positive about the merits of CFD so far and about the term at which CFD will replace and improve existing mixing correlations. To quote him: ‘‘It may be easier to ‘measure’ the local and transient details of the turbulent ﬂows in stirred vessels and the spatial distributions in e.g. mixing rates and bubble, drop and crystal sizes computationally than by means of experimental techniques!’’ When it comes to the design of chemical reactors the authors admit that CFD is certainly not a panacea. ‘‘Scale-up of many chemical reactors, in particular the multi-phase types, is still surrounded by a fame of mystery indeed.’’ The importance of chemical-reaction kinetics and the interaction of the latter with transport phenomena is the central theme of the contribution of Fox from Iowa State University. The chapter combines the clarity of a tutorial with the presentation of very recent results. Starting from simple chemistry and singlephase ﬂow the reader is lead towards complex chemistry and two-phase ﬂow. The issue of SGS modeling discussed already in Chapter 2 is now discussed with respect to the concentration ﬁelds. A detailed presentation of the joint Probability Density Function (PDF) method is given. The latter allows to account for the interaction between chemistry and physics. Results on impinging jet reactors are shown. When dealing with particulate systems a particle size distribution (PSD) and corresponding population balance equations are intro-

PREFACE

xiii

duced. The author emphasizes that a balance between the degree of detail or complexity of the chemistry and that of the physics should be maintained. The last contribution comes from Dixon (Worcester Polytechnic Institute), and Nijemeisland and Stitt (Johnson Matthey). The subject is another classic of reactor engineering: the catalytic ﬁxed-bed reactor. Heat transfer issues on both reactor scale and catalyst pellet scale are addressed. Steam reforming is used as a typical example of a strongly endothermic reaction requiring high-heat ﬂuxes through the reactor walls. The presence of the tube wall causes changes in bed structure, ﬂow patterns, transport rates and the amount of catalyst per unit volume, and is usually the location of the limiting heat-transfer resistance. Special attention is given to the modeling of the ‘‘structure’’ of a packed bed. The importance of wall functions, to be applied not only at the reactor wall but also at the external pellet surface, is stressed. The authors show ample results of their own work without neglecting the contributions of others. At the end of this chapter the reader will be convinced of the importance of the local nonuniformities in the temperature ﬁeld not only within a catalyst pellet but also from one pellet to the other. Let me conclude by thanking the authors for their willingness to contribute, despite health problems for some of them, and for their ﬂexibility with respect to timing. Guy B. Marin Ghent, Belgium April 2006

3-D DIRECT NUMERICAL SIMULATION OF GAS–LIQUID AND GAS–LIQUID–SOLID FLOW SYSTEMS USING THE LEVEL-SET AND IMMERSED-BOUNDARY METHODS Yang Ge and Liang-Shih Fan Department of Chemical and Biomolecular Engineering, The Ohio State University, Columbus, OH 43210, USA I. Introduction II. Front-Capturing and Front-Tracking Methods A. Level-Set Method B. Immersed Boundary Method III. System 1: Flow Dynamics of Gas–Liquid–Solid Fluidized Beds A. Numerical Procedure for Solving the Gas–Liquid Interface B. Governing Equations for the Gas–Liquid–Solid Flow C. Modeling the Motion and Collision Dynamics of Solid Particles in Gas–Liquid–Solid Fluidization D. Results and Discussions IV. System 2: Deformation Dynamics of Liquid Droplet in Collision with a Particle with Film-Boiling Evaporation A. Simulation of Saturated Droplet Impact on Flat Surface in the Leidenfrost Regime B. Simulation of Subcooled Droplet Impact on Flat Surface in Leidenfrost Regime C. Simulation of Droplet–Particle Collision in the Leidenfrost Regime V. Concluding Remarks References

2 4 6 9 11 12 13 14 16 27 29 38 49 58 61

Abstract The recent advances in level-set and Immersed Boundary methods (IBM) as applied to the simulation of complex multiphase ﬂow systems are described. Two systems are considered. For system 1, a computational scheme is conceived to describe the three-dimensional (3-D) bubble Corresponding author. Tel.: +1-614-688-3262(o). E-mail: [email protected]

1 Advances in Chemical Engineering, vol. 31 ISSN 0065-2377 DOI 10.1016/S0065-2377(06)31001-0

Copyright r 2006 by Elsevier Inc. All rights reserved

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YANG GE AND LIANG-SHIH FAN

dynamics in gas–liquid bubble columns and gas–liquid–solid ﬂuidized beds. This scheme is utilized to simulate the motion of the gas, liquid, and solid phases, respectively, based on the level-set interface tracking method, the locally averaged time-dependent Navier–Stokes equations coupled with the Smagorinsky subgrid scale stress model, and the Lagrangian particle motion equations. For system 2, the hydrodynamics and heat-transfer phenomena of a liquid droplet in motion and during the impact process with a hot ﬂat surface, as well as with a particle, are illustrated. The 3-D level-set method is used to portray the droplet surface deformation whilst in motion and during the impact process. The IBM is employed so that the particle–ﬂuid boundary conditions are satisﬁed. The governing equations for the droplet and the surrounding gas phase are solved utilizing the ﬁnite volume method with the Arbitrary Lagrangian Eulerian (ALE) technique. To account for the multiscale effect due to lubrication-resistance induced by the vapor layer between the droplet and solid surface or solid particle formed by the ﬁlm-boiling evaporation, a vapor-ﬂow model is developed to calculate the pressure and velocity distributions along the vapor layer. The temperature ﬁelds in all phases and the local evaporation rate on the droplet surface are illustrated using a full-ﬁeld heat-transfer model.

I. Introduction Gas–liquid–solid (three-phase) ﬂow systems involve a variety of operating modes of gas, liquid, and solid phases, including those with solid particles and/or liquid droplets in suspended states. Commercial or large-scale operations using three-phase ﬂow systems are prevalent in physical, chemical, petrochemical, electrochemical, and biological processes (Fan, 1989). In the gas–liquid–solid ﬂuidization systems with liquid as the continuous phase, the systems are characterized by the presence of gas bubbles, which induce signiﬁcant liquid mixing and mass transfer. The ﬂow structure in the systems is complex due to intricate coalescence and breakup phenomena of bubbles. The fundamental dynamics of solids suspensions in the systems is closely associated with the particle–particle collision and particle–bubble interactive behavior. For three-phase ﬂows that occur in the feed nozzle area of a ﬂuid catalytic cracking (FCC) riser in gas oil cracking, on the other hand, the gas phase is continuous where oil is injected from the nozzle with the mist droplets formed from the spray in contact with high-temperature catalyst particles (Fan et al., 2001). The droplets may splash, rebound, or remain on the catalyst particle surface after the impact, and the oil is evaporated and cracked into lighter hydrocarbons. Such contact phenomena are also prevalent in the condensed mode operation of the Unipol process for

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

3

polypropylene or polyethylene production, where droplet–particle collisions in the feed nozzle are also accompanied by intense liquid evaporation. In this study, both systems involving three-phase ﬂuidization and evaporative droplet and particle collisions are simulated using CFD based on the 3-D level-set and immersed boundary method (IBM). CFD is a viable means for describing the ﬂuid dynamic and transport behavior of gas–liquid–solid ﬂow systems. There are three basic approaches commonly employed in the CFD for study of multiphase ﬂows (Feng and Michaelides, 2005): the Eulerian–Eulerian (E-E) method, the Eulerian–Lagrangian (E-L) method, and direct numerical simulation (DNS) method. In the E-E method (Anderson and Jackson, 1967; Joseph and Lundgren, 1990; Sokolichin and Eigenberger, 1994, 1999; Zhang and Prosperetti, 1994, 2003; Mudde and Simonin, 1999), both the continuous phase and the dispersed phase, such as particles, bubbles, and droplets, are treated as interpenetrating continuous media, occupying the same space as does the continuous phase with different velocities and volume fractions for each phase. In this method, the closure relationships such as the stress and viscosity of the particle phase need to be formulated. In the E-L method, or discrete particle method (e.g., Tsuji et al., 1993; Lapin and Lu¨bbert, 1994; Hoomans et al., 1996; Delnoij et al., 1997), the continuous ﬂuid phase is formulated in the Eulerian mode, while the position and the velocity of the dispersed phase, particles, or bubbles, is traced in the Lagrangian mode by solving Lagrangian motion equations. The grid size used in the computation for the continuous-phase equations is typically much larger than the object size of the dispersed phase, and the object in the dispersed phase is treated as a point source in the computational cell. With this method, the coupling of the continuous phase and the dispersion phase can be made using the Particle-Source-In-Cell method (Crowe et al., 1977). The closure relationship for the interaction forces between phases requires to be provided in the E-L method. In the DNS (Unverdi and Tryggvason, 1992a,b; Feng et al., 1994a,b; Sethian and Smereka, 2003), the grid size is commonly much smaller than the object size of the dispersed phase, and the moving interface can be represented by implicit or explicit schemes in the computational domain. The velocity ﬁelds of the ﬂuid phase are obtained by solving the Navier–Stokes equation considering the interfacial forces, such as surface tension force or solid–ﬂuid interaction force. The motion of the object of the dispersed phase is represented in terms of a time-dependent initial-value problem. With the rapid advances in the speed and memory capacity of the computer, the DNS approach has became important in characterizing details of the complex multiphase ﬂow ﬁeld. This paper is intended to describe recent progress on the development of the level-set method and IBM in the context of the advanced front-capturing and front-tracking methods. The paper is also intended to discuss the application of them for the 3-D DNS of two complex three-phase ﬂow systems as described earlier.

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YANG GE AND LIANG-SHIH FAN

II. Front-Capturing and Front-Tracking Methods In the DNS of multiphase ﬂow problems, there are various methods available for predicting interface position and movement, such as the moving-grid method, the grid-free method (Scardovelli and Zeleski, 1999) and the ﬁxed-grid front-tracking/front-capturing method. In the moving-grid method, which is also known as the discontinuous-interface method, the interface is a boundary between two subdomains of the grid (Dandy and Leal, 1989). The grid may be structured or unstructured and even near-orthogonal, moving with the interface (Hirt et al., 1974). It treats the system as two distinct ﬂows separated by a surface. When the interface moves or undergoes deformation, new, geometrically adapted grids need to be generated or remeshed (McHyman, 1984). The remeshing can be a very complicated, time-consuming process, especially when it involves a signiﬁcant topology change, and/or a 3-D ﬂow. Methods in which grids are not required include the marker particle method (Harlow and Welch, 1965) and the smoothed particle hydrodynamics method (Monaghan, 1994). The ﬁxed-grid method, which is also known as the continuous-interface method, employs structured or unstructured grids with the interface cutting across the ﬁxed grids. It treats the system as a single ﬂow with the density and viscosity varying smoothly across a ﬁnite-thickness of the interface. The numerical techniques used to solve the moving interface problem with ﬁxed, regular grids can be categorized by two basic approaches: the front-tracking method (e.g., Harlow and Welch, 1965; Peskin, 1977; Unverdi and Tryggvason 1992a, b; Fukai et al., 1995) and the front-capturing method (e.g., Osher and Sethian, 1988; Sussman et al., 1994; Kothe and Rider, 1995; Bussmann et al., 1999). For a 3-D multiphase ﬂow problem, the ﬁxed-grid method is the most frequently used due to its efﬁciency and relative ease in programming. The front-tracking method explicitly tracks the location of the interface by the advection of the Lagrangian markers on a ﬁxed, regular grid. The marker-andcell (MAC) method developed by Harlow and Welch (1965) was the ﬁrst fronttracking technique applied in DNS, e.g., it was used by Harlow and Shannon (1967) to simulate the droplet impact on a ﬂat surface without considering the viscosity and the surface-tension forces in the momentum-conservation equation. Fujimoto and Hatta (1996) simulated the impingement process of a water droplet on a high-temperature surface by using a single-phase 2-D MAC type solution method. The no-slip and free-slip boundary conditions are iteratively adopted on the liquid–solid interface for the spreading and recoiling process, respectively. Fukai et al. (1995) developed the adaptive-grid, ﬁniteelement method to track the droplet free surface in collision with a surface while considering the wettability on the contact line. The front-tracking method developed by Unverdi and Tryggvason (1992a, b) and Tryggvason et al. (2001) leads to many applications in the simulation of droplet or bubble ﬂow. In this method, the location of the interface is expressed by discrete surface-marker

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

5

particles. High-order interpolation polynomials are employed to ensure a high degree of accuracy in the representation of the interface. An unstructured surface grid connecting the surface-marker particles is introduced within a volumetric grid to track the bubble front within the computational domain. Thus, discretization of the ﬁeld equations is carried out on two sets of embedded meshes: (a) the Eulerian ﬂuid grid, which is 3-D, cubical, staggered structured, and nonadaptive; and (b) the Largrangian front grid, which is 2-D, triangular, unstructured, and adaptive (Unverdi and Tryggvason 1992a, b). The inﬁnitely thin boundary can be approximated by a smooth distribution function of a ﬁnite thickness of about three to four grid spacing. The variable density Navier–Stokes equations can then be solved by conventional Eulerian techniques (Unverdi and Tryggvason 1992a, b). This method can be numerically stiff as the density ratio of the two ﬂuids increases, and may pose difﬁculties when the appearance, the connection, the detachment, and the disappearance of the gas–liquid interface are encountered. Such interface behavior occurs in the coalescence, breakup, or formation of bubbles and droplets in an unsteady ﬂow. The front-tracking method is therefore computationally intensive. Agresar et al. (1998) extended the front-tracking method with adaptive reﬁned grids near the interface to simulate the deformable circulation cell. Sato and Richardson (1994) developed a ﬁnite-element method to simulate the moving free surface of a polymeric liquid. The IBM proposed by Peskin (1977) in studying the blood ﬂow through heart valves and the cardiac mechanics also belongs to the class of front-tracking techniques. In the IBM method, the simulation of the ﬂuid ﬂow with complex geometry was carried out using a Cartesian grid, and a novel procedure was formulated to impose the boundary condition at the interface. Some variants and modiﬁcations of this method were proposed in simulating various multiphase ﬂow problems (Mittal and Iaccarino, 2005). An introduction to the IBM method is given in Section II.B. The front-capturing method, on the other hand, is the Eulerian treatment of the interface, in which the moving interface is implicitly represented by a scalarindicator function deﬁned on a ﬁxed, regular mesh point. The movement of the interface is captured by solving the advection equation of the scalar-indicator function. At every time step, the interface is generated by piecewise segments (2-D) or patches (3-D) reconstructed by this scalar function. In this method, the interfacial force, such as the surface-tension force, is incorporated into the ﬂowmomentum equation as a source term using the continuum surface force (CSF) method (Brackbill et al., 1992). This technique includes the volume of ﬂuid (VOF) method (Hirt and Nichols, 1981; Kothe and Rider, 1995), the marker density function (MDF) (Kanai and Mtyata, 1998), and the level-set method (Osher and Sethian, 1988; Sussman et al., 1994). In the VOF method, an indicator function is deﬁned as: 0 for a cell with pure gas, 1 for a cell with pure liquid, and 0 to 1 for a cell with a mixture of gas and liquid. An interface exists in those cells that give a VOF value of neither 0 nor 1. Since the indicator function is not explicitly associated with a particular front

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YANG GE AND LIANG-SHIH FAN

grid, an algorithm is needed to reconstruct the interface. This is not an easy task, especially for a complex dynamic interface requiring 3-D calculation. Pasandideh-Ford et al. (1998) used a modiﬁed SOLA-VOF method to solve the momentum and heat-transfer equations for droplet deposition on a steel surface. Bussmann et al. (1999, 2000) developed a 3-D model to simulate the droplet collision onto an incline surface and its splash on the surface, utilizing a volume-tracking methodology. Mehdi-Nejad et al. (2003) also used the VOF method to simulate the bubble-entrapment behavior in a droplet when it impacts a solid surface. Karl et al. (1996) simulated small droplet (100–200 mm) impact onto the wall in the Leidenfrost regime using a VOF method. A free-slip boundary condition and a 1801 contact angle were applied on the solid surface. Harvie and Fletcher (2001a,b) developed an axisymmetric, 2-D VOF algorithm to simulate the volatile liquid droplet impacting on a hot solid surface. The vapor ﬂow between the droplet and solid surface was solved by a 1-D, creeping ﬂow model, which neglects the inertial force of the ﬂow. This model, despite being accurate at a lower We, failed to reproduce the droplet dynamics at a higher Weber number. Other front-capturing methods include the constrained interpolation proﬁle (CIP) method (Yabe, 1997), and the phase-ﬁeld method (Jamet et al., 2001). In the level-set method, the moving interface is implicitly represented by a smooth level-set function (Sethian and Smereka, 2003). The level-set method has proved capable of handling problems in which the interface moving speed is sensitive to the front curvature and normal direction. A signiﬁcant advantage of the level-set method is that it is effective in 3-D simulation of the conditions with large topological changes, such as bubble breaking and merging, droplet–surface collisions with evaporation. In this study, the level-set technique (Sussman et al., 1994) is employed to describe the motion of 3-D gas–liquid interfaces. In the following section a description of this technique is given.

A. LEVEL-SET METHOD The level-set method, which was ﬁrst derived by Osher and Sethian (1988), is a versatile method for capturing the motion of a free surface in 2-D or 3-D on a ﬁxed Eulerian grid. While similar to the VOF method, the level-set method also uses an indicator function to track the gas–liquid interface on the Eulerian grid. Instead of using the marker particles or points to describe the interface, a smooth level-set function is deﬁned in the ﬂow ﬁeld (Sussman et al., 1994). Consider a nonbody conformal Cartesian grid which is used to simulate the ﬂow with a deformable interface G, as shown in Fig. 1. The whole computational domain is separated by the interface into two regions: O and O+. The value of the level-set function is negative in the O region and positive in the O+ region, while the interface G is simply described as the zero level set of

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

7

+

Γ Ω-

Ω+

FIG. 1. The level sets of distance function for a smooth interface over a Cartesian grid.

the level-set function f, i.e., G ¼ xjfðx; tÞ ¼ 0

(1)

where x represents the position vector and t the time. Taking fo0 as being inside the interface G (in O) and f40 as being outside the interface G (in O+), the level-set function has the form: 8 > < o0; x 2 O (2) fðx; tÞ ¼ 0; x 2 G > : 40; x 2 O þ The evolution of f in a ﬂow ﬁeld is given by the so-called weak-form equation: @f þ V rf ¼ 0 @t where V is the velocity of ﬂuid, and is given 8 V ; > < V ¼ V ¼ V þ; > : V ; þ

(3)

by x 2 O x2G

(4)

x 2 Oþ

For gas–liquid bubble ﬂow, V and V+ are the gas and liquid velocities, respectively, and the zero-level set of f marks the bubble interface, which moves with time. For gas-droplets ﬂows, on the other hand, V and V+ represent the

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YANG GE AND LIANG-SHIH FAN

velocity of the liquid and gas phases, respectively, and the zero-level set of f deﬁnes the droplet surface (Ge and Fan, 2005). To compute the motion of two immiscible and incompressible ﬂuids such as a gas–liquid bubble column and gas-droplets ﬂow, the ﬂuid-velocity distributions outside and inside the interface can be obtained by solving the incompressible Navier–Stokes equation using level-set methods as given by Sussman et al. (1994): @r þ r ðrVÞ ¼ 0 @t

(5)

@rV þ r ðrVVÞ ¼ rp þ r s þ rg þ F s @t

(6)

where Fs is the surface tension force which is calculated by (Brackbill et al., 1992): F s ¼ skðfÞdðfÞrf

(7)

k(f) is the curvature which can be estimated as r (rf/|rf|). A smooth d function is deﬁned as (Sussman et al., 1998; Sussman and Fatemi, 1999): ( 1 fob dH b ðfÞ 2 ð1 þ cosðpf=bÞÞ=b; ¼ db ðfÞ (8) df 0; otherwise where Hb(f) follows the Heaviside formulation (Sussman et al., 1998; Sussman and Fatemi, 1999) given by

H b ðfÞ ¼

8 > <

1 0

f4b fo b

p

otherwise

> : 1 ð1 þ f þ 1 sinðpf=bÞÞ 2

b

(9)

The surface-tension force Fs in Eq. (7) is smoothed and distributed into the thickness of the interface. In order to circumvent numerical instability, the ﬂuid properties such as density and viscosity in the interface region are determined with a continuous transition: rðfÞ ¼ r þ ðr rþ ÞH b ðfÞ

(10)

mðfÞ ¼ m þ ðm mþ ÞH b ðfÞ

(11)

Since the values for r(f), m(f), and the surface-tension force could be distorted if the variation of rf along the interface is very large, the thickness of the interface needs to be maintained uniformly, i.e. rf ¼ 1 (Sussman et al., 1998). In the algorithm developed, the general level set function f(x,t) is replaced by a

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

9

distance function d(x,t), whose value represents the signed normal distance from x to the interface. d(x,t) would satisfy jrd j ¼ 1 and d ¼ 0 for xAG (Sussman et al., 1998). Even if the initial value of the level-set function f(x,0) is set to be the distance function, the level set function f may not remain as a distance function at t40 when the advection equation, Eq. (3), is solved for f. Thus, a redistance scheme is needed to enforce the condition of rf ¼ 1. An iterative procedure was designed (Sussman et al., 1998) to reinitialize the level-set function at each time step so that the level-set function remains as a distance function while maintaining the zero level set of the level-set function. This is achieved by solving for the steady-state solution of the equation (Sussman et al., 1994, 1998; Sussman and Fatemi, 1999): @d ¼ sinðfÞð1 rfÞ @t

(12)

dðx; 0Þ ¼ fðxÞ

(13)

jrd j ¼ 1 þ OðD2 Þ

(14)

until

where the sin function is deﬁned as 8 > < 1; 0; sinðfÞ ¼ > : 1;

fo0 f¼0 f40

(15)

In Eq. (12), t is an artiﬁcial time that has the unit of distance. The solutions for Eq. (12) are signed distances and only those within a thickness of 3–5 grid sizes from the interface are of interest (Sussman et al., 1994, 1998; Sussman and Fatemi, 1999). Equation (12) needs to be integrated for 3–5 time steps using a time step Dt ¼ 0.5D.

B. IMMERSED BOUNDARY METHOD The IBM was originally proposed by Peskin (1977) to model the blood ﬂow through heart valves. Since then, this method has been extensively modiﬁed and extended to simulate various ﬂuid ﬂows in a complex geometrical conﬁguration using a ﬁxed Cartesian mesh (Unverdi and Tryggvason, 1992a,b; Udaykumar, et al., 1997; Ye et al., 1999; Fadlun et al., 2000; Lai and Peskin, 2000; Kim et al., 2001). In the IBM, the presence of the solid object in a ﬂuid ﬁeld is represented by a virtual-body force ﬁeld, which is applied on the computational grid in the vicinity of the solid–ﬂow interface through a Dirac delta function (Lai and

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YANG GE AND LIANG-SHIH FAN

Peskin, 2000). Various schemes have been proposed to calculate the virtual force density in the literature. Goldstein et al. (1993) developed a virtual boundary formulation to simulate the startup ﬂow over a cylinder. In their formation, the virtual force ﬁeld is calculated in a feedback manner in order to satisfy the boundary condition at the solid surface. Mohd-Yusof (1997) developed an alternative direct forcing scheme to evaluate the virtual force based on the N-S equation at discrete times. Fadlum et al. (2000) extended the direct forcing scheme of Mohd-Yusof (1997) to a 3-D ﬁnite-difference method. Instead of evaluating and applying the virtual force, the velocity at the ﬁrst grid point outside the solid boundary is estimated through a linear interpolation of the moving velocity of the boundary and the velocity at the second external grid point. Conceptually, this velocity interpolation scheme is equivalent to applying the momentum force inside the ﬂow ﬁeld (Kim et al., 2001). This scheme is more efﬁcient in 3-D because it has no adjustable constant and has no extra restriction on the scale of the time step, which is required in the feedback-forcing scheme. Kim et al. (2001) simulated the ﬂow over complex geometry in a ﬁnitevolume approach with staggered meshes. The momentum force and mass source were applied on the immersed boundary to satisfy the no-slip boundary condition and the ﬂow continuity. The basic idea of the IBM is that the presence of the solid boundary (ﬁxed or ~p applied on moving) in a ﬂuid can be represented by a virtual body force ﬁeld F the computational grid at the vicinity of solid–ﬂow interface. Thus, the Navier– Stokes equation for this ﬂow system in the Eulerian frame can be given by @rV þ r ðrVVÞ ¼ rp þ r s þ rg þ F~p @t

(16)

~p depends not only on the unsteady It is noted that the virtual body force F ﬂuid velocity, but also on the velocity and location of the particle surface, which is also a function of time. There are several ways to specify this boundary force, such as the feedback forcing scheme (Goldstein et al., 1993) and direct forcing scheme (Fadlun et al., 2000). In 3-D simulation, the direct forcing scheme can give higher stability and efﬁciency of calculation. In this scheme, the discretized momentum equation for the computational volume on the boundary is given as V tþ1 ¼ V t þ DtðRHS t þ F tp Þ

(17)

where RHS refers to all the terms in the right-hand side of Eq. (16) except the virtual body force F~p . The virtual body force Ftp is used to maintain the ﬂuid velocity to be equal to the particle velocity at the particle surface (i.e., no-slip boundary condition), which is V tþ1 ¼ V p ðtÞ

(18)

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

11

where Vp is the particle velocity. Thus, the discrete virtual force can be deﬁned as F tp ¼ ðV p V t Þ=Dt RHS t

(19)

Since the computational grids are generally not coincident with the location of the particle surface, a velocity interpolation procedure needs to be carried out in order to calculate the boundary force and apply this force to the control volumes close to the immersed particle surface (Fadlun et al., 2000). Other than the virtual momentum force F~p , a virtual mass source/sink should also be applied to the particle surface to satisfy the continuity for the control volume containing the particle surface or the particle (Kim et al., 2001). The mass source can be calculated by qt ¼

1 X ~t ai V i ~ ni Dsi DV i

(20)

where DV is the volume of the computation cell (control volume) and Dsi the surface area of surface i of this cell. For a 3-D case, i ¼ 1, 2, y, 6. ~ ni is ~t the ﬂuid velocity at each face of the the normal vector of each face of the cell. V i cell. ai the ﬂag to indicate whether the virtual body force is applied to face i of the cell or not. ai ¼ 1 when the force is applied, otherwise it is zero. Therefore, the continuity equation of the incompressible ﬂuid can be written as (Kim et al., 2001): ~¼q rV

(21)

III. System 1: Flow Dynamics of Gas–Liquid–Solid Fluidized Beds The ﬂows in a gas–liquid–solid ﬂuidized bed or a gas–liquid bubble column are represented by two regimes, the homogeneous and the heterogeneous. In the homogenous regime, the coalescence of bubbles does not occur and there is little variation of bubble sizes. However, this is not the case in the heterogeneous regime. The ﬂow structure in the heterogeneous regime is complex due to substantial coalescence and breakup of bubbles. Both the E-E and the E-L methods have proven to be more effective in modeling the homogenous regime than the heterogeneous regime of gas–liquid ﬂow. In the simulation of the heterogeneous regime of gas–liquid ﬂows using either the E-E or the E-L method, the challenge lies in the establishment of an accurate closure relationship for the interphase momentum exchange. The interphase momentum exchange is induced through the drag force that liquid exerts on the bubble surface, the virtual mass force due to the bubble and liquid inertial motion, and the lift force caused by the shear ﬂows around the bubbles. In gas–liquid bubble columns and gas–liquid–solid

12

YANG GE AND LIANG-SHIH FAN

ﬂuidized systems, the interstitial forces under the bubble coalescence and breakup conditions are not well established. A computational model based on the level-set methods given below provides some information on the much needed closure relationship of the interphase momentum exchange noted above.

A. NUMERICAL PROCEDURE

FOR

SOLVING

THE

GAS– LIQUID INTERFACE

The level-set technique described in Section II.A is employed to capture the motion of 3-D gas–liquid interfaces. The numerical procedures for solving the gas–liquid interface include ﬁnding the solution for the time-dependent Eqs. (3), (5), and (6). Given fnand Vndeﬁned at cell centers at one time instant tn, fn+1, and Vn+1 can be solved over a time increment at a new time instant tn+1 ¼ tn+Dt following the procedures given below: Solve Eqs. (5) and (6) to obtain the velocity distribution in the ﬂow ﬁeld Vn+1 using the Arbitrary-Lagrangian-Eulerian (ALE) scheme (Kashiwa et al., 1994). Solve Eq. (3) to obtain fn+1 using the second-order TVD-Runge-Kutta method presented as follows: ¯ nþ1 ¼ fn þ Dtftn f fnþ1 ¼ fn þ

Dt ¯ ðf þ ftn Þ 2 tnþ1

(22) (23)

where ftn ¼ VnDfn and the time steps are the same as that used in calculating Vn+1, which is determined by restrictions due to the Courant–Friedrichs–Levy (CFL) condition, gravity, viscosity, and surface tension. Solve Eq. (12) to perform the redistancing. Although, in principle, Eq. (12) would not alter the location of the zero-level set of f, in practice, with numerical computation it may not be true. A redistance operation is needed to maintain the volume conservation. Therefore, Eq. (12) is modiﬁed to (Sussman et al., 1998): @d ¼ sinðfÞð1 rfÞ þ lij f ðfÞ Lðf; dÞ þ lij f ðfÞ @t

(24)

where R O lij ¼ Rij

H0 ðfÞLðf; dÞ

Oij

H0 ðfÞf ðfÞ

(25)

and f ðfÞ H 0 ðfÞrf

(26)

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

B. GOVERNING EQUATIONS

FOR THE

13

GAS– LIQUID– SOLID FLOW

The gas–liquid–solid ﬂow is characterized by a wide range of physical length scales, including small to large eddies in the bubble wake, and size in the millimeter range for solid particles and in the millimeter/centimeter range for gas bubbles. The accurate description of the gas bubble surface and bubbling ﬂow requires the use of ﬁne grids, while the tracking of the motion of solid particles needs the grid size to be much larger than the particle sizes. For simulation of a gas–liquid–solid ﬂuidized bed, the locally averaged Navier–Stokes equations (Anderson and Jackson, 1967) are used to describe the liquid phase ﬂow outside the gas bubble, and the gas phase ﬂow inside the gas bubble. Due to the large grid size used, the liquid phase turbulence needs to be considered. In this study, a modiﬁed coefﬁcient that illustrates the effect of the bubble-induced turbulence for a subgrid scale (SGS) stress model is employed. The level-set method and the numerical procedures described in Sections II.A and III. are used to simulate the motion and the topological variation of the gas bubble. The locally averaged governing equations of Eqs. (5) and (6) for liquid ﬂow outside the bubble and gas ﬂow inside the bubble are given as: @r þ r ðrVÞ ¼ 0 @t @rV þ r ðrVVÞ ¼ rp þ r s r ssg þ rg þ F D þ F s @t

(27) (28)

e represents the void fraction of liquid or gas and satisﬁes: þ p ¼ 1

(29)

where ep is the void fraction of solid particles. tsg the SGS stress term. It is modeled by the Smagorinsky (1963) model written as @V i @V j sg þ sij ¼ nT (30) @xj @xi where nT is deﬁned as nT ¼ ðC s lÞ2 jS j

(31)

nT ¼ C s f ðyÞl 2 jSj

(32)

for bulk ﬂow, and

for walls with a wall function f(y). Cs is the Smagorinsky coefﬁcient, l ¼ D, and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jSj 2S ij S ij (33)

14

YANG GE AND LIANG-SHIH FAN

The volumetric ﬂuid–particle interaction force FD in Eq. (28) is calculated from the forces acting on the individual particles in a cell: P FD ¼

f kd , DOij

(34)

where fd is the ﬂuid–particle interaction force for a single particle and DO the cell volume.

C. MODELING THE MOTION AND COLLISION DYNAMICS GAS– LIQUID– SOLID FLUIDIZATION

OF

SOLID PARTICLES

IN

The motion of a particle in the ﬂow ﬁeld can be described in the Lagrangian coordinate with the origin placed at the center of the moving particle. There are two modes of particle motion, translation and rotation. Interparticle collisions result in both the translational and the rotational movement, while the ﬂuid hydrodynamic forces cause particle translation. Assuming that the force acting on a particle can be determined exclusively from its interaction with the surrounding liquid and gas, the motion of a single particle without collision with another particle can be described by Newton’s second law as dxp ¼ Vp dt mp

dV p p ¼ mp g þ d 3p ðrp þ r s r ssg Þ þ f d þ f am þ f s 6 dt

(35)

(36)

where xp and Vp are the particle position and particle velocity, respectively, and dp the diameter of the particle. The ﬁve terms on the right-hand side of Eq. (36) represent, respectively, the gravity force, the ﬂuid stress gradient force, the total drag force, the added mass force and the bubble-surface-tension-induced force. The Saffman, the Magnus, and the Basset forces are ignored. Note that the lubrication effect due to particle collisions in liquid is signiﬁcant. The liquid layer dynamics pertaining to the lubrication effect was examined by Zenit and Hunt (1999). Zhang et al. (1999) used a Lattice-Boltzmann (LB) simulation to account for a close-range particle collision effect and developed a correction factor for the drag force for close-range collisions, or the lubrication effect. Such a term has been incorporated in a 2-D simulation based on the VOF method (Li et al., 1999). Equation (36) does not consider the lubrication effect. Clearly, this is a crude assumption. However, in the threephase ﬂow simulation, this study is intended to simulate only the dilute solids suspension condition (ep ¼ 0.42–3.4%) with the bubble ﬂow time of less than 1 s starting when bubbles are introduced to the solids suspension at a prescribed ep.

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

15

The particle collision effect under this simulation condition, therefore, would be small. Note that depending on the manner in which the drag force and the buoyancy force are accounted for in the decomposition of the total ﬂuid–particle interactive force, different forms of the particle motion equation may result (Jackson, 2000). In Eq. (36), the total ﬂuid–particle interaction force is considered to be decomposed into two parts: a drag force (fd) and a ﬂuid stress gradient force (see Eq. (2.29) in Jackson, 2000)). The drag force can be related to that expressed by the Wen–Yu equation, fWen–Yu, by f d ¼ f WenYu

(37)

The Wen and Yu (1966) equation is given by 1 f WenYu ¼ pd 2p C D 2 rV V p ðV V p Þ 8

(38)

where the effective drag coefﬁcient CD is calculated by C D ¼ C D0 4:7

(39)

In Eq. (39), CD0 is a function of the particle Reynolds number, Rep ¼ rd p jV V p j=m. For rigid spherical particles, the drag coefﬁcient CD0 can be estimated by the following equations (Rowe and Henwood, 1961): ( 24 0:687 Þ; Rep o1000 Rep ð1 þ 0:15ðRep Þ C D0 ¼ (40) 0:44; Rep 1000 The added mass force accounts for the resistance of the ﬂuid mass that is moving at the same acceleration as the particle. Neglecting the effect of the particle concentration on the virtual-mass coefﬁcient, for a spherical particle, the volume of the added mass is equal to one-half of the particle volume, so that 1 DV DV p 3 f am ¼ pd p r (41) 12 Dt Dt When particles approach the gas–liquid interface, the surface-tension force acts on the particles through the liquid ﬁlm. The bubble-surface-tension induced force can be described by fs ¼

p 3 d sKðfÞdðfÞrf 6 p

(42)

When the particle inertia overcomes the surface-tension-induced force, the particle will penetrate the bubbles. Recognizing that particle penetration may not lead to bubble breakage, details of bubble instability due to particle collision are given in Chen and Fan (1989a, b).

16

YANG GE AND LIANG-SHIH FAN

To simulate the particle–particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefﬁcient of 0.9 and a friction coefﬁcient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum ﬂight time of particles before any collision. Compared with the soft-sphere particle–particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation; thus, only the translational motion equation is required to describe the ﬂuid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation; therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996; Crowe et al., 1998).

D. RESULTS

AND

DISCUSSIONS

The computation performed in this study is based on the model equations developed in this study as presented in Sections II.A, III.A, III.B, and III.C These equations are incorporated into a 3-D hydrodynamic solver, CFDLIB, developed by the Los Alamos National Laboratory (Kashiwa et al., 1994). In what follows, simple cases including a single air bubble rising in water, and bubble formation from a single nozzle in bubble columns are ﬁrst simulated. To verify the accuracy of the model, experiments are also conducted for these cases and the experimental results are compared with the simulation results. Simulations are performed to account for the bubble-rise phenomena in liquid–solid suspensions with single nozzles. Finally, the interactive behavior between bubbles and solid particles is examined. The bubble formation and rise from multiple nozzles is simulated, and the limitation of the applicability of the models is discussed. 1. Single Air Bubble Rising in Water The simulation for a single air bubble rising in water (density: 0.998 kg/cm3; viscosity: 0.01 Pa s; surface tension: 0.0728 N/m) is performed in a 4 4 8 cm3 3-D column. A uniform grid size of 0.05 cm is used for three dimensions which generates 80 80 160 ( ¼ 1.024 106) grid points in the computational domain. Initially, a spherical air bubble is positioned at rest in this domain with its center located 0.5 cm above the bottom and the liquid is quiescent. The freeslip boundary conditions are imposed on all six walls. Note that the dimension of the computational domain is selected based on numerical experiments. It is found that, under both free-slip and no-slip wall boundary conditions, when the distance of the bubble interface to the wall is more than twice as large as the

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

17

bubble diameter, there is practically little effect of the wall boundary conditions on the simulation results. Thus, with an exception of the cases involving several bubbles which migrate to the near-wall region through the zigzag motion that will be discussed in a later section, the simulation results obtained in this study are not affected by these wall boundary conditions. The time for an air bubble of 0.8 cm in diameter rising from the initial position to the outlet of the column is about 0.4 s, which corresponds to a bubble rise velocity of 18.75 cm/s. The simulation was conducted using a Cray SV1 supercomputer at the Ohio Supercomputer Center (OSC). The CPU time to compute the entire process of bubble rising is 4 h. The simulation results for the positions and the shape changes of the 0.8 cm air bubble rising in water are shown in Fig. 2. The value for the Eotvos number (E0 ¼ gDrd2e /s) and the Morton number (M ¼ gm4Dr/r2s3) are 8.5 and 2.5 107, respectively. The time increment between two bubble images in Fig. 2 is 0.05 s. As seen in the ﬁgure, the bubble shape undergoes continuous changes from a sphere initially to an oblate ellipsoidal cap, and ﬂuctuates between an oblate ellipsoidal cap and a spherical cap. The rectilinear motion of a bubble in water exhibited in the ﬁgure, occurs for the ﬁrst several fractions of a second of a single bubble rising (with symmetric wakes) in a quiescent liquid even though the bubble Reynolds number (Reb) is in the wake shedding regime (Reb4400). The computed results obtained in this study capture such

FIG. 2. Simulated positions and shape variations of a rising bubble in a water column. Initial bubble diameter 0.8 cm and time increment 0.05 s.

18

YANG GE AND LIANG-SHIH FAN

phenomena well. The liquid-ﬁeld disturbance would eventually induce an asymmetric wake, yielding wake-shedding phenomena of the rising bubble. The mesh-reﬁnement studies are conducted to examine the mesh effect on the computation results. Simulations of a 0.8 cm air bubble rising in a 4 4 8 cm3 water column (as shown in Fig. 2) are repeated at three different mesh resolutions: from a lower resolution of 40 40 80 grid points with a grid size of 0.1 cm to a higher resolution of 100 100 200 grid points with a grid size of 0.04 cm. The simulated variations of bubble rise velocity and bubble aspect ratio (height/width) with time are shown in Figs. 3a and b, respectively. Fig. 3a indicates that the bubble-rise velocity measured based on the displacement of the top surface of the bubble (Ubt) quickly increases and approaches the terminal bubble rise velocity in 0.02 s. The small ﬂuctuation of Ubt is caused by numerical instability. The bubble-rise velocity measured based on the displacement of the bottom surface of the bubble (Ubb) ﬂuctuates signiﬁcantly with time initially and converges to Ubt after 0.25 s. The overshooting of Ubb can reach 45–50 cm/s in Fig. 3a. The ﬂuctuation of Ubb reﬂects the unsteady oscillation of the bubble due to the wake ﬂow and shedding at the base of the bubble. Although the relative deviation between the simulation results of the 40 40 80 mesh and 100 100 200 mesh is notable, the deviation is insigniﬁcant between the results of the 80 80 160 mesh and those of the 100 100 200 mesh. The agreement with experiments at all resolutions is generally reasonable, although the simulated terminal bubble rise velocities (20 cm/s) are slightly lower than the experimental results (2125 cm/s). A lower bubble-rise velocity obtained from the simulation is expected due to the no-slip condition imposed at the gas–liquid interface, and the ﬁnite thickness for the gas–liquid interface employed in the computational scheme. The aspect ratio shown in Fig. 3b describes the change of the bubble shape with time during the bubble rising. The simulation and the experimental results generally agree well, as shown in the ﬁgure. It can also be seen that the simulation results are only sensitive to the mesh size of 40 40 80 mesh and the deviation between the results of 80 80 160 mesh and 100 100 200 mesh is small. Thus, Figs. 3a and b indicate that a reasonable accuracy can be reached in this bubble-rise simulation with a 80 80 160 mesh (grid size of 0.05 cm). The simulation results on bubble velocities, bubble shapes, and their ﬂuctuation shown in Fig. 3 are consistent with the existing correlations (Fan and Tsuchiya, 1990) and experimental results obtained in this study. Bubble rise experiments were conducted in a 4 cm 4 cm Plexiglas bubble column under the same operating conditions as those of the simulations. Air and tap water were used as the gas and liquid phases, respectively. Gas is introduced through a 6 mm nozzle. Note that water contamination would alter the bubble-rise properties in the surface tension dominated regime. In ambient conditions, this regime covers the equivalent bubble diameters from 0.8 to 4 mm (Fan and Tsuchiya, 1990). All the air–water experiments and simulations of this study are carried out under the condition where most equivalent bubble diameters exceed

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

19

4 mm.These ﬂow conditions correspond to the bubble inertial regime, and thus, the extent of water contamination plays a negligible role in the determination of the bubble-rise properties. The thickness of the gas–liquid interface is set as 3D based on the parameters used in the case of Sussman (1998), with the same density-ratio on the interface and similar Reynolds number. An interface thickness of 5D is also examined in the simulation and no signiﬁcant improvement is observed. The accurate prediction of the bubble shape (shown in Figs. 3 and 4) can be attributed, in part, to the manner in which the surface-tension force is treated as a body force in the computation scheme. Speciﬁcally, since the surface-tension force acting on a solid particle is considered only when a solid particle crosses the gas–liquid interface and the solid particle is considered as a point, the accuracy of the calculation of this force can be expected if the surface tension is interpreted as a body force acting on each grid node near the interface. 2. Bubble Formation from an Orifice The air-bubble formation from a single oriﬁce in water is simulated. The computational domain is 2 2 4 cm3. A uniform grid size of 0.025 cm and 80 80 160 grid points are used to obtain convergent solutions for the bubble formation process. This mesh-size effect is examined by comparing the simulation results on the bubble-formation processes with experimental measurements. As shown in Fig. 4, decreasing the mesh size or increasing the mesh resolution from 40 40 80 to 80 80 160 improves the accuracy of the prediction results on the bubble shape. Further increase in mesh resolution does not practically change the simulation results. Simulations are then performed for gas bubbles emerging from a single nozzle with 0.4 cm I.D. at an average nozzle velocity of 10 cm/s. The experimental measurements of inlet gas injection velocity in the nozzle using an FMA3306 gas ﬂow meter reveals an inlet velocity ﬂuctuation of 3–15% of the mean inlet velocity. A ﬂuctuation of 10% is imposed on the gas velocity for the nozzle to represent the ﬂuctuating nature of the inlet gas velocities. The initial velocity of the liquid is set as zero. An inﬂow condition and an outﬂow condition are assumed for the bottom wall and the top walls, respectively, with the free-slip boundary condition for the side walls. Fig. 5 shows the simulated air-bubble formation and rising behavior in water. For the ﬁrst three bubbles, the formation process is characterized by three distinct stages of expansion, detachment, and deformation. In comparison with the bubble formation in the air–hydrocarbon ﬂuid (Paratherm) system, the coalescence of the ﬁrst two bubbles occurs much earlier in the air–water system. Note that the physical properties of the Paratherm are rl ¼ 870 kg/m3, ml ¼ 0.032 Pa s, and s ¼ 0.029 N/m at 25 1C and 0.1 MPa. This is due to the fact that, compared to that in the air–Paratherm system, the ﬁrst bubble in the air–water system is much larger in size and hence higher in rise velocity leading

20

YANG GE AND LIANG-SHIH FAN

FIG. 3. (a) Comparison of the simulation results and experimental results of the bubble rise velocity. (b) Comparison of the simulation and experimental results of the bubble aspect ratio.

to a longer time for its coalescence with the second bubble. Beginning with the third bubble, the formation and rising behavior of air bubbles in water shows strongly asymmetric behavior. As is evident from Fig. 5, the bubble rises in a spiral path or a zigzag path.

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

21

FIG. 4. Comparison of the experimental measurement and the simulation results with different resolutions of air-bubble formation in water.

In order to verify the simulation results, experiments on bubble behavior in bubble columns are carried out under conditions similar to the simulations. A 3-D rectangular bubble column with the dimension of 8 8 20 cm3 is used for the experiments. Four nozzles with 0.4 cm I.D. and a displacement of 2.4 cm are designed in the experiments. For single-nozzle experiments, air is injected into the liquid bed through one of the oriﬁces while the others are shut off. The outlet air velocity from the nozzle is approximated using the measured bubbling

22 YANG GE AND LIANG-SHIH FAN

FIG. 5. Simulation results of air-bubble formation from a single nozzle in water. Nozzle size 0.4 cm I.D. and nozzle gas velocity 10 cm/s.

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

23

frequency and the initial bubble size. A high-speed video camera (240 frames/s) is used to obtain the images of bubbles emerging from the oriﬁce in the liquid. A common dimensionless number used to characterize the bubble formation from oriﬁces through a gas chamber is the capacitance number deﬁned as: Nc ¼ 4Vcgrl/pD20Ps. For the bubble-formation system with inlet gas provided by nozzle tubes connected to an air compressor, the volume of the gas chamber is negligible, and thus, the dimensionless capacitance number is close to zero. The gas-ﬂow rate through the nozzle would be near constant. For bubble formation under the constant ﬂow rate condition, an increasing ﬂow rate signiﬁcantly increases the frequency of bubble formation. The initial bubble size also increases with an increase in the ﬂow rate. Experimental results are shown in Fig. 6. Three different nozzle-inlet velocities are used in the air–water experiments. It is clearly seen that at all velocities used for nozzle air injection, bubbles rise in a zigzag path and a spiral motion of the bubbles prevails in air–water experiments. The simulation results on bubble formation and rise behavior conducted in this study closely resemble the experimental results.

FIG. 6. Experimental results on air-bubble formation and bubble rising in water. Nozzle size 0.4 cm I.D. and nozzle gas velocity (a) 6.0 cm/s; (b) 10.0 cm/s; (c) 14 cm/s.

24

YANG GE AND LIANG-SHIH FAN

3. Gas– Liquid– Solid Fluidization As noted earlier, to simulate the bubble motion in a gas–liquid bubble column accurately, ﬁne grid sizes, 0.025 cm for air–water and 0.05 cm for air–Paratherm system, should be used in the computation. This ﬁne-grid computation yields essentially the results of DNS. These grid sizes are smaller than the size of the solid particle usually employed for the three-phase ﬂuidized bed operation. For the particle size of 0.08 cm used in the present simulation of a three-phase ﬂuidized bed, the computational grid size is required to be no less than 0.2 cm in order to track both the bubble ﬂow and the particle motion. Note that the system simulated in this study is a dilute liquid–solid bed with a minimum of particle–particle collisions and uniform particle distribution. Although a grid size of D410 Dp as generally used in the Lagrangian simulation of ﬂuid–particle ﬂows is preferable, the grid size used under the current simulation of three-phase ﬂows is acceptable. There were no numerical stability or convergence problems encountered in the computation. For simulation of the bubble formation in a gas–liquid bubble column, a coarse grid size of 0.2 cm in a 4 4 8 cm3 domain with 21 21 41 grid points is used in this study. However, due to this large grid size used, without any turbulence model, the simulation cannot accurately track the discrete bubble-formation process. Speciﬁcally, simulation without consideration of the turbulent effects, the bubbles with distorted wake structure are seen to be connected like a jet above a nozzle. An SGS stress model is thus employed and incorporated into the code for subsequent simulation. The simulation of the gas–liquid bubble column system indicates that experimental results on a bubble formation in an air–Paratherm medium can be well described when Cs values are in a range of 1.0 to 1.2 with a grid size of 0.2 cm.Note that the values for the Smagorinsky coefﬁcient for single phase ﬂow are 0.1–0.2. The results are shown in Fig. 7(a). Subsequently, simulations are performed for the air–Paratherm–solid ﬂuidized bed system with solid particles of 0.08 cm in diameter and 0.896 g/cm3 in density. The solid particle density is very close to the liquid density (0.868 g/ cm3). The boundary condition for the gas phase is inﬂow and outﬂow for the bottom and the top walls, respectively. Particles are initially distributed in the liquid medium in which no ﬂows for the liquid and particles are allowed through the bottom and top walls. Free slip boundary conditions are imposed on the four side walls. Speciﬁc simulation conditions for the particles are given as follows: Case (b) 2,000 particles randomly placed in a 4 4 8 cm3 column; Case (c) 8,000 particles randomly placed in a 4 4 8 cm3 column; and Case (d) 8,000 particles randomly placed in the lower half of the 4 4 8 cm3 column. The solids volume fractions are 0.42, 1.68, and 3.35%, respectively for Cases (b), (c), and (d). The bubble-formation process at different solids concentrations is shown in Figs. 7(b)–(d) and is compared with that without particles as shown in Fig. 7(a). For the ﬁrst 0.3 s, little change is observed in the bubble-formation process

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

25

FIG. 7. Simulation results of bubble formation and rising in Paratherm NF heat-transfer ﬂuid with and without particles. Nozzle size 0.4 cm I.D., liquid velocity 0 cm/s, gas velocity 10 cm/s, and particle density 0.896 g/cm3. (a) No particle; (b) 2000 particles; (c) 8000 particles; (d) 8000 particles.

for the three solids concentrations used in this simulation. After 0.4 s, however, signiﬁcant changes can be found for the cases with high solids concentrations. This can be seen from the ﬁrst bubble in each case. When the solid concentration is low or no solids are present, the ﬁrst bubble grows on the

26

YANG GE AND LIANG-SHIH FAN

FIG. 7 (Continued)

oriﬁce and connects to the second bubble. For the high solids concentration cases, the ﬁrst bubble is not well connected to the second bubble. This is particularly true for Case (d) when the bubble rises into the solids-free region or freeboard region of the bed. The solid particle entrainment is clearly observed in Case (d).

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

27

IV. System 2: Deformation Dynamics of Liquid Droplet in Collision with a Particle with Film-Boiling Evaporation The phenomena of evaporative liquid droplets impacting onto solid objects at high temperatures are of relevance to many engineering problems, such as sprinkler systems in the iron making or metal-casting processes, ink-jet spraypainting, impingement of oil droplets on turbine engines, meteorology, and spray coating of substrates. An evaporative liquid jet in gas–solid ﬂow systems is also of interest to current technology applications in chemical, petroleum, and materials processing industries, such as FCC, polyethylene synthesis (Kunii and Levenspiel, 1991; Fan et al., 2001) and microelectronic materials manufacturing. In FCC riser reactors, for example, gas oil at a low temperature is injected into the riser from feed nozzles located at the bottom of the riser and the mist droplets formed from the spray contact with high-temperature ﬂuidized catalyst particles. The vaporized oil then carries the catalyst particle up through the riser. In the feed nozzle region, the size of the droplet can be comparable or signiﬁcantly smaller (or larger) than the size of particle. The droplet can always have a different momentum, thus the collision between the catalytic particles and oil droplet may have various modes. Fig. 8 shows some of the collision modes existing in a feed-nozzle region (Zhu et al., 2000). Smaller droplets may rebound from the surface of larger particles upon impact, and smaller particles

Large Droplet

Small Droplet

FIG. 8. Various modes of droplet–particle collisions.

28

YANG GE AND LIANG-SHIH FAN

may penetrate through or penetrate but retain inside the larger droplets. Larger droplets may break into smaller drops during the impact and/or remain attached to the particle surface after the collision, which may intensify the particle aggregation. Clearly, understanding the droplet and particle collision mechanics are crucial to an accurate account of the momentum and heat transfer between the droplet and solid object, which is important for prediction of hydrocarbon product distributions in light of catalytic and the thermal-cracking reactions in the riser. It is also relevant to the design of feed nozzles that provide desired droplet properties for optimum droplet contact with catalyst particles in the reactor. In most of the applications, the solid objects (e.g., the catalyst particles in FCC reactor) are always under high temperature, and the droplet impact processes are accompanied with intensive evaporation. The nature of the collision of the droplet with the superheated objects exhibits a great diversity in hydrodynamic and thermodynamic properties, such as droplet splash and rebound, wetting or nonwetting contact, nucleate boiling or ﬁlm boiling, and Marangoni effect. Further, the droplet shape, the contact area and the cooling effectiveness during the impact not only depend on such hydrodynamic forces as the inertia, pressure, surface tension, and viscous forces but also on the degrees of the surface superheating and the droplet subcooling (Inada et al., 1985). As the solid temperature rises to superheated conditions, the characteristics of liquid–solid contact signiﬁcantly change and the evaporation rate affects the droplet hydrodynamics. Under this condition, the nonwetting contact may develop during the collision, and the evaporation is under the ﬁlm-boiling regime, or so called Leidenfrost regime (Gottfried et al., 1966). In the Leidenfrost regime, the vapor pressure generated from the droplet evaporation prevents the direct contact of the droplet with the solid objects. The heat transfer from the hot objects to the droplet is also hindered due to the resistance of the vapor layer existing between the droplet and the solid surface. In this work, a 3-D numerical model is developed and the simulation is conducted to account for the behavior of the droplet–particle collision in the Leidenfrost regime. Experimental and numerical studies of droplets impacting onto a ﬂat surface of varied temperatures have been extensively reported in the literature. The effects of the initial droplet temperature on ﬁlm-boiling impact are signiﬁcant (Inada et al., 1985; Harvie and Fletcher, 2001b). Depending on the initial droplet temperature, there are two types of droplet impact: saturated impact and subcooled impact. The saturated impact involves the initial temperature at the boiling point of the liquid or saturation temperature of the liquid. The subcooled impacts, on the other hand, involve the droplet initial impact temperature below the liquid-saturation temperature. The experimental results for these two types of impact are brieﬂy described below. The modeling and numerical approaches used for the droplet impingement onto isothermal or heated ﬂat wall are also given.

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

A. SIMULATION OF SATURATED DROPLET IMPACT LEIDENFROST REGIME

ON

FLAT SURFACE

29

IN THE

Wachters and Westerling (1966) ﬁrst presented a classiﬁcation of the dynamic regimes of the impact based on their experiments in which water drops with a diameter of 2.3 mm impact on a polished gold surface at temperatures between 200 1C and 400 1C. They studied the saturated impact of water droplets and found that for the impact with Weo30, where We ¼ 2rlV2R/s, the surface tension of the droplet dominates the impact process, and the droplet recoils and rebounds from the surface without disintegration. At 30oWeo80, the droplet undergoes a similar spreading and recoiling process as that for Weo30. In the rebounding process, the droplet may disintegrate into several smaller droplets (secondary droplets) and the shape of the droplet may then become unstable. For the impact with We480, the impact inertial force (or kinetic energy) is so large that splashing occurs during the early stage of the impact, while the droplet breaks up into a number of small droplets. Based on the measured heat ﬂux on the solid surface, Wachters and Westerling (1966) also estimated the relative volume decrease of the droplet during the impact. It was found that, when the solid temperature is higher than 200 1C, the averaged evaporation rate of the droplet decreases with an increase in the surface temperature. At the nonwetting condition when the surface temperature reaches 400 1C, the volume (mass) change of the droplet due to the evaporation during the impact is slight (0.2–0.3%) for a wide range of We. Groendes and Mesler (1982) studied the saturated ﬁlm boiling impacts of a 4.7 mm water droplet on a quartz surface of 460 1C. The ﬂuctuation of the surface temperature was detected using a fast-response thermometer. The maximal temperature drop of the solid surface during a droplet impact was reported to be about 20 1C. Considering the lower thermal diffusivity of quartz, this temperature drop implies a low heat-transfer rate on the surface. Biance et al. (2003) studied the steady-state evaporation of the water droplet on a superheated surface and found that for the nonwetting contact condition, the droplet size cannot exceed the capillary length. Ge and Fan (2005) developed a 3-D numerical model based on the level-set method and ﬁnite-volume technique to simulate the saturated droplet impact on a superheated ﬂat surface. A 2-D vapor-ﬂow model was coupled with the heattransfer model to account for the vapor-ﬂow dynamics caused by the Leidenfrost evaporation. The droplet is assumed to be spherical before the collision and the liquid is assumed to be incompressible. 1. Hydrodynamic Model and Numerical Solution In the level-set method, the free surface of the droplet is taken as the zero in the level-set function fð~ x; tÞ as given in Eq. (2). The motion of the interface is

30

YANG GE AND LIANG-SHIH FAN

traced by solving the Hamilton–Jacobi-type convection equation, given as Eq. (3), in the computational domain. The mass loss of the droplet due to evaporation during the impact process is neglected in surface-tracking based on the experimental results of Wachters and Westerling (1966). With the level-set method, the equation of motion of the ﬂuid follows the Navier–Stokes equation as given by Eqs. (5) and (6). The density and viscosity are deﬁned by Eqs. (10) and (11). The computational code used in solving the hydrodynamic equation is developed based on the CFDLIB, a ﬁnite-volume hydro-code using a common data structure and a common numerical method (Kashiwa et al., 1994). An explicit time-marching, cell-centered Implicit Continuous-ﬂuid Eulerian (ICE) numerical technique is employed to solve the governing equations (Amsden and Harlow, 1968). The computation cycle is split to two distinct phases: a Lagrangian phase and a remapping phase, in which the Arbitrary Lagrangian Eulerian (ALE) technique is applied to support the arbitrary mesh motion with ﬂuid ﬂow. ~n ¼ V ~ð~ Let fn ¼ fð~ x; tn Þ and V x; tn Þ be the cell-centered level-set function and n velocity at time t , respectively. The numerical procedures to solve the velocity ~nþ1 , and the level-set function fn+1 at tn+1 ¼ tn+Dt can be described ﬁeld V below: ~nþ1 by solving the governing equation, Eqs. (1) Compute the velocity ﬁeld V (5–6), using the cell-centered ICE technique and ALE technique (Kashiwa et al., 1994). ¯ nþ1 . The (2) Solve the convection equation of fð~ x; tÞ (Eq. (3)) to obtain the f high order (3rd order) essentially non-oscillatory (ENO) upwind scheme ~G rf based (Sussman et al., 1994) is used to calculate the convective term V nþ1 ~ on the updated velocity ﬁeld V . The time advancement is accomplished using the second-order total variation diminishing (TVD) Runge-Kutta method (Chen and Fan, 2004). ¯ nþ1 as the initial (3) Perform the redistance procedure to obtain the fn+1 using f value. The detail of the redistance computation is given by Sussman et al. (1998). (4) Calculate the density and the viscosity of the ﬁeld using Eqs. (10)–(11) with the updated level-set function fn+1. ~nþ1 and the fn+1 are the same, which The time steps (Dt) for calculating the V is determined by the CFL condition and under constraints of the viscous and surface tension (Sussman et al., 1994). Considering a surface temperature which is higher than the Leidenfrost temperature of the liquid in this study, it is assumed that there exists a microscale vapor layer which prevents a direct contact of the droplet and the surface. Similar to Fujimoto and Hatta (1996), the no-slip boundary condition is adopted at the solid surface during the droplet-spreading process and the free-slip

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

31

condition is applied for the recoiling and rebounding periods. The velocity at the grid point inside the solid surface is solved together with whole domain but is reset according to the relative boundary condition (Ge and Fan, 2005). 2. Vapor-Flow Model As the thickness of the vapor layer (5–20 mm) is several orders of magnitude smaller than the macroscale of the ﬂow ﬁeld (i.e., the diameter of the droplet), it would be impractical to use the same computation mesh for both macroﬂow and vapor-layer ﬂow (Harvie and Fletcher, 2001a). Thus, a 2-D model is developed to simulate the dynamics of the vapor ﬂow between the droplet and the surface. For the ﬁlm-boiling impact problem, the vapor-layer model would allow determination of the evaporation-induced pressure in the vapor layer without neglecting the inertial force of the vapor ﬂow. In the symmetrical coordinates (x,l) shown in Fig. 9, assuming that the gas in the vapor layer is only saturated vapor and neglecting the temporal term, the continuity and momentum equations for incompressible vapor ﬂows with gravitation terms neglected are given by @ux ux @ul þ þ ¼0 @x x @l

(43)

2 @ux @ux @ P @ u x @2 u l ux þ ul ¼ þn @x r @x @l @l2 @x@l

(44)

2 @ul @ul @ P @ ul @2 ux 1 @ul 1 @ux þ ul ¼ þ þn ux @l r @x @l @x2 @x@l x @x x @l

(45)

where ux, ul are the vapor-ﬂow velocities in x and l direction, respectively, n is the kinematical viscosity of the vapor. To determine the relative signiﬁcance of λ

droplet

ξ

Vapor flow O

Solid surface

FIG. 9. Coordinates for the vapor-layer model.

32

YANG GE AND LIANG-SHIH FAN

each term in these motion equations, an order of magnitude analysis is made by considering the following dimensionless groups: x x¯ ¼ ; R

l Z¼ ; d

u¯ x ¼

ux ; Ux

u¯ l ¼

ul ; Ul

p¯ ¼

p ; rU 2x

¯t ¼

tU x ; R

Red ¼

dðxÞuld n (46)

where R is the droplet radius; d the vapor-layer thickness; Red the local evaporation Reynolds number; uld(x) the local vapor velocity; and Ux, Ul the velocity scalars in x, l directions, respectively. Two assumptions can be made in accounting for the collision process: (a) The vapor-layer thickness is much smaller than the radius of the droplet; (b) The velocity for the vapor ﬂow is much larger than the rates of variation of the vapor-layer thickness and breadth. Based on these assumptions and the order of magnitude analysis, the x momentum equation can be simpliﬁed to: @ux @ux @ P @2 u x ux þ ul ¼ (47) þn 2 @x r @x @l @l The boundary conditions are: l ¼ 0;

ux ðx; 0Þ ¼ ul ðx; 0Þ ¼ 0

l ¼ d; ux ðx; dÞ ¼ ul ðxÞ; ul ðx; dÞ ¼ uld ðxÞ @ x ¼ 0; @x ¼ 0; x ¼ xb ; p ¼ pb

(48)

where pb is the pressure of the ambient gas at the outside edge of the vapor layer. In the impact process that involves large temperature differences (DT) between the surface and the droplet, such as the ones considered in this study (e.g., DTﬃ300–500 1C), the value for Red is about 0.5–1.0. Thus, the inertial force of the vapor ﬂow would be of the same order of magnitude as the viscous force, and cannot be neglected in Eq. (47) for the vapor-ﬂow model. To solve Eq. (47), a variable transformation is considered: ul ðx; ZÞ ¼ Zuld ðxÞ

ux ðx; ZÞ ¼ OðxÞFðZÞ

(49)

uld (x) can be calculated through the energy-balance equation at the vapor–droplet interface. O(x) and F(Z) are single-variable functions. With this transformation, the solution for the x momentum can be converted into that of an ordinary differential equation (ODE) of O(x): F00 ðZÞ þ Red ZF0 ðZÞ Red FðZÞ ¼ jðxÞ

jðxÞ ¼

d2 @ ðP=rÞ nOðxÞ @x

(50)

(51)

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

33

The general solution of the Eq. (50) can be obtained in power series form. Under the condition that Red Oð1Þ, F(Z) can be approximated by only including the ﬁrst three terms in the power series with good accuracy: FðZÞ ¼ Fd Z jðxÞZ

1 Red 2 24

2 Z Red 4 Z þ jðxÞ 2 24

(52)

The averaged vapor-ﬂow velocity is given by Z

1

u¯ x ðxÞ ¼ O 0

1 d2 3 @ p 1 Red Fd dZ ¼ uld ðxÞ 2 20 @x r 12g

(53)

The vapor-continuity equation can be expressed by u¯ x ðxÞ ¼

1 xdðxÞ

Z

x

x0 uld ðx0 Þdx0

(54)

0

The pressure distribution in the vapor layer can be obtained by solving Eqs. (53) and (54) using a piecewise integration method (Ge and Fan, 2005). In this procedure, the thickness of the vapor layer d(x) is obtained from the level-set function. The uld(x) is calculated by _ v uld ðxÞ ¼ m=r

@dðxÞ @t

(55)

where the local evaporation rate m˙ is deﬁned by the heat-transfer model. The vapor-pressure force simulated by this model is applied as an interfacial force to the droplet bottom surface. 3. Heat-Transfer Model Heat transfer occurs not only within the solid surface, droplet and vapor phases, but also at the liquid–solid and solid–vapor interface. Thus, the energybalance equations for all phases and interfaces are solved to determine the heattransfer rate and evaporation rate. Inside the solid surface, the heat-conduction equation in 3-D coordinates is @T s @ @T s @ @T s @ @T s ¼ as þ þ (56) @x @x @y @y @z @z @t where Ts(x,y,z) is the solid temperature and as the thermal diffusivity of solid. The heat transfer within the droplet is described by the following thermalenergy transport equation with neglecting viscous dissipation: 2 @T d @T d @T d @T d @ T d @2 T d @ 2 T d þu þv þw ¼ ad þ þ (57) @t @x @y @z @x2 @y2 @z2

34

YANG GE AND LIANG-SHIH FAN

Using the same assumptions that were made in the vapor-layer model, the energy-conservation equation for the incompressible 2-D vapor phase can be simpliﬁed to a 1-D equation in boundary layer coordinates: @2 T v ¼0 @Z2

(58)

The radiative heat transfer across the vapor layer is neglected under the condition that the solid temperature is lower than 700 1C (Harvie and Fletcher, 2001a,b). On the liquid–vapor interface, the energy-balance equation is kv

T ss T ds _ c ¼ mL d

(59)

where kv is the thermal conductivity of the vapor; Tss and Tds are the temperatures of the solid surface and the droplet surface. The thermal boundary condition at the solid–vapor interface is kv

T ss T ds @T s ¼ ks d @Z

(60)

where the heat ﬂux in the Z direction is assumed to be much larger that that in the x direction. The numerical method used for solving the heat-transfer equation is similar to that for solving the momentum equation, which is a ﬁnite-volume, ALE method (Kashiwa et al., 1994). 4. Results and Discussion To validate the model developed in the present study, the simulations are ﬁrst conducted and compared with the experimental results of Wachters and Westerling (1966). In their experiments, water droplets impact in the normal direction onto a hot polished gold surface with an initial temperature of 400 1C. Different impact velocities were applied in the experiment to test the effect of the We number on the hydrodynamics of the impact. The simulation of this study is conducted for cases with different Weber numbers, which represent distinct dynamic regimes. The simulation shown in Fig. 10 is an impact of a saturated water droplet of 2.3 mm in diameter onto a surface of 4001C with an impact velocity of 65 cm/s, corresponding to a Weber number of 15. This simulation and all others presented in this study are conducted on uniform meshes (Dx ¼ Dy ¼ Dz ¼ D). The mesh resolution of the simulation shown in Fig. 10 was 0.08 mm in grid size, although different resolutions are also tested and the results are compared in Figs. 11 and 12. The average time-step in this case is around 5 ms. It takes 4000 iterations to simulate a real time of 20 ms of the impact process. The simulation

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

35

FIG. 10. Water droplet impacts on a ﬂat surface. The initial droplet diameter is 2.3 mm and the surface temperature is 400 1C. We ¼ 15.

36

YANG GE AND LIANG-SHIH FAN

FIG. 11. Simulated 3-D views of the impact for We ¼ 15 as a function of the mesh resolution.

code is run on the cray-SV1 supercomputer at the OSC. The computing time of this case is about 12 h. Comparing the 3-D images simulated and the experimental photographs in Fig. 10, it can be seen that the droplet shapes are well reproduced by the present model. During the ﬁrst 3.5 ms of the impact (frames 1–3), a liquid ﬁlm with ﬂattened disc shape is formed immediately after the impact. The inertial force drives the liquid to continue spreading on the solid surface, while the surface tension and the viscous forces resist the spreading of the liquid ﬁlm. As a result, the droplet spreading speed decreases and the ﬂuid mass starts to accumulate at

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

37

2.0 0.150mm grid size 0.120mm grid size 0.100mm grid size 0.075mm grid size 0.060mm grid size 0.050mm grid size

1.8 1.6 1.4

Experiment of Wachters and Westerling 6

R/R0

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

2

4

6

8

101

2

141

61

8

Time after impact(ms) FIG. 12. Spread factor of the droplet verse time for the impact condition given in Fig. 10 at different mesh resolutions.

the leading edge of the liquid ﬁlm (2.5–3.2 ms). After the droplet spreads to the maximum extent, the liquid ﬁlm starts to shrink back to its center (frames 4 and 5) due to the surface-tension force at the edge of the ﬁlm. At 3.55 ms (frame 4), the simulated droplet shows a concave structure with a void in the center, which is also shown in the experimental photograph of 3.85 ms. This structure, also called a ring structure, has also been widely reported in the literature. In the cross-sectional images at 3.55 ms, the velocity ﬁeld shows that the inward ﬂow ﬁrst starts from the outer edge of the liquid ﬁlm, which conﬁrms that recoiling ﬂow is driven by the surface tension. After 4.4 ms, the droplet continues to recoil and forms an upward ﬂow in the center of the droplet (frames 5 and 6), and this leads to a bouncing of the droplet up from the surface (frame 7). The peanutshape droplet (also called dumb-bell shape by Harvie and Fletcher (2001b) shown in the experimental photograph at 14.56 ms is reproduced in the simulation. The impact process shown in Fig. 10 is also simulated at six different grid resolutions, i.e., 0.150, 0.120, 0.100, 0.075, 0.060, and 0.050 mm in mesh sizes, with the corresponding cells per droplet radius (CPR) of 7.6, 9.6, 11.5, 15.3, 19.0, and 23.0, respectively. The comparison of the 3-D images among three resolutions is shown in Fig. 11. The corresponding CPRs of these resolutions are 9.6, 11.5, and 15.3. It can be found that the simulated droplet shapes are similar at all three resolutions during the spreading process and even the early stage of the recoiling process (2–6 ms). The deviation appears in the late stage of

38

YANG GE AND LIANG-SHIH FAN

the recoiling process (8 ms), while the droplet generated on coarser mesh (0.12 mm) tends to be more uniform in structure and less elongated in the vertical direction. The difference in the droplet shape for 0.1 mm mesh and 0.075 mm mesh is relatively small. Fig. 12 shows the spread factors simulated on meshes with different resolutions along with the measurement value of Wachters and Westerling (1966). The spread factor is deﬁned as the radius of the droplet on the solid surface divided by the initial radius of the droplet. Although the convergence is not perfect, the agreement between the experiment and the simulations is relatively good for all resolutions. Consistent with the results of Fig. 11, the effect of the mesh resolution on spread factor becomes notable after 8 ms since the moment of impact, and the coarser resolution tends to yield a slower rebounding process. The simulations were also performed under same conditions as the case of Fig. 10 but for higher impact velocities. The simulated-droplet dynamics and heat-transfer rate at the solid surface at different impact velocities are given in Ge and Fan (2005).

B. SIMULATION OF SUBCOOLED DROPLET IMPACT LEIDENFROST REGIME

ON

FLAT SURFACE

IN

Subcooled impacts, in which the initial temperature of the droplet is below the liquid saturation temperature, are of primary interest in experiments since the condition of the spray liquids is often of the ambient temperature in practical applications (Inada et al., 1985; Chandra and Avedisian, 1991; Chen and Hsu, 1995). Chandra and Avedisian (1991) studied the collision dynamics of a 24 1C n-heptane droplet impacting on a metallic surface with a Weber number equal to 43. The transition from the nucleate boiling to the ﬁlm boiling was identiﬁed when the surface temperature rises from the boiling point (170 1C) to above the Leidenfrost temperature (200 1C) of n-heptane. They found that under the ﬁlm-boiling condition, the liquid–solid contact is hindered by the vapor layer, as evidenced by the disappearance of the bubbles inside the liquid droplet under the nucleate boiling condition. The contact angle of the liquid to the surface was also reported to increase with an increase in the surface temperature, and reached 1801 in the ﬁlm boiling condition. Qiao and Chandra (1996) measured the temperature drop of a stainless steel surface during the impact of the subcooled water and the n-heptane droplets in low gravity. They found that when the surface temperature is above the superheat limit, the temperature drop of the surface is relatively small for the impact of n-heptane droplet (less than 20 1C). But for the impact of the water droplet, the temperature drop of the surface can reach 150 1C, which implies a high heat ﬂux and the intermittent contact of the liquid and the solid surface. Hatta et al. (1997) found that at low impact We number, the dynamics of water droplet is almost independent of

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

39

the surface materials when the surface temperature is above the Leidenfrost temperature. The effects of the subcooling degree of the droplet on the ﬁlm-boiling impact are studied by Inada et al. (1985). They found that the heat-transfer rate on the solid surface during an impact of a 4-mm water droplet increases signiﬁcantly with a decrease in the initial droplet temperature. The boiling regimes were classiﬁed to represent different droplet dynamics and heat-transfer modes at various droplet and surface temperatures. Inada et al. (1988) also measured the thickness of the vapor ﬁlm between the impinging droplet and the surface at various degrees of subcooling. Chen and Hsu (1995) measured the transient local heat ﬂux at the surface of a presuperheated plate, which undergoes the impingement of subcooled water droplets. A fast-response microthermocouple was designed to capture the instantaneous changes of the solid-surface temperature. Although the droplet dynamics of the impact process was not presented, they concluded that both the surface temperature and the degree of the droplet subcooling are crucial to the intermittent contact mode at the solid surface. At the ﬁlm-boiling regime with the surface temperature superheated at 400 1C, a subcooled droplet tends to disintegrate during the impact at We ¼ 55. In subcooled impact, the initial droplet temperature is lower than the saturated temperature of the liquid of the droplet, thus the transient heat transfer inside the droplet needs to be considered. Since the thickness of the vapor layer may be comparable with the mean free path of the gas molecules in the subcooled impact, the kinetic slip treatment of the boundary condition needs to be applied at the liquid–vapor and vapor–solid interface to modify the continuum system.

1. Hydrodynamic Model The ﬂow ﬁeld of the impacting droplet and its surrounding gas is simulated using a ﬁnite-volume solution of the governing equations in a 3-D Cartesian coordinate system. The level-set method is employed to simulate the movement and deformation of the free surface of the droplet during impact. The details of the hydrodynamic model and the numerical scheme are described in Sections II.A and 1 V.A.1. During the subcooled droplet impact, the droplet temperature will undergo signiﬁcant changes due to heat transfer from the hot surface. As the liquid properties such as density rl(T), viscosity ml(T), and surface tension s(T) vary with the local temperature T, the local liquid properties can be quantiﬁed once the local temperature can be accounted for. The droplet temperature is simulated by the following heat-transfer model and vapor-layer model. Since the liquid temperature changes from its initial temperature (usually room temperature) to the saturated temperature of the liquid during the impact, the linear

40

YANG GE AND LIANG-SHIH FAN

variation of liquid properties with the temperature is assumed, which is gðTÞ ¼ g0 þ

T T0 ðg g0 Þ T sa T 0 sa

g ¼ rl ; ml ; s

(61)

where T0 and Tsa are the initial and saturated temperatures of the liquid respectively; g0 and gsa are the liquid property at T0 and Tsa, respectively. The boundary condition adopted at the solid surface is described in Section IV.A.1. 2. Heat Transfer Inside the Droplet and Across the Vapor Layer For the subcooling impact, especially for the high subcooling degree case in which the droplet initial temperature is much lower than the saturated temperature, the heat transfer within the droplet is signiﬁcant and hence affects the droplet evaporation rate. Neglecting the viscous dissipation, the equation of the conservation of the thermal energy inside the droplet is given by @T ~ þ V rT ¼ al r rT @t

(62)

where a1 is the thermal diffusivity of liquid. At other free surfaces of the droplet, the adiabatic boundary condition is applied which is given by ~ nG r d T ¼ 0

(63)

where ~ nG is the normal vector of the droplet surface, which can be calculated based on the level-set function: rf ~ nG ¼ rf

(64)

rdT is the temperature gradient which is evaluated only on the droplet side. The heat-conduction equation inside the solid surface is given in Section IV.A.3. The heat transfer across the vapor layer and the temperature distribution in the solid, liquid, and vapor phases are shown in Fig. 13. In the subcooled impact, especially for a droplet of water, which has a larger latent heat, it has been reported that the thickness of the vapor layer can be very small and in some cases, the transient direct contact of the liquid and the solid surface may occur (Chen and Hsu, 1995). When the length scale of the vapor gap is comparable with the free path of the gas molecules, the kinetic slip treatment of the boundary condition needs to be undertaken to modify the continuum system. Consider the Knudsen number deﬁned as the ratio of the average mean free path of the vapor to the thickness of the vapor layer: Kn ¼

l d

(65)

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

41

Td

ql

Droplet

Td1 Td2

δ

Vapor layer

Ts1

qs

Ts2

Solid surface Ts FIG. 13. Temperature distribution and heat ﬂux across the vapor layer.

where l is the mean free path of molecule. Harvie and Fletcher (2001c) analyzed the kinetic of the molecular behavior at the solid and evaporative surface for 0.01oKno0.1. Based on their simple kinetic theory, the effective temperature discontinuity at liquid–vapor surface and solid–vapor surface can be given by T s2 T s1 ¼ C T;s ðT s1 T d2 Þ

(66)

T d1 T d2 ¼ C T;l ðT s1 T d1 Þ

(67)

where Ts1,Ts2,Td1,Td2 are the interface temperatures of solid surface and droplet shown in Fig. 13. CT is deﬁned by 9 5 2 st C T ¼ Kn g 4 4 st

(68)

where st is the thermal accommodation coefﬁcient deﬁned by Harvie and Fletcher (2001c). At the liquid–vapor interface, the energy balance equation is given by kv

T s1 T d1 _ c ¼ ql þ mL d

(69)

where kv is the thermal conductivity of the vapor; Lc the latent heat of the liquid; q1 the heat ﬂux at the droplet surface at the liquid side, which is given by q1 ¼ k1 rd T.

42

YANG GE AND LIANG-SHIH FAN

3. Vapor-Layer Model with Kinetic Treatment at Boundary The vapor-layer model developed in Section IV.A.2 is based on the continuum assumption of the vapor ﬂow. This assumption, however, needs to be modiﬁed by considering the kinetic slip at the boundary when the Knudsen number of the vapor is larger than 0.01 (Bird, 1976). With the assumption that the thickness of the vapor layer is much smaller than the radius of the droplet, the reduced continuity and momentum equations for incompressible vapor ﬂows in the symmetrical coordinates (x,l) are given as Eqs. (43) and (47). When the Knudsen number of the vapor ﬂow is between 0.01 and 0.1, the ﬂow is in the slip regime. In this regime, the ﬂow can still be considered as a continuum at several mean free paths distance from the boundary, but an effective slip velocity needs to be used to describe the molecular interaction between the gas molecules and the boundary. Based on the simple kinetic analysis of vapor molecules near the interface (Harvie and Fletcher, 2001c), the boundary conditions of the vapor ﬂow at the solid surface can be given by l ¼ 0; ux ðx; 0Þ ¼ F s

@ux ðx; 0Þ; @l

ul ðx; 0Þ ¼ 0

(70)

and the boundary conditions at the droplet surface is l ¼ d;

ux ðx; dÞ ux;l ðxÞ ¼ F l

@ux ðx; dÞ; ul ðx; dÞ ¼ ul;l ðxÞ @l

(71)

where ux,l(x), ul,l (x) are the velocities of the droplet surface in x and l direction, respectively; Fs and Fl are deﬁned by F s ¼ ls

2 sv;s sv;s

F l ¼ ll

2 sv;l sv;l

(72)

where ls, ll are the mean free path of the gas molecules near solid and droplet surface. Equations (43) and (47) can be solved by using the similar procedure as given by Section IV.A.2 with the boundary condition given by Eqs. (70) and (71). Thus the vapor pressure can be determined by Z

xb

pðxÞ ¼ p0 þ

rjðxÞ dx

(73)

x

where x0 is the radius of the extent of the vapor layer andjðrÞis given by jðxÞ ¼

u¯ 2x u¯ x ull 2nC 1 u¯ x ðB1 B2 Þ þ B1 2 d x d

(74)

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

43

where C 21 C 1 C 2 2C 1 C 3 þ C 22 þ þ þ C 1 C 3 þ C 23 5 2 3 2C 21 C 1 C 2 C 22 2 1 þ þ þ C1C3 þ C2C3 ¼ 3 2 15 3 6 2 þ 2kl þ 2ks yl 2ks yl ¼ 3 1 þ kl þ 4ks þ 6kl ks 6 þ 6kl 2yl ¼ 1 þ kl þ 4ks þ 6kl ks ¼ ks C 2

B1 ¼ B2 C1 C2 C3

ð75Þ

where kl ¼ F l =d; ks ¼ F s =d; yl ¼ ux;l =¯ux . The averaged velocity of the vapor is expressed by Eq. (54). The pressure distribution in the vapor layer can be obtained by solving Eqs. (54) and (73)– (75) by a piecewise integration method. Details of the solving procedure and how to use the vapor pressure in ﬂow ﬁeld calculation are given in Section IV.A.2. 4. Results and Discussion Three different subcooled impact conditions under which experiments were conducted and reported in the literature are simulated in this study. They are: (1) n-heptane droplets (1.5 mm diameter) impacting on the stainless steel surface with We ¼ 43 (Chandra and Avedisian, 1991), (2) 3.8 mm water droplets impacting on the inconel surface at a velocity of 1 m/s (Chen and Hsu, 1995), and (3) 4.0 mm water droplets impacting on the copper surface with We ¼ 25 (Inada et al., 1985). The simulations are conducted on uniform Cartesian meshes (Dx ¼ Dy ¼ Dz ¼ D). The mesh size (resolution) is determined by considering the mesh reﬁnement criterion in Section V.A. The mesh sizes in this study are chosen to provide a resolution of CPR ¼ 15. Fig. 14 shows the comparison of the photographs from Chandra and Avedisian (1991) with simulated images of this study for a subcooled 1.5 mm n-heptane droplet impact onto a stainless-steel surface of 200 1C. The impact velocity is 93 cm/s, which gives a Weber number of 43 and a Reynolds number of 2300. The initial temperature of the droplet is room temperature (20 1C). In Fig. 14, it can be seen that the evolution of droplet shapes are well simulated by the computation. In the ﬁrst 2.5 ms of the impact (frames 1–2), the droplet spreads out right after the impact, and a disk-like shape liquid ﬁlm is formed on the surface. After the droplet reaches the maximum diameter at about 2.1 ms, the liquid ﬁlm starts to retreat back to its center (frame 2 and 3) due to the surface-tension force induced from the periphery of the droplet. Beyond 6.0 ms, the droplet continues to recoil and forms an upward ﬂow in the center of the

44

YANG GE AND LIANG-SHIH FAN

FIG. 14. n-Heptane droplet collision with surface at 200 1C. Experimental images (right) are presented by Chandra and Avedisian (1991). We ¼ 45. The size of the last frame is reduced.

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

45

droplet (frames 3 and 4), which leads to the bouncing of the droplet up from the surface (frame 5). The photograph at 8.0 ms shows that the tip of the elongated droplet separates from the main body of the droplet and the main body of the droplet then breaks up into smaller secondary drops (frame 5). This phenomenon was reproduced accurately in the simulation. Fig. 15 shows the detailed structure of the droplet from a viewing angle of 601. Experimental images show that a hole is formed in the center of the droplet for a short time period (3.4–4.8 ms) and the center of the liquid droplet is a dry circular area. The simulation also shows this hole structure although a minor variation exists over the experimental images. As the temperature of the surface is above the Leidenfrost temperature of the liquid, the vapor layer between the droplet and the surface diminishes the liquid–solid contact and thus yields a low surface-friction effect on the outwardly spreading liquid ﬂow. When the droplet periphery starts to retreat due to the surface-tension effect, the liquid in the droplet center still ﬂows outward driven by the inertia, which leads to the formation of the hole structure. The impact process of a 3.8 mm water droplet under the conditions experimentally studied by Chen and Hsu (1995) is simulated and the simulation results are shown in Figs. 16 and 17. Their experiments involve water-droplet impact on a heated Inconel plate with Ni coating. The surface temperature in this simulation is set as 400 1C with the initial temperature of the droplet given as 20 1C. The impact velocity is 100 cm/s, which gives a Weber number of 54. Fig. 16 shows the calculated temperature distributions within the droplet and within the solid surface. The isotherm corresponding to 21 1C is plotted inside the droplet to represent the extent of the thermal boundary layer of the droplet that is affected by the heating of the solid surface. It can be seen that, in the droplet spreading process (0–7.0 ms), the bulk of the liquid droplet remains at its initial temperature and the thermal boundary layer is very thin. As the liquid ﬁlm spreads on the solid surface, the heat-transfer rate on the liquid side of the droplet–vapor interface can be evaluated by qdrop ¼ kl

T d2 T d;bulk dT

(76)

where dT is the thickness of the thermal boundary layer; Td2 and Td,bulk are the droplet temperature at the surface and in the bulk, respectively. The thermal boundary layer thickness can be estimated by (Pasandideh-Ford et al., 2001): dT ¼

2d 0 0:5 Re Pr0:4

(77)

where Re is the Reynolds number of the impinging liquid ﬂow and is deﬁned by Re ¼ rVd 0 =m; and V the liquid ﬁlm velocity. At the early stage of the spreading, V is close to the initial impact velocity of the droplet, and thus, it gives a thin thermal boundary layer as shown in frames 1 and 2 of Fig. 16. When

46

YANG GE AND LIANG-SHIH FAN

FIG. 15. Experimental photos (left) by Chandra and Avedisian (1991) and simulated images (right) of the spreading droplet on surface at 200 1C. The formation of a hole in the center of the liquid is captured.

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

47

FIG. 16. Simulated temperature ﬁeld in the liquid and solid phases.

the droplet spreads to the maximum extent and starts to recoil, the liquid velocity diminishes to zero and the thermal boundary layer is disrupted. Until then, the temperature rise inside the droplet becomes signiﬁcant (frames 3 and 4 of Fig. 16). The simulated temperature distribution inside the solid surface is also shown in Fig. 16, in which the isotherm of 399 1C is chosen to represent the area of temperature drop during the impact of the droplet. The simulation of droplet impact shown in Fig. 16 is conducted under perfectly symmetrical conditions, which is not easy to achieve in the experiments.

48

YANG GE AND LIANG-SHIH FAN

FIG. 17. Droplet impacts on the ﬂat surface with a small tangential velocity. Other conditions are the same as those in Fig. 16.

As the droplet is released from the nozzle and moves toward the superheated surface, some uncontrollable factors such as the angle of dropping, obliquity of the surface, and perturbation in the ambient conditions render it difﬁcult to maintain a perfectly normal collision between the droplet and solid surface. The 3-D simulation of this study is capable of reproducing the imperfect droplet–surface impact condition represented by asymmetrical collision. Fig. 17 shows the simulated 3-D images of the droplet under the same impact condition as that shown in Fig. 16 but with a 5 cm/s tangential velocity. The simulated solid surface temperature is shown in Fig. 18 with comparisons to the experimental measurements of Chen and Hsu (1995). It can be seen that with a small obliquity, there is a signiﬁcant effect on the movement behavior of the droplet. Speciﬁcally, in this case, the droplet moves away from the impact center during the recoiling process, which leads to a faster recovery of the solid surface temperature. The simulations are further conducted under the experimental conditions of Inada et al. (1985). In their experiments, 4.0 mm water droplets impact on a heated platinum surface at a temperature up to 420 1C. The subcooling degree

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

49

temperature of solid surface at r=0 (°C)

500

400

300

200 at z= 0.00mm at z=-0.12mm at z=-0.24mm at z=-0.36mm Experiment of Chen and Hsu(1995)

100

0 0

5

10

15

20

25

30

35

40

45

50

Time (ms) FIG. 18. Simulated solid surface temperatures compared with the experiments of Chen and Hsu (1995). The droplet impacts on the ﬂat surface with a small tangential velocity (5 cm/s) as shown in Fig. 17.

of the droplet (dTsub) can be varied from 2 1C to 88 1C. The droplet falls down 20 mm before it impacts on the surface, thus the velocity of the impact can be estimated as 64 cm/s, which gives a Weber number of about 25. In this simulation, the initial temperature of surface is ﬁxed on 420 1C, which ensure the nonwetting contact between the droplet and the surface. Fig. 19 shows the simulated solid temperature compared with the measured value at two locations. T1 and T2 were measured at 0.28 and 0.74 mm depth beneath the surface (Inada et al., 1985). Simulated surface temperatures agreed well with the measured T1 and T2; both decrease until 13–14 ms after the impact. The simulated temperature at the surface of the solid (Tw) is also shown in Fig. 19. The maximum temperature drop at the surface is about 60 1C, which occurs at about 13 ms after the impact.

C. SIMULATION

OF

DROPLET– PARTICLE COLLISION

IN THE

LEIDENFROST REGIME

An efﬁcient numerical model to describe the unsteady, 3-D ﬂuid ﬂow during the droplet–particle collision with evaporation is developed (Ge and Fan, 2005). From the numerical point of view, the droplet and solid surface need to be

50

YANG GE AND LIANG-SHIH FAN

440

Simulated TW Simulated T1 Measured T1 Simulated T2 Measured T2

430

Surface temperature (°C)

420 410 400 390 380 370 360 350 0

5

10

15

20

time after impact (ms) FIG. 19. Simulated solid surface temperatures compared with the experiments of Inada et al. (1985). T1 and T2 are the temperatures at locations inside the surface with Z1 ¼ 0.28 mm and Z2 ¼ 0.74 mm.

captured in the ﬂow ﬁeld and the solid-ﬂow boundary condition at the particle surface needs to be satisﬁed. The numerical method adopted is a combination of the level-set approach and the IBM. The evaporation effect is accounted for by the vapor pressure force, which is calculated in a dynamics vapor-ﬂow model. Energy balance equations in each phase are solved with interface boundary condition to give the temperature distribution and evaporation rate. 1. Hydrodynamic Model with Level-Set Method In this model, two level-set functions (fd, fp) are deﬁned to represent the droplet interface (fd) and the moving particle surface (fp), respectively. The free surface of the droplet is taken as the zero in the droplet level-set function fd ð~ x; tÞ, and the advection equation (Eq. (3)) of the droplet level-set function (fd) is solved to capture the motion of the droplet surface. The particle level-set function (fp) is deﬁned as the signed distance from any given point ~ x in the Eulerian system to the particle surface: fp ¼ ~ x~ x0 ðtÞ Rp (78) where ~ x0 ðtÞ is the position vector of the center of the particle and Rp the particle radius. With this deﬁnition, fp40 when ~ x is outside the particle, fpo0 when ~ x is inside the particle.

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

51

The momentum equation for this 3-phase ﬂow system in the Eulerian frame can be given by ! ~ @V ~V ~ ¼ rp þ r~ ~ þ skðfd Þdðfd Þrfd þrV g þ r ð2mDÞ r @t ~vapor þ F~p ðfp Þ þ F

ð79Þ

In this equation, the presence of the solid particle in the ﬂuid is represented by a virtual boundary body force ﬁeld, F~p ðfp Þ, deﬁned by the IBM which will be ~vapor is vapor pressure force exerting on the discussed in Section IV.C.2. F droplet–particle contact area due to the effect of the evaporation, which will be discussed in vapor-layer model of Section IV.C.3. 2. Immersed-Boundary/Level-Set Method for Particle– Flow Interaction In the IBM, the presence of the solid boundary (ﬁxed or moving) in the ﬂuid can be represented by a virtual body force ﬁeld F~p ðfp Þ applied on the computational grid at the vicinity of solid–ﬂow interface. Considering the stability and efﬁciency in a 3-D simulation, the direct forcing scheme is adopted in this model. Details of this scheme are introduced in Section II.B. In this study, a new velocity interpolation method is developed based on the particle level-set function (fp), which is shown in Fig. 20. At each time step of the simulation, the ﬂuid–particle boundary condition (no-slip or free-slip) is imposed on the computational cells located in a small band across the particle surface. The thickness of this band can be chosen to be equal to 3D, where D is the mesh size (assuming a uniform mesh is used). If a grid point (like p and q in Fig. 20), where the velocity components of the control volume are deﬁned, falls into this band, that is Dx4fp 4 Dx

(80)

the velocity at these points will be redeﬁned using linear interpolation based on the velocity and particle level-set function (fp) at the neighboring grid point. At grid point p in Fig. 20, which is located in the band and outside the particle surface, the ﬂuid velocity is determined by (for no-slip boundary condition): U p ¼ V p ðtÞ þ

n fp X U p0 ;i V p ðtÞ fp0 ;i n i

(81)

where Up0 and fp0 are the ﬂuid velocity and particle level-set function at grid point p0 (in Fig. 20). It should be noted that only the neighboring grid points located outside the band are chosen to interpolate the velocity at point p. The velocities at these points (p0 ) are obtained by solving the N-S equation like other points far from the interface. n is the total number of these neighboring points. It can be seen that when fp ¼ 0, meaning that the grid point p is right on the particle surface, the velocity at p is equal to the particle velocity: Up ¼ Vp(t).

52

YANG GE AND LIANG-SHIH FAN

∆x

∆x P’

P’ Up’

p

P’

q Up q’

Fluid

Particle

interface FIG. 20. Velocity interpolation scheme based on the particle level-set function.

The velocity at grid point q (in Fig. 20), which is located inside the small band and on the particle side, is determined by interpolating the ﬂuid velocity at the neighboring grid points outside the particle surface: U q ¼ V p ðtÞ þ

m fq X U p;i V p ðtÞ fp;i m i

(82)

where m is the total number of neighboring grid points which are located outside the particle surface.

3. Vapor Layer and Heat-Transfer Model As the vapor ﬂows in the direction along the spherical surface of the particle, a boundary layer coordinate (x, l, o) given in Fig. 21 is employed to describe the vapor-layer equation. In this coordinate, the continuity and momentum equations for incompressible vapor ﬂows with gravitation terms neglected

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

O

53

P

Particle surface ξ

λ

ω

FIG. 21. Boundary layer coordinates.

are given by @ux cotðx=RÞ 2 @ul ux þ ul þ þ ¼0 R R @x @l

(83)

2 @ux @ux ux ul @ P @ ul 2 @ux @2 ux ux þ ul þ ¼ 2 n @x r @x @l R @x@l R @l @l

(84)

ux

2 @ul @ul u2x @ P n @ ul @ux @ 2 ux R 2 þ ul ¼ R þ @l r R @x @l R @x @x@l @x x @ul ux @ux þ cot R @x R @l

ð85Þ

where ux, ul are the vapor-ﬂow velocities in x and l direction, respectively. Based on the assumptions and the order of magnitude analysis described in Section IV.A.2, and following the similar procedure of solution, the averaged vapor-ﬂow velocity is given by Z u¯ x ðxÞ ¼ O 0

1

1 d2 3 @ p 1 Red Fd dZ ¼ uld ðxÞ 2 20 @x r 12g

(86)

The vapor continuity equation can be expressed by u¯ x ðxÞ ¼

1 xdðxÞ

Z

x

x0 uld ðx0 Þ dx0 0

(87)

54

YANG GE AND LIANG-SHIH FAN

Droplet Surface

i+1

i φp,i

Pi+1 δ

Particle Surface

i-1

-φd,i, δi Pi

Pi-1

i+2 i’+1

i’+3

i’

i’-1

i’+2

FIG. 22. Scheme for determining the thickness of the vapor layer.

The uld(x) is calculated using Eq. (55). The pressure distribution in the vapor layer can be obtained by solving Eqs. (86) and (87). In this procedure, the thickness of the vapor layer (d(x)) and the extent of the layer (xb) are obtained through a vapor-layer identiﬁcation scheme based on the particle and droplet level-set function. Fig. 22 illustrates the details of the scheme. As the particle is in motion, at every time step, a series of grid points near the particle surface are ﬁrst identiﬁed to measure the vapor layer. As shown in Fig. 22, these grids points are in a small band around the surface and can be outside the surface (y,i1, i, i+1, i+2,y) or inside the surface (y, i0 1, i0 , i0 +1, i0 +2,y). If the droplet surface is represented by points (y, Pi1, Pi, Pi+1,y) in Fig. 22 and point Pi is located on the mesh line between the mesh knots i and i0 , the vapor-layer thickness at Pi can be calculated based on the values of the level-set function at i and i’ deﬁned as (fd,i,fp,i) and (fd,i0 ,fp,i0 ), respectively. Since the level-set function is the signed distance from the computation knots to the droplet and particle surface after the redistance process is performed, the vapor-layer thickness (di) at Pi can be estimated by di ¼

fd;i þ fp;i þ fd;i0 þ fp;i0 2

(88)

The heat-transfer model described in Sections IV.A.3 and IV.B.2 can be applied to calculate the temperature distribution inside the droplet and energy

55

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

balance at the interface. Inside the particle, the heat-conduction equation is @T ¼ as r rT @t

(89)

where as is the thermal diffusivity of particle. When the particle surface is in contact with the droplet through the vapor layer, the boundary condition at the particle surface is given by kv

T ss T ds ¼ ks~ np r p T s d

(90)

where ~ np is the normal vector of the particle surface, rpT the temperature gradient which is evaluated only on the particle side. This boundary condition, Eq. (90) is imposed on the computational cells, which are located in a small band near the particle surface on the solid side. 4. Results and Discussion Droplet–particle collision conditions under which experiments are conducted are simulated in this study. In the experiment, the brass particle is heated on a heating plate with adjustable temperature settings, and a high-speed camera capable of capturing 500 frames per second will be used to record the droplet–particle collision process. A droplet is formed by the use of the syringe with various needle sizes. In the experiments, acetone droplets with 1.6–2.2 mm diameter impact onto the brass particles of sizes 5.5 mm or 3.6 mm.The particle temperature is 200 1C–300 1C, which is much higher than the boiling point of acetone and ensures that the contact is nonwetting. The simulation is conducted on uniform Cartesian meshes (Dx ¼ Dy ¼ Dz ¼ D). The mesh sizes in this study are chosen to give a resolution of CPR ¼ 15. Fig. 23 shows the comparison of the photographs with simulated images for a 2.1 mm acetone droplet impact onto a 5.5 mm brass particle of 250 1C. The impact velocity is 45 cm/s. The initial temperature of the droplet is close to the boiling point of acetone (56 1C). A comparison of the images shows that the impact process is predicted well by this model. Similar to the impact on a ﬂat surface, the droplet spreads on the particle surface at the ﬁrst stage (0–5.5 ms), then recoils due to the vapor pressure force and surface tension force (7.5–13.5 ms), and eventually rebounds away from the particle (17.5–27.5 ms). The droplet ﬁlm reaches the maximum extent at about 7.0 ms, by which time the radius of droplet–particle contact area is about 1.5 times that of the original radius of the droplet. The total contact time is about 17 ms, which is relevant to the ﬁrst-order vibration period of the oscillating drop. Fig. 24 shows the collision process between a moving particle (1.5 mm in diameter) and a large water droplet. A water droplet 2.5 mm in diameter has an initial velocity of 25 cm/s and an initial temperature of 100 1C. The initial velocity and temperature of the particle are 25 cm/s and 400 1C, respectively.

56

YANG GE AND LIANG-SHIH FAN

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0 0

0.5 0.1 0.2 0.3

0.4

0.5

0.6 0

-0.5ms

0.5

0.6 0

0.4

0.5

0.6 0

0.4

0.5

0.6 0

0.4

0.5

0.6 0

0.5 0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.5 0.1 0.2 0.3

0.4

0.5

0.6 0

1.5ms

0 0

0.5 0.1 0.2 0.3

17.5ms 0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.5 0.1 0.2 0.3

0.4

0.5

0.6 0

5.5ms

0 0

0.5 0.1 0.2 0.3

21.5ms 0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0 0

0.4

0.6

0.5

0 0

0.5 0.1 0.2 0.3

13.5ms

0.6

0 0

0 0

0.5 0.1 0.2 0.3

7.5ms

0.4

0.5

0.6 0

0 0

0.5 0.1 0.2 0.3

27.5ms

FIG. 23. Experimental photos (left) and simulated images (right) of the 2.1 mm acetone droplet impact on 5.5-mm particle at 250 1C. Impact velocity V ¼ 45 cm/s.

The physical properties of the particle are the same as FCC particles. The simulation is conducted using a 140 140 200 rectangular mesh covering a 7 mm 7 mm 10 mm computational domain. Both the 3-D images and the temperature ﬁeld are shown in this ﬁgure. It can be seen in this ﬁgure that since the inertia (mass) of the particle is smaller than that of the droplet, the particle velocity decreases rapidly as soon as the particle collides with the droplet. After the collision, the surface tension of

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

57

400

t = 0.0ms

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

200

0.3

0.2

150

350 300 250

0.2

0.1

0.1 0.2

0.4

0.6

0.8

1

1.2

0

0.6 0.4 0.2

0

100 0

0.2

0.4

0.6

0.8

1

50 0 400

t = 10ms

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

200

0.3

0.2

150

350 300 250

0.2

0.1

0.1 0.2

0.4

0.6

0.8

1

1.2

0

0.6 0.4 0.2

0

100 0

0.2

0.4

0.6

0.8

1

50 0 400

t = 14ms

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

200

0.2

150

350 300 250

0.3 0.2

0.1

0.1 0.2

0.4

0.6

0.8

1

1.2

0

0.6 0.4 0.2

0

100 0

0.2

0.4

0.6

0.8

1

50 0 400

t = 17ms

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

200

0.3

0.2

150

350 300 250

0.2

0.1

0.1 0.2

0.4

0.6

0.8

1

1.2

0

0.6 0.4 0.2

0

100 0

0.2

0.4

0.6

0.8

1

50 0 400

t = 24ms

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

200

0.3

0.2

150

350 300 250

0.2

0.1

0.1 0.2

0.4

0.6

0.8

1

1.2

0

0.6 0.4 0.2

0

100 0

0.2

0.4

0.6

0.8

1

50 0

FIG. 24. Particle and droplet collision at the same initial velocity. The images on the right show the temperature ﬁeld.

58

YANG GE AND LIANG-SHIH FAN

the droplet and the vapor-layer pressure induced by the evaporation drive the particle to the inverse direction. Since the Weber number of the collision is not very large (We ¼ 15), the droplet only undergoes small deformation without splashing. After collision, the droplet still moves along its original path with a decreased velocity, while the rebounding velocity of the particle is larger than that of the droplet. It can also be found that the temperature drop of the particle is not signiﬁcant in this condition.

V. Concluding Remarks The most signiﬁcant advantage of the level-set method and the IBM in solving 3-D, three-phase ﬂow problems is that the governing equations can conveniently be discretized and solved on ﬁxed, regular structured grids, rather than resorting to the classic body-ﬁtted mesh approach. These methods are efﬁcient in utilizing computational resources while retaining the accuracy of the computational results. The level-set method implicitly captures the motion of the interface by solving the advection equation of the level-set function, and can be easily implemented for 3-D interface tracking. The level-set methods are particularly effective in handling ﬂow problems that involve changes in the topology of evolving interfaces, and in which the speed of the interface is sensitive to local surface geometry, such as curvature and normal vector. Although, like other front-capture techniques, the level-set method may suffer from some problems associated with the preservation of mass conservation (Puckett et al., 1997), the method is able to accurately compute interfacial ﬂows with surface tension force and other complex physical forces acting on the front. In the conventional IBM, the moving front is represented by a ﬁnite number of discrete Lagrangian markers along the interface, which move in the ﬂow. In general, this arrangement has the property of preservation of the clear interface position and hence mass conservation; however, with this arrangement, it is difﬁcult to trace the interface with complicated shape and topological change in three dimensions. The combination of the level-set method and the IBM as described in this study can allow accuracy in interface movement, as well as conservation in moving object mass when simulating the complex three-phase interaction systems. In system 1, the 3-D dynamic bubbling phenomena in a gas–liquid bubble column and a gas–liquid–solid ﬂuidized bed are simulated using the level-set method coupled with an SGS model for liquid turbulence. The computational scheme in this study captures the complex topological changes related to the bubble deformation, coalescence, and breakup in bubbling ﬂows. In system 2, the hydrodynamics and heat-transfer phenomena of liquid droplets impacting upon a hot ﬂat surface and particle are analyzed based on 3-D level-set method and IBM with consideration of the ﬁlm-boiling behavior. The heat transfers in

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each phase are solved using a microscale vapor-ﬂow model which is applied to determine the vapor pressure force during the contact process between the droplet and the superheated surface. The simulation model developed in this study is capable of reproducing the droplet impaction process with signiﬁcant heat transfer and phase change (evaporation). Both the saturated impact and the subcooling impact are considered. The simulation results are found to be in good agreement with the experimental measurements of the droplet deformation process and the surface-temperature variation.

Nomenclature NOTATION CD CD0 Cs D~ d dp fam fd fdr fs g Hb k Kn l Lc mp _ m n p Pr q R Re S T t U u

modiﬁed drag coefﬁcient drag coefﬁcient Smagorinsky coefﬁcient stress tensor distance function (or diameter) diameter of particle added mass force ﬂuid–particle interaction force drag force surface tension force gravitational acceleration heaviside function heat conductivity Knudsen number length scale latent heat mass of particle mass evaporation rate normal vector pressure Prandtl number heat ﬂux (or virtual mass source) radius of the droplet particle Reynolds number shear strain rate temperature time velocity scale of the vapor ﬂow velocity of the vapor ﬂow

60 V We x

YANG GE AND LIANG-SHIH FAN

velocity (or volume) Weber number position vector

GREEK LETTERS a D G F O b d e f k l m nT r s t x Z n

thermal diffusivity grid size gas–liquid interface function deﬁned in vapor-layer model (4.7) cell volume (or function deﬁned in vapor-layer model (4.7)) half of the thickness of the interface d function (or the thickness of the vapor-layer or thermal boundary layer) void fraction level-set function curvature of the surface constant deﬁned in Eq. (24) (or coordinate deﬁned in vapor-layer model) molecular viscosity kinematic turbulent viscosity density surface tension (or thermal accommodation coefﬁcient) viscous stress tensor, artiﬁcial time coordinate deﬁned in vapor-layer model dimensionless variable deﬁned in Eq. (46) kinematic viscosity of vapor ﬂow

SUBSCRIPTS AND SUPERSCRIPTS b d g l o p r s v

boundary droplet gas phase liquid phase initial particle radial direction solid vapor phase

SIMULATION OF GAS– LIQUID AND GAS– LIQUID– SOLID FLOW SYSTEMS

x l G ij k sa sg

61

vapor-layer coordinate vapor-layer coordinate gas–droplet interface cell index particle index saturated sub-grid scale

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MULTISCALE MODELING OF GAS-FLUIDIZED BEDS M.A. van der Hoef1, M. Ye1, M. van Sint Annaland1, A.T. Andrews IV2, S. Sundaresan2 and J.A.M. Kuipers1, 1

Department of Science & Technology, University of Twente, 7500 AE Enschede, The Netherlands 2 Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA I. Introduction A. Gas-Fluidized Beds B. Numerical Models for Gas and Solid Flows C. The Multi Level Modeling Approach for Gas–Solid Flows II. Lattice Boltzmann Model A. From Lattice-Gas to Lattice-Boltzmann B. The Lattice BhatnagarGrossKrook Model C. Modeling Solid Particles D. Results for the Gas–Solid Drag Force III. Discrete Particle Model A. Introduction B. Particle Dynamics: The Soft-Sphere Model C. Gas Dynamics D. Interphase Coupling E. Energy Budget F. Results for the Excess Compressibility IV. Two-Fluid Model A. Introduction B. GOVERNING Equations C. General Kinetic Theory D. Kinetic Theory of Granular Flow E. Numerical Solution Method F. Application to Geldart A Particles V. Towards Industrial-Scale Models A. The Limits of the Two-Fluid Model B. State-of-the-Art on Dealing with Unresolved Structures C. A Different Approach: The Discrete Bubble Model VI. Outlook References

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Corresponding author. Tel: +31 53 489 3039; Fax: +31 53 489 2882 E-mail: j.a.m.kuipers@tnw.

utwente.nl 65 Advances in Chemical Engineering, vol. 31 ISSN 0065-2377 DOI 10.1016/S0065-2377(06)31002-2

Copyright r 2006 by Elsevier Inc. All rights reserved

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Abstract Numerical models of gas-ﬂuidized beds have become an important tool in the design and scale up of gas-solid chemical reactors. However, a single numerical model which includes the solid-solid and solid-ﬂuid interaction in full detail is not feasible for industrial-scale equipment, and for this reason one has to resort to a multiscale approach. The idea is that gas-solid ﬂow is described by a hierarchy of models at different length scales, where the particle-particle and ﬂuid-particle interactions are taken into account with different levels of detail. The results and insights obtained from the more fundamental models are used to develop closure laws to feed continuum models which can be used to compute the ﬂow structures on a much larger (engineering) scale. Our multi-scale approach involves the lattice Boltzmann model, the discrete particle model, and the continuum model based on the kinetic theory of granular ﬂow. In this chapter we give a detailed account of each of these models as they are employed at the University of Twente, accompanied by some illustrative computational results. Finally, we discuss two promising approaches for modeling industrial-size gas-ﬂuidized beds, which are currently being explored independently at the Princeton University and the University of Twente. I. Introduction A. GAS-FLUIDIZED BEDS Gas-ﬂuidized beds consist of ﬁne granular material (usually smaller than 5 mm) that are subject to a gas ﬂow from below, large enough so that the gas drag on the particles can overcome the gravity and the particles can ﬂuidize. When in the ﬂuidized state, the moving particles work effectively as a mixer resulting in a uniform temperature distribution and a high mass transfer rate, which are beneﬁcial for the efﬁciency of many physical and chemical processes. For this reason, gas-ﬂuidized beds are widely applied in the chemical, petrochemical, metallurgical, environmental, and energy industries in large-scale operations involving adhesion optimized coating, granulation, drying, and synthesis of fuels and base chemicals (Kunii and Levenspiel, 1991). Lack of understanding of the fundamentals of dense gas–particle ﬂows in general has led to severe difﬁculties in the design and scale-up of these industrially important gassolid contactors (van Swaaij, 1985). In most cases, the design and scale-up of ﬂuidized bed reactors is a fully empirical process based on preliminary tests on pilot-scale model reactors, which is a very time consuming and thus expensive activity. Clearly, computer simulations can be a very useful tool to aid this design and scale-up process. Basically, such simulations can be used for two different

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purposes. Firstly, to contribute to our understanding—that is, the simulations are used to obtain a fundamental insight into the complex dynamic behavior of dense ﬂuidized suspensions, which should lead to an understanding based on elementary physical principles such as drag, friction, dissipation, etc.; this also includes the testing of elementary assumptions in theoretical models, such as the Maxwell velocity distribution of the particles. Secondly, the simulations can be used as a design tool, where the ultimate goal is to have a numerical model with predictive capabilities for the dense gas–particle ﬂows encountered in engineering-scale equipment. Clearly, it will not be possible to have one single simulation method that can achieve this, but one rather needs a hierarchy of methods for modeling the ﬂow phenomena on different length and time scales. Obviously, these two items are not strictly separated; in contrast, the most fruitful approach is when they are simultaneously followed, so that they can mutually beneﬁt from each other. In this chapter, we want to focus on the use of simulation methods as a design tool for gas-ﬂuidized bed reactors, for which we consider gas–solid ﬂows at four distinctive levels of modeling. However, before discussing the multilevel scheme, it is useful to ﬁrst brieﬂy consider the numerical modeling of the gas and solid phase separately. B. NUMERICAL MODELS

FOR

GAS

AND

SOLID FLOWS

1. Gas Phase The description of a gas ﬂow is well established from the micro- to macroscales. The length scale of a gas ﬂow can be characterized by the local Knudsen number Kn, which is deﬁned as Kn ¼

l L

where l is the mean free path of the molecules and L is the characteristic length scale of the ﬂow. The models to describe a gas ﬂow are schematically shown in Fig. 1. For large-scale systems with Kno0.01, the gas ﬂow can be described by ordinary ﬂuid dynamics where the macroscopic ﬁelds (such as density and velocity) are formulated by Navier–Stokes equations in a three-dimensional (3D) coordinate space, together with no-slip boundary conditions. A number of welldeveloped numerical algorithms and meshing techniques in computational ﬂuid dynamics (CFD) can be used to handle very complex geometries (Anderson, 1995). If the system becomes smaller, say 0.01oKno0.1, the Navier–Stokes equations still hold, but caution must be exercised for the boundary conditions because partial slip might exist between the gas–solid interfaces. For a rareﬁed gas where Kn40.1, the continuum assumption breaks down, and the so-called kinetic theory of (dense) molecular gases should be applied. Kinetic theory differs from the ordinary ﬂuid dynamics as there is just one ﬁeld (the density of molecules) in the phase space. The basic equation in kinetic theory, in the

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M.A. VAN DER HOEF ET AL.

FIG. 1. The various levels of modeling gas ﬂow.

simplest form, is the Boltzmann equation that describes the evolution of the density function f in a six-dimensional (6D) space (three coordinates and three velocity components) (Chapman and Cowling, 1970). At this scale, computational techniques such as molecular dynamics (MD) (Allen and Tildesley, 1990) and direct simulation Monte Carlo (DSMC) approach (Bird, 1976) can be very efﬁcient. In these techniques, the motion of molecules is traced on an individual basis. Gas pressure and other transport coefﬁcients, including gas viscosity and thermal conductivity, are obtained by methods from statistical mechanics (Chapman and Cowling, 1970). Molecules can be treated either as hard spheres or points with certain interaction potentials, depending on their physical properties. In the extreme case where the mean free path is very large compared to the system sizes (i.e., Kn410), the molecules move freely and just collide with the walls. This is the limit of free molecular flow, where the system behaves as an ideal gas. Clearly, there are two quite different types of models for a gas ﬂow: the continuum models and the molecular models. Although the molecular models can, in principle, be used to any length scale, it has been almost exclusively applied to the microscale because of the limitation of computing capacity at present. The continuum models present the main stream of engineering applications and are more ﬂexible when applying to different macroscale gas ﬂows; however, they are not suited for microscale ﬂows. The gap between the continuum and molecular models can be bridged by the kinetic theory that is based on the Boltzmann equation. 2. Solid Phase The methods used for modeling pure granular ﬂow are essentially borrowed from that of a molecular gas. Similarly, there are two main types of models: the continuous (Eulerian) models (Dufty, 2000) and discrete particle (Lagrangian) models (Herrmann and Luding, 1998; Luding, 1998; Walton, 2004). The continuum models are developed for large-scale simulations, where the controlling equations resemble the Navier–Stokes equations for an ordinary gas ﬂow. The discrete particle models (DPMs) are typically used in small-scale simulations or

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in the investigation of the detailed physics of granular ﬂow. A kinetic theory of granular ﬂow (KTGF) has also been proposed to connect the microscale picture of granular ﬂow to the macroscale description (Jenkins and Savage, 1983; Lun et al., 1984). However, a granular ﬂow differs signiﬁcantly from a molecular gas ﬂow. The collisions between molecules are elastic, and the kinetic energy is conserved in isothermal systems. For the molecular gas, there is a well-deﬁned equilibrium state in the absence of external energy sources, and one can deﬁne a thermal temperature based on the internal kinetic energy. The interaction between macroscopic particles, however, is far more complicated. The collision between two macroscopic particles will come with surface friction and elastic–plastic deformation, which leads to the dissipation of kinetic energy. This inelasticity forms the primary feature of granular ﬂow that differentiates it from a molecular gas (Campbell, 1990). Clearly, without any external energy sources, a granular system will continuously ‘‘cool down,’’ and an equilibrium state can never be reached. To model granular systems, DPMs using the same techniques as MD methods can be used, where it is assumed that the particle motion can be well described by the Newtonian equations. However, in order to establish a continuum description, a number of serious difﬁculties are encountered when one tries to describe the ﬁelds in phase space. First, an energy source term and a dissipative term should be included in the Boltzmann equations, which complicates the (approximate) solution. Also, particle sizes may show a certain distribution even for the same type of materials. It is well known that a difference in particle sizes will result in the segregation of granular materials (e.g., the Brazil nut effect). Furthermore, in most granular ﬂows the effect of gravity cannot be ignored, which introduces an anisotropy in the velocity ﬂuctuation of particles. Clearly, the deﬁnitions of the particle-phase pressure and other transport coefﬁcients are not straightforward because normally a homogeneous equilibrium state does not exist. For these reasons, the construction of a reliable hydrodynamical model for granular ﬂow offers a great challenge for both scientists and engineers (Goldhirsch, 1999). 3. The Interphase Coupling The prime difﬁculty of modeling two-phase gas–solid ﬂow is the interphase coupling, which deals with the effects of gas ﬂow on the motion of solids and vice versa. Elgobashi (1991) proposed a classiﬁcation for gas–solid suspensions based on the solid volume fraction es, which is shown in Fig. 2. When the solid volume fraction is very low, say eso106, the presence of particles has a negligible effect on the gas ﬂow, but their motion is inﬂuenced by the gas ﬂow for sufﬁciently small inertia. This is called ‘‘one-way coupling.’’ In this case, the gas ﬂow is treated as a pure ﬂuid and the motion of particle phase is mainly controlled by the hydrodynamical forces (e.g., drag force, buoyancy force, and so

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FIG. 2. Interphase coupling. Based on Elgobashi, Appl. Sci. Res. (1991).

on), while the particle–particle interaction is assumed to be irrelevant. With increasing solid volume fraction up to eso103, the effects of the particle phase on the gas-phase ﬂow pattern will become important. In this region, turbulent structures encountered in gas ﬂows can be modiﬁed by the presence of particles. It is commonly accepted that particle–particle interactions still do not play a dominant role in this regime, which we normally refer to as ‘‘two-way coupling.’’ For even higher solid volume fractions (es4103), the momentum of particles will be transported not only by the free-ﬂight mechanism but also by the collisions between particles and particles with the conﬁning walls. This means that the particle–particle interaction will be very important and ‘‘fourway coupling’’ should be taken into account. Note that it is precisely this denseparticle regime that is important for the industrial applications of two-phase ﬂows. However, a numerical model that includes the solid–solid and solid–ﬂuid interactions in full detail is not feasible for industrial-scale equipment, and for this reason one has to resort to a multilevel approach. C. THE MULTI LEVEL MODELING APPROACH

FOR

GAS– SOLID FLOWS

As mentioned previously, the construction of reliable models for large-scale gas–solid contactors is seriously hindered by the lack of understanding of the fundamentals of dense gas–particle ﬂows (van Swaaij, 1985). In particular, the phenomena that can be related to the effective gas–particle interaction (drag forces), particle–particle interactions (collision forces), and particle–wall interaction are not well understood (Kuipers and van Swaaij, 1998; Kuipers et al., 1998). The prime difﬁculty here is the large separation of scales: the largest ﬂow structures can be of the order of meters, and yet, these structures are directly inﬂuenced by details of the particle–particle and particle–gas interactions, which take place on the scale of millimeters, or even micrometers. As shown above, for both the gas and particle phase, continuum-(Eulerian) and discrete-(Lagrangian) type of models can be applied, depending on the length scales involved. Thus, in order to model gas–solid two-phase ﬂows at different scales, one can

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71

FIG. 3. Multilevel modeling scheme.

choose appropriate combinations of the gas- and solid-phase models, where in all cases a four-way coupling is used either directly or effectively, depending on the scale of the simulation. The basic idea is that the smaller scale models, which take into account the various interactions (ﬂuid–particle, particle–particle) in detail, are used to develop closure laws that can represent the effective ‘‘coarsegrained’’ interactions in the larger scale models. Note that it is not guaranteed that some subtle correlations between small- and large-scale processes exist, which cannot be captured by effective interactions. However, experience has shown that in many cases the main characteristics of gas–solid ﬂows can be well described by the use of closure relations. In this chapter, we discuss three levels of modeling: the lattice Boltzmann model (LBM), the DPM and the two-ﬂuid model (TFM) based on the KTGF. In Fig. 3, we show a schematic representation of the three models, including the information that is abstracted from the simulations, which is incorporated in higher scale models via closure relations, with the aid of experimental data or theoretical results. We will next give a brief description of each of these models. 1. Two-Fluid Model At the largest scale, a continuum description is employed for both the solid phase and the gas phase, and a CFD-type Eulerian code is used to describe the time evolution of the local mass and momentum density of both phases (see Refs. Kuipers et al., 1992 and Gidaspow, 1994 amongst others). In a more sophisticated model, based on the KTGF, also the local granular temperature of the solid phase is a dynamical variable, and thus included in the update. With modernday computers, the TFM model can predict the ﬂow behavior of gas–solid ﬂows of systems with a linear dimension of the order of 1 m, denoted as the ‘‘engineering’’ scale, corresponding typically to the size of pilot plants, which is in between the laboratory scale (0.1 m) and the industrial scale (10 m). The TFM relies heavily on closure relations for the effective solid pressure and viscosity, and gas–solid drag. The basic idea of the multiscale modeling is that

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these relations are obtained from kinetic theory and from numerical data collected in the more detailed scale models. 2. Discrete Particle Model At one level higher in detail (and thus smaller in scale), we have the DPM. Here the continuum description for the solid phase is replaced by a description with discrete particles, which are modeled by spheres (Hoomans et al., 1996, 2000). The ﬂow ﬁeld is still continuous and updated by the same methods as in the TFM, where the scale at which the gas ﬂow ﬁeld is described is an order of magnitude larger than the particles (a CFD-grid cell typically contains O(102)–O(103) particles). The motion of the particles is governed by Newton’s law, where the forces on the particles are integrated using standard schemes for ordinary differential equations (ODEs). These forces follow from the interaction with the ﬂuid phase and collisions with the other particles. Therefore, both a drag-force closure and a collision model have to be speciﬁed for this level of modeling. The advantage of this model is that it can account for the particle–wall and particle–particle interactions in a realistic manner. This model allows one to validate (and modify) the viscosity and pressure closures derived from the KTGF, which are used in the TFM simulations. Still, a closure law for the effective momentum exchange between the two phases has to be speciﬁed for this model. The system sizes that can be studied are of the order of O(105) particles, which corresponds (for millimeter-sized particles) to systems that have a linear dimension of the order of 0.1 m (i.e., laboratory scale). 3. Lattice Boltzmann Model At the most detailed level of description, the gas ﬂow ﬁeld is modeled at scales smaller than the size of the solid particles. The interaction of the gas phase with the solid phase is incorporated by imposing ‘‘stick’’ boundary conditions at the surface of the solid particles. This model thus allows one to measure the effective momentum exchange between the two phases, which is a key input in all the higher scale models. A particularly efﬁcient method to solve the ﬂow ﬁeld between the spheres is the LBM (Ladd, 1994; Ladd and Verberg, 2001), although in principle other direct numerical simulation (DNS) techniques can also be used. The number of particles in such a simulation is typically around 500, which is sufﬁciently high to account for swarm effects. The goal of these simulations is to construct drag laws for dense gas–solid systems. For low Reynolds numbers (Re), the functional form of the drag law can be derived from theory using the Carman–Kozeny approximation, where the simulation data is then used to determine the unknown parameters such as the Kozeny constant. For higher Reynolds numbers, a theoretical evaluation of the functional form is not possible and the drag law is simply constructed as the best possible ﬁt to the simulation data, where the functional form is dictated by a compromise between

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simplicity and accuracy. For both low and high Reynolds numbers, the drag laws are validated (and possibly adjusted) on the basis of pressure-drop data. A graphical representation of the multilevel approach is shown in Fig. 4. All three models are now commonly accepted and are widely used by a number of research groups (both academic and industrial) around the world. In a recent paper, we have given an overview of the three models as they are employed at the University of Twente, together with some illustrative examples (Van der Hoef et al., 2004). In this chapter, we will focus on the technical details of each of the models, much of which has not been published elsewhere. The development of detailed closure relations from the simulations, as indicated in Fig. 3, is still ongoing. Some preliminary results for both the drag-force closures and solid pressure will be presented in the Sections II and III. In this chapter, we will

FIG. 4. Graphical representation of the multilevel modeling scheme. The arrows represent a change of model. On the left is a ﬂuidized bed on a life-size scale, a section of which is modeled by the two-ﬂuid model (TFM) (see enlargement), where the shade of grey of a cell indicates the solidphase volume fraction. On the right, the same section is modeled using discrete particles. The gas phase is solved on the same grid as in the TFM. The bottom graph shows the most detailed level, where the gas phase is solved on a grid much smaller than the size of the particles. Note that in reality, the separation in scales is much more extreme, and also that the section that can be modeled by the TFM of the industrial-scale ﬂuidized bed is much smaller.

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only consider monodisperse systems, but nevertheless we will formulate the softsphere model in Section III.B for general polydisperse systems where a particle a has an individual radius Ra. Apart from Section III.B, the size of the particles are indicated by a single radius R or a diameter d. Finally, we note that the TFM can simulate ﬂuidized beds only at engineering scales corresponding typically to the size of pilot plants, and the industrial-scale ﬂuidized bed reactors (diameter 1–5 m, height 3–20 m) are still far beyond its capabilities. In Section V, we discuss two promising approaches for modeling large-scale gas–solid ﬂow, which are currently being explored independently at the Princeton University and the University of Twente. We stress, however, that these approaches are still under development and that they should be recognized as only preliminary.

II. Lattice Boltzmann Model As mentioned in Section I, there are two fundamentally different types of models to describe a gas ﬂow: the continuum models and the molecular models. In principle, the molecular models can be applied at any length scale; however, in practice this is limited to microscopic scales only because of the limitation of computer time. The continuum models present the main stream of engineering applications and are more ﬂexible when applying to different macroscale gas ﬂows. The gap between the continuum and molecular models can be bridged, however, by the lattice Boltzmann (LB) simulation model that applies at a ‘‘mesoscopic’’ scale, which is in between fully microscopic and macroscopic scales. The LB model that is currently the most widely used—the lattice BhatnagarGrossKrook (BGK) model—is nothing but a ﬁnite difference version of the continuous, macroscopic BGK equation introduced in 1954 (Bhatnagar et al., 1954). Historically, however, this LB model has evolved from the microscopic lattice-gas simulation models for ﬂuids, and we will also follow this route here.

A. FROM LATTICE-GAS

TO

LATTICE-BOLTZMANN

1. Lattice-Gas Models As mentioned earlier, in principle, one can model the dynamics of a simple classical ﬂuid by means of MD simulations. This technique, although straightforward, is relatively time-consuming, and therefore not suitable for observation of large-scale ‘‘macroscopic’’ phenomena in the ﬂuid. However, one often does not need such a detailed description of the microdynamics as provided by MD. In such cases, it would be more efﬁcient to strip the MD model down to its barest essentials, where the only requirement is that the model behaves like a ﬂuid macroscopically, but is still atomistic in character—i.e., the mechanism underlying the ﬂuid motion is the movement of particles. From the derivation of

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the ﬂuid dynamics equations it is clear that a key ingredient in such a model must be local conservation of mass and momentum. The simplest model to think of would be one with a single species of particles moving on a lattice with discrete velocities. In 1985, such a model was introduced for two-dimensional (2D) ﬂuid ﬂow by Frisch et al. (1986), where particles move on a triangular lattice. Such a lattice has just enough symmetry to guarantee isotropic, macroscopic equations of motion. The rules and basic idea of the FHP model are illustrated in Fig. 5. Later, the model has been extended to three dimensions as well. From the update rules it is clear that lattice-gas cellular automata (LGCA) make an efﬁcient simulation scheme, in particular on a parallel computer, since the rules are completely local. Moreover, stability of the algorithm is guaranteed, since the update involves only bit manipulation, i.e., the update is exact with no round-off errors. We will continue with a more formal description of general LGCA models.

2. Definitions and Equation of Motion In LGCA models, time and space are discrete; this means that the model system is deﬁned on a lattice and the state of the automaton is only deﬁned at regular points in time with separation dt. The distance between nearest-neighbor sites in the lattice is denoted by dl. At discrete times, particles with mass m are situated at the lattice sites with b possible velocities ci, where i A {1, 2, y, b}. The set ci can be chosen in many different ways, although they are restricted by the constraint that r0 ¼ r þ ci dt

(1)

FIG. 5. Example of the time evolution in a small section of the FHP model. In the ﬁgure, each arrow represents the velocity of a single particle. The particles are situated at the lattice sites. The update of the lattice consists of two steps. First there are local collisions at all sites, simultaneously, and such that locally the number of particles and momentum is conserved. Note that only some cases lead to a new conﬁguration. The next step is a propagation step: all the particles move simultaneously according to their velocities to neighboring sites. Particles do not interact during this step. Note that the ﬁgure only represents a small section of the lattice; therefore, we can only give the complete ﬁnal conﬁguration of the central site, as the state of the other sites after the propagation depends on neighboring sites that are not shown. Next there will again be a collision step, etc.

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where r and r0 must be lattice sites. Apart from Eq. (1), it has been proved essential to have additional symmetry requirements on the velocity set in order to get isotropic, macroscopic behavior from these models. These requirements turn out to be that the even-rank tensors that can be constructed from the velocity set are isotropic up to 4th rank, and the odd-rank tensors are zero. The time evolution of the LGCA consists of two steps: 1. Propagation: All particles move in one time step dt from their initial lattice position r to a new lattice position r0 ¼ r+cidt. 2. Collision: The particles at all lattice sites undergo a collision that conserves the total number of particles and the total momentum at each site. The collision rules may or may not be deterministic. The state of the automaton at time t can be completely determined by the ‘‘boolean’’ variable ni(r,t), which is equal to 1(0) if a particle is present (absent) on site r with velocity ci. From this it follows that the local microscopic density r~ and ﬂow velocity u~ at site r are given by X X ~ tÞ ¼ m ~ tÞuðr; ~ tÞ ¼ m pðr; ni ðr; tÞ; pðr; ni ðr; tÞci (2) i

i

with m as the mass of the particles. The update of ni(r,t) (from propagation and collision) can formally be written by the following equation of motion: ni ðr þ ci dt; t þ dtÞ ¼ ni ðr; tÞ þ Di ðnðr; tÞÞ with Di ðnðr; tÞÞ ¼

X s;s0

(3)

ðs0i si Þxðs; s0 Þ P nsjj ð1 nj Þ1sj j

With the sum is over all possible states s and s0 of a single site, and x(s, s0 ) is a collision function that is equal to one for states s, s0 where s goes over into s0 in a collision, and zero for all other possible pairs of states. Note that expression in Eq. (3) is the formal expression for the update. In a numerical code, the state ni(r,t) of the systems is represented by a b—bit word for every site r at time t. The collision process can then be done by a very quick table lookup; whereas, for the propagation step the bits from one word have to be put at the same bit positions in the words describing the states of the neighboring sites. Despite its extremely simpliﬁed microdynamical behavior, it turns out that the analogy of these models with the ‘‘real’’ ﬂuid models is very close, such that the theoretical framework of statistical mechanics of simple ﬂuids can be applied, to a great extent, to these discrete ﬂuids. That is, starting from the formal expression in Eq. (3), it can be proven that the macroscopic equations of motions are, in a welldeﬁned limit, equivalent to the Navier–Stokes equations (Frisch et al., 1987; Ernst and Dufty, 1989). This solid theoretical basis makes the LGCA method not just a toy model of computer scientists but also a numerical scheme that can be seriously

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considered for the study of hydrodynamic ﬂow phenomena. However, such an application is seriously hindered by two big drawbacks of LGCA. Firstly, their inherent noisiness, which means that massive averaging is required to get accurate numbers. And secondly, it turns out that the method is not suitable for modeling ﬂuid ﬂow at Reynolds numbers above Re ¼ 100, which is related to the fact that the viscosity cannot be made lower than a certain value, since it is dictated by the collision function x. It is for these reasons that the current class of LGCA methods cannot compete with CFD methods for modeling large-scale ﬂuid ﬂow. 3. Averaged Equation of Motion The two drawbacks mentioned above can be overcome, however, by considering the ensemble-averaged version of the microscopic equation of motion, Eq. (3): f i ðr þ ci dt; t þ dtÞ ¼ f i ðr; tÞ þ C i ðf ðr; tÞÞ

(4)

with fi(r,t) ¼ /ni(r,t)S—the average occupation number of link i at site r and time t, which is now a ﬂoating number between zero and one, and X Y s C i ðf Þ ¼ hDi ðnÞi ¼ ðs0i si Þ xðs; s0 Þ f j j ð1 f j Þ1sj (5) s;s0

j

where in the second step the assumption is made that the particle occupation numbers on a single site are not correlated, so that the average of the product can be written as the product of the average. The ensemble averaged density r and ﬂow velocity u follow from Eq. (2): X X ~ ¼m ~ ui ¼ m r ¼ hri f i; ru ¼ hr~ f i ci (6) i

i

where we have omitted the space and time dependence. In its present form, the collision operator in Eq. (5) is not very useful for simulations, since the update of fi at each site requires the double sum over all possible states, where there are over 16 million states (224) for the 3D models. This problem can be circumvented by expanding the collision function about the equilibrium distribution eq function feq i , for which it holds that C(fi ) ¼ 0: b X

f i ðr þ ci dt; t þ dtÞ ¼ f i ðr; tÞ þ

Lij ðf j ðr; tÞ f eq j ðr; tÞÞ

j¼1

where L is the linearized collision operator: Lij ¼

@C i @f j

! f j ¼ f eq j

(7)

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which can be evaluated directly from Eq. (5). Note that L, which has to be calculated only once for a given set of collision rules, is now a small b b matrix, compared to x that is a 2b 2b matrix. Equation (7) can be directly converted into an algorithm for simulation purpose. The advantage of an LB simulation is that the system is essentially free of noise. Also, the linearized collision operator need not necessarily be evaluated from an existing set of microscopic collision rules x(s, s0 ). One is free to deﬁne any operator L, which has the correct symmetry and conserves momentum and number of particles. As an example, for the 2D hexagonal lattice, one can derive from the requirements of symmetry and conservation that L should have the following form: 0

a Bb B B Bc L¼B Bd B B @c b

b a

c b

d c

c d

b c

a b

b a

c b

d

c

b

a

1 b cC C C dC C cC C C bA

c

d

c

b

a

(8)

with a¼

4 1 þ l; 21 3

b¼

1 1 l; 7 6

c¼

4 1 l; 21 6

d¼

1 1 þ l 7 3

where l can take any value between 0 and 2 and is related to the kinematic viscosity via 1 1 dl 2 m¼ r 4l 8 dt This gives the possibility to make the viscosity arbitrarily small, so that simulations can be performed also at Reynolds numbers higher than 100.

B. THE LATTICE BHATNAGARGROSSKROOK MODEL In the linearized LB equation Eq. (7), the ensemble averaged effect of the particle–particle collision is now represented by a relaxation of the distribution function fi to the equilibrium function feq i , where the matrix Lij does not necessarily have to correspond to an existing set of collision rules. The question now arises if L can be simpliﬁed even further to the form Lij ¼ adij, so that the LB equation takes the form (with t ¼ –dt/a) f i ðr þ ci dt; t þ dtÞ ¼ f i ðr; tÞ

dt ðf ðr; tÞ f eq i ðr; tÞÞ t i

(9)

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79

At ﬁrst sight this does not seem possible because the speciﬁc form of L, for instance Eq. (8), which followed from the requirement of symmetry. However, it turns out that b, c, and d can be put to zero, if an equilibrium function fieq with different weights is used for the different directions, so that the lack of symmetry can be remedied (Qian et al., 1992; Ladd, 1994; Succi, 2001). Speciﬁcally, the fieq should take the form

ci f eq i ¼a r

1þ

c i u ð c i uÞ 2 u2 þ 2 c2s 2c4s 2cs

(10)

where the weight aci only depends on the magnitude ci of the velocity ci connected to the link direction i, and cs is the speed of sound. For the popular D3Q19 model (3D, p 19ﬃﬃﬃ velocities), there are 6 velocities with ci ¼ 1dl/dt, 12 velocities with ci ¼ 2dl=dt; and one zero velocity, ci ¼ 0 (see Fig. 6). The 0 parameters that yield pthe ﬃﬃ proper equilibrium distribution in Eq. (10) are a ¼ 1 2 1/3, a ¼ 1/36, and a ¼ 1=18, and the speed of sound is usually set to cs ¼ 1/3(dl/dt) (Ladd and Verberg, 2001; Succi, 2001). To ﬁrst order, expression in Eq. (9) represents the ﬁnite difference form of the well-known BGK equation:

@ þ r ci @t

fi ¼

1 f i f eq i t

FIG. 6. The D3Q19 model.

(11)

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M.A. VAN DER HOEF ET AL.

It can be shown that this equation will yield the familiar conservation of mass and momentum equations @ r þ r ðruÞ ¼ 0; @t

@ ðruÞ þ r ðruuÞ ¼ r p¯ @t

(12)

where the pressure tensor is equal to 2 T ¯ p¯ ¼ pI m ðruÞ þ ðruÞ l m ðr us Þ I¯ 3 with I¯ the unit tensor, (ru)ab ¼ raub, ðruÞTab ¼ rbua, and the pressure p, shear viscosity m, and bulk viscosity l given by p ¼ c2s r;

m ¼ tc2s r;

l¼0

(13)

As said, the lattice BGK in Eq. (9) is the ﬁnite difference version of Eq. (11) to ﬁrst order in dt. To second order in dt, however, Eq. (9) represents the ﬁnite difference version of a slightly different expression:

dt @ @ 1 þ r ci f i ¼ f i f eq þ r c þ f i f eq i i i @t t 2t @t

(14)

In the route to Navier–Stokes, it turns out that fieq in the second term on the right-hand side (RHS) does not play a role, so that we can rewrite Eq. (14) as a normal BKG equation with a different prefactor on the RHS:

@ þ r ci @t

fi ¼

dt f f eq i t dt=2 i

We thus ﬁnd that the lattice BGK model describes, to second order in dt, the ﬂuid according to the Navier–Stokes equation with a viscosity 1 m ¼ t dt c2s r 2 where the extra term –(1/2)dt is due to the ﬁnite difference scheme. As a demonstration of how well the simple lattice BGK scheme in Eq. (9) can describe ﬂuid ﬂow, we show in Fig. 7 the velocity proﬁle from a lattice BGK simulation for forced Poiseuille ﬂow and shear ﬂow; it can be seen that excellent agreement is found with the theoretical results that follow from the Navier–Stokes equation. Note that for the simulations shown in Fig. 7 we used stick-boundary

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Shear flow

Forced Poiseuille flow

Lattice Boltzmann Theory

Lattice Boltzmann Theory

v0 Fluid velocity

Fluid velocity

FIG. 7. Velocity proﬁle for Poiseuille ﬂow and shear ﬂow. The points are the LBM data, and the solid lines are the theoretical proﬁles. For the simulations, we used the D3Q18 model with 25 lattice sites across the channel.

conditions at the walls of the channel. The implementation of such conditions is similar to the boundary conditions that are required to model large, solid objects in the LB model, which is described in Section II.C. C. MODELING SOLID PARTICLES In order to simulate large, moving particles in the LB model, we should deﬁne additional rules that describe the interaction of the LB gas with the surface of the solid particles. One essential ingredient of the moving-boundary rules is that these rules result, on average, in a dissipative force on the suspended particle. An obvious choice of rules is those according to which the gas next to the solid particle moves with the local velocity of the surface of the solid particle. In this way, one models the hydrodynamic ‘‘stick’’ boundary condition; for a spherical particle suspended in an inﬁnite 3D system, moving with velocity v, this will give rise to a frictional force on the particle F ¼ 3pmdv, at least in the limit of low Reynolds numbers Re ¼ rdv/m, where d is the hydrodynamic diameter of the particle and m is the shear viscosity. A particular efﬁcient and simple way to enforce stick-boundary rules was introduced by Ladd (1994). First the boundary nodes are identiﬁed, which are deﬁned as the points halfway two lattice sites that are inside and outside the particle and closest to the actual surface (see Fig. 8, left graph, solid squares). For a static particle, the boundary rule is simply that a distribution moving such that it would cross the boundary is ‘‘bounced back’’ at the boundary node. Since this node is halfway the link, the bounce-back rule has the effect so (see Fig. 8): f i ðr; t þ dtÞ ¼ f i0 ðr; tÞ

(15)

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boundary node

i i’

site

site r

s

FIG. 8. Left graph: example of a boundary node map for a disc in a 2D hexagonal lattice. Right graph: illustration of the bounce-back rule on an enlarged section of the boundary. The distribution at site r that moves at time t into direction i0 , instead of arriving at the (virtual) site s, is bounced at the boundary node, and thus arrives back at site r at time t+dt, but is now headed in the opposite direction i.

where i and i0 are opposite links. This rule ensures that the ﬂuid velocity at the boundary node indeed vanishes; the momentum at the boundary node at time t+(1/2)dt is given by rb ub ¼ f i0 ðr; tÞ ci0 þ f i ðr; t þ dtÞ ci

(16)

Inserting Eq. (15) and using ci ¼ –ci0 gives rbub ¼ 0. For nonstatic particles, the local ﬂuid velocity must be set equal to the local boundary velocity vb. This can be achieved by a simple modiﬁcation of the bounce-back rule: f i ðr; t þ dtÞ ¼ f i0 ðr; tÞ þ a ci

(17)

where a is chosen such that ub ¼ vb. Note that only the component of vb in the direction of the link can be set in this way. For more details we refer to the papers by Ladd (1994) and Ladd and Verberg (2001). In Fig. 9 (right graph), we show the LBM simulation results for the velocity of a single free-falling sphere in an (effectively) unbounded ﬂuid. As can be seen from Fig. 9, the boundary rule results in a terminal velocity according to the Stokes–Einstein friction force. Note that the actual plateau value of the velocity is slightly smaller than the theoretical prediction. This can be attributed to the fact that the radius of the particle is not well deﬁned because of the irregular shape of boundary–node surface of the sphere. In fact, the free-falling sphere experiment (or a similar experiment with periodic boundary conditions) is used for calibration purposes,

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Sedimentation of a single sphere

Particle velocity

0.006

0.004

Simulation

0.002

vterm = mg/3πµd

0.000

0

1000

2000

3000

4000

5000

time FIG. 9. Velocity of a single sphere in a 3D LB gas. The black line is the data from LBM, which has the proper functional form v(t) ¼ vN(1 – exp(-gt/vN)). The grey line is theoretical terminal velocity, which is slightly higher than vN.

i.e., the effective hydrodynamic radius of the spheres is obtained from the terminal velocity, where it is assumed that the Stokes–Einstein relation holds. D. RESULTS

FOR THE

GAS– SOLID DRAG FORCE

The drag force from the gas phase on an assembly of spheres can, in principle, be obtained from the terminal velocity in a sedimentation experiment. However, the drag force can also be directly measured in the simulation from the change in gas momentum due to the boundary rules. The change in gas momentum per unit time, required to maintain stick-boundary conditions for particle a, is equal to minus the total force Fg-s,a that the gas phase exerts on particle a. This total force has two contributions (see also Section III.D), namely the drag force Fd,a due the ﬂuid–solid friction at the surface of the spheres and a force Fp,a ¼ –Varp due to the static pressure gradient rp, which drives the gas ﬂow past the spheres (Va ¼ pd3/6 P is the volume of the sphere). From a balance of forces it follows that Vrp ¼ aFg-s,a, with V the total volume of the system; eliminating rp from the expressions gives Fd,a ¼ eFg-s,a with e the volume fraction of the gas phase. The procedure to obtain the drag force for monodisperse systems in an LB simulation is then as follows. N particles with diameter d are distributed randomly in a box of nx ny nz lattice sites, so that the gas fraction equals e ¼ 1Npd3/(6nxnynz dl3). Some typical values are N ¼ 54 and d ¼ 25dl, where periodic boundary conditions are used. All spheres are forced to move with the same constant velocity vsim in some arbitrary direction, so that the array of spheres moves as a static conﬁguration through the system. A uniform force is applied to the gas phase, to balance the total force

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P N a¼1 Fg!s;a from the moving particles on the gas phase. From this it follows that in a frame of reference where the particles are static, the superﬁcial ﬂow velocity is equal to –vsim, so that Re ¼ pd|vsim|/m, where m is the P viscosity. Once an equilibrium state is obtained, the average value F¯ g!s ¼ h N a¼1 Fg!s;a =Nit is determined, with / St a time average. The momentum exchange coefﬁcient b, as deﬁned in Eq. (44), is then determined via b¼

F¯ g!s ð1 Þ2 vsim V a

since the relative velocity u–va in Eq. (44) corresponds to –vsim/e in the LB simulations and Fd,a ¼ eFg-s,a. Note that the dimensionless quantity bd2/m will only depend on the Reynolds number and the gas fraction e. Ergun (1952) showed that when the experimental data for b for different e and Re is plotted in a single {log(x), log(y)} graph with x¼

Re 1

y¼

bd 2 1 mð1 Þ Re

all data fall onto a single curve y ¼ 150/x+1.75, which corresponds to bd 2 ¼ 150ð1 Þ þ 1:75Re mð1 Þ

(18)

which is the famous Ergun equation. In Fig. 10, we show the data from extensive LB simulations (Van der Hoef et al., 2004; Beetstra et al., 2006) for a range of gas fractions and Reynolds numbers, on the same {log(x), log(y)} graph. We ﬁnd that our LBM data deviates substantially from the Ergun equation: for low Re numbers the Ergun equation underestimates the drag force, whereas for high Re numbers the Ergun equation overestimates the drag force. A simple remedy would be to use different coefﬁcients in the Ergun equation, for instance 180 instead of 150 and 1.0 instead of 1.75. However, it can be seen from Fig. 10 that not all data obey the functional form y ¼ A/x+B. Note also that the Ergun equation was derived for packed beds, and is not expected to be valid for high gas fractions; for that range, normally the Wen and Yu Eq. (46) is used. However, we ﬁnd that this equation signiﬁcantly underpredicts the drag force at higher Reynolds numbers (see Fig. 10). Based on our data from the LB simulations, we suggest the following new drag-force correlation that we write in the form of an Ergun-type equation (Beetstra et al., 2006; Van der Hoef et al., 2005): bd 2 ¼ Að1 Þ þ BRe mð1 Þ

(19)

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

85

3

(βd2/µ) ∗ ε/((1−ε)Re)

10

Ergun WenYu (ε=0.8) Hill et al (ε=0.5) Equation (19) (ε=0.5) LBM (ε=0.4−0.8)

102

101

100 10−1

101

103

105

Re / (1 − ε) FIG. 10. Normalized drag force at arbitrary Reynolds numbers and gas fractions. The symbols represent the simulation data, the solid line the Ergun correlation Eq. (18), the dashed line the Wen–Yu correlation Eq. (46) for e ¼ 0.8, and the grey line the correlation by Hill et al. (2001a,b) Eq. (47) and the long-dashed line Eq. (19), both for e ¼ 0.5.

only with coefﬁcients that depend on both e and Re: A ¼ 180 þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 184 1 þ 1:5 1 1

and B¼

0:31ð1 þ 3ð1 Þ þ 8:4Re0:343 Þ 1 þ 103ð1Þ Re22:5

Expression in Eq. (19) is within 8% of all simulation data up to Re ¼ 1000. Since this relation has been derived very recently (Beetstra et al., 2006), it has not been applied yet in the higher scale models discussed in Sections III and IV. However, the expression by Hill et al. in Eq. (47) derived from similar type of LBM simulations is consistent with our data, in particular when compared to the large deviations with the Ergun and Wen and Yu equations. So, we expect that the simulation results presented in Section IV.F using the Hill et al. correlation will not be very different from the results that would be obtained with expression in Eq. (19). A more detailed account of the derivation of expression in Eq. (19) and a comparison with other drag-force relations can be found in Ref. Beetstra et al. (2006).

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III. Discrete Particle Model A. INTRODUCTION DPMs offer a viable tool to study the macroscopic behavior of assemblies of particles and originate from MD methods. Initiated in the 1950s by Alder and Wainwright (1957), MD is by now a well-developed method with thousands of papers published in the open literature on just the technical and numerical aspects. A thorough discussion of MD techniques can be found in the book by Allen and Tildesley (1990), where the details of both numerical algorithms and computational tricks are presented. Also, Frenkel and Smit (1996) provide a comprehensive introduction to the ‘‘recipes’’ of classical MD with emphasis on the physics underlying these methods. Nearly all techniques developed for MD can be directly applied to discrete particles models, except the formulation of particle–particle interactions. Based on the mechanism of particle–particle interaction, a granular system may be modeled either as ‘‘hard-spheres’’ or as ‘‘soft-spheres.’’ 1. Hard-Sphere Model In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. Campbell and Brennen (1985) reported the ﬁrst hard-sphere discrete particle simulation used to study granular systems. Since then, the hard-sphere models have been applied to study a wide range of complex granular systems. Hoomans et al. (1996) used the hard-sphere model, in combination with a CFD approach for the gas-phase conservation equations, to study gas–solid twophase ﬂows in gas-ﬂuidized beds. By using this model, they studied the effect of particle–particle interaction on bubble formation (Hoomans et al., 1996) and the segregation induced by particle-size differences and density differences (Hoomans et al., 2000). This model has been further used in connection with the kinetic theory of granular dynamics by Goldschmidt et al. (2001), highpressure ﬂuidization by Li and Kuipers (2002), and circulating ﬂuidized beds by Hoomans (2000). Similar simulations have been carried out by other research groups. Ouyang and Li (1999) developed a slightly different version of this model. Helland et al. (1999) recently developed a DPM in which hard-sphere collisions are assumed, but where a time-driven scheme (typically found in the soft-sphere model) is used to locate the collisional particle pair. Effect of the gas turbulence has also

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been taken into account in some hard-sphere models by Helland et al. (2000), Lun (2000), and Zhou et al. (2004). At high-particle number densities or low coefﬁcients of normal restitution e, the collisions will lead to a dramatical decrease in kinetic energy. This is the socalled inelastic collapse (McNamara and Young, 1992), in which regime the collision frequencies diverge as relative velocities vanish. Clearly in that case, the hard-sphere method becomes useless. 2. Soft-Sphere Model In more complex situations, the particles may interact via short- or longrange forces, and the trajectories are determined by integrating Newtonian equations of motion. The soft-sphere method originally developed by Cundall and Strack (1979) was the ﬁrst granular dynamics simulation technique published in the open literature. In soft-sphere models, the particles are allowed to overlap slightly and the contact forces are subsequently calculated from the deformation history of the contact using a contact-force scheme. The soft-sphere models allow for multiple particle overlap, although the net contact force is obtained from the addition of all pair-wise interactions. The soft-sphere models are essentially time-driven, where the time step should be carefully chosen in calculating the contact force. The soft-sphere models that can be found in literature mainly differ from each other with respect to the contact-force scheme that is used. A review of various popular schemes for repulsive interparticle forces is presented by Scha¨fer et al. (1996). Walton and Braun (1986) developed a model that uses two different spring constants to model the energy dissipation in the normal and tangential directions. In the force scheme proposed by Langston et al. (1994), a continuous potential of an exponential form is used, which contains two unknown parameters: the stiffness of the interaction and an interaction constant. A 2D soft-sphere approach was ﬁrst applied to gas-ﬂuidized beds by Tsuji et al. (1993), where the linear spring–dashpot model—similar to the one presented by Cundall and Strack (1979)—was employed. Xu and Yu (1997) independently developed a 2D model of a gas-ﬂuidized bed. However in their simulations, a collision detection algorithm that is normally found in hard-sphere simulations was used to determine the ﬁrst instant of contact precisely. Based on the model developed by Tsuji et al. (1993), Iwadate and Horio (1998) incorporated van der Waals forces to simulate ﬂuidization of cohesive particles. Kafui et al. (2002) developed a DPM based on the theory of contact mechanics, thereby enabling the collision of the particles to be directly speciﬁed in terms of material properties such as friction, elasticity, elasto-plasticity, and auto-adhesion. It is also interesting to note that soft-sphere models have also been applied to other applications such as gas–particle heat transfer by Li and Mason (2000) and coal combustion by Zhou et al. (2003). Clearly, these methods open a new way to study difﬁcult problems in ﬂuidized bed reactors.

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3. Comparison between Hard- and Soft-Sphere Models Although both hard- and soft-sphere models have been used in the simulation of granular ﬂow, each has its own characteristics that make them very efﬁcient in some cases, while inefﬁcient in others. The two types of models are compared in Table I. Hard-sphere models use an event-driven scheme because the interaction times are (assumed to be) small compared to the free-ﬂight time of particles, where the progression in physical time depends on the number of collisions that occur. In contrast, in the soft-sphere models a time step that is signiﬁcantly shorter than the contact time should be used. This directly implies that the computational efﬁciency of the soft-sphere model (compared to the hard-sphere model) decreases when the ratio of the free-ﬂight time to the contact time increases, which is the case when the system becomes less dense. In the soft-sphere models, a slight deformation of particles is allowed, so that multiple contacts between several pairs of particles are possible, which should never happen in the event-driven models. As mentioned above, a lower coefﬁcient of normal restitution may lead to the inelastic collapse in hard-sphere simulations. Incorporation of cohesive forces, especially the pair-wise forces, is quite straightforward in soft-sphere models. This is because the collisional process in the soft-sphere model is described via the Newtonian equations of motion of individual particles, that is, in terms of forces. In the hard-sphere system, the update is not via forces (since they are, loosely speaking, either zero or inﬁnite), but via a momentum exchange at contact. This means that for short-range forces, such as the cohesive force, a kind of hybrid method for the interaction at close encounters has to be devised, which is not straightforward. In contrast, for systems with different size particles, it is the soft-sphere model that poses some difﬁculties. In a soft-sphere system using a linear spring–dashpot scheme, the spring stiffness is dependent on the particle size. This means that in principle a different spring stiffness should be used for calculating the contact forces between particles with different sizes, otherwise the computing efﬁciency will drop substantially.

COMPARISON

BETWEEN

TABLE I HARD- AND SOFT-SPHERE MODELS. THE SYMBOLS INDICATE GOOD (++), NORMAL (+), AND NOT SUITABLE ()

Computing efﬁciency Multiple contacts Dense systems Incorporation of cohesive force Energy conservation during collisions Multiple particle sizes

Hard-sphere

Soft-sphere

++ — — + ++ ++

+ ++ ++ ++ + +

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In the following, we focus on the soft-sphere method since this really is the ‘‘workhorse’’ of the DPMs. The reason is that it can in principle handle any situation (dense regimes, multiple contacts), and also additional interaction forces—such as van der Waals or electrostatic forces—are easily incorporated. The main drawback is that it can be less efﬁcient than the hard-sphere model. B. PARTICLE DYNAMICS: THE SOFT-SPHERE MODEL 1. The Equations of Motion The linear motion of a single spherical particle a with mass ma and coordinate ra can be described by Newton’s equation: ma

d 2 ra ¼ Fcontact;a þ Fpp;a þ Fext;a dt2

(20)

where the RHS is the total force on the particle, which has three basic contributions: (i) The total contact force Fcontact,a is the sum of the individual contact forces exerted by all other particles in contact with the particle a, which are naturally divided into a normal and a tangential component: X Fcontact;a ¼ ðFab;n þ Fab;t Þ b2contactlist

(ii) The total external force Fext,a: Fext;a ¼ Fg;a þ Fd;a þ Fp;a which includes the gravitational force Fg,a ¼ mag, and forces exerted by the surrounding gas phase: the drag force Fd,a and a force Fp,a from the pressure gradient. (iii) The sum of all other particle–particle forces Fpp,a that can include shortrange cohesive forces Fcoh,a, which follow from the van der Waals interaction between the molecules that the particles are made up of, as well as long-range electrostatic forces. In this chapter, we will only consider the cohesive forces. Note that for liquid–solid systems, Eq. (20) should also include the shortrange lubrication forces and the effects of other forces such as the ‘‘virtual mass’’ force. But this is beyond the scope of this chapter. Finally, the rotational motion of particle a is given by Ia

doa ¼ Ta dt

(21)

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M.A. VAN DER HOEF ET AL.

where Ia is moment of inertia, oa the angular velocity, and Ta the torque, which depends only on the tangential component of the individual contact forces: X Ta ¼ ðRa nab Fab;t Þ b2contactlist

with Ra as the radius of particle a. We will next give a more detailed description of the contact force, the cohesive force, and the integration of the equations of motion—Eqs. (20) and (21). The description of the forces resulting from interaction with the gas phase is given in Section III.D, whereas the dynamics of the gas phase itself is described in Section III.C. 2. Contact Force The calculation of the contact force between two particles is actually quite involved. A detailed model for accurately computing contact forces involves complicated contact mechanics (Johnson, 1985), the implementation of which is extremely cumbersome. Many simpliﬁed models have therefore been proposed, which use an approximate formulation of the interparticle contact force. The simplest one was originally proposed by Cundall and Strack (1979), where a linear-spring and dashpot model is employed to calculate the contact forces (see Fig. 11 and 12). In this model, the normal component of the contact force between two particles a and b can be calculated by Fab;n ¼ kn dn nab Zn vab;n

(22)

where kn is the normal spring stiffness, nab the normal unit vector, Zn the normal damping coefﬁcient, and vab,n the normal relative velocity. The overlap dn is given by dn ¼ ðRb þ Ra Þ jrb ra j

FIG. 11. Graphical representation of the linear spring–dashpot soft-sphere model. From Hoomans, Ph.D. thesis, University of Twente (2000).

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91

FIG. 12. The coordinate system used in the soft-sphere model.

with Ra and Rb denoting the radii of the particles. The normal unit vector is deﬁned as rb ra (23) nab ¼ j rb ra j The relative velocity of particles a and b is vab ¼ ðva vb Þ þ ðRa oa þ Rb ob Þ nab

(24)

where va and vb are the particle velocities, and oa and ob the angular velocities. The normal component of the relative velocity between particle a and b is vab;n ¼ ðvab nab Þnab

(25)

For the tangential component of the contact force, a Coulomb-type friction law is used: ( Fab;t ¼

kt dt Zt vab;t

mf Fab;n tab

for Fab;t mf Fab;n

for Fab;t 4mf Fab;n

(26)

where kt, dt, Zt, and mf are the tangential spring stiffness, tangential displacement, tangential damping coefﬁcient, and friction coefﬁcient, respectively. In

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Eq. (26), the tangential relative velocity vab,t, and tangential unit vector tab are deﬁned as vab;t

vab;t ¼ vab vab;n tab ¼

vab;t

The calculation of the tangential displacement dt requires some special attention and will be addresses in Section III.B.3. 3. Tangential Displacement Suppose that the tangential displacement at to is equal to dt0 ; then one would expect that the displacement dt at time t follows by simply integrating the tangential velocity (Hoomans, 2000): Z

t

vab;t dt

d t ¼ d t0 þ t0

This expression, however, is only justiﬁed for 2D systems, where the particles are represented essentially by disks, which are conﬁned in a single plane and the particle–particle contact occurs along a line, as shown in Fig. 13. So, the tangential component of the relative velocity is always in the same plane and no coordinate transformation is required. In a 3D system, however, it becomes more complicated. The particle–particle contact now occurs in a plane. The tangential component of the relative velocity is always in this plane and vertical to the normal unit vector according to the deﬁnition. Since the normal unit vector is not necessarily situated in the same plane at any time, it is desirable to transfer the old tangential displacement to the new contact plane before we calculate the new tangential displacement. To this end, a 3D rotation of the old tangential displacement should be applied. As

FIG. 13. The rotation of the contact plane during particle–particle collisions.

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93

the tangential velocity vector is always vertical to the normal unit vector, the 3D rotation can be done around the vector determined by nab n0,ab, as shown in Fig. 14. So in a 3D situation, the tangential displacement is determined by Z t d t ¼ d t0 H þ vab;t dt (27) t0

where the rotation matrix is 0

qh2x þ c

B qh h þ shz H¼B @ x y

qhx hz shy

qhx hy shz qh2y þ c qhy hz þ shx

qhx hz þ shy

1

C qhy hz shx C A qh2z þ c

(28)

with h, c, s, and q are deﬁned as h ¼

and

nab n0;ab

; nab n0;ab

c ¼ cos j;

s ¼ sin j;

q¼1c

j ¼ arcsin nab n0;ab

Are Eqs. (27) and (28) sufﬁcient to describe the tangential displacement during particle–particle contact? In the absence of friction, the answer is yes. When we consider friction during particle–particle contact—as pointed out by Brendel

FIG. 14. The transformation of tangential displacement vector.

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and Dippel (1998)—the use of Eqs. (27) and (28) may give rise to unphysical behavior for dense systems due to the allowance of an arbitrarily large tangential displacement (Eq. 28). In a dilute system, this will not be a problem since the multiple-particle contacts do not happen frequently. In this case if the contacts ends, the tangential displacements will be set to zero. In contrast for dense systems, multiple-particle contacts are very common and the contact history for a speciﬁc particle could be very long. The long contact history causes a relatively large tangential displacement, which means that an extra friction force should be taken into account. This problem can be overcome, however, by using the method proposed by Brendel and Dippel (1998), where the tangential displacement during the friction is calculated by dt ¼ mf|Fab,n|tab/kt, so that 8

R < dt0 H þ t vab;t dt for Fab;t mf Fab;n

t0

dt ¼ : mf Fab;n tab =kt for Fab;t 4mf Fab;n

(29)

4. Collision Parameters To solve the Eqs. (20) and (21), we have to specify ﬁve parameters: normal and tangential spring stiffness kn and kt, normal and tangential damping coefﬁcient Zn and Zt, and the friction coefﬁcient mf . In order to get a better insight into how these parameters are related, it is useful to consider the equation of motion for the overlap in the normal direction dn: ::

:

meff dn ¼ kn dn Zn dn

(30)

which follows from Eq. (20) when only the normal contact force is taken into account. In Eq. (30), meff is the reduced mass of the two interacting particles a and b: 1 1 1 ¼ þ meff ma mb Equation (30) is the well-known differential equation of the damped harmonic oscillator, the solution of which is dn ðtÞ ¼ ðu0 =OÞ expðCtÞ sinðOtÞ

(31)

dn ðtÞ ¼ ðu0 =OÞ expðCtÞðC sinðOtÞ þ O cosðOtÞÞ

(32)

:

with u0 ¼ d_ n ð0Þ as the initial relative velocity, and O¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ O20 C2 O0 ¼ kn =meff

C ¼ Zn =ð2meff Þ

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The duration of a contact can be determined from dn(tcontact,n) ¼ 0, which gives tcontact,n ¼ p/O, so that the relative velocity just after contact equals :

dn ðtcontact;n Þ ¼ u0 expðCtcontact;n Þ According to the deﬁnition, the coefﬁcient of normal restitution is given by :

e¼

dn ðtcontact;n Þ :

¼ expðpC=OÞ

(33)

dn ð0Þ Thus, we can calculate the normal damping coefﬁcient via pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ln e meff kn Zn ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p2 þ ln2 e

ðea0Þ

Note that for e ¼ 0 we get O ¼ 0 according to Eq. (33), so that in that case pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Zn ¼ 2 kn meff . We can follow a similar procedure for the tangential spring–dashpot system. So, the tangential damping coefﬁcient is determined by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ln et m0eff kt Zt ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p2 þ ln2 et

ðet a0Þ

where m0 eff ¼ 2 meff/7 is the reduced mass of the two-particle system interaction via a tangential linear spring. Note that m0 eff is different from meff, since in a tangential direction both the rotational and translational momentum must be considered. In the case of particle–wall contact, we shall simply treat particle b as a big particle with an inﬁnite radius, so that we have meff ¼ ma

2 m0eff ¼ ma 7

The contact force between two particles is now determined by only ﬁve parameters: normal and tangential spring stiffness kn and kt, the coefﬁcient of normal and tangential restitution e and et, and the friction coefﬁcient mf. In principle, kn and kt are related to the Young modulus and Poisson ratio of the solid material; however, in practice their value must be chosen much smaller, otherwise the time step of the integration needs to become impractically small. The values for kn and kt are thus mainly determined by computational efﬁciency and not by the material properties. More on this point is given in the Section III.B.7 on efﬁciency issues. So, ﬁnally we are left with three collision parameters e, et, and mf, which are typical for the type of particle to be modeled.

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5. Cohesive Force Cohesion between particles can arise from a variety of sources including van der Waals forces, liquid bridging (i.e., capillary forces), sintering, and so on. Of these forces, which become increasingly important as the particle size decreases, the van der Waals force is generally accepted as the dominating cohesive force in gas-ﬂuidized beds of ﬁne particles (Geldart A and C particles), and will be considered next. The van der Waals force is present between any two molecules (polar or nonpolar) and follows from the interaction of the ﬂuctuating dipole moments on the molecules. According to the London theory, the potential energy of two molecules i and j at distance rij, due to the van der Waals interaction, is equal to jðrÞ ¼ C 6 r6 ij . The total energy between two macroscopic bodies a and b, made of the same material, then equals: Z Z X X 2 r6 V ¼ C 6 ¼ C r dra drb jra rb j6 (34) 6 ij v v a b fi on ag f j on bg R where in the last step we replaced Si on a by V a dra r(and similarly for b), where r is the density of the material, which is justiﬁed since the number of molecules present in the particles is very large. For two spheres with radii Ra and Rb, where the centers are at position ra and rb, respectively, expression Eq. (34) can be evaluated analytically (Hunter, 1986; Israelachvili, 1991): 2 r¯ ab 4 A 2 2 V ðrab Þ ¼ þ ln þ 6 r¯ 2ab 4 r¯2ab r¯ 2ab with r¯2ab ¼

r2ab ðRa Rb Þ2 ; Ra Rb

rab ¼ jrb ra j;

A ¼ p2 r2 C 6

(35)

The parameter A is known as the Hamaker constant. The force on sphere a then follows via Fcoh;a ¼

@V ðrab Þ 32 A rab nab nab ¼ @rab 3 Ra Rb r¯4ab ð¯r2ab 4Þ2

(36)

with nab deﬁned by Eq. (23). When the spheres are nearly touching (rabRa+Rb), and for Ra ¼ Rb ¼ R, the force in Eq. (36) can be simpliﬁed to Fcoh;a ¼

AR nab 12s2ab

sab ¼ rab 2R

Note that Eq. (36) exhibits an apparent numerical singularity in that the van der Waals interaction diverges if the surface distance between two particles approaches zero. In reality, such a situation will never occur because of the

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short-range repulsion between particles. In the present model, we have not included such a repulsion; however, we can avoid the numerical singularity by deﬁning a cut-off (maximal) value of the van der Waals force between two spheres. In practice, it is more convenient to use the equivalent cut-off value for the intersurface distance, s0ab, instead of for the interparticle force. The Hamaker constant A can, in principle, be determined from the C6 coefﬁcient characterizing the strength of the van der Waals interaction between two molecules in vacuum. In practice, however, the value for A is also inﬂuenced by the dielectric properties of the interstitial medium, as well as the roughness of the surface of the spheres. Reliable estimates from theory are therefore difﬁcult to make, and unfortunately it also proves difﬁcult to directly determine A from experiment. So, establishing a value for A remains the main difﬁculty in the numerical studies of the effect of cohesive forces, where the value for glass particles is assumed to be somewhere in the range of 10–21 joule. 6. Integrating the Equations of Motion In the following section, we only consider the integration of the equation of linear motion Eq. (20); the procedure for the equation of rotational motion, Eq. (21), will be completely analogous. Mathematically, Eq. (20) represents an initial-value ordinary differential equation. The evolution of particle positions and velocities can be traced by using any kind of method for ordinary differential equations. The simplest method is the ﬁrst-order integrating scheme, which calculates the values at a time t+dt from the initial values at time t (which are indicated by the superscript ‘‘0’’) via: ð0Þ va ¼ vð0Þ a þ aa dt;

ra ¼ rð0Þ a þ va dt

(37)

where aa is the acceleration: aa ¼

Fcontact;a þ Fpp;a þ Fext;a ma

(38)

The ﬁrst-order integration scheme, however, will introduce a drift in the energy; from Eq. (37), we have 2 2 2 ð0Þ ð0Þ ð0Þ 2 ðva að0Þ a dtÞ ¼ ðva Þ þ ðaa dtÞ ð2va aa dtÞ ¼ ðva Þ

so 1 1 2 2 1 ð0Þ 2 ðva Þ2 ðva að0Þ a dtÞ þ aa dt ¼ ðva Þ 2 2 2

(39)

The ﬁrst term on the left of Eq. (39) is the reduced kinetic energy of the particle at time t+dt, the second term is the work due to all kinds of external forces, and the ﬁrst term on the right is the reduced kinetic energy at time t. The remaining

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term a2a dt2 =2 is always positive, and this energy is introduced into the system solely due to the numerical method, for each time step. In the past decades, a large number of methods have been proposed to achieve better energy conservation: for example, the Gear family of algorithms and the family of Verlet algorithms (Frenkel and Smit, 1996). In our 3D code, we have incorporated yet another type of method developed by Beeman, which has a somewhat better energy conservation than the Verlet algorithm (Frenkel and Smit, 1996). In the Beeman method, the position and velocity of particle a are calculated via 2 ð0Þ 1 ð1Þ ra a aa þ þ ðdtÞ2 3 a 6 1 5 ð0Þ 1 ð1Þ ð0Þ a aa va va þ aa dt þ ðdtÞ 3 6 a 6 rð0Þ a

vð0Þ a dt

where the superscript (–1) denotes the values at time t–dt. Note that the Beeman–Verlet algorithm is not self starting, so it requires the storage of the old value of the acceleration a(–1).

7. Efficiency Issues: Spring Constants and Neighbor Lists To perform simulations of relatively large systems for relatively long times, it is essential to optimize the computational strategy of discrete particle simulations. Obviously, the larger the time step dt, the more efﬁcient the simulation method. For the soft-sphere model, the maximum value for dt is dictated by the duration of a contact. Since there are two different spring–dashpot systems in our current model, it is essential to assume that tcontact,n ¼ tcontact,t, so that sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p2 þ ðln eÞ2 p2 þ ðln et Þ2 ¼ kn =meff kt =m0eff If we further assume that e ¼ et, then the relation between the normal and tangential spring stiffness is kt meff 2 ¼ ¼ kn m0eff 7 Based on the discussion in previous sections, we can calculate the time step by 1 1 dt ¼ tcontact;n ¼ KN KN

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p2 þ ðln eÞ2 kn =meff

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99

where KN is the minimum number of steps during one contact. Our experience is that KN must not be less than 5, and is normally in the range 15–50. It can thus be seen that a smaller spring stiffness kn leads to a larger time step, and therefore it is useful to ﬁrst perform a number of test simulations with different values for : kn. Another issue is the maximum overlap dmax, which occurs at dðtÞ ¼ 0. From Eq. (31) it follows that dmax ¼ ðu0 =O0 Þ exp ðC=OÞ arcsinðO=O0 Þ which must typically be less than 1% of the particle diameter. A second way of speeding up the simulation is the use of neighbor lists and cell list, which was originally developed for MD simulations (Allen and Tildesley, 1990). The neighbor list contains a list of all particles within the cut-off sphere of a particular particle, so that the separations do not need to be calculated at each step, which is shown in Fig. 15. The neighbor list cut-off scutoff should be deﬁned with care. A too small cut-off value may result in some neighboring particles to be excluded from the list. In contrast, however, a big cut-off value will greatly reduce the computational efﬁciency. To speed up the searching for neighbors, the particles in each ﬂuid cell in this research are put into a corresponding list. All neighbors of a particle will then be found either in the cell containing the particle or in an adjacent cell.

FIG. 15. The scheme of neighbor list and cell lists. The particle of interest is black; the grey particles are within the neighbor list cut-off.

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C. GAS DYNAMICS In the DPM the gas phase is treated as a continuum phase, the dynamics of which can be described by a set of volume-averaged Navier–Stokes equations (Kuipers et al., 1992). From mass conservation, we have @ðrÞ þ r ðruÞ ¼ 0 @t

(40)

where r is the gas density, e the local porosity, and u the gas velocity. Momentum conservation gives that @ðruÞ þ r ðruuÞ ¼ rp S p r ð¯tÞ þ rg @t

(41)

where p is the gas phase pressure, t¯ the viscous stress tensor, g the gravitational acceleration, and Sp a source term that describes the momentum exchange with the solid particles present in the control volume: Sp ¼

1 V

Z N part X bV a ðu va Þdðr ra ÞdV 1 a¼1

(42)

Here V represents the local volume of a computational cell and Va the volume of particle a. The d-function ensures that the drag force acts as a point force at the (central) position of this particle. In Eq. (42), b is the momentum transfer coefﬁcient, which will be discussed in more detail in Section III.D. The gas phase density r is calculated from the ideal gas law: r¼

pM RT

where R is the universal gas constant (8.314 J/(mol K)), T the temperature, and M the molecular mass of the gas. The equation of state of the ideal gas can be applied for most gases at ambient temperature and pressure. The viscous stress tensor t¯ is assumed to depend only on the gas motion. For gasﬂuidized beds, the general form for a Newtonian ﬂuid (Bird et al., 1960) can be used: 2 t¯ ¼ l m ðr uÞI¯ þ mðru þ ðruÞT Þ 3

(43)

with l the gas phase bulk viscosity, m the gas phase shear viscosity, and I¯ the unit tensor. Normally, the bulk viscosity of the gas phase can be set equal to zero (Bird et al., 1960). Note that no turbulence modeling is taken into

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101

account. For dense gas–solid ﬂuidization, this can be justiﬁed since the turbulence is completely suppressed in the particle bed due to the high solids volume fraction. The numerical method for solving the set of Eqs. (40) and (41) is similar to the method that is used in the TFM, which is discussed in detail in Section IV.E. The time step by which the gas-phase is updated is typically one order of magnitude larger than the time step dt that is used for updating the soft-sphere system. The boundary conditions are taken into account by utilizing ﬁctitious cells at the boundaries and a ﬂag-matrix concept, which allows different boundary conditions to be applied for each single cell. A variety of boundary conditions can be applied by speciﬁcation of the value of the cell ﬂag fl(i, j, k), which deﬁnes the relevant boundary condition for the corresponding cell (i, j, k). A typical set of boundary conditions used in a 2D simulation is shown in Fig. 16. In Table II, we explain the meaning of each type of boundary condition. Normally, the bottom distributor is deﬁned as inﬂux cells formulated by fl(i, j, k) ¼ 4, where the void fraction is set to a constant value of 0.4.

FIG. 16. The typical set of boundary conditions used in 2D simulations.

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VALUES

FOR THE

TABLE II CELL FLAG, WHICH DEFINE

THE

BOUNDARY CONDITIONS

fl(i, j, k)

The type of cell

1 2 3 4 5 6 7

Interior cell, no boundary conditions have to be speciﬁed Impermeable wall, free-slip boundaries Impermeable wall, no-slip boundaries Inﬂux cell, velocities have to be speciﬁed Prescribed pressure cell, free-slip boundaries Continuous outﬂow cell, free-slip boundaries Corner cell, no boundary conditions have to be speciﬁed

D. INTERPHASE COUPLING For dense gas–solid two-phase ﬂows, a four-way coupling is required; however, the coupling between particles is managed in a natural way in DPMs. The task is, therefore, only to ﬁnd a two-way coupling between the gas and the solid phases, which satisﬁes Newton’s third law. Basically, the gas phase exerts two forces on particle a: a drag force Fd,a due the ﬂuid–solid friction at the surface of the spheres, and a force Fp,a ¼ –Va rp due to the pressure gradient rp in the gas phase. We will next describe these forces in more detail, along with the procedure to calculate void fraction, which is an essential quantity in the equations for the gas–solid interaction. 1. Drag Force The drag force that the gas phase exerts on a particle a, consistent with the source term Sp in expression Eq. (41), reads Fd;a ¼

V ab ðu va Þ 1

(44)

where b is the momentum exchange coefﬁcient. The commonly used drag correlations for b in the simulation of gas-ﬂuidized beds are the Ergun (1952) equation for denser beds (eo0.8): bd 2 ¼ 150ð1 Þ þ 1:75Rea mð1 Þ

(45)

and the Wen and Yu (1966) equation for dilute systems (e40.8): bd 2 3 ¼ C d Rea 1:65 mð1 Þ 4

(46)

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103

with Rea ¼ reda|u–va|/m the Reynolds number of particle a and Cd the drag coefﬁcient, for which the expression by Schiller and Nauman (1935) is used: ( Cd ¼

Þ=Rea 24ð1 þ 0:15Re0:687 a

Rea o103

0:44

Rea 4103

Note that the validity of both the Ergun and Wen and Yu equations has recently been questioned on the basis of LB data, and alternative drag-force correlations have been proposed. From LB simulations, Hill et al. (2001a, b) suggest the following relation for Stokes ﬂow (lim Rea-0): bd 2 ¼ Ao ð1 Þ mð1 Þ with

Ao ¼

8 > < 180

183 1þp3ﬃ ð1Þ1=2 þ135 64 ð1Þþ16:14ð1Þ

2 > : ð1Þþ0:681ð1Þ2 8:48ð1Þ3 þ8:16ð1Þ4

o0:6 40:6

whereas for Rea440, they found that the drag force increases linearly with Rea: bd 2 ¼ A2 ð1 Þ þ 0:60573 þ 1:9083 ð1 Þ þ 0:2092 Rea mð1 Þ

(47)

In the paper by Hill et al. (2001b), values for A2 are only given for a ﬁnite number of gas fractions1; however, A2 is nearly the same as Ao (Koch and Hill, 2001). Note that in Section II.D we suggest a different expression for b, also on the basis of lattice Boltzmann simulations. 2. Force from the Pressure Gradient The force on particle a due to the pressure gradient rp in the gas phase is equal to Fp;a ¼ V a rp Note that the reaction of this force (thus the two-way coupling) is incorporated in the momentum conservation equation of the gas phase in the ﬁrst term on the RHS of Eq. (41). The local value for rp at ra is obtained from a volumeweighted averaging technique using the values of the pressure gradients at the eight surrounding grid nodes. The volume-weighted averaging technique used to 1

In Ref. Hill et al. (2001b), values are listed for F2, which relates to A2 via F2 ¼ A2(1e)/(18e3).

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¯ of a quantity Qijk from the eight surrounding obtain the local-averaged value Q computational nodes is shown in Fig. 17. The local-averaged value is calculated as follows: ¯ ¼ Q1 V 8 þ Q2 V 7 þ Q3 V 6 þ Q4 V 5 þ Q5 V 4 þ Q6 V 3 þ Q7 V 2 þ Q8 V 1 Q DX DY DZ where V 1 ¼ dx d y dz V 5 ¼ dx dy d~ z

V 2 ¼ d~ x dy dz V 6 ¼ d~ x dy d~ z

V 3 ¼ dx d~ y dz V 7 ¼ dx d~ y d~ z

V 4 ¼ d~ x d~ y dz V 8 ¼ d~ x d~ y d~ z

with d~ x ¼ DX dx; d~ y ¼ DY dy; d~ z ¼ DZ dz, and the distances dx, dy, and dz—necessary in this averaging technique—are calculated from the position of the particle in the staggered grid (see also Fig. 24). Note that the same technique of volume weighting is also used to obtain local gas velocities and local void fractions at the position of the center of the particle. 3. Void Fraction Calculation From the position of each particle, we can calculate its contribution to the local solid volume fraction es in any speciﬁed ﬂuid cell. This local void fraction, e ¼ 1 – es, is one of the key parameters that controls the momentum exchange between the phases and should be determined with care.

FIG. 17. The scheme of volume-weighted averaging.

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105

For a 2D situation, the void fraction e(i, j) can be calculated on the basis of the area occupied by the particles in the cell of interest. A particle can be present in multiple cells, however, as shown in Fig. 18. Hoomans et al. (1996) developed a method to account for the multiple cell overlap. The area of Aii,jj is given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Aii;jj 1 2 2 d 1 d þ d 1 d arccos d ¼ d d þ arcsin d 1 2 1 2 1 2 1 2 2 R2a

(48)

and area Ai,jj by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ai;jj 1 1 2 2 p d d ¼ d þ þ arccosd 1 d þ d 1 d arccosd 1 2 1 2 1 2 1 2 2 2 R2a

(49)

with d1 ¼ d1/Ra and d2 ¼ d2/Ra (see Fig. 18). The area Aii,j can be calculated by an equation similar to Eq. (49). However, the void fraction calculated in this way is based on a 2D distribution of disks, whereas the empirical drag-force correlations are derived for 3D systems. To correct for this inconsistency, the void fraction calculated on the basis of area (e2D) is transformed into a 3D void fraction (e3D) using the following equation: 2 3D ¼ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ p 3ð1 2D Þ3=2

FIG. 18. The multiple cell overlap of a single particle. From Hoomans, Ph.D. thesis, University of Twente (2000).

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M.A. VAN DER HOEF ET AL.

In a true 3D situation, we can calculate the void fraction on the basis of actual volume of the particles. However, no analytical expression is available for volume Vii,jj. Hoomans et al. (1996) suggested the approximation V ii V jj Va Va Va

V ii;jj

(50)

with Vii ¼ Vii,jj+Vii,j and Vjj ¼ Vii,jj+Vi,jj. The volume of the sphere caps Vii and Vjj can be calculated exactly by V jj 1 ¼ ð1 d2 Þ2 ð2 þ d2 Þ Va 4

V ii 1 ¼ ð1 d1 Þ2 ð2 þ d1 Þ Va 4

with di ¼ di/Ra being the distance from the center of the particle to the cell boundary relative to the radius of the particle and Va ¼ 4pRa3/3 the volume of the particle. The error in the calculation of the porosity that is introduced by the approximation in Eq. (50) is negligibly small when the particle radii are an order of magnitude smaller than the size of the CFD-grid cell, which is required in any case in order to have a grid-independent value of the porosity. In this context, it is noteworthy that recently a new method has been developed that can generate a grid-independent estimate of e, even when the size of the particles is of the order of the size of the grid cells (Link et al., 2005). E. ENERGY BUDGET To relate the discrete particle simulations to the KTGFs, it is very useful to analyze the detailed information of the energy evolution in the system. The total energy balance of the system is obtained by calculating all relevant forms of energy as well as the work performed due to the action of external forces. rot Translational kinetic energy Etrans kin and rotational kinetic energy Ekin

N

E trans kin ¼

part 1X ma ðva va Þ 2 a¼1

N

E rot kin ¼

part 1X I a ðoa oa Þ 2 a¼1

(51)

Potential energy from gravity

Ep ¼

N part X

ma ðg ra Þ

a¼1

Potential energy of the normal spring and tangential spring N

Es ¼

part X 1X ðkn d2ab;n þ kt d2ab;t Þ 2 a¼1 b

where b4a and b A the contactlist of a, and dab,n, dab,t the overlap and relative tangential displacement, respectively, of particle a and b.

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

107

The work done by the external forces and the cohesive force in one time step

dt N part X

W ext ¼ dt

ðFd;a þ Fp;a þ Fcoh;a Þ va

a¼1

Also, the energy dissipated during the particle–particle contact process has to be considered and is determined by the following: Energy dissipated by the normal and tangential spring in one time step dt

E ds ¼ dt

N part X X

ðZn ðvab;n vab;n Þ þ Zt ðvab;t vab;t ÞÞ

a¼1

b

Energy dissipated by the friction between particles in one time step dt

E df ¼ dt

N part X X a¼1

ðmf Fab;n tab;n vab;t Þ

b

where b4a and b A the contactlist of a. The total energy of the system is then equal to rot E tot ¼ E p þ E trans kin þ E kin þ E s þ E st W ext þ E ds þ E df

F. RESULTS

FOR THE

EXCESS COMPRESSIBILITY

In previous work, we have mainly used the DPM model to investigate the effects of the coefﬁcient of normal restitution and the drag force on the formation of bubbles in ﬂuidized beds (Hoomans et al., 1996; Li and Kuipers, 2003, 2005; Bokkers et al., 2004; Van der Hoef et al., 2004), and not so much to obtain information on the constitutive relations that are used in the TFMs. In this section, however, we want to present some recent results from the DPM model on the excess compressibility of the solids phase, which is a key quantity in the constitutive equations as derived from the KTGF (see Section IV.D.). The excess compressibility y can be obtained from the simulation by use of the virial theorem (Allen and Tildesley, 1990). y¼

XX 1 Fab rab ¯ 6mN part y a b

(52)

where the sums are over all particles, with the restriction that a6¼b. In Eq. (52), Fab is the interaction force between particles a and b. For the soft-sphere model

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M.A. VAN DER HOEF ET AL.

as presented in the previous sections, in the absence of cohesive forces, Fab is equal to the sum of Fab,n and Fab,t, as given by Eqs. (26) and (21), and then only when particles a and b are in contact. Furthermore, in Eq. (52), y¯ is the average granular temperature, which can be deﬁned as the average over the total volume of the local granular temperature deﬁned by Eq. (59). In the absence of any drift trans velocity, my¯ ¼ 2E trans kin =ð3NÞ, with E kin as the total translational kinetic energy given by Eq. (51). Note that for the hard-sphere model there are no forces, and a different procedure is required. In that case, the solids pressure (and thus the excess compressibility) can be obtained from the average number of collisions per unit time (Allen and Tildesley, 1990). We have performed simulations for 500 particles with periodic boundary conditions and no gas phase present. Owing to the inelastic collisions, the particles will continuously dissipate energy, which would eventually cause the particles to come to a quiescent state. In this work, we therefore drive the system by two different techniques: (1) rescaling the particle velocities every time step, according to the desired granular temperature; (2) accelerating the particles randomly. Method (2) is most robust but less efﬁcient. The rescaling procedure, however, does not attain an equilibrium state for high solid fractions. For this reason, the random acceleration procedure is used to simulate the denser system with a solid fraction higher than 0.45, while the rescaling procedure is used for lower solid fractions. For more details on the procedures, we refer to a recent paper (Ye et al., 2005). All the parameters are normalized by the particle radius, particle density, and granular temperature. First, we should check whether the soft-sphere model gives results comparable to those from the hard-sphere model, since the approximate theories of granular ﬂow are based on the latter model. To this end, we carried out several sets of simulations with particles starting from either random positions or facecentered cubic (FCC) positions. The thermodynamic properties of the hardsphere system for these two conﬁgurations have been well documented by many researchers (Alder and Wainwright, 1957; Carnahan and Starling, 1969; Hoover and Ree, 1969; Erpenbeck and Wood, 1984). In Fig. 19, we show our simulation results for smooth, elastic, and cohesiveless spheres in periodic boundary domains, where at the start of the simulation the particles are placed in an FCC grid. For such systems, Hoover and Ree (1969) observed a phase transition from the ﬂuid state to the solid state at y ¼ 7.27. As can be seen, both the hardsphere and soft-sphere simulations clearly display this transition point. For the ﬂuid state, our simulation data from both models is in very good agreement with the Carnahan–Starling equation of state (Carnahan and Starling, 1969). 4s 22s (53) ð1 s Þ3 The conclusion is that the soft-sphere model can be used as an alternative for the hard-sphere model, as far as the calculation of the excess compressibility is concerned. yES ¼

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109

FIG. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan–Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC conﬁguration. Spring stiffness is K ¼ 70,000, granular temperature is y ¼ 1.0, and coefﬁcient of normal restitution is e ¼ 1.0. The system is driven by rescaling.

Next, we consider a system of inelastic spheres (ISs). As can be seen from Eq. (81), the KTGF predicts that the excess compressibility yIS of ISs is a linear function of the coefﬁcient of normal restitution e, yIS ¼

ð1 eÞ ES y 2

(54)

where yES is the excess compressibility of elastic spheres (ESs). In Fig. 20, we show our simulation results for the excess compressibility of ISs, both for the soft-sphere and the hard-sphere model. The solid fraction in the initial conﬁguration is ﬁxed at 0.05. It is shown that for this dilute system, the simulation results of both models are in very good agreement with the prediction in Eq. (54) from the KTGF (solid line). Note that the Eq. (54) is derived under the assumption that the particles are only slightly inelastic, i.e., e1.0. In Fig. 21, the excess compressibility is shown as a function of the solid fraction for different coefﬁcients of normal restitution e. These results are compared with the Eq. (54), where the excess compressibility yES is taken from either the Ma–Ahmadi correlation (Ma and Ahmadi, 1986) or the Carnahan–Starling correlation. As can be seen, the excess compressibility agrees well with both correlations for a solid fraction es up to 0.55. For extremely dense systems, i.e., es40.55, the Ma–Ahmadi correlation presents a much better estimate of the excess compressibility, which is also the case for purely elastic particles (see Fig. 23).

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M.A. VAN DER HOEF ET AL.

1.0 yIS = yES (1+e)/2

yIS / yES

0.9

Hard−sphere Soft−sphere

0.8

0.7

0.6

0.5 0.0

0.2

0.4

0.6

0.8

1.0

e FIG. 20. Excess compressibility yIS for a system of inelastic hard spheres, as function of the coefﬁcient of normal restitution, for one solid fraction (es ¼ 0.05). The excess compressibility has been normalized by the excess compressibility yES of the elastic hard spheres system. Other simulation parameters are as in Fig. 19.

Up to this point, we have neglected the cohesive van der Waals forces between the particles, which is only justiﬁed if particles are larger than say 1 mm. Presently, the van der Waals forces have not been included in the KTGF; a ﬁrst step would be to consider the effect of such forces on the excess compressibility by also including the interparticle force of Eq. (36) in Fab of Eq. (52). In Fig. 22, the results for the excess compressibility for different Hamaker constants A are shown. For simplicity, a coefﬁcient of normal restitution e ¼ 1.0 is used. We consider two different Hamaker constants: A ¼ 3.0 1012 and A ¼ 3.0 1010 (in units where rs ¼ 1, R ¼ 1, and y ¼ 1). From Fig. 22, we see that for these two Hamaker constants, the simulation results differ only slightly from the prediction in Eq. (54), where yES is calculated from the Ma–Ahmadi correlation, which suggests that cohesion has only a weak inﬂuence on the excess compressibility—at least for the values of Hamaker constant that we studied. In this context, it should be noted that the quantiﬁcation of the cohesive force is not straightforward since there is no reference force (such as gravitational force) in these systems. We consider these systems as slightly cohesive since the ratio of the cohesive potential and the average kinetic energy per particle is small, i.e., j ¼ 6.25 1086.25 106. At the same time, the ratio between the cohesive force and contact force ranges from 1.11 105 to 1.11 103. If a strong cohesive force is present, particles in the system may form complicated structures, whereas a homogeneous state is one of the basic assumptions underlying

111

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

40

40

e = 1.00 30

yIS

yIS

30 20 10

0.2 0.4 Solid fraction εs

0 0.0

0.6

40

0.2 0.4 Solid fraction εs

0.6

40

e = 0.90

30

yIS

yIS

20 10

0 0.0

30

e = 0.95

20 10 0 0.0

e = 0.80

20 10

0.2 0.4 Solid fraction εs

0.6

0 0.0

0.2 0.4 Solid fraction εs

0.6

FIG. 21. The excess compressibility from soft-sphere simulations, with random initial particle positions, for different coefﬁcients of normal restitution e: (a) e ¼ 1.0 (top-right); (b) e ¼ 0.95 (topleft); (c) e ¼ 0.90 (bottom-right); (d) e ¼ 0.80 (bottom-left). The simulation results (symbols) are compared with Eq. (54) based on the Ma–Ahmadi correlation (solid line) or the Carnahan–Starling correlation (dashed line). The spring stiffness is set to kn ¼ 70,000.

the KTGF. A more detailed analysis of the effect of the cohesive force on the excess compressibility can be found in Ref. Ye et al. (2005).

IV. Two-Fluid Model A. INTRODUCTION In the Euler–Euler models, i.e., the TFMs, it is assumed that both the gas and the solid phase are interpenetrating continua. This continuous approach is especially useful and computationally cost-effective when the volume fractions of the phases are comparable, or when the interaction within and between the phases plays a signiﬁcant role in determining the hydrodynamics of the system. As discussed before, it is relatively straightforward to model the gas phase, for instance by the use of well-established CFD techniques. The challenge is to establish an accurate ‘‘hydrodynamic’’ description of the particulate phase.

112

M.A. VAN DER HOEF ET AL.

FIG. 22. The effect of the cohesive force on the excess compressibility. The coefﬁcient of normal restitution is e ¼ 1.0, and granular temperature is T ¼ 1.0. The Hamaker constant is A ¼ 3.0 1012 (circles) and 3.0 1010 (crosses).

Anderson and Jackson (1967, 1968, 1969) and Ishii (1975) have separately derived the governing equations for TFMs from ﬁrst principles. Although the details of constructing the averaged equations are different, the ﬁnal equations are essentially the same. The TFMs differ signiﬁcantly from each other as different closures for the solid stress tensor are used. There are basically three types of approaches to deﬁne the solid stress tensor, or more speciﬁcally the solid viscosity. In the early hydrodynamic models— developed by Jackson and his co-workers (Anderson and Jackson, 1967; Anderson et al., 1995), Kuipers et al., (1992), and Tsuo and Gidaspow (1990)—the viscosity is deﬁned as an empirical constant, and also the dependence of the solid phase pressure on the solid volume fraction is determined from experiments. The advantage of this model is its simplicity, the drawback is that it does not take into account the underlying characteristics of the solid phase rheology. In another class of models, pioneered by Elghobashi and Abou-Arab (1983) and Chen (1985), a particle turbulent viscosity, derived by extending the concept of turbulence from the gas phase to the solid phase, has been used. This is the so-called k model, where the k corresponds to the granular temperature and is a dissipation parameter for which another conservation law is required. By coupling with the gas phase k turbulence model, Zhou and Huang (1990) developed a k model for turbulent gas–particle ﬂows. The k models do not

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113

include the effect of particle–particle collisions, and so these models are restricted to dilute gas–particle ﬂows. Signiﬁcant contributions to the modeling of gas–solid ﬂows have been made by Gidaspow and co-workers (1994), who combined the kinetic theory for the granular phase with continuum representations for the particle phase. There are a number of other studies using this approach. Sinclair and Jackson (1989) predicted the core-annular regime for steady developed ﬂow in a riser. Ding et al. (1990) simulated a bubbling ﬂuidized bed. Transient simulations and comparisons to data were done by Samuelsberg and Hjertager (1996). Nieuwland et al. (1996) investigated a circulating ﬂuidized bed using the KTGF. Detamore et al. (2001) have performed an analysis of scale-up of circulating ﬂuidized beds using kinetic theory. One of the strengths of the KTGF, although still under development, is that it can offer a very clear physical picture with respect to the key parameters (e.g., particle pressure, particle viscosity, and other transport coefﬁcients) that are used in the TFMs. The TFMs based on KTGF requires less ad hoc adjustments compared to the other two types of models. Therefore, it is the most promising framework for modeling engineering-scale ﬂuidized beds. B. GOVERNING EQUATIONS In the TFM, both the gas phase and the solid phase are described as fully interpenetrating continua using a generalized form of the Navier–Stokes equations for interacting ﬂuids. The continuity and momentum equations for the gas phase are given by expressions identical to Eqs. (40) and (41), except for the gas–solid interaction term: @ðrÞ þ ðr ruÞ ¼ 0 @t

(55)

@ðruÞ þ ðr ruuÞ ¼ rp bðu us Þ r ð¯sÞ þ rg (56) @t with t¯ as the viscous stress tensor of the gas phase given by Eq. (43). The continuity and momentum equations for the particle phase are given by a similar set of equations: @ðs rs Þ þ ðr s rs us Þ ¼ 0 @t @ðs rs us Þ þ ðr s rs us us Þ ¼ s rp rps þ bðu us Þ r s¯ s þ s rs g @t

(57) (58)

where es ¼ 1e and us is the velocity of the solid phase. Note that rs is the material density of the solid phase, so that the local mass per unit volume is equal to rses. Obviously, the numerical scheme for updating the solid phase is now analogous to (and synchronous with) that of the gas phase, the details of

114

M.A. VAN DER HOEF ET AL.

which are given in Section IV.E. Since the concept of particles has disappeared completely in such a modeling, the effect of particle–particle interactions can only be included indirectly, i.e., via the effective solid pressure ps and the effective solid stress tensor s¯ s . A description that allows for a more detailed description of particle–particle interactions follows from the KTGF, which expresses the pressure and the solid stress tensor as a function of the local granular temperature y, which is deﬁned from the ﬂuctuation in the velocity of the individual solid particles. More precisely, the granular temperature at r is deﬁned as * + Nr 1 1 X 2 y¼ ðva us Þ 3 N r a¼1

(59)

where /.S is an ensemble average, and the sum is over all Nr particles in a small control volume dV around r. Note that also the solid density and velocity as they appear in Eqs. (57) and (58) can be deﬁned from the positions and momenta of the individual particles by similar type of averages2: * s rs ¼

Nr 1 X ma dV a¼1

+

* s rs us ¼

Nr 1 X m a va dV a¼1

+ (60)

For particles of equal mass, we thus have esrs ¼ mn with n the local number density of particles. From the KTGF, the time evolution of the granular temperature is given by 3 @ ðs rs yÞ þ r ðs rs yus Þ ¼ ðps I¯ þ t¯ s Þ : rus r qs 3by g 2 @t

(61)

with qs the kinetic energy ﬂux and g the dissipation of kinetic energy due to inelastic particle collisions. In Eqs. (58) and (61), there are three unknown quantities (pressure, stress tensor, and energy ﬂux), which must be expressed in terms of the three basic hydrodynamic variables (density, velocity, and temperature), in order to get a closed set of equations. This is the subject of the KTGF, and the resulting closures will be presented in Section IV.D. However, before doing so, we will ﬁrst give a brief description of the general principles of kinetic theory.

2 Note that for dV-0 the local density and momentum density can be written as P P esps ¼ amad(r– ri) and espsus ¼ amavad (r– ri), which are the expressions that are usually found in literature.

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115

C. GENERAL KINETIC THEORY In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function fs(r, v), which can be deﬁned as the number density of the solid particles in the 6D coordinate and velocity space. That is, fs(r, v, t) dv dr is the average number of particles to be found in a 6D ‘‘volume’’ dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by Z 1 r¯ s ðr; tÞ ¼ m f s ðr; v; tÞdv (62) 1

and Z

1

r¯ s ðr; tÞus ðr; tÞ ¼

mv f s ðr; v; tÞdv

(63)

1

where the local density is deﬁned as r¯ s ¼ rs s with rs as the material density of the solid particles. The granular temperature, deﬁned by Eq. (59), follows from Z 1 1 r¯ s ðr; tÞyðr; tÞ ¼ mðv us Þ2 f s ðr; v; tÞdv (64) 3 1 The evolution of the one-particle distribution function fs can be described by the Boltzmann equation @ f ðr; v; tÞ þ v rf s ðr; v; tÞ ¼ C @t s

(65)

which is basically a continuity equation, where the second term on the left-hand side (LHS) represents the change of fs in time due to streaming and the collision function C on RHS represents the change of fs due to particle–particle interactions. Conservation of mass, momentum, and energy in a collision gives that C should satisfy Z 1 Z 1 Z 1 Cdv ¼ 0; Cvdv ¼ 0; Cu2 dv ¼ 0 1

1

R

1

R

R Taking the same integrals ( ydv, yvdv, and yu2dv) of the Boltzmann equation Eq. (65), making use of Eqs. (62) and (63), yields @ r¯ þ r ðr¯ s us Þ ¼ 0 @t s

(66)

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M.A. VAN DER HOEF ET AL.

@ ðr¯ us Þ þ r ðr¯ s us us Þ ¼ r p¯ @t s

(67)

3 @ ðr¯ s yÞ þ r ðr¯ s yus Þ ¼ ¯p : rus r qs 2 @t

(68)

with Z

Z

1

mVVf s ðr; v; tÞdv;

p¯ ¼

q¼

1

mV2 Vf s ðr; v; tÞdv; 1 2 1

V ¼ v us

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p¯ and heat ﬂux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman–Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic ﬁelds in time and space. To second order, one arrives at the familiar expression for p¯ and q p¯ ¼ ps I¯ þ s¯ s ;

qs ¼ ks ry

(69)

with I¯ is the unit tensor, and 2 s¯ s ¼ ms ðrus Þ þ ðrus ÞT ls ms ðr us ÞI¯ 3

(70)

where ðruÞab ¼ ra ub ; ðruÞTab ¼ rb ua . Inserting the above expression for p¯ and q into Eqs. (67) and (68) will give the Navier–Stokes equations, where the parameters ks, ls, ms, and ps can be calculated (at least in princeiple) when the collision function C is known. For the simple BGK equation Eq. (11), this will result in the relations of Eq. (13). For an accurate description of the solid phase, however, one requires a much more detailed expression for C, which contains the details of the particle–particle interactions. Although this is a laborious route, it opens a possibility for making a link between the ‘‘microscopic’’ details of particle collisions and the ‘‘macroscopic’’ transport coefﬁcients. Apart from the details of the particle–particle interactions, C does also depend on the joint probability function f ð2Þ s (r1, v1, r2, v2, t), provided that the interactions between the particles are pair-wise additive (generally for n-body interactions, C will ð2Þ depend on f(n) s ). In order to get a closed equation, f s should be described in term of fs. If the velocities v1 and v2 are not correlated, one can write f ð2Þ s ðr1 ; v1 ; r2 ; v2 ; tÞ ¼ gðr12 ; s Þ f s ðr1 ; v1 ; tÞ f s ðr2 ; v2 ; tÞ

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

117

where g(r12, es) is the pair distribution function, which depends only on the distance r12 ¼ jr2 r1 j and the solid fraction. For sufﬁciently low density, g ¼ 1, the collision function takes the form C¼

1 m2

Z

Z dOsðOÞ

dv0 ðv v0 Þ f s ð~vÞf s ð~v0 Þ f s ðvÞf s ðv0 Þ

(71)

where we have omitted the r, t argument of fs. In Eq. (71), v~ , v~ 0 , are the velocities of the two particles involved after the collision, which can be constructed from the initial velocities v, v0 from conservation of energy and momentum: v~ ¼ v þ aða ðv v0 ÞÞ

v~ 0 ¼ v0 þ aða ðv0 vÞÞ

with a as the unity vector along the line connecting the two centers of the particle before the collision. Furthermore, in Eq. (71), s(O) represents the ‘‘cross-section’’ and O is the solid angle in which the particle is scattered. More details on these concepts can be found in the standard literature (Chapman and Cowling, 1970; Liboff, 1998). Using this form of the collision function, it can be shown that pressure ps, shear viscosity ms, and thermal conductivity ks in Eqs. (69) and (70) are given by mid s

5 ¼ prs d 96

rﬃﬃﬃ y ; p

kid s

75 pr d ¼ 384 s

rﬃﬃﬃ y ; p

pid s ¼ s rs y

(72)

where d is the diameter of the particles, and the superscript id (ideal) indicates that the expressions are for the limit of a dilute gas, for which the pressure is given by the ideal gas law. For high densities, g cannot be set equal to one, and the collision function becomes much more complex and so is not given here. It turns out, however, that instead of using the full radial distribution function, it is sufﬁcient to use the value at contact r ¼ R, so that we deﬁne a new function: wðs Þ ¼ gðR; s Þ In the standard Enskog theory (SET), the shear viscosity and thermal conductivity of ESs are found to be equal to3 mES s

¼

mid s

1 4 þ þ 0:7614wbrs brs wbrs 5

(73)

3 See Chapman and Cowling (1970). Note that the true expression for mES reads s id 1 4 4 12 mES s ¼ c1 ms ðxbrs þ 5 þ 25 ð1 þ pc2 Þwbrs Þbrs , with c1 ¼ c2 ¼ 1.016. In most expressions in literature, c1 is set equal to 1; in expression Eq. (82) of Gidaspow, both c1 ¼ c2 ¼ 1, which is the cause of the slightly different coefﬁcient 0.771, compared to 0.7614 in Eq. (75). For practical purposes, the difference is negligible. Similar remarks can be made about k.

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M.A. VAN DER HOEF ET AL.

id kES s ¼ ks

1 6 þ þ 0:7574wbrs brs wbrs 5

(74)

with b ¼ 2espd3/3 m, so that bps ¼ 4es. Note that the pressure of a dense system is directly related to the radial distribution function at contact (Chapman and Cowling, 1970; Hansen and McDonald, 1986): id ES pES s ¼ ps ð1 þ y Þ

yES ¼ wbrs ¼ 4ws

with yES the excess compressibility of the elastic hard-sphere system. Thus in the Enskog theory, the transport coefﬁcients are completely determined by the yES: 1 4 id ES þ 0:7614y ¼ 4m þ mES (75) s s s yES 5 1 6 ES þ 0:7574y þ (76) yES 5 Various expressions for yES have been proposed in literature based on the virial coefﬁcients and simulation data. Most of these have the following general form: P cn ð4s Þnþ1 ES y ðs Þ ¼ n¼0 (77) ð1 ðs =cp Þa Þb id kES s ¼ 4ks s

with ecp the ‘‘close-packed’’ solid fraction, at which the pressure diverges. In Table III, we summarize the parameters found by different authors. A comparison of expression in Eq. (77) with the MD simulation data from Alder and Wainwright (1960) and Woodcock (1981) is shown in Fig. 23. In our current VALUES

ecp a b co c1 c2 c3 c4 c5 c6 c7 c8

TABLE III PARAMETERS

FOR THE

IN

EQ. (77)

CS

MA

SSM

TC

1 1 3 1 1/8 0 0 0 0 0 0 0

0.64356 3 0.67802 1 0.625 0.2869 0.070554 0 0 0 0 0

0.6435 1 0.76 1 0.3298 0.08867 0.01472 0.0005396 0.0003574 0.0005705 –0.0001212 –0.0001151

0.6875 1 1 1 0.2613 0.05968 0.005905 –0.001191 –0.0004455 –0.0004818 –0.00003636 –0.00008182

Note: CS: Carnahan and Starling (J. Chem. Phys. 51, 635 (1969)); MA: Ma and Ahmadi (J. Chem. Phys. 84, 3449 (1986)); SSM: Song, Stratt and Mason (J. Chem. Phys. 88, 1126 (1988)); TC: Tobochnik and Chapin (J. Chem. Phys. 88, 5824 (1988)).

119

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

80 Carnahan−Starling Ma−Ahmadi Song−Stratt−Mason MD (Woodcock) MD (Alder & Wainwright)

yES

60

40

20

0 0.4

0.5

0.6

0.7

solid fraction εs FIG. 23. Comparison of the expressions from Eq. (77) and Table III with data from MD simulations.

version of the TFM, we use the expression by Ma and Ahmadi (1986) (see also Fig. 21). Alder et al. (1970) have also measured the shear viscosity in MD simulations of dense hard-sphere systems. It was found that the Enskog approximation in Eq. (75) is very accurate up to es ¼ 0.3; however, for higher solid fractions the theory signiﬁcantly underestimates the shear viscosity up to a factor of two for esE0.5. D. KINETIC THEORY

OF

GRANULAR FLOW

In the KTGF, the dissipation of energy in collisions is included in the Enskog theory. Currently, only the effect of the coefﬁcient of normal restitution has been considered, although it is anticipated that friction also plays an important role. The derivation of the constitutive equations for ISs can be found in the book by Gidaspow (1994) and the papers by Jenkins and Savage (1983), Lun et al. (1984), Ding and Gidaspow (1990), and Nieuwland et al. (1996). Here, we will present the expressions for ps, ms, and Ks from the book of Gidaspow (1994) (Eqs. (T.9.1), (9.183), (9.250), (9.262), (9.268), and (9.272)): pIS s ¼ ½1 þ 2ð1 þ eÞs g s rs y mIS s

5 ¼ prs d 96

rﬃﬃﬃ rﬃﬃﬃ 2 y 2 4 4 2 y 1 þ ð1 þ eÞs g þ s rs dgð1 þ eÞ p ð1 þ eÞg 5 5 p

(78)

(79)

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M.A. VAN DER HOEF ET AL.

kIS s

75 pr d ¼ 384 s

rﬃﬃﬃ rﬃﬃﬃ 2 y 2 6 y 2 1 þ ð1 þ eÞs g þ 2s rs dgð1 þ eÞ p ð1 þ eÞg 5 p

(80)

where g is the value of the radial distribution function of a hard-sphere ﬂuid at contact and e is the coefﬁcient of normal restitution. Note that from Eq. (78) it follows that the excess compressibility of the IS system is equal to yIS ¼ 2ð1 þ eÞs g ¼

ð1 þ eÞ ES y 2

(81)

that is, the dissipation in the collisions reduces the excess compressibility by a factor of (1+e)/2. Replacing 2(1+e)esg in Eqs. (79) and (80) by yIS and using expression in Eq. (72) for msid and ksid gives " # 1 2 IS 2 48 IS 1 4 id id IS y y þ 0:771y þ þ m ¼ ms 4s IS 1 þ ¼ ms 4s (82) y 5 25p yIS 5 " k¼

kid s 4s

# 1 3 IS 2 32 IS 1 6 id IS y 1þ y þ þ þ 0:767y ¼ ks 4s yIS 5 25p yIS 5

(83)

which are of the same form as the Enskog expressions in Eqs. (75) and (76), with yES replaced by yIS.3 It thus turns out, like for the elastic hard spheres, that the constitutive equations are completely determined by the excess compressibility, and that the general form of the Enskog equations is not affected by the dissipation of energy in the collisions. Note that in the granular temperature equation Eq. (61), there is one extra term that is absent in the SET, namely the dissipation of ﬂuctuating kinetic energy g. From the KTGF follows that " rﬃﬃﬃ # 3 4 y IS r us g ¼ ð1 eÞy rs s y 2 d p E. NUMERICAL SOLUTION METHOD Owing to the tendency of inelastic particles to contract in high-density clusters, and the strong nonlinearity of the particle pressure near the maximum packing density, special attention has to be paid to the numerical implementation of the model equations. Most ‘‘classic’’ constant property TFMs are solved using computational methods based on the implicit continuous Eulerian (ICE) method pioneered by Harlow and Amsden (1975). The implementation is based on a ﬁnite difference technique and the algorithms closely resemble the SIMPLE algorithm (Patankar and Spalding, 1972), whereby a staggered grid is employed to reduce numerical instability. A detailed discussion on the application of this numerical technique to TFMs for gas-ﬂuidized beds is presented by Kuipers et al. (1992).

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

121

In principle, this numerical solution method can be straightforwardly applied to ‘‘modern’’ TFMs with closure laws according to the KTGF. However, when doing so, the numerical stability of the TFM is highly affected by the value of the coefﬁcient of normal restitution. Problems that can be handled with acceptable time steps of 10– 4 s for ideal particles (e ¼ 1) require time steps of 10– 5 s when the coefﬁcient of normal restitution is taken to be 0.97, and unacceptably small time steps of 10– 6 s have to be taken when the coefﬁcient of normal restitution is reduced below 0.93. This extreme sensitivity to the value of the coefﬁcient of normal restitution is caused by the fact that particle volume fractions at the next time level are estimated without taking into account the strong nonlinear dependence of the particle pressure on the particle volume fraction. A new numerical algorithm, which estimates the new particle volume fraction taking the compressibility of the particulate phase more directly into account, is presented in this section.

1. Discretization of the Governing Equations The set of conservation equations, supplemented with the constitutive equations, boundary, and initial conditions cannot be solved analytically, and a numerical method must be applied to obtain an approximate solution. Therefore, the domain of interest is divided into a number of ﬁxed Eulerian cells through which the gas–solid dispersion moves. A standard ﬁnite difference technique is applied to discretize the governing equations.4 The cells are labeled by indices i, j, and k located at their centers, and a staggered grid conﬁguration is applied. According to this conﬁguration the scalar variables are deﬁned at the cell centers, whereas the velocities are deﬁned at the cell faces, as indicated in Fig. 24. Furthermore, different control volumes have to be applied for mass and granular energy conservation on one hand and the momentum conservation equations on the other. The control volumes for mass and granular energy conservation coincide with the Eulerian cells, whereas the control volumes for momentum conservation in all three directions are shifted half a cell with respect to the Eulerian cells. Applying ﬁrst-order time differencing and fully implicit treatment of the convective ﬂuxes, the discretized form of continuity equation for the solid phase, Eq. (57), becomes o dt n nþ1 hs rs us;x inþ1 hs rs us;x ii 1 1 ;j;k ;j;k iþ 2 2 dx o dt n nþ1 hs rs us;y inþ1 þ h r u i 1 1 s s;y s i;jþ2;k i;j2;k dy n o dt nþ1 nþ1 hs rs us;z ii;j;kþ þ ¼0 1 hs rs us;z i 1 i;j;k 2 2 dz

n ðs rs Þnþ1 i;j;k ðs rs Þi;j;k þ

4

This part is based upon Chapter 2 of the thesis of Goldschmidt (2001).

ð84Þ

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M.A. VAN DER HOEF ET AL.

FIG. 24. Positions at which the key variables are evaluated for a typical computational cell in the staggered-grid conﬁguration.

where the superscripts n and n+1 indicate that the quantities are at the old and the new time, respectively. For the discretization of all convective mass, momentum, and ﬂuctuating kinetic energy ﬂuxes the second-order accurate Barton scheme (Centrella and Wilson, 1984; Hawley et al., 1984) is applied. A schematic representation of this scheme for the convective transport of a quantity D (e.g., er) by a velocity Vi+1/2 (e.g., ux) is given in Fig. 25. In the discretization of the momentum Eq. (58), the terms associated with the gas and solid pressure gradients are treated fully implicitly. The interphase momentum transfer term is treated in a linear implicit fashion, and all other terms are treated explicitly. The discretization of the solid phase momentum in Eq. (58) for the x-direction is given by n o nþ1 dt nþ1 nþ1 n ðs rs us;x Þnþ1 ðpÞ ¼ A ð Þ ðpÞ 1 1 1 s iþ1;j;k i;j;k iþ2;j;k iþ2;j;k iþ2;j;k dx n o dt nþ1 n nþ1 ðps Þiþ1;j;k ðps Þnþ1 ð85Þ i;j;k þ dtbiþ12;j:k ðux us;x Þiþ12;j;k dx where momentum convection, viscous interaction, and gravity have been collected in the explicit term An. The equation for the y-direction is obtained by

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

123

FIG. 25. Schematic representation of the Barton scheme for the convective ﬂux of a quantity D by velocity Vi+1/2 in the x-direction.

substituting y for x, B for A, and a change of subscripts: ð. . . Þiþ12;j;k ) ð. . . Þi;jþ12;k

ð. . . Þiþ1;j;k ) ð. . . Þi;jþ1;k

and the equation for the z-direction is obtained by the substituting z for x, C for A, and a change of subscripts ð. . . Þiþ12;j;k ) ð. . . Þi;j;kþ12

ð. . . Þiþ1;j;k ) ð. . . Þi;j;kþ1

Note that the mass and momentum equations for the gas phase can simply be obtained by replacing es-e, rs-r, us-u in Eqs. (84) and (85), and dropping the terms concerning the particle-pressure gradient. The granular energy equation is solved in a fully implicit manner. The solution of the equation however proceeds through a separate iterative procedure that solves the granular temperature equations for the whole computational domain when this is required by the main solution procedure discussed in the next paragraph. In this separate iterative procedure, the terms regarding convective transport and generation of ﬂuctuating kinetic energy by viscous shear are explicitly expressed in terms of the most recently obtained granular temperature y*. The granular energy dissipation term is treated in a semi-implicit manner, whereas all other terms are treated fully implicitly. The

124

M.A. VAN DER HOEF ET AL.

applied discretization of the granular temperature equation is given by g 3 3 nþ1 ðs rs yÞnþ1 ðs rs yÞni;j þ D i;j;k dt3bni;j;k yi;j;k dt ynþ1 i;j;k ¼ 2 2 y i;j;k i;j;k n o nþ1 dt ðus;x Þ iþ1;j;k ðus;x Þ i1;j;k ðps Þi;j;k 2 2 dx n o nþ1 dt ðus;y Þ i;jþ1;k ðuk;y Þ i;j1;k ðps Þi;j;k 2 2 dy n o dt nþ1 ðus;z Þ i;j;kþ1 ðuk;y Þ i;j;k1 ðps Þi;j;k 2 2

dx o o dt 1 n nþ1 1 n nþ1 nþ1 nþ1 ðks Þiþ1;j;k yiþ1;j;k yi;j;k ðks Þ i1;j;k yi;j;k yi1;j;k þ 2 2 dx dx dx

n o o dt 1 1 n nþ1 nþ1 ðks Þ i;jþ1;k yi;jþ1;k yi;j;k ynþ1 ynþ1 ðks Þ i;jþ1;k þ i;j;k i;j1;k 2 2 dx dy dy

n o o dt 1 1 n nþ1 nþ1 nþ1 ynþ1 Þ y ð86Þ ðk þ ðks Þ i;j;kþ1 yi;j;kþ1 y 1 s i;j;k i;j;k1 i;j;k2 2 dz dz dz i;j;k

In this equation, the superscript (*) indicates that a term is computed based upon the most recent information, which complies with the (n+1)th time level when all iterative loops have converged. Further, the convective transport and viscous generation of ﬂuctuating kinetic energy have been collected in the explicit term D*. The iterative solution procedure for the granular energy equations continues until the convergence criteria nþ1 ynþ1 i;j;k yi;j;k oey yi;j;k

(87)

are simultaneously satisﬁed for all cells within the computational domain. For a typical value of ey ¼ 10– 6, this takes only a couple of iterations per time step. 2. Solution Procedure of the Finite Difference Equations The numerical solution of the discretized model equations evolves through a sequence of computational cycles, or time steps, each of duration dt. For each computational cycle, the advanced (n+1)-level values at time t+dt of all key variables have to be calculated for the entire computational domain. This calculation requires the old n-level values at time t, which are known from either the previous computational cycle or the speciﬁed initial conditions. Then each computational cycle consists of two distinct phases: calculation of the explicit terms An, Bn, and Cn in the momentum equations

for all interior cells and implicit determination of the pressure, volume fraction, and granular

temperature distributions throughout the computational domain with an iterative procedure. The implicit phase consists of several steps.

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

125

The ﬁrst step involves the calculation of the mass residuals of the solid phase (Ds)i,j,k and the gas phase (Dg)i,j,k from the continuity equations, for all interior cells: ðDs Þ i;j;k ¼ ðs rs Þ i;j;k ðs rs Þni;j;k o dt n hs rs us;x i iþ1;j;k hs rs us;x i i1;j;k þ 2 2 dx o dt n hs rs us;y i i;jþ1;k hs rs us;y i i;j1;k þ 2 2 dy n o dt hs rs us;z i i;j;kþ1 hs rs us;z i i;j;k1 þ 2 2 dz ðDg Þ i;j;k ¼ ðrÞ i;j;k ðrÞni;j;k o dt n hrux i iþ1;j;k hrux i i1;j;k þ 2 2 dx o dt n hruy i i;jþ1;k hruy i i;j1;k þ 2 2 dy n o dt hruz i i;j;kþ1 hruz i i;j;k1 þ 2 2 dz

ð88Þ

ð89Þ

If the convergence criteria ðDg Þ i;j;k oeg ðrÞ i;j;k

(90)

ðDs Þ i;j;k oes ðs rs Þ i;j;k

(91)

are not satisﬁed for all computational cells (typically eg ¼ es ¼ 10– 6), a whole ﬁeld pressure correction is calculated, satisfying ðDg Þ i;j;k ¼ ðJ g Þni;j;k ðdpÞi;j;k ðJ g Þni1;j;k ðdpÞi1;j;k ðJ g Þniþ1;j;k ðdpÞiþ1;j;k ðJ g Þni;j1;k ðdpÞi;j1;k ðJ g Þni;jþ1;k ðdpÞi;jþ1;k ðJ g Þni;j;k1 ðdpÞi;j;k1 ðJ g Þni;j;kþ1 ðdpÞi;j;kþ1

ð92Þ

where (Jg)n represents the Jacobi matrix for the gas phase. This matrix contains the derivatives of the defects Dg with respect to the gas phase pressure, for which explicit expressions can be obtained from the continuity equation for the gas phase in combination with the momentum equations. To save computational effort, the elements of the Jacobi matrix are evaluated at the old time level. The banded matrix problem corresponding to Eq. (92) is solved using a standard ICCG sparse matrix technique. Once new pressures have been obtained, the corresponding new gas phase densities are calculated. So far, the solution procedure has been exactly the same as the SIMPLE procedure that is usually applied for the solution of the ‘‘classic’’ TFMs with

126

M.A. VAN DER HOEF ET AL.

constant property closure equations. In the next step however, the standard procedures continue with the computation of the new velocities from the coupled momentum equations, after which the new volume fractions are obtained from the solid phase mass balances, and only then the new solid pressures are determined. This regularly leads to excessive compaction and extremely high particle pressures in areas where the particle packing densities are close to random close packing. Therefore, the new solution procedure computes the particle volume fractions, taking the compressibility of the solid phase more directly into account. Similar to the pressure correction for the gas phase, a whole ﬁeld particle volume fraction correction is computed, satisfying ðDs Þ i;j;k ¼ ðJ s Þni;j;k ðds Þi;j;k ðJ s Þni1;j;k ðds Þi1;j;k ðJ s Þniþ1;j;k ðds Þiþ1;j;k ðJ s Þni;j1;k ðds Þi;j1;k ðJ s Þni;jþ1;k ðds Þi;jþ1;k ðJ s Þni;j;k1 ðds Þi;j;k1 ðJ s Þni;j;kþ1 ðds Þi;j;kþ1

ð93Þ

In this Eq. (Js)n is the Jacobi matrix for the solid phase, which contains the derivatives of the mass residuals for the particulate phase to the solid volume fraction. Explicit expressions for the elements of the Jacobi matrix can be obtained from the continuity for the solid phase and the momentum equations. For example for the central element, the following expression is obtained from the solid phase continuity equation, in which the convective terms are evaluated with central ﬁnite difference expressions: ðJ s Þni;j;k

( ) @ðs rs us;x Þ iþ1;j;k @ðs rs us;x Þ i1;j;k @ðDs Þ i;j;k dt 2 2 ¼ ¼ ðrs Þ i;j;k þ dx @ðs Þ i;j;k @ðs Þ i;j;k @ðs Þ i;j;k ( ) dt @ðs rs us;y Þi;jþ12;k @ðs rs us;y Þi;j12;k þ dy @ðs Þ i;j;k @ðs Þ i;j;k ( ) dt @ðs rs us;z Þi;j;kþ12 @ðs rs us;z Þi;j;k12 þ dz @ðs Þ i;j;k @ðs Þ i;j;k

ð94Þ

The derivatives of the mass ﬂuxes to the solid volume fractions can subsequently be obtained from the solid phase momentum equations. From Eq. (85), the discretized x-momentum equation, the derivatives of the mass ﬂuxes in the x-direction can easily be obtained, e.g., @ðs rs us;x Þ iþ1;j;k 2

@ðs Þ i;j;k

o dt @p 1 dt n s ðpÞiþ1;j;k ðpÞi;j;k þ ¼ 2 dx dx @s i;j;k þ

dtbniþ1;j;k 2

@ðux us;x Þ iþ1;j;k 2

@ðs Þ i;j;k

ð95Þ

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

127

The second term on the RHS of this equation shows that the compressibility of the solid phase is taken directly into account in the estimation of the new particle volume fractions. Furthermore, the expression for the derivatives of the velocities to the solid pressure can be obtained by combination with the xmomentum equation for the gas phase that results in @ðrux Þ iþ1;j;k 2

@ðs Þ i;j;k

o @ðux us;x Þ iþ1;j;k 1 dt n n 2 ðpÞiþ1;j;k ðpÞi;j;k dtbiþ1;j;k ¼ 2 2 dx @ðs Þ i;j;k

(96)

Together with Eq. (93), this equation forms a set of equations from which explicit expressions for the derivatives of the velocities can readily be obtained. Expressions for the y- and z-direction and for the other elements of the Jacobi matrix are obtained in a similar manner. After the new solid volume fractions have been obtained from Eq. (93), new particle pressures are calculated, where after new velocities can be obtained from the coupled momentum equations. Next, new granular temperatures are calculated from the granular energy equations by an iterative procedure described in Section IV.E.1. Finally, the new mass residuals (Dg)i,j,k and (Ds)i,j,k are computed and the convergence criteria are checked again. Though this new algorithm still requires some time step reﬁnement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10– 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum ﬂuidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. F. APPLICATION

TO

GELDART A PARTICLES

A great challenge in CFD modeling of gas–solid two-phase ﬂows is to obtain realistic predictions of the ﬂuidization behavior of small particles such as Geldart A particles (Geldart, 1973), for which the standard TFM has so far failed to predict even the bubbling ﬂuidization. Ferschneider and Mege (1996) found a major overprediction of bed expansion in a bubbling bed of FCC particles, and Bayle et al. (2001) obtained the same results in a turbulent bed of FCC particles. Recently, Lettieri et al. (2003) used a particle–bed model, originally developed by Chen et al. (1999), to investigate the homogeneous ﬂuidization of Geldart A particles. It has been demonstrated that a

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homogeneous expansion can be obtained in this particle–bed model. However in this model, an artiﬁcial particle-phase elasticity force is required. McKeen and Pugsley (2003) used the two-ﬂuid CFD code MFIX to simulate a freely bubbling bed of FCC catalyst for U0 ¼ 0.05–0.2 m/s and compared their simulation results with ECT data. In accordance with ﬁndings of Ferschneider and Mege (1996), McKeen and Pugsley (2003) also found that the standard CFD model greatly overpredicted bed expansion without any modiﬁcations of the drag closures. By using a scale factor of 0.25 for the commonly used gas–solid drag laws, they found that their simulation results are in accordance with experimental observations. They argued that this is due to the formation of clusters with a size smaller than the CFD grid size. Such small-scale clusters have not been reported before, in particular for particles with a size of 75 mm. Although the van der Waals force can play a role in the ﬂuidization of Geldart A particles, it is not clear how this force affects the gas–solid drag. The inﬂuence of the cohesion on the KTGF has not been carefully checked so far. Recently, Kim and Arastoopour (2002) tried to extend the kinetic theory to cohesive particles; however, their ﬁnal expression for the particular phase stress is very complex. A simpler route would be to assume that the Enskog expressions in Eqs. (75)–(76) still hold for cohesive particles, only with a modiﬁed excess compressibility. However at present, it proves difﬁcult to give an accurate estimate of the deviation of y due to the cohesive force (see Fig. 22). Moreover, as discussed in Section III.F, also the magnitude of the cohesive force itself (i.e., the Hamaker constant) is not known. For this reason, we will only study the effect of the gas–particle drag in this section, where we use two different models: (i) the ‘‘ab-initio’’ drag model in Eq. (47) derived from detailed scale LB simulations and (ii) the empirical drag model in Eq. (46). Note that for the latter model, the literature values for the exponent n are extremely scattered (Morgan et al., 1971). In Table IV, we show the results for n from different experiments for Geldart A particles, which are clearly much higher than the value n ¼ 4.65, originally obtained by Richardson and Zaki (1954). In this section, we show results using the Wen and Yu expression with the commonly used value n ¼ 4.65, and with the highest reported value n ¼ 9.6, from the experiments by Lettieri et al. (2002). For the simulations we use a 2D TFM as described in the previous sections. The simulation conditions are speciﬁed in Table V. The gas ﬂow enters at the bottom through a porous distributor. The initial gas volume fraction in each ﬂuid cell is set to an average value of 0.4 and with a random variation of 75%. Also for the boundary condition at the bottom, we use a uniform gas velocity with a superimposed random component (10%), following Goldschmidt et al. (2004). The simulations show that for low gas velocities (U0 ¼ 0.009 m/s), the commonly used exponent n ¼ 4.65 does not yield a realistic bed expansion dynamics for Geldart A particles. By using a large exponent (n ¼ 9.6), which was determined by gas ﬂuidization of Geldart A particles, we can get a bed expansion

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EXPONENT

TABLE IV GELDART A PARTICLES

N FOR

dp (mm)

n

Lettieri et al. (2002), Newton and Gates (2002): 71 57 49 Massimilla et al. (1972): 60 53 45 Lewis and Bowerman (1952): 86 Whitmore (1957): 65

Gas-ﬂuidization 9.6 9.0 8.2 Gas-ﬂuidization () 7.12 6.86 6.1 Liquid-ﬂuidization 8.3 Liquid-sedimentation 9.5

TABLE V SIMULATION CONDITIONS Parameters Gas shear viscosity Gas temperature Gas pressure Gas constant CFD time step

Value 5

1.8 10 Pa s 293 K 1.01 105 Pa 8.314 J/(mol K) 1.0 104 s

Parameter

Value

CFD cells Size of the cell Particle diameter Particle density Coefﬁcient of restitution

30 45 5 5 mm2 75 mm 1,500 kg/m3 0.97

around 31% of the initial bed height, which is much closer to the experimental results (Geldart, 1973). Basically, a larger exponent n in Eq. (46) will lead to a higher drag at the same gas velocity. It can thus be argued that at low-gas velocities the drag force is underestimated by the commonly used drag models. The question arises what the physical origin is of such large exponents. One possibility is that they are caused by microstructures that form from small-scale instabilities and perhaps other mechanisms. Also, the experiments by Lettieri et al. (2002) showed a much larger apparent terminal velocity, which is indicative of a much larger effective size. If such microstructures cannot be captured by the CFD grid, then the use of a modiﬁed drag function, such that the experimental bed expansion is obtained, is a possible way to go about. It should be stressed, however, that this type of approach is rather ad hoc and not in the spirit of the multiscale modeling strategy. It has been reported by several researchers (Ferschneider and Mege, 1996; Bayle et al., 2001; McKeen and Pugsley, 2003) that an overestimated bed expansion was found at a high-gas velocity (0.2 m/s). We also carried out several simulations for a high gas velocity, U0 ¼ 0.2 m/s. We still use the drag model given by Eq. (46) with an exponent n ¼ 4.65. The simulation domain, however,

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is enlarged so that a high bed expansion can be accommodated. The computational domain is composed of 30 70 cells, and the size of each cell still remains as 5 5 mm2. With such a high gas velocity the bed in fact is in the turbulent ﬂuidization regime. In Fig. 26, we show the results obtained at different points in time when the bed reaches a dynamical equilibrium. Clearly, the particle phase displays a turbulence-like ﬂow pattern. Also, an overestimation of bed height is found in the simulations, which is around 100% of the initial bed height. We also carried out a set of simulations using Eq. (47) as a drag model, which was based on the data of LB simulations. The results are shown in Fig. 27. As can be seen, no big differences can be observed compared to the results from the drag model given by Eq. (46) with an exponent n ¼ 4.65. A similar simulation was also carried out by McKeen and Pugsley (2003). They also found an overestimation of the bed height, compared to their experimental results. They argued that a factor should be used to scale down the

FIG. 26. The bed expansion dynamics of Geldart A particles from the TFM. The superﬁcial gas velocity U0 is set to 0.2 m/s. The exponent n of the Wen and Yu equation is set to 4.65. No cohesion is considered here. The results are, from the right to left, taken at t ¼ 9.6, 9.7, 9.8, 9.9, and 10.0 s.

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131

FIG. 27. The same as in Fig. 26, but now using the LB drag model in Eq. (47) from Hill et al. (2001b), with A2 ¼ Ao.

drag force in this regime in order to obtain a better agreement with the experiments. In Fig. 28, we show the results of our simulations with a drag force (n ¼ 4.65) scaled down by a factor 0.15. A signiﬁcant decrease of the bed height is found, with a bed expansion that is around 16% of the initial bed height, close to the experimental observations (McKeen and Pugsley, 2003).

V. Towards Industrial-Scale Models In Section I, we mentioned that the TFM can simulate ﬂuidized beds at engineering scales (height 1–2 m), and that the large-scale industrial ﬂuidizedbed reactors (diameter 1–5 m, height 3–20 m) are still far beyond its capabilities. Clearly, it would be highly desirable to predict the properties of gas–solid ﬂows at the industrial scale; however at present, there is no fully evolved model— based on fundamental principles—which is capable of this. In this section, we outline some new ideas in this direction that have been developed both at the

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FIG. 28. The same as in Fig. 26, but with the drag force scaled by a factor of 0.15.

Princeton University and at the University of Twente. Before doing so; however, it is ﬁrst important to understand why the current class of TFMs is not suitable for describing large-scale gas–solid ﬂows. A. THE LIMITS

OF THE

TWO-FLUID MODEL

Let us step back and examine the TFM and the closures we described thus far in the chapter. Recall that the details of ﬂow at the level of individual particles are erased by the averaging process leading to the TFM equations, and that their consequences appear in the averaged equations through terms which have to be closed. The size of the averaging region was not explicitly considered anywhere in the derivation of the TFM equations or the closures, and it was implicitly assumed that a separation of scale exists—namely, the size of the averaging region is much larger than the particle size—but is much smaller than the scale of the macroscopic ﬂow structure that we wish to study by solving the TFM equations. The assumption of such a separation of scales underlies the very formulation of continuum models.

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133

Furthermore, the closures for the ﬂuidparticle drag and the particlephase stresses that we discussed were all derived from data or analysis of nearly homogeneous systems. In what follows, we refer to the TFM equations with closures deduced from nearly homogeneous systems as the microscopic TFM equations. The kinetic theory based model equations fall in this category. We illustrated how these equations are discretized over an appropriate numerical grid and also showed some sample results. One can readily appreciate that one must choose the grid sizes in the numerical solution of the TFM equations to be smaller than the shortest length scale at which the TFM equations afford inhomogeneities. This requirement leads to a practical difﬁculty when one tries to solve these microscopic TFM equations for gas–particle ﬂows, as discussed below. Gas–particle ﬂows in ﬂuidized beds and riser reactors are inherently unstable and they manifest inhomogeneous structures over a wide range of length and time scales. There is a substantial body of literature where researchers have sought to capture these ﬂuctuations through numerical simulation of microscopic TFM equations, and it is now clear that TFMs for such ﬂows do reveal unstable modes whose length scale is as small as ten particle diameters (e.g., see Agrawal et al., 2001; Andrews et al., 2005). This is illustrated in Fig. 29. Transient simulations of a ﬂuidized suspension of ambient air and typical ﬂuid catalytic cracking catalyst particles were performed (using MFIX (Syamlal et al., 1993; Syamlal, 1998, which is an open-domain code for solving multiphase ﬂow problems) in a 2D periodic domain at different grid resolutions. These simulations employed kinetic theory-based (microscopic) TFM equations; see Agrawal et al. (2001) for a summary of the equations, closures, and parameter values used in the simulations. Although there are some slight differences between the closure expressions used by these authors and those described (as illustrative examples) in this article, the differences are only quantitative and not qualitative, so there is no need to present these closures here. A pressure drop that is commensurate with the weight of the gas–particle mixture in the periodic box was applied across the box in the vertical direction, which provided the driving force for the upﬂow of the ﬂuidizing gas. The simulations revealed that an initially homogeneous suspension gave way to an inhomogeneous state with persistent ﬂuctuations. Snapshots of the particle volume fraction ﬁelds obtained in simulations with different number of spatial grids are shown in Fig. 29. It is readily apparent that ﬁner and ﬁner structures get resolved as the number of spatial grids is increased. Statistical quantities, such as average slip velocity between the gas and particle phases, obtained by averaging over the whole domain, were found to depend on the grid resolution employed in the simulations and they became nearly grid-size independent only when grid sizes of the order of a few (E10) particle diameters were used. Thus, if one sets out to solve microscopic TFM equations, grid sizes of the order of few particle

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FIG. 29. Snapshots of particle volume fraction ﬁelds obtained while solving a kinetic theory-based TFM. 75 mm ﬂuid catalytic particles in ambient air. Simulations were done over a 16 32 cm periodic domain. The average particle volume fraction in the domain is 0.05. Dark (light) color indicates regions of high (low) particle volume fractions. (See Refs. Agrawal et al., 2001; Andrews et al., 2005) for other parameter values.) Source: Andrews and Sundaresan (2005).

diameters are required; such ﬁne spatial grids (and the fact that inhomogeneous structures extend down to this ﬁne scale) limit the time steps that can be taken as well. For most devices of practical (commercial) interest, such extremely ﬁne spatial grids and small time steps are unaffordable (e.g., see Sundaresan, 2000). Indeed, gas–particle ﬂows in large ﬂuidized beds and risers are often simulated by solving discretized versions of the TFM equations over coarse spatial grids. Such coarse-grid simulations do not resolve the small-scale (i.e., subgrid scale) spatial structures that, according to the microscopic TFM equations, do indeed exist. The effect of these unresolved structures must be brought to bear on the structures resolved in coarse-grid simulations through appropriate modiﬁcations to the closures—for example, the effective drag coefﬁcient in the coarse-grid simulations will be smaller than that in the original TFM to reﬂect the tendency of the gas to ﬂow around the unresolved clusters. Qualitatively, this is equivalent to an effectively larger apparent size for the particles. One can readily pursue this line of thought and examine the inﬂuence of these unresolved structures on the effective interphase transfer and dispersion coefﬁcients that should be used in coarse-grid simulations. Inhomogeneous distribution of particles will promote by passing of the gas around the particle-rich regions and this will necessarily decrease the effective interphase mass and energy transfer rates. Similarly, ﬂuctuations associated with the small-scale inhomogeneities will contribute to the rate of dispersion of the particles and the gas, but they will be unaccounted for in the coarse-grid simulations of the microscopic TFM equations.

MULTISCALE MODELING OF GAS-FLUIDIZED BEDS

B. STATE-OF-THE-ART

ON

DEALING

WITH

135

UNRESOLVED STRUCTURES

Researchers have approached this problem of treating unresolved structures through various approximate schemes. O’Brien and Syamlal (1993) and Boemer et al. (1994) pointed out the need to correct the drag coefﬁcient to account for the consequence of clustering and proposed a correction for the very dilute limit. Some authors have used apparent cluster size in an effective drag-coefﬁcient closure as a tuning parameter; for example see McKeen and Pugsley (2003), who attribute the larger apparent size to interparticle attractive forces, and others have deduced corrections to the drag coefﬁcient using energy minimization multiscale approach (see Yang et al., 2004). The concept of particlephase turbulence has also been explored to introduce the effect of the ﬂuctuations associated with clusters and streamers on the particle-phase stresses (Dasgupta et al., 1994; Hrenya and Sinclair, 1997). However, a systematic approach that combines the inﬂuence of the unresolved structures on the drag coefﬁcient and the stresses that can arise even when interparticle forces are not important has not yet emerged. One can summarize the multiscale character of TFM simulations using coarse spatial grids as follows. When confronted with the task of performing simulation of gas–particle ﬂows in large process vessels, one faces constraints on affordable grid resolution; this can lead to unresolved subgrid structures that would have been obtained if only the TFM equations were solved on a ﬁne spatial grid. The consequence of these subgrid structures on the ﬂow pattern resolved by the coarse-grid simulations should be brought in through appropriate corrections to the closure relations. If one simply uses the closures in the microscopic TFM without adding the corrections, then there is no guarantee that the obtained solution is a true solution for the TFM equations that one sets out to solve. This is well known in other contexts, such as single-phase turbulence. Large eddy simulations introduce corrections to the ﬂuid-phase stress through subgrid models; for example, Smagorinsky, in his pioneering work (Smagorinsky, 1963), introduced a model for subgrid viscosity correction. Agrawal et al. (2001) pointed out that, in gas–particle ﬂows such as those encountered in ﬂuidized beds and riser ﬂows, one should include subgrid corrections for not only the effective particle and ﬂuid-phase stresses but also the effective drag. They showed that the effective drag law and the effective stresses obtained by averaging (the results gathered in highly resolved simulations of a set of microscopic TFM equations, such as that corresponding to the most resolved snapshot in Fig. 29 over the whole (periodic) domain were very different from those used in the microscopic TFM and that they depended on size of the domain over which simulations were carried out (Agrawal et al., 2001). They also found that all the effects seen in the 2D simulations persisted when simulations were repeated in three dimensions (3D) and that both 2D and 3D simulations revealed the same qualitative trends.

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Andrews et al. (2005) performed many highly resolved simulations of ﬂuidized gas–particle mixtures in a 2D periodic domain, whose size coincided with that of the grid size in an anticipated large-scale riser ﬂow simulation. Through such highly resolved simulations, they constructed ad hoc subgrid models for the effects of the ﬁne-scale ﬂow structures on the drag force and the stresses, and examined the consequence of these subgrid models on the outcome of the coarse-grid simulations of gas–particle ﬂow in a large-scale vertical riser. They have demonstrated that these subgrid scale corrections can affect the predicted ﬂow patterns profoundly. Thus, there is no doubt that one must carefully examine whether the microscopic TFM equations must be modiﬁed to introduce the effects of unresolved structures before embarking on coarse-grid simulations of gas–particle ﬂows in chemical reactors. At the same time, the ad hoc method employed by Andrews et al. (2005), namely performing highly resolved simulations in periodic domains whose linear dimensions are the same as those of the grids, is not a rigorous approach to take either; for example, one can anticipate that the periodic boundary conditions imposed in such highly resolved simulations would place some restrictions (on the small-scale ﬂow structure) that would be absent in the real, large-scale ﬂow. Thus, alternate approaches to constructing closures suitable for coarse-grid computations must be developed. Adopting the approach pursued in large eddy simulations, one can start with the TFM equations and perform a ﬁltering operation, where the averaging is done over a ‘‘ﬁlter’’ length scale that is somewhat larger than the grid size to be used in the coarse-grid simulation of large-scale process vessels and over high (temporal) frequencies. The mathematical steps involved in ﬁltering any version of the microscopic TFM are conceptually straightforward (e.g., see Zhang and VanderHeyden, 2002) and will not be presented here. We simply note that the dominant terms in the ﬁltered equations can be recast in exactly the same form as the original TFM equations; however, effective stresses, interphase interaction force term, etc. will now involve additional contributions resulting from the ﬁltering process. (It is because of this similarity that one can use the same platform such as MFIX to perform integration of the ﬁltered equations as well.) Insight into these closures for the additional contributions resulting from the ﬁltering process can be gained through analysis of computational data gathered through highly resolved simulations in sufﬁciently large domains, while ensuring that the overall ﬂow domain simulated is considerably larger than the region over which the ﬁltering operation is performed. This is illustrated below by some results obtained by Andrews and Sundaresan (2005). Consider a highly resolved simulation of a set of microscopic TFM equations for a ﬂuidized suspension of particles in a large periodic domain. The ﬁltering operation does not require a periodic domain; however, as each location in a periodic domain is statistically equivalent to any other location, statistical averages can be gathered much faster when simulations are done in periodic domains. After an initial transient period that depends on the initial conditions,

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137

persistent, time-dependent, and spatially inhomogeneous structures develop. Fig. 30 shows an instantaneous snapshot of the particle volume fraction ﬁeld in one such 2D simulation (performed using MFIX) and the cells (i.e., ﬁne grids) used in the simulations. One can then zoom in any region of desired size and average any quantity of interest over all the cells inside that region, and obtain region-averaged (ﬁltered) values. Note that one can choose a large number of regions inside the overall domain and thus several region-averaged values can be constructed for any quantity of interest from each instantaneous snapshot. When the system is in a statistical steady state, one can construct tens of thousands of such averages by repeating the analysis at various time instants. Returning to Fig. 30, note that the averages over different regions at any given time are not equivalent; for example, at the given instant, different regions (of the same size) will correspond to different region-averaged particle volume fractions, particle and ﬂuid velocities, and so on. Thus, one cannot simply lump the results obtained over all the regions; instead, the results must be grouped

FIG. 30. Snapshot of particle volume fraction ﬁelds obtained while solving a kinetic theory-based TFM. Fluid catalytic particles in air. Simulations were done over a 16 16 cm periodic domain. 128 128 cells (shown in the ﬁgure). The average particle volume fraction in the domain is 0.05. Dark (light) color indicates regions of high (low) particle volume fractions. Squares of different sizes illustrate regions (i.e., ﬁlters) of different sizes over which averaging over the cells is performed. Source: Andrews and Sundaresan (2005).

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1x1 cells = 0.125cm 2x2 cells = 0.250cm 4x4 cells = 0.50cm 8x8 cells = 1.0cm 16x16 cells = 2.0cm

90 80

effective drag coefficient

70 60 50 40 30 20 10 0

0

0.05

0.1

0.15

0.2 φs bin

0.25

0.3

0.35

0.4

FIG. 31. Filtered drag coefﬁcient (in CGS units) extracted from simulations over 16 16 cm domain using 128 128 cells. Source: Andrews and Sundaresan (2005).

into bins based on various markers and perform statistical averages within each bin to get useful information. The 2D simulations of Andrews and Sundaresan (2005) revealed that the single most important marker for regions is the average particle volume fraction in that region. Therefore, in order to expose the effects of particle volume fraction on the ﬁltered (i.e., region-averaged) quantities, they classiﬁed the region-averaged data into bins of particle volume fraction and evaluated the ﬁltered slip velocity, ﬂuid–particle interaction force, etc., and averaged each of these quantities within each bin. From such bin statistics, they calculated the ﬁltered drag coefﬁcient, ﬁltered particle-phase pressure, and ﬁltered particle-phase viscosity as functions of ﬁltered particle volume fraction. Fig. 31 shows the variation of the ﬁltered drag coefﬁcient as a function of the ﬁltered (i.e., region-average) particle volume fraction for various ﬁlter sizes.5 Each point represents the average of many realizations in a bin. (Here, the ﬁltered drag coefﬁcient is deﬁned as the region-average drag force divided by the region-average slip velocity.) It is clear from Fig. 31 that the ﬁltered drag 5

Strictly speaking, one should use 2D bins involving particle volume fraction and a Reynolds number based on slip velocity to classify the ﬁltered drag coefﬁcient; however in these simulations, the Reynolds number effect was found to be weak and hence the data were collapsed to just volume fraction bins.

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coefﬁcient depends on the size of the ﬁlter used in the analysis. This ﬁgure includes results obtained from three different simulations corresponding to three different average particle volume fractions in the domain (0.05, 0.15, and 0.40). The larger the ﬁlter size the smaller is the drag coefﬁcient , the reason being that the averaging (i.e., ﬁltering) is being performed over larger and larger clusters—the larger the clusters, the greater is the bypassing of the gas around the clusters and hence lower is the apparent drag coefﬁcient. Fig. 32 shows the variation of ﬁltered particle-phase pressure as a function of the ﬁltered particle volume fraction for various ﬁlter sizes. Here the ﬁltered particle-phase pressure includes the pressure arising from the streaming and collisional parts captured by the kinetic theory and the sub-ﬁlter-scale Reynolds-stress like velocity ﬂuctuations (see Agrawal et al., 2001 for further details). Indeed, the contributions resulting from the sub-ﬁlter-scale velocity ﬂuctuations swamp the kinetic theory pressure, indicating that at the coarse-grid scale one can even ignore the kinetic theory contributions to the pressure! This ﬁgure clearly shows that the ﬁltered particle-phase pressure increases with ﬁlter size, and this is a direct consequence of the fact that the energy associated with the

200 1x1 cells = 0.125cm 2x2 cells = 0.250cm 4x4 cells = 0.50cm 8x8 cells = 1.0cm 16x16 cells = 2.0cm

180 160 140

ps

120 100 80 60 40 20 0 0

0.05

0.1

0.15

0.2 φs bin

0.25

0.3

0.35

0.4

FIG. 32. Filtered particle phase pressure (in CGS units) extracted from simulations over 16 16 cm domain using 128 128 cells. Source: Andrews and Sundaresan (2005). The ﬁltered particlephase pressure includes the Reynolds stress-like ﬂuctuations and the kinetic theory pressure.

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time and domain−averaged slip velocity

velocity ﬂuctuations increases with ﬁlter length (analogous to what one has in single-phase turbulence). The trends presented in Figs. 31 and 32 qualitatively similar to those presented earlier by Agrawal et al. (2001) and Andrews et al. (2005) who, for the sake of simplicity, did simulations on much smaller domains and let the ﬁlter size be the same as the domain size. This shows clearly that the effects leading to the type of results presented in Figs. 31 and 32 are robust. Andrews and Sundaresan (2005) have also extracted the ﬁltered particlephase viscosity from these simulations and found that at low particle volume fractions (0.0–0.25), the ﬁltered viscosity varies nearly linearly with particle volume, and that it increases monotonically (and nearly linearly) with ﬁlter size. A ﬁnal piece of the proof-of-concept calculations is to compare the predictions obtained by solving the ﬁltered TFM equations with highly resolved simulations of the microscopic TFM equations. For this purpose, Andrews and Sundaresan (2005) performed simulations of the microscopic TFM equations in a 16 32 cm periodic domain at various resolutions (e.g., see Fig. 29). From these simulations, they extracted domain-average quantities in the statistical steady state (see Agrawal et al., 2001 for a discussion of how these data are gathered). Fig. 33 shows the domain-average slip velocity between the gas and particle phases at various grid resolutions (shown by the squares connected by

100

80

60

40

20 kinetic theory model filter size = 2.0cm 0

0

10

20

30 40 50 60 grids in lateral direction

70

80

90

FIG. 33. Comparison of the domain-average slip velocity (in cm/s) determined by solving a microscopic TFM and the corresponding ﬁltered TFM. 16 32 cm periodic domain. Domain-average particle volume fraction ¼ 0.05. Number of grids in the vertical direction is twice that in the lateral direction. Source: Andrews and Sundaresan (2005).

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141

the bold solid line in this ﬁgure). After sufﬁcient grid resolution, this quantity clearly levels off, indicating convergence in a statistical sense. They also performed computations with the ﬁltered TFM, using the computationally generated closures (e.g., drag and particle-phase pressure closures shown in Figs. 31 and 32, and particle-phase viscosity closure, not shown) for a 2 2 cm ﬁlter. The domain-average slip velocity obtained by solving the ﬁltered equations at different grid resolutions are shown in Fig. 33 as triangles (connected by the thin solid line). Fig. 33 reveals two important features. Firstly, at coarse resolutions, the domain-average slip velocity obtained by solving the microscopic TFM changes appreciably with grid resolution; in contrast, the grid-size dependence of the slip velocity computed by solving the ﬁltered TFM is much weaker. Secondly, at sufﬁciently high-grid resolution, both approaches yield comparable predictions, and this is an important ﬁrst step in validating the ﬁltered TFM approach. Another result that is not evident in Fig. 33 concerns the computational times required for gathering the statistical steady-state values of various quantities (such as the slip velocity shown in Fig. 33); at comparable grid resolutions, the computational time required to solve the ﬁltered equations is much smaller than that for the microscopic equations. This can be attributed to the fact that the structures obtained in the solution of the ﬁltered equations are comparatively coarser than those for the microscopic TFM equations. C. A DIFFERENT APPROACH: THE DISCRETE BUBBLE MODEL An alternative scheme to tackle the problem of large-scale ﬂow structures is being pursued at Twente University. In this model the bubbles, as observed in the DPM and TFM models of gas-ﬂuidized beds, are considered as discrete entities. This is the so-called discrete bubble model, which has been successfully applied in the ﬁeld of gas–liquid bubble columns (Delnoij et al., 1997). The idea to apply this model to describe the large-scale solids circulation that prevail in gas–solid reactors is new, however, and involves some slight modiﬁcations of the equivalent model for gas–liquid systems (Bokkers et al., 2005a). To this end, the emulsion phase is modeled as a continuum—like the liquid in a gas–liquid bubble column—and the larger bubbles are treated as discrete bubbles. Note that granular systems have no surface tension, so in that respect there is a pronounced difference with the bubbles present in gas–liquid bubble columns. For instance, the gas will be free to ﬂow through a bubble in gas–solid systems, which is not the case for gas–liquid systems. As far as the numerical part is concerned, the DBM strongly resembles the DPM as outlined in Section III, since it is also of the Euler–Lagrange type with the emulsion phase described by the volume-average Navier–Stokes equations: @ðe re Þ þ r ðe re ue Þ ¼ 0 @t

(97)

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with re, ee, and ue the density, volume fraction, and ﬂow velocity, respectively, of the emulsion phase. Momentum conservation gives that @ðe re ue Þ þ r ðe re ue ue Þ ¼ e rp SE r ð¯se Þ þ e re g @t

(98)

where the symbols take their usual meaning, and the subscript ‘‘e’’ indicates emulsion phase. The term SE accounts for the two-way coupling between the dispersed phase and the continuous phase. The bubbles are considered as discrete elements that are tracked individually according to Newton’s second law of motion: dvb ¼ Ftot mb (99) dt where Ftot is the sum of different forces acting on a single bubble: Ftot ¼ Fg þ Fp þ Fd þ FW þ FVM

(100)

As in the DPM model, the total force on the bubble has contributions from gravity (Fg), pressure gradients (Fp), and drag from the interaction with emulsion phase (Fd). The sum of Fg and Fp is equal to (pe – p) Vbg, with Vb the volume of the bubble. For the drag force on a single bubble (diameter db), the correlations for the drag force on a single sphere are used, only p with ﬃﬃﬃﬃﬃﬃﬃﬃa modiﬁed drag coefﬁcient Cd, such that it yields the relation vbr ¼ 0:711 gd b by Davies and Taylor (1950) for the rise velocity of a single bubble. Note that in Eq. (100), there are two forces present that are not included in the DPM, namely the wake force FW and the virtual mass force FVM. The wake force, accounting for the acceleration of a bubble in the wake of a leading bubble, is neglected in this application; whereas for the virtual mass force, the relation by Auton (1983) is used: FVM

DI þ I ru ; ¼ Dt

I ¼ 0:5re V b ðvb ue Þ

An advantage of this approach to model large-scale ﬂuidized bed reactors is that the behavior of bubbles in ﬂuidized beds can be readily incorporated in the force balance of the bubbles. In this respect, one can think of the rise velocity, and the tendency of rising bubbles to be drawn towards the center of the bed, from the mutual interaction of bubbles and from wall effects (Kobayashi et al., 2000). In Fig. 34, two preliminary calculations are shown for an industrial-scale gas-phase polymerization reactor, using the discrete bubble model. The geometry of the ﬂuidized bed was 1.0 3.0 1.0 m (w h d). The emulsion phase has a density of 400 kg/m3, and the apparent viscosity was set to 1.0 Pa s. The density of the bubble phase was 25 g/m3. The bubbles were injected via 49 nozzles positioned equally distributed in a square in the middle of the column.

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FIG. 34. Snapshots of the bubble hold-up in the DBM model without bubble coalescence, and the time average vector plot of the emulsion phase after 100 s of simulation; (a)+(b) u0 ¼ 0.1.

In Figs. 34a and 34c, snapshots are shown of the bubbles that rise in the ﬂuidized bed with a superﬁcial gas velocity of 0.1 m/s (a) and 0.3 m/s (c). It is clearly shown that the bubble holdup is much larger with a superﬁcial gas velocity of 0.3 m/s. However, the number of bubbles in this case is too large, since bubble coalescence has not been accounted for in these simulations. In Figs. 34b and 34d, time-averaged plots are shown of the emulsion velocity after 100 s of simulation. The large convection patterns, upﬂow in the middle and downﬂow along the wall, are clearly demonstrated. Coalescence, which is a highly prevalent phenomenon in ﬂuidized beds, is not included in the simulations described above. However, since all the bubbles are tracked individually, it is relatively straightforward to include this in the DBM. In the latest version of the DBM (Bokkers et al., 2005a), a simple coalescence model is included, which was found to have a large effect on the macroscale circulation pattern. In this model, all the bubbles that ‘‘collide’’ will coalesce till a maximum size, where the largest bubbles start to breakup.

VI. Outlook In this chapter, we have discussed three levels of modeling for dense gas–solid ﬂows, with the emphasis on the technical details of each of the models, which have not been published elsewhere. Up till now, the models have mainly been used to obtain qualitative information, that is, to acquire insight into the mechanisms underlying the gas–solid ﬂow structures. However, the ultimate objective of the multiscale approach is to obtain quantitative

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information that can be fed into the higher level models. Such a program has just started at the University of Twente, and in Sections II.D and III.F we have shown some ﬁrst results from this line of research. Much remains to be done, however; speciﬁcally for each level: (i) For the LBM, we have presented a new drag-force correlation for ideal systems: homogeneous (unbounded) static arrays of perfect monodisperse spheres; in ﬂuidized beds, however, such systems are hardly ever encountered. Therefore, the next step(s) would be to consider the effects of heterogeneity, mobility, and polydispersity. With respect to the latter, our LB studies for binary systems at very low Reynolds numbers showed that the drag-force relations that are currently used for such systems are wrong by up to a factor of 5, for relatively moderate diameter ratios of 1:4 (Van der Hoef et al., 2005). For higher Reynolds numbers (100 and 500), the results show a similar trend (Beetstra et al., 2006). Some preliminary results on size segregation indicate that our new drag-force relations have a large effect on the segregation phenomena in binary ﬂuidized systems (Beetstra et al., 2006). With respect to heterogeneous structures, it will be obvious that the drag force of clusters of spheres will be very different from that of a homogeneous suspension. We have recently performed an LBM study for small clusters (close-to-sphere and H-shaped), and the conclusion was that the effective drag of the cluster was equivalent to that of a large sphere that has the same (projected) surface area as the cluster, perpendicular to the direction of ﬂow (Beetstra et al., 2006). (ii) In the DPM, future work will be focused on measuring the particulate pressure and viscosity, where it will be of particular interest to test how well the general Enskog relation in Eq. (75) between the viscosity and excess compressibility holds, which follows from kinetic theory, and if necessary adjust the equations on the basis of the simulation data. The next step would be to include the effect of the gas phase (drag), particle friction, van der Waals interactions, and also polydispersity. Note that the determination of the viscosity is not straightforward, and has to our knowledge not been measured previously in discrete particle simulations of ﬂuidized beds. One option is to use methods from MD simulations (Allen and Tildesley, 1990)): statistical mechanics of nonequilibrium systems gives that the shear viscosity is equal to the time integral (Green–Kubo integral) of the stress–stress correlation function /sxy (0)sxy(t)S, where sxy is the ‘‘microscopic’’ stress tensor, which can be written in terms of the particle positions, velocities, and forces (Hansen and McDonald, 1986; Allen and Tildesley, 1990). The second method is by measuring the velocity decay of the impact of a large sphere (diameter D) in a ﬂuidized bed. When we assume that the collisions of the large intruder with the small bed particles take the effect of a Stokes–Einstein drag force 3pmsDu on the large particle, with u the velocity of the intruder, then the effective solids viscosity

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can be obtained directly. Our discrete particle simulations of such an impact in a bed of monodisperse particles are reported in Ref. Lohse et al. (2004). (iii) With respect to the TFM, the main challenges are the simulations of polydisperse systems and ﬁne powders. For the latter, we saw in Section IV.F that without an ad hoc scaling of the drag force, the current class of TFMs cannot predict the ﬂuidization properties of A-powders. For even smaller particles, cluster-like structures might be formed, so that the application of the current version of KTGF should be seriously questioned in any case. In other words, it would be unlikely that a simple reduction of only gas–particle drag, as suggested by McKeen and Pugsley (2003), would sufﬁce in that case. For multidisperse ﬂuidized beds, the current class of TFMs still lacks the capability of describing quantitatively particle mixing and segregation rates. Recently, we have extended the KTGF to bidisperse systems (Bokkers, 2005; Bokkers et al., 2006), and the next challenge would be to extend this to general polydisperse systems. Also in the current KTGF, the effect of particle–particle friction is not incorporated. A recent simulation study using the DPM showed that particle friction has a large inﬂuence on the mixing behavior when a single bubble is injected into the system (Bokkers et al., 2006). It was also found that the effects of lack of friction could not be remedied by using a smaller coefﬁcient of normal restitution, which implies that friction should be taken into account explicitly in the KTGF. (iv) The models for describing industrial-scale gas–solid ﬂow are clearly still in the preliminary stage. In this chapter, we have outlined two promising approaches and noted that much remains to be done before their usefulness as tools for the design and scale-up of chemical reactors can be ascertained. In Section V.B, we have demonstrated the potential value of the ﬁltering approach. Many challenges still remain. It must be veriﬁed that results of the type shown in Figs. 31 and 32 persist in 3D; simple predictive theories to capture the ﬁlter-size dependence of the ﬁltered drag coefﬁcient and stresses must be developed, and the viability of this approach should be validated by comparison with experimental data. For the discrete bubble model described in Section V.C, future work will be focused on implementation of closure equations in the force balance, like empirical relations for bubble-rise velocities and the interaction between bubbles. Clearly, a more reﬁned model for the bubble–bubble interaction, including coalescence and breakup, is required along with a more realistic description of the rheology of ﬂuidized suspensions. Finally, the adapted model should be augmented with a thermal energy balance, and associated closures for the thermophysical properties, to study heat transport in largescale ﬂuidized beds, such as FCC-regenerators and PE and PP gas-phase polymerization reactors.

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THE DETAILS OF TURBULENT MIXING PROCESS AND THEIR SIMULATION Harry E.A. Van den Akker Department of Multi-Scale Physics, Faculty of Applied Sciences, Delft University of Technology, Delft, The Netherlands I. Introduction A. The Role of Turbulence B. The Role of Computational Fluid Dynamics C. The Scope of this Review II. Various types of ﬂuid ﬂow simulations A. Direct Numerical Simulations B. Large Eddy Simulations C. Reynolds Averaged Navier–Stokes Simulations D. The Simulation of Processes in a Turbulent Single-Phase Flow E. The Computational Fluid Dynamics of Two-Phase Flows III. Computational Aspects A. Finite Volume Techniques B. The Size of the Computations C. Lattice-Boltzmann Techniques D. A Mutual Comparison of Finite Volume and Lattice-Boltzmann IV. Boundary Conditions A. Moving Boundaries B. Curved Boundaries C. The Domain and the Grid V. Simulations of Turbulent Flows in Stirred Vessels A. Turbulence Properties B. Validation of Turbulent Flow Simulations VI. Operations and Processes in Stirred Vessels A. Mixing and Blending B. Suspending Solids C. Dissolving Solids D. Precipitation and Crystallization VII. Stirred Gas–Liquid and Liquid–Liquid Dispersions A. Bakker’s GHOST! Code B. Venneker’s DAWN Code C. Further Simulations D. A Promising Prospect

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Corresponding author E-mail: [email protected]

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HARRY E. A. VAN DEN AKKER VIII. Chemical Reactors A. Mechanistic Micromixing Models B. A Lagrangian Approach C. A Eulerian Probabilistic Approach D. A Promising Prospect IX. Summary and Outlook A. The Various Computational Fluid Dynamics Options B. The Promises of Direct Numerical Simulations and Large Eddy Simulations C. An Outlook Acknowledgements References

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Abstract This chapter is devoted to turbulent mixing processes carried out in - mainly - stirred vessels. It reviews ﬁrst a number of turbulent ﬂow characteristics as far as relevant to a wide variety of single-phase and two-phase mixing processes and, secondly and most importantly, the details of the advanced Computational Fluid Dynamics (CFD) techniques required for simulating such processes with a large degree of conﬁdence. The processes considered comprise blending, solids suspension, dissolution, precipitation, crystallization, chemical reactions, and dispersing gases and immiscible liquids. The emphasis in this chapter is on the fruitful application of Large Eddy Simulations for reproducing the local and transient ﬂow conditions in which these processes are carried out and on which their performance depends. In addition, examples are given of using Direct Numerical Simulations of ﬂow and transport phenomena in small periodic boxes with the view to ﬁnd out about relevant details of the local processes. Finally, substantial attention is paid throughout this chapter to the attractiveness and success of exploiting lattice-Boltzmann techniques for the more advanced CFD approaches.

I. Introduction Mixing is an operation inherent to numerous processes encountered in the chemical process industries. Mixing devices such as stirred tanks occur abundantly in plants and processing facilities. This review focuses on stirred vessels being operated under turbulent-ﬂow conditions. Their design and scale-up, their operation, and their often seemingly conﬂicting performances under varying

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conditions have always troubled plant engineers and fascinated researchers. The turbulent-ﬂow phenomena and mixing rates encountered in stirred vessels as well as the processes being dependent on proper and effective mixing have been the subject of numerous research studies since the early 1950s (e.g., Rushton et al., 1950; Kramers et al., 1953). The early studies often documented Power Number–Reynolds Number relationships and related to mixing and circulation times (Holmes et al., 1964; Voncken et al., 1964). Many scale-up rules in terms of nondimensional numbers were derived for a variety of operations such as blending, aeration (Westerterp et al., 1963), and suspending solids (Zwietering, 1958), usually on the basis of integral investigations for speciﬁc stirrer/vessel combinations. One was just interested in the global ﬂow ﬁeld characteristics and in the overall performance of the stirred vessel as a whole. This already was a major step forward in comparison with the ﬁction of a continuous stirred tank (reactor) widely used in the ﬁeld of chemical engineering. Current industrial interests, however, are far beyond this basic level of understanding. Companies are continuously looking for process improvements and for options of debottlenecking plants, in the context of improving performance, proﬁtability, competitiveness, and sustainability. The result is an increasing demand on local ﬂow information since in real life many processes exhibit substantial spatial variations in, e.g., bubble or drop size (Tsouris and Tavlarides, 1994; Luo and Svendsen, 1996; Schulze et al., 2000; Venneker et al. 2002) or in crystal size (Ten Cate et al., 2000; Hollander et al., 2001a,b; Rielly and Marquis, 2001). Further, the yield and selectivity of many chemical reactors may depend on the rate the various chemical species involved are brought into intimate contact and this rate may vary spatially as well (Bakker and Fasano, 1993; Bakker, 1996; Akiti and Armenante, 2004). Such spatial variations in, e.g., mixing rate, bubble size, drop size, or crystal size usually are the direct or indirect result of spatial variations in the turbulence parameters across the ﬂow domain. Stirred vessels are notorious indeed, due to the wide spread in turbulence intensity as a result of the action of the revolving impeller. Scale-up is still an important issue in the ﬁeld of mixing, for at least two good reasons: ﬁrst, usually it is not just a single nondimensional number that should be kept constant, and, secondly, average values for speciﬁc parameters such as the speciﬁc power input do not reﬂect the wide spread in turbulent conditions within the vessel and the nonlinear interactions between ﬂow and process. Colenbrander (2000) reported experimental data on the steady drop size distributions of liquid–liquid dispersions in stirred vessels of different sizes and on the response of the drop size distribution to a sudden change in stirred speed. Knowledge of spatial variations in bubble, drop, and crystal sizes is often desired or required, but extremely hard to obtain experimentally. Intrusive measuring and sampling probes may disturb ﬂow and process locally. Taking samples may affect the sizes: in the sampling procedure, samples may experience

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ﬂow conditions different from those at the sampling position as a result of which particle size may change. Optical techniques may provide a solution but under speciﬁc conditions (optical accessability and density) and in speciﬁc geometries only. Knowledge of local ﬂow variables and local mixing and transfer rates is hard to obtain experimentally as well, since the turbulent ﬂow in most stirred vessels in industry is inhomogeneous and often time dependent, and comprises a wide range of spatial scales and associated temporal scales. CFD might provide a way of elucidating all these spatial variations in ﬂow conditions, in species concentrations, in bubble drop and particle sizes, and in chemical reaction rates, provided that such computational simulations are already capable of reliably reproducing the details of turbulent ﬂows and their dynamic effects on the processes of interest. This Chapter reviews the state of the art in simulating the details of turbulent ﬂows and turbulent mixing processes, mainly in stirred vessels. To this end, the topics of turbulence and CFD both need a separate introduction.

A. THE ROLE

OF

TURBULENCE

The spatial scales in turbulent mixing range from the size of the impeller (blade) down to the typical size of the smallest, so-called Kolmogorov eddies in which the turbulent kinetic energy is dissipated into heat due to the action of friction. In many applications, mixing at the small scales is of paramount importance since many rate-limiting phenomena take place within these Kolmogorov scale and are dominated by the dynamics of these Kolmogorov eddies. Kresta and Brodkey (2004) present a valuable discussion on the role of turbulence in mixing applications and on the role of time and length scales and of scale-up rules. One could say that in turbulent mixing the details really matter! Deriving local rates of energy dissipation within these Kolmogorov eddies from experimental velocity data is at its best a tedious activity not viable for chemical engineers in industry. Computational simulations may provide a way out. The turbulent-ﬂow structures in stirred tanks are highly 3-D and complex because of the complex geometry of the device. Vortical structures, high turbulence levels, and high rates of energy dissipation particularly in the vicinity of the impeller dominate the turbulent ﬂow in stirred tanks. Under the action of the revolving impeller, the ﬂuid is circulated through the tank. Bafﬂes along the tank wall prevent the liquid from carrying out a solid-body rotation about the impeller axis and enhance mixing, partly via vessel-size macroinstabilities. Turbulent ﬂows in stirred vessels are very complex indeed. As a result, the turbulent-ﬂow ﬁeld in a stirred vessel may be far from isotropic and homogeneous. Some of the cornerstones of turbulence theory, however, start from the assumption that production and dissipation of turbulent kinetic energy balance locally. In many chemical engineering ﬂows, this

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assumption may not be satisﬁed everywhere. In the complex ﬂows of most process equipment, turbulence intensity and other turbulence properties vary spatially. These spatial variations may induce ﬂows, diffusion, and dispersion across the ﬂow domain (see, e.g., Ducci and Yianneskis, 2006). This implies that turbulence is not necessarily dissipated where it has been produced: it ﬁrst may be transported toward another part of the ﬂow domain to get dissipated there. As a result, strictly speaking, the common concepts of turbulence theory may not be valid and applicable. Turbulence research still focuses on canonical cases of high Reynolds Number ﬂows in order to deepen their understanding of the subject (‘‘the most important unsolved problem of classical physics’’, according to the great physicist Richard Feynman). Typical examples are grid/isotropic turbulence (in the absence of walls), channel and pipe ﬂow, and free shear layers and jets (see, e.g., Dimotakis, 2005) or rather simple hydraulic problems (Rodi, 1984). At the same time, turbulence research usually refrains from considering practical chemical engineering problems such as the complications and practicalities of stirred vessels. Chemical engineers, however, have to ﬁnd practical ways for dealing with turbulent ﬂows in ﬂow devices of complex geometry. It is their job to exploit practical tools and ﬁnd practical solutions, as spatial variations in turbulence properties usually are highly relevant to the operations carried out in their process equipment. Very often, the effects of turbulent ﬂuctuations and their spatial variations on these operations are even crucial. The classical toolbox of chemical engineers falls short in dealing with these ﬂuctuations and its effects. Computational Fluid Dynamics (CFD) techniques offer a promising alternative approach.

B. THE ROLE

OF

COMPUTATIONAL FLUID DYNAMICS

Nowadays, the effects of, e.g., vessel design and operation conditions can be assessed by means of computational simulations indeed. To mention just a few examples: standard CFD simulations of stirred vessels are perfectly suited to sort out the effect of varying impeller clearance in terms of eliminating dead zones or reducing their size, or the effect of impeller speed on the degree of solids suspension. Various commercial software vendors offer efﬁcient CFD packages, which in many cases can be seen as real workhorses for industrial applications. CFD has tempestuously developed into a very versatile tool, not only in the hands of ﬂuid ﬂow experts, but also in those of chemical engineers. Since computational resources have increased substantially at sharply decreasing costs, detailed computational information about the ﬂows can nowadays be obtained even at a fraction of the cost of the corresponding experiments. Consequently, computational modeling of the ﬂow has become an attractive alternative route of describing ﬂows and improving and scaling up operations in stirred tanks. Papers such as Ditl and Rieger (2006) presenting

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design correlations drafted on the basis of experimental data collected in long series of tests will soon be a thing of the past. In the meantime, CFD can even do more. It may be easier to ‘measure’ the local and transient details of the turbulent ﬂows in stirred vessels and the spatial distributions in, e.g., mixing rates and bubble, drop, and crystal sizes computationally than by means of experimental techniques! A good example is the spatial distribution of the kinetic energy dissipation rate e (see Micheletti et al., 2004). Such a quest—from commercial interests (market pull)—in the details of turbulent ﬂows and associated processes then urges for really powerful CFD techniques. On the reverse, the expectations as to the potential of pursuing a computational route toward a better understanding of turbulent ﬂows and processes are also fed by the remarkable progress made recently in the ﬁeld of CFD (technology push). This paper deals with the advanced CFD of turbulent stirred vessels. The advances attained in recent years in the ﬁeld of CFD really matter for the degree of accuracy and conﬁdence at which the performance of stirred reactors and of other operations carried out in stirred vessels can be simulated, as these processes and operations may strongly depend on the details of the physics and chemistry involved. The latter details require the more advanced CFD techniques indeed. Recently, Paul et al. (2004) compiled many engineering design principles developed in the ﬁeld of mixing over the last 30 years into the NAMF (North American Mixing Forum) Handbook of Mixing. This ‘Bible of Mixing’ provides within its 1,375 pages a wealth of information and design guidelines for the practicing engineer who needs to both identify and solve mixing problems. It also contains a few chapters and sections reviewing achievements and promises of CFD for the ﬁeld of mixing. In Chapter 5, Marshall and Bakker (2004) present an overview on Computational Fluid Mixing; their review, however, is rather limited in scope and in computational methods covered. Patterson et al. (2004) devote some 30% of their Chapter 13 to computational simulations of ‘Mixing and Chemical Reactions.’ In the remaining Chapters of the Handbook, CFD is largely presented as an immature technique which may become relevant for the practicing engineer in due course only. The present author thinks one can and should be much more positive about the merits of CFD so far and about the term at which CFD will replace and improve existing mixing correlations. This message was already passed on at earlier occasions (Van den Akker, 1997, 2000). Substantial progress has been made in exploiting CFD not just for simulating the turbulent-ﬂow ﬁeld but also with the view of representing the various processes carried out in stirred vessels. Owing to the enormous growth in computer power and the proliferation of models and numerical methods, CFD nowadays is very powerful and versatile. This does not mean that every ﬂow or process can be mimicked in all detail, since Direct Numerical Simulations (DNS) resolving all turbulent ﬂuctuations involved in ﬂow and process will keep requiring insurmountable amounts of

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computer power and simulation times for the next decades. Current computer power, however, already offers various ways out which are of inestimable value to practicing engineers indeed. First of all, the increased computer power makes it possible to switch to transient simulations and to increase spatial resolution. One no longer has to be content with steady ﬂow simulations on relatively coarse grids comprising 104–105 nodes. Full-scale Large Eddy Simulations (LES) on ﬁne grids of 106–107 nodes currently belong to the possibilities and deliver realistic reproductions of transient ﬂow and transport phenomena. Comparisons with quantitative experimental data have increased the conﬁdence in LES. The present review stresses that this does not only apply to the hydrodynamics but relates to the physical operations and chemical processes carried out in stirred vessels as well. Examples of LES-based simulations of such operations and processes are due to Hollander et al. (2001a,b, 2003), Venneker et al. (2002), Van Vliet et al. (2005, 2006), and Hartmann et al. (2006). A further option is to forget about simulating the ﬂow and the processes in the whole vessel and to zoom into local processes by carrying out a DNS for a small box. The idea is to focus on the ﬂow and transport phenomena within such a small box, such as mass transport and chemical reactions in or around a few eddies or bubbles, or the hydrodynamic interaction of a limited number of bubbles, drops, and particles including their readiness to collisions and coalescence. Examples of such detailed studies by means of DNS are due to Ten Cate et al. (2004) and Derksen (2006b). The effect of this small box being immersed in the dynamic ambiance of the turbulent ﬂow is mimicked by using periodic boundary conditions, which protect the inside of the box against restricting effects of the boundaries. By imposing typical (averaged) conditions of, e.g., ﬂow, rate of energy dissipation, temperature, species concentrations, and/or volume fractions, we then are capable of studying how the ﬂow and the various processes of interest evolve as a result of the governing models and equations. Where experimental techniques often fail in elucidating the mechanisms behind certain phenomena and processes, computational simulations are perfectly capable of doing this. This is a most welcome aspect of computational simulations indeed. As a matter of fact, one may think of a multiscale approach coupling a macroscale simulation (preferably, a LES) of the whole vessel to meso or microscale simulations (DNS) of local processes. A rather simple, off-line way of doing this is to incorporate the effect of microscale phenomena into the full-scale simulation of the vessel by means of phenomenological coefﬁcients derived from microscale simulations. Kandhai et al. (2003) demonstrated the power of this approach by deriving the functional dependence of the singleparticle drag force in a swarm of particles on volume fraction by means of DNS of the ﬂuid ﬂow through disordered arrays of spheres in a periodic box; this functional dependence now can be used in full-scale simulations of any ﬂow device.

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One may object that such computational simulations are too advanced and too much time consuming for practicing engineers. Similar doubts as to the usefulness of CFD for chemical engineers were raised in the early days of CFD. Nowadays, however, CFD software is used throughout the chemical process industries for a wide variety of applications indeed. This is at least partly due to the development of very versatile and robust commercial CFD software that has turned into a valuable tool for chemical engineers in industry. In addition, in the last 15 years, the size of the computational simulations has increased substantially, keeping more or less equal pace with Moore’s law saying the number of transistors on integrated circuits doubles every 2 years. There is no reason why computer power and CFD—in a type of playing leapfrog process—would not keep growing at the same pace. This means that what is advanced and time consuming today will be a routine job in 3 or 5 years. Of course, academia may have the lead in this development.

C. THE SCOPE

OF THIS

REVIEW

This review paper is restricted to stirred vessels operated in the turbulent-ﬂow regime and exploited for various physical operations and chemical processes. The developments in the ﬁeld of computational simulations of stirred vessels, however, are not separated from similar developments in the ﬁelds of, e.g., turbulent combustion, ﬂames, jets and sprays, tubular reactors, and multiphase reactors and separators. Fortunately, there is a strong degree of synergy and mutual cross-fertilization between these various ﬁelds. This review paper focuses on aspects speciﬁc to stirred vessels (such as the revolving impeller, the resulting strong spatial variations in turbulence properties, and the macroinstabilities) and on the processes carried out in them. Because of the above interactions, it is impossible to distinctly mark the start of the CFD of stirred vessels. The ﬁrst papers on CFD simulations of stirred vessels were due to Harvey and Greaves (1982), Placek and Tavlarides (1985), Placek et al. (1986), Middleton et al. (1986), and Ranade et al. (1989). Patterson was certainly one of the pioneers exploring the options for computational simulations of turbulent reactors (see, e.g., Patterson, 1985). From about 1990 onwards, the triennial European Conferences on Mixing, the bi-annual meetings of the NAMF and the regular IChemE Fluid Mixing Events provided the ﬂoor for numerous oral presentations on computational simulations in the mixing ﬁeld as well as for beneﬁcial exchanges of ideas between academic researchers and industrial users. The results of some of the early simulations in the ﬁeld of stirred vessels are still reported in this review paper. They may serve to illustrate the substantial progress made since the early days. Several of the strong simpliﬁcations of those days are no longer required indeed. Most importantly, incorporating physical

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operations and chemical processes into the ﬂuid ﬂow simulations has become quite viable, as this review intends to demonstrate. This review strongly focuses on the potential of LES and DNS for reproducing not only the hydrodynamics of turbulent stirred vessels but also for providing a basis for simulating a wide variety of physical and chemical processes in this equipment. The ﬁrst journal paper presenting simulation results obtained by means of LES for a stirred vessel was due to Eggels (1996), who was also the ﬁrst to exploit a lattice-Boltzmann (LB) technique to this purpose. Of course, LES and LB do not necessarily go along, LB being just an attractive solution technique perfectly suited for parallel computing. Since, in the wake of this pioneering Eggels’ paper, the topics of LES and LB have become leading themes in the research group of the present author, this review contains many references to work of Derksen et al. (from 1996 onwards) and to many PhD theses and associated papers from this group. Gradually, however, LES (in a Finite Volume, FV, context) is also receiving attention from other research groups and from the commercial software vendors.

II. Various types of fluid flow simulations Computational ﬂuid dynamics techniques are not capable of fully resolving the highly turbulent ﬂow in most industrial applications within a reasonable time span. In a DNS, all scales of the motion are simulated by means of the classical Navier– Stokes (NS) equations. For simulating turbulent ﬂows, this would require that the spacing of the computational grid be sufﬁciently ﬁne to even resolve the smallest eddies the name of Kolmogorov is associated with. As 3/4 the Kolmogorov length scale, ZK, is as small as Re times some macroscopic 9/4 dimension (see, e.g., Tennekes and Lumley, 1972), a 3-D DNS scales with Re (e.g., Wilcox, 1993; Moin and Kim, 1997). This really limits the applicability of DNS to rather low Reynolds numbers. Even nowadays, a DNS of the turbulent ﬂow in, e.g., a lab-scale stirred vessel at a low Reynolds number (Re ¼ 8,000) still takes approximately 3 months on 8 processors and more than 17 GB of memory (Sommerfeld and Decker, 2004). Hence, the turbulent ﬂows in such applications are usually simulated with the help of the Reynolds Averaged Navier– Stokes (RANS) equations (see, e.g., Tennekes and Lumley, 1972) which deliver an averaged representation of the ﬂow only. This may lead, however, to poor results as to small-scale phenomena, since many of the latter are nonlinearly dependent on the ﬂow ﬁeld (Rielly and Marquis, 2001). The exponential increase in computational power along with a drastic drop in price for computer hardware has led to the ability to solve industrial ﬂows by means of LES. The major advantage of LES is that closure is applied to just the

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ﬂuctuations that are smaller than the grid spacing, whereas the large turbulent scales are solved explicitly. This means that grid spacing also acts as a low pass ﬁlter: ﬂuctuations smaller than the grid spacing are ﬁltered out, i.e., not resolved in the simulation. It is then assumed that due to this separation in scales, the so-called subgrid scale (SGS) modeling is largely geometry independent because of the universal behavior of turbulence at the small scales. The SGS eddies are therefore more close to the ideal concept of isotropy (according to which the intensity of the ﬂuctuations and their length scale are independent of direction) and, hence, are more susceptible to the application of Boussinesq’s concept of turbulent viscosity (see page 163). Large eddy simulations ask for a ﬁne grid to realize the above separation in scales. LES are therefore computationally more expensive than RANS-based simulations. It is not an option to take refuge to coarse grids in order to reduce simulation times, as LES on a coarse grid does not make sense physically. The grid spacing should be such that at least (a substantial) part of the inertial subrange can be resolved. Large eddy simulations yield much more detailed simulation results indeed, not only because the grids used easily comprise millions of grid cells—these numbers being larger than common in RANS simulations by at least one order of magnitude—but also because the simulations are inherently transient and reproduce the dynamics of a large proportion of the wide spectrum of eddies. In fact, LES are positioned somewhere in between DNS and RANS. 7 Derksen (2003) carried out LES on a grid of 1.4 10 nodes while a DNS 12 would have required a grid of as many as 10 nodes, the latter number being far beyond computational capabilities both currently and in the foreseeable future. We will now treat the various CFD options in some more detail. A. DIRECT NUMERICAL SIMULATIONS The term ‘direct’ in ‘Direct Numerical Simulations’ indicates that the ﬂow is fully resolved by solving, without any modeling, the classical NS equations @v 1 þ v rv ¼ rp þ nr2 v @t r

(1)

and that not any motion or eddy at whatever scale is ignored, provided that the calculation grid is sufﬁciently ﬁne. The latter condition implies that reﬁning the grid would (hardly) change the ﬂow ﬁeld resulting from the simulation. As a matter of fact, laminar ﬂows belong to the type of ﬂows excellently viable to DNS. Other important targets for DNS are the turbulent ﬂows at Reynolds numbers up to say 10,000 in simple geometries (such as straight channels or curved

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pipes), the local ﬂow ﬁeld around a single particle, or the turbulent two-phase ﬂow in a periodic box of limited size, again under the proviso that the calculation grid is sufﬁciently ﬁne to capture all details of the ﬂow at the scale of particle or box.

B. LARGE EDDY SIMULATIONS In LES, it is accepted that the ﬂow is not fully resolved: turbulent eddies smaller than the grid spacing D are not solved explicitly. These small eddies do contribute, however, to the redistribution of momentum within the ﬂow ﬁeld. The resulting set of NS equations then runs as @~v 1 þ v~ r~v ¼ rp~ þ nr2 v~ r s @t r

(2)

where v~ and p~ now denote the variables to be resolved in the simulation. Note that the latter term in this set of equations represents the effect of SGS stresses that are not calculated explicitly. After having introduced the decomposition 1 s ¼ s0 þ trðsÞI 3

(3)

the ﬁrst term at the right-hand side of Eq. (3) is modeled with the help of an effective SGS viscosity coefﬁcient ne: (4) s0 ¼ ne r~v þ ðr~vÞT while the second term at the right-hand side of Eq. (3) is incorporated into the pressure term of Eq. (2). This implies that all eddies larger than the grid size are explicitly resolved in a LES. A ﬂow ﬁeld obtained by means of LES therefore is inherently transient and 3-D. To take advantage of the concept of LES, which is particularly aimed at resolving a great deal of the time and length scales of a turbulent-ﬂow ﬁeld, ﬁne grids should be used. This implies that LES applied to a ﬂow domain of some size are computationally quite demanding. Restricting LES to 2-D in view of saving computational time does not make sense physically, as the dynamics of turbulent ﬂows in process equipment is inherently 3-D. In dealing with the SGS terms, Revstedt et al. (1998, 2000) and Revstedt and Fuchs (2002) did not use any model; rather, they assumed these terms were just as small as the truncation errors in the numerical computations. This heuristic approach lacks physics and does not deserve copying. A most welcome aspect of LES is that the SGS stresses may be conceived as being isotropic, i.e., insensitive to effects of the larger scales, to the way the turbulence is induced and to the complex and varying boundary conditions of the ﬂow domain. Exactly this

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feature renders modeling attractive, in contrast with modeling all turbulent stresses as done in RANS-based simulations (see the next section). The most widely used model for the SGS stresses is due to Smagorinsky (1963) and involves a SGS eddy viscosity, ne, which is related to the local ~ resolved deformation rate S: ne ¼ c2s D2

qﬃﬃﬃﬃﬃ 2 S~

(5)

with 1 2 S~ ¼ r~v þ ðr~vÞT : r~v þ ðr~vÞT 2

(6)

While the theoretical value (based on homogeneous, isotropic turbulence) of the Smagorinsky coefﬁcient cs amounts to 0.165 (Mason and Callen, 1986), in many simulation studies lower values for cs proved to result in a better reproduction of experimental data. This may have to do with the abundant presence of shear ﬂows in process equipment. Derksen (2003) reported that varying cs values in the range 0.08–0.14 does not have a large impact on the simulation results. A value of 0.12 is recommended. At the basis of the Smagorinsky SGS model is the assumption of equilibrium between production and dissipation of turbulent kinetic energy in the inertial subrange of eddy sizes. In stirred tanks, however, there is hardly any position where this equilibrium prevails. Furthermore, there is not a good reason why the Smagorinsky coefﬁcient should be constant across the ﬂow domain. Other more speciﬁc SGS models have therefore been proposed and also investigated as to their impact on the resulting ﬂow ﬁelds and turbulence characteristics. Hartmann et al. (2004a) assessed the usability of an SGS model due to Voke (more geared to low-Reynolds number turbulence), while Derksen (2001) investigated a so-called structure function SGS model for a turbulent viscosity that depends on eddy size. Derksen (2006a) supplemented the standard Smagorinsky SGS model with wall-damping functions to bring the eddy viscosity explicitly to zero at solid walls, since in physical reality velocity ﬂuctuations and subgrid stresses are zero at walls. Recently, FLUENT 6.2 came with several new SGS models. Further reﬁnements in SGS modeling may be expected to improve the accuracy of LES. An inherent property of the LES approach is that the simulated ﬂow ﬁeld is no longer steady, but exhibits a transient character due to the presence and motion of large-scale eddies. The LES methodology has proven to be a powerful tool for studying and visualizing stirred tank ﬂows (Eggels, 1996; Derksen et al. 1999; Bakker et al., 2000; Derksen, 2001; Bakker and Oshinowo, 2004), as it inherently takes the unsteady and periodic behavior of the ﬂow (around impeller and bafﬂes) into account.

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C. REYNOLDS AVERAGED NAVIER– STOKES SIMULATIONS The focus of RANS simulations is on the time-averaged ﬂow behavior of turbulent ﬂows. Yet, all turbulent eddies do contribute to redistributing momentum within the ﬂow domain and by doing so make up the inherently transient character of a turbulent-ﬂow ﬁeld. In RANS, these effects of the full range of eddies are made visible via the so-called Reynolds decomposition of the NS equations (see, e.g., Tennekes and Lumley, 1972, or Rodi, 1984) of the ﬂow variables into mean and ﬂuctuating components. To this end, a clear distinction is required between the temporal and spatial scales of the mean ﬂow on the one hand and those associated with the turbulent ﬂuctuations on the other hand. Via this Reynolds decomposition and after subsequent averaging all terms of the NS equations, the so-called turbulent or Reynolds stresses ui uj emerge in the transport equations, where these stresses represent the additional averaged momentum transport due to the eddies. These stresses may be resolved explicitly from separate transport equations which in sufﬁx notation (usual in the ﬁeld of turbulence) look as follows: @ui uj @ui uj þ uk ¼ Pij þ Dij ij þ Pij @t @xk

(7)

in which the ﬁrst three terms of the right-term side denote the production, diffusion, and dissipation of the turbulent stresses, respectively, while the last term is the so-called pressure-strain term that represents the redistribution of the turbulent kinetic energy among the three coordinate directions that makes the turbulence more isotropic (Rodi, 1984). Several of these terms need modeling for which a gamut of choices is available. In principle, this approach implies the need of solving nine more partial differential equations per grid cell. As a result, CPU times required for computational simulations on the basis of some Reynolds Stress Model (RSM) are relatively high. A cure against these longer CPU times is the Algebraic Stress Model (ASM) described by, e.g., Rodi (1984) and used and recommended by, e.g., Bakker (1992) and Bakker (1996). Most commercial codes do no longer support an ASM. Usually, however, the stresses are modeled with the help of a single turbulent viscosity coefﬁcient that presumes isotropic turbulent transport. In the RANSapproach, a turbulent or eddy viscosity coefﬁcient, nt, covers the momentum transport by the full spectrum of turbulent scales (eddies). Frisch (1995) recollects that as early as 1870 Boussinesq stressed turbulence greatly increases viscosity and proposed an expression for the eddy viscosity. The eventual set of equations runs as @V 1 þ V rV ¼ rP þ ðn þ nt Þr2 V @t r

(8)

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In its turn, the turbulent viscosity may be position dependent and generally may be modeled in terms of a model, very usually a k–e model: nt ¼ C m

k2

(9)

where k is the concentration of turbulent kinetic energy in J/kg (or m2/s2) and e is the rate of dissipation (in W/kg, or m2/s2/s) of this turbulent kinetic energy. These two concentrations k and e are generally conceived as the most important parameters describing a turbulent-ﬂow ﬁeld. In their turn, their spatial distributions within the turbulent-ﬂow domain may be calculated from the following transport equations for k and e, respectively: @k @k þ ui ¼ Pk þ Dk þ Pk @t @xi

(10)

the right-hand side of which contains similar terms as the above transport equations for the turbulent stresses, and @ @ þ ui ¼ P þ D O @t @xi

(11)

in which the last term denotes the destruction of e. The interested reader is referred to, e.g., Rodi (1984) for the meaning of all these right-hand terms and their modeling. The assumption often used in classical turbulence theory that production and dissipation of turbulent kinetic energy balance locally, is found by putting in Eq. (10) all terms but the ﬁrst and third at the right-hand side equal to zero. These two transport equations for k and e form an inherent part of any k–e model of RANS-simulations. As the result of ‘closing’ the turbulence modeling such that no further unknown variables and equations are introduced, the e-equation does contain some terms that are still the result of modeling, albeit at the very small scales (e.g., Rodi, 1984). The (isotropic) eddy viscosity concept and the use of a k–e model are known to be inappropriate in rotating and/or strongly 3-D ﬂows (see, e.g., Wilcox, 1993). This issue will be addressed in more detail in Section IV. Some researchers prefer different models for the eddy viscosity, such as the k–o model (where o denotes vorticity) that performs better in regions closer to walls. For this latter reason, the k–e model and the k–o model are often blended into the so-called Shear-Stress-Transport (SST) model (Menter, 1994) with the view of using these two models in those regions of the ﬂow domain where they perform best. In spite of these objections, however, RANS simulations mostly exploit the eddy viscosity concept rather than the more delicate and time-consuming RSM turbulence model. They deliver simulation results of—in many cases—reasonable or sufﬁcient accuracy in a cost-effective way.

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RANS-based simulations exploiting the eddy viscosity concept just reproduce the average ﬂow ﬁeld and the spatial distribution of turbulence properties such as k and e. As such, RANS-based simulations are excellently suited for identifying dead zones, recirculatory ﬂows, short-circuiting between entrance and exit, and further undesired ﬂow features. Even in transient RANS-based simulations, however, it is not a priori clear which part of the ﬂuctuations is temporally resolved and which part is taken care of by the turbulence model. This inherent property of RANS-based simulations especially raises concerns in the case of ﬂows exhibiting no clear spectral separation between low-frequency coherent motions (such as macroinstabilities, precessing vortices, and trailing vortices) and turbulent ﬂuctuations (making part of the cascade of eddies or whirls typical of turbulence); and: ‘‘A mechanistic picture of turbulence cannot be treated on the average since such ﬂows are dynamic.’’ (Praturi and Brodkey, 1978). Yet (steady) RANS-based simulations are attractive as they relatively cheaply deliver a quick impression of the overall ﬂow ﬁeld in the vessel. Effects on the overall ﬂow ﬁeld of varying the position of impeller, feed pipe, withdrawal pipe, and/or heat coil can easily be explored. Note that the Eqs. (1), (2), and (8) are really and essentially different due to the absence or presence of different turbulent transport terms. Only by incorporating dedicated formulations for the SGS eddy viscosity can one attain that LES yield the same ﬂow ﬁeld as DNS. RANS-based simulations with their turbulent viscosity coefﬁcient, however, essentially deliver steady ﬂow ﬁelds and as such are never capable of delivering the same velocity ﬁelds as the inherently transient LES or DNS, irrespectively of the reﬁnement of the computational grid!

D. THE SIMULATION

OF

PROCESSES

IN A

TURBULENT SINGLE-PHASE FLOW

For simulating computationally the spatial and temporal evolution of both physical and chemical processes in mixing devices operated in a turbulent singlephase mode, two essentially different approaches are available: the Lagrangian approach and the Eulerian technique. These will be explained brieﬂy. In the Lagrangian approach, individual parcels or blobs of (miscible) ﬂuid added via some feed pipe or otherwise are tracked, while they may exhibit properties (density, viscosity, concentrations, color, temperature, but also vorticity) that distinguish them from the ambient ﬂuid. Their path through the turbulent-ﬂow ﬁeld in response to the local advection and further local forces (if applicable) is calculated by means of Newton’s law, usually under the assumption of ‘one-way coupling’ that these parcels do not affect the ﬂow ﬁeld. On their way through the tank, these parcels or blobs may mix or exchange mass and/or temperature with the ambient ﬂuid or may adapt shape or internal velocity distributions in response to events in the surrounding ﬂuid.

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In real life, the parcels or blobs are also subjected to the turbulent ﬂuctuations not resolved in the simulation. Depending on the type of simulation (DNS, LES, or RANS), the wide range of eddies of the turbulent-ﬂuid-ﬂow ﬁeld is not necessarily calculated completely. Parcels released in a LES ﬂow ﬁeld feel both the resolved part of the ﬂuid motion and the unresolved SGS part that, at best, is known in statistical terms only. It is desirable that the forces exerted by the ﬂuid ﬂow on the particles are dominated by the known, resolved part of the ﬂow ﬁeld. This issue is discussed in greater detail in the next section in the context of tracking real particles. With a RANS simulation, the turbulent velocity ﬂuctuations remaining unresolved completely, the effect of the turbulence on the tracks is to be mimicked by some stochastic model. As a result, particle tracking in a RANS context produces less realistic results than in an LES-based ﬂow ﬁeld. An early example of tracking ﬂuid parcels in a stirred tank can be found in Bouwmans (1992) and Bouwmans et al. (1997). Bakker (1996) used a tracking routine with the view to provide a Lagrangian description of micromixing in a stirred tank chemical reactor. Lapin et al. (2004) recently described a computational strategy for ‘travelling along the lifelines’ of single cells (i.e., tracking them) in stirred bioreactors in order to characterize the dynamics of the heterogeneous cell population and to study the impact of spatial and dynamic variations in concentrations of substrates and products across the reactor. The motions of the individual ﬂuid parcels may be overlooked in favor of a more global, or Eulerian, description. In the case of single-phase systems, convective transport equations for scalar quantities are widely used for calculating the spatial distributions in species concentrations and/or temperature. Chemical reactions may be taken into account in these scalar transport equations by means of source or sink terms comprising chemical rate expressions. The pertinent transport equations run as @T þ v rT ¼ kr2 T þ q @t

(12)

@c þ v rc ¼ Dr2 c þ r @t

(13)

and

In Eq. (13), r stands for the production (or consumption) of the species of interest due to a chemical reaction, while in Eq. (12) q represents the heat production, e.g., due to one of more chemical reactions. Equation (13) is often referred to as the Convection-Diffusion-Reaction (CDR) equation. Since turbulent ﬂuctuations not only occur in the velocity (and pressure) ﬁeld but also in species concentrations and temperature, the convection diffusion equations for heat and species transport under turbulent-ﬂow conditions also comprise cross-correlation terms, obtained by properly averaging products of

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velocity–concentration, velocity–temperature, concentration–temperature, and concentration–concentration ﬂuctuations, on the analogy of the Reynolds stresses in the NS equations (e.g., Patterson, 1985; Ranade, 2002). The challenge is still to ﬁnd appropriate closure relations for these cross-correlation terms: these may be either phenomenological or mechanistic (‘micromixing models’) or probabilistic (exploiting probability density functions, PDFs). In stirred chemical reactors, unlike in combustion and with other gas-phase reactions, these closure terms should take into account that for liquids the Schmidt number (Sc ¼ n=D) is in the order 100–1,000, and, hence, the role of species diffusion at scales within the Kolmogorov eddies should explicitly be taken into account (Kresta and Brodkey, 2004). Essential is that diffusion of chemical species is governed by the Batchelor length scale ZB which obeys to ZB ¼ ZK Sc1=2

(14)

which for large Sc numbers is much smaller than the Kolmogorov length scale ZK indeed. Such so-called micromixing processes have to be described by means of micromixing models which will be dealt with in some greater detail in Section VIII. These convective transport equations for heat and species have a similar structure as the NS equations and therefore can easily be solved by the same solver simultaneously with the velocity ﬁeld. As a matter of fact, they are much simpler to solve than the NS equations since they are linear and do not involve the solution of a pressure term via the continuity equation. In addition, the usual assumption is that spatial or temporal variations in species concentration and temperature do not affect the turbulent-ﬂow ﬁeld (another example of oneway coupling).

E. THE COMPUTATIONAL FLUID DYNAMICS

OF

TWO-PHASE FLOWS

On the analogy of simulating the process of adding blobs of a miscible liquid, two-phase ﬂow in stirred tanks in a RANS context may be treated in two ways: Euler–Lagrange or Euler–Euler, with the second, dispersed phase treated according to a Lagrangian approach and from a Eulerian point of view, respectively. 1. Euler– Lagrangian Approach The Euler–Lagrangian approach is very common in the ﬁeld of dilute dispersed two-phase ﬂow. Already in the mid 1980s, a particle tracking routine was available in the commercial CFD-code FLUENT. In the Euler–Lagrangian approach, the dispersed phase is conceived as a collection of individual particles (solid particles, droplets, bubbles) for which the equations of motion can be solved individually. The particles are conceived as point particles which move

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across the ﬂow domain in response to the turbulent-ﬂow ﬁeld of the carrier phase. The consequence of treating particles as point particles is that the detailed ﬂow between the particles in response to the presence and motion of the particles is not resolved. For the hydrodynamics forces acting on the particles, mostly single-particle expressions are used; this implies that hydrodynamic interactions between particles are ignored completely. In many cases, direct interactions of particles owing to collisions are ignored as well. All this—along with the computational burden that increases linearly with the number of particles being tracked—may limit the practical applicability of the method to dilute systems with relatively low volume fractions of dispersed phase and/or to ﬂow domains of small size. By feeding back the reactive forces exerted by the particles on the continuous carrier phase, two-way coupling between the two phases is obtained. Without much discussion, one may anticipate that particle inertia, gravity, and drag force need to be part of the equations of motion describing the motion and paths of the particles. Since a stirred tank is very inhomogeneous and exhibits strong gradients in velocity, pressure, and stress ﬁelds, it is difﬁcult to estimate a priori if more exotic ﬂuid-particle forces such as the Saffman lift force, the Magnus force, and the history force may play a role of signiﬁcance either globally or locally. For a concise summary about these forces and for expressions for these forces, the reader is referred to, e.g., Derksen (2003). Added mass may be important for bubbly ﬂows. It is obvious that in such a Lagrangian approach distributions in particle size may easily be taken into account. As discussed earlier in the context of tracking miscible parcels or blobs, particles travel through the resolved average or ﬂuctuating velocity ﬁeld as well as feel the unresolved velocity ﬂuctuations. Since the major ﬂuid-particle force may be the drag force, the ﬂuid’s velocity ﬁeld is of primary importance, the turbulent velocity ﬂuctuations inclusively, whether or not they are resolved in the simulation; ﬂuctuations in pressure and stresses may be secondary. Supplying stochastic variations on top of the resolved velocity ﬁeld mimics the unresolved ﬂuctuations and brings the expected seemingly erratic paths of the particles about. Of course, the role of the artiﬁcially introduced stochastics for mimicking the effect of all eddies in a RANS-based particle tracking is much more pronounced than that for mimicking the effect of just the SGS eddies in a LES-based tracking procedure. In addition, the random variations may suffer from lacking the spatial or temporal correlations the turbulent ﬂuctuations exhibit in real life. In RANS-based simulations, these correlations are not contained in the steady spatial distributions of k and e and (if applicable) the Reynolds stresses from which a typical turbulent time scale such as k/e may be derived. One may try and cure the problem of missing the temporal coherence in the velocity ﬂuctuations by picking a new random value for the ﬂuid’s velocity only after a certain period of time has lapsed.

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In LES-based simulations, just the SGS part of the turbulence spectrum needs to be mimicked by stochastics. The idea is that the resolved eddies have the biggest impact on the paths of the particles indeed. This requires not only that the resolved velocity ﬂuctuations should be stronger than the (estimated) SGS ﬂuctuations, but also that the particle relaxation time should be larger than the time-step applied in the LES. Meeting the latter criterion implies that the time step of the LES is capable of keeping up with the time scale the particle needs to respond to changes in the ﬂow ﬁeld of the surrounding ﬂuid. In the context of LES, picking a new random velocity only after (part of) a time ksgs/e has lapsed may cure the problem of missing the coherence in the SGS eddies when mimicking the effect of the SGS eddies on the particle tracks. Here, ksgs stands for the kinetic energy associated with the SGS eddies only and has to be estimated. All these issues have been extensively discussed by, e.g., Derksen (2003, 2006a).

2. Euler– Euler Approach In the complete Eulerian description of multiphase ﬂows, the dispersed phase may well be conceived as a second continuous phase that interpenetrates the real continuous phase, the carrier phase; this approach is often referred to as two-fluid formulation. The resulting ‘simultaneous’ presence of two continua is taken into account by their respective volume fractions. All other variables such as velocities need to be averaged, in some way, in proportion to their presence; various techniques have been proposed to that purpose leading, however, to different formulations of the continuum equations. The method of ensemble averaging (based on a statistical average of individual realizations) is now generally accepted as most appropriate. In the two-ﬂuid formulation, the motion or velocity ﬁeld of each of the two continuous phases is described by its own momentum balances or NS equations (see, e.g., Rietema and Van den Akker, 1983 or Van den Akker, 1986). In both momentum balances, a phase interaction force between the two continuous phases occurs predominantly, of course with opposite sign. Two-ﬂuid models therefore belong to the class of two-way coupling approaches. The continuum formulation of the phase interaction force should reﬂect the same effects as experienced by the individual particles and discussed above in the context of the Lagrangian description of dispersed two-phase ﬂow. One therefore has to decide here which components of the phase interaction force (drag, virtual mass, Saffman lift, Magnus, history, stress gradients) are relevant and should be incorporated in the two sets of NS equations. The reader is referred to more speciﬁc literature, such as Oey et al. (2003), for reports on the effects of ignoring certain components of the interaction force in the two-ﬂuid approach. The question how to model in the two-ﬂuid formulation (lateral) dispersion of bubbles, drops, and particles in swarms is relevant

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as well: various models are available. See also the discussion on page 204 as to (Eq. (19)). Another important issue in two-ﬂuid models is about modeling the turbulent stresses under two-phase conditions (e.g., Van den Akker, 1998). At this moment, there is still no consensus on a universal two-phase turbulence model. Generally, turbulence in the continuous phase may be generated by shear due to large-scale velocity gradients felt by the continuous phase itself (just like in single-phase ﬂows) as well as by the presence and relative motion of the dispersed phase particles. The ratio at which these two mechanisms contribute to the generation of turbulence may be an important factor in drafting a universal model. In addition, the dispersed phase may exhibit a turbulent-ﬂow behavior in response to the turbulent motions of the continuous phase in which it is embedded; this response may depend on several time scales and their interaction (Oey et al., 2003). Lance et al. (1991) suggested that the motion of bubbles promotes a return to isotropy (see also Van den Akker, 1998). A universal model, however, is not available right now. In dense systems such as encountered in solids suspension, particle–particle interaction may be important as well. Then, the closure of solid-phase stresses is an important issue for which kinetic theory models and solids phase viscosity may be instrumental (see, e.g., Curtis and Van Wachem, 2004). As a matter of fact, in comparison with the Euler–Lagrangian approach, the complete Eulerian (or Euler–Euler) approach may better comply with denser two-phase ﬂows, i.e., with higher volume fractions of the dispersed phase, when tracking individual particles is no longer doable in view of the computational times involved and the computer memory required, and when the physical interactions become too dominating to be ignored. Under these circumstances, the motion of individual particles may be overlooked and it is wiser to opt for a more superﬁcial strategy that, however, still has to take the proper physics into account. Precisely owing to the continuum description of the dispersed phase, in Euler–Euler models, particle size is not an issue in relation to selecting grid cell size. Particle size only occurs in the constitutive relations used for modeling the phase interaction force and the dispersed-phase turbulent stresses. In the case of droplets and bubbles, particle size and number density may respond to variations in shear or energy dissipation rate. Such variations are abundantly present in turbulent-stirred vessels. In fact, the explicit role of the revolving impeller is to produce small bubbles or drops, while in substantial parts of the vessel bubble or drop size may increase again due to locally lower turbulence levels. Particle size distributions and their spatial variations are therefore commonplace and unavoidable in industrial mixing equipment. This seriously limits the applicability of common Euler–Euler models exploiting just a single value for particle size. A way out is to adopt a multiﬂuid or multiphase approach in which various particle size classes are distinguished, with mutual transition paths due to particle break-up and coalescence. Such models will be discussed further on.

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III. Computational Aspects As the continuity equation, the NS equations, and the transport equations for the turbulent variables are highly nonlinear, any CFD-calculation is essentially iterative. Generally, the convergence rate of simulations depends on the number of grid points and on the number of equations to be solved. The number of grid points is associated with the desired or required degree of detail and accuracy of the simulations given the type of simulation selected. In running a DNS (provided it is doable) one is interested in a fully resolved ﬂow ﬁeld and the grid should be sufﬁciently ﬁne to catch all motions. In running a LES, the grid should be sufﬁciently ﬁne for the subgrid scales to become independent of the macroﬂow. The number of equations to be solved is, among other things, related to the turbulence model chosen (in comparison with the k– e model, the RSM involves ﬁve more differential equations). The number of equations further depends on the character of the simulation: whether it is 3-D, 21/2-D, or just 2-D (see below, under ‘The domain and the grid’). In the case of two-phase ﬂow simulations, the use of two-ﬂuid models implies doubling the number of NS equations required for single-phase ﬂow. All this may urge the development of more efﬁcient solution algorithms. Recent developments in computer hardware (faster processors, parallel platforms) make this possible indeed. Various numerical techniques are available for discretizing the set of partial differential equations to be solved. Discretizing essentially is the method of converting the (partial) differential into algebraic equations by transforming (partial) derivatives into ﬁnite-difference formulations. First of all, most (commercial) ﬂow simulation codes exploit the ﬁnite-volume (or ﬁnite-difference) method that has been discussed extensively by Shyy (1994). An introduction to the concept can be found with, e.g., Abbott and Basco (1989) and Shaw (1992). Many more techniques are available for discretization, such as ﬁnite-element, spectral, arc-length, and ﬁlter-scheme methods, which are beyond the scope of the present review. The result is a (large) set of algebraic equations anyway: one algebraic equation per grid point for each ﬂow variable that connects the value of a particular ﬂow variable at a particular grid point with those at a number of neighboring grid points.

A. FINITE VOLUME TECHNIQUES Most commercial CFD-codes are based upon the ideas and numerical methods developed back in the 1970s at Imperial College London by Spalding, Patankar, Gosman, and others: the FV formulation (the natural balance formulation of the NS equations and

the continuity equation),

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the staggered grid concept (velocity components and scalar quantities such as

pressure are not deﬁned at the same mesh points), the common discretization schemes (central, upwind, quadratic upwind), the pressure–velocity coupling according to the Semi-Implicit Method for

Pressure Linked Equations (SIMPLE) or related algorithms, and the matrix solvers for the resulting sets of algebraic equations.

Useful reviews of these basic elements of CFD can be found with Patankar (1980), Abbott and Basco (1989), Shaw (1992), and Ranade (2002). In the meantime, substantial progress has been realized in developing more effective and powerful numerical techniques. Several of them have made it into the common commercial CFD packages. Just as an example, several of the commercial vendors have incorporated the option of collocated grids. A few more important issues should be highlighted here. First of all, the current iterative solution procedures solve the various momentum equations successively; although this highly segregated solution technique does not require large computer memory capacity, it does result in slow convergence and, hence, in long CPU times. Reducing the degree of separation, i.e., coupling several momentum equations and solving them simultaneously, substantially speeds up the convergence rate of the calculation and reduces the number of iterations required (Van Santen et al., 1996); the larger matrices do make greater demands on computer memory. Second, the various discretization schemes common to all commercial software may suffer from the appearance of spurious oscillations (wiggles) in regions of strong gradients and from numerical or false diffusion, or smearing (Shyy, 1994). Finding the optimum scheme to avoid numerical diffusion is often not very easy, especially in convection-dominated ﬂows, when direction of ﬂow and grid orientation do not match everywhere. Wiggles may pose a serious problem in solving turbulent ﬂows, multiphase ﬂows, and species transport as the pertinent equations contain variables that are inherently positive, such as k, e, phase fractions and concentrations. For these variables, wiggles may not be tolerated as they could give rise to negative values which may result in divergence of the algorithm. Although adaptive grid techniques, e.g., local grid reﬁnement, may cure the problem of oscillations, the most promising among the modern schemes seems to be the antidiffusion concept of Total Variation Diminishing (TVD) that has proliferated in quite a few variants. By adding subgrid points, TVD schemes do increase the calculational burden. In FLUENT 6.2, a bounded central differencing scheme is switched on for LES, replacing the second-order upwind scheme which by default discretizes the convective terms in RANS-based simulations. Third, writing the discretized equations in matrix form results in sparse matrices, often of a tri-diagonal form, which traditionally are solved by successive under- or over-relaxation methods using the tri-diagonal matrix algorithm

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(TDMA). In the 1990s, however, two novel classes of methods have entered the scene, viz. Krylov subspace methods (such as Conjugate Gradient CG, the improved

BiCGSTAB, and GMRES) in combination with preconditioners for matrix manipulations aimed at enhanced convergence, and Multigrid methods in which convergence is improved by solving the equations in, e.g., a nested iteration on multiple grids, starting, e.g., on a coarse grid and then moving to a ﬁner grid, et cetera. For a more detailed description of these more sophisticated solution methods the reader is referred to, e.g., Vuik (1993) and Shyy (1994). Adopting these more sophisticated solution techniques becomes more important with increasing number of partial differential equations to be solved, such as in two-phase ﬂow CFD (see, e.g., Lathouwers and Van den Akker, 1996; Van Santen et al., 1996; Lathouwers, 1999). In the last decade, most new algorithms, schemes, solvers, and preconditioners have found their way into most commercial software packages. Multigrid solvers are also available. Furthermore, all CFD vendors have developed powerful pre- and post processing routines.

B. THE SIZE

OF THE

COMPUTATIONS

The number of grid points used for CFD-studies of stirred tanks strongly varies with the type of CFD and generally increases in time. While for Bakker and Van den Akker (1994a, b) the maximum number of grid points amounted to some 25,000, these days hundreds of thousands grid points are not uncommon for RANS-simulations. Usually, DNS and LES are carried out with the view to arrive at very ﬁne spatial and temporal resolutions; then, substantially more grid points are needed than with RANS-based simulations. For his LES, 6 6 Eggels (1996) used two uniform grids comprising 1.73 10 and 13.8 10 6 nodes, respectively. Derksen and Van den Akker (1999) used up to 6 10 grid 6 points, Hollander et al. (2000) some 2 10 , Derksen (2001), Lu et al. (2002), 6 and Hartmann et al. (2006) up to some 13.8 10 , while Ten Cate et al. (2000) 6 went as high as 35 10 grid points for an industrial crystallizer. In addition, adding more transport equations for simulating physical or chemical processes and running CFD for multiphase ﬂows increases the size of the computational jobs, both in the number of equations to be solved and in terms of the difﬁculty of getting the solution converged. Advanced CFD simulations (both in terms of numbers of grid points and partial differential equations) therefore require increasing amounts of computer memory and CPU-time. Chemical engineers increasingly get familiar with the idea of exploiting CFD, though still mostly of the RANS-type. Gradually, the

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advantages of LES become more obvious. These days, it becomes attractive to speed up CFD simulations by running them on parallel computer platforms: a cluster of pc’s operating under, e.g., LINUX, or a massive parallel machine (e.g., a CRAY). In this way, larger CFD simulations can be run overnight or over the weekend. In addition, computer hardware (processors, memory) keep becoming faster and cheaper. Just to illustrate what currently is becoming feasible: Derksen (2006a) carried out his LES job on a platform of 12 CPU’s, and needed 7 h of wall-clock time for a single impeller revolution; as a mater of fact, such a simulation yields an incredible degree of detail. Parallel computing requires software made suitable for operating on parallel processors. Decomposition of the ﬂow domain and attributing each domain to a separate processor is the common procedure. A fast communication between the various processors is crucial, not to partly spoil the gain obtained by going parallel. Although commercial CFD vendors make versions of their software suited for parallel computing, it is precisely the promises of parallel computing that turn the Lattice Boltzmann techniques into an attractive option. The locality of the collision operation in this technique (see below) along with a high computational efﬁciency allows for simulating complex ﬂow systems with high spatial and temporal resolution. Furthermore, the efﬁciency of the scheme hardly depends on the complexity of the ﬂow domain. Compared to the conventional FV solvers of the current commercial CFD codes, which should be robust and suited for a wide variety of ﬂow problems and ﬂow conditions, a LB solver is faster by at least one order of magnitude (see, e.g., Table I). All these properties make LB very attractive for LES in complex geometries and even very competitive in comparison with the conventional FV techniques.

TABLE I NUMERICAL SETTINGS OF TWO SIMULATIONS ON A GAS CYCLONE WITH DIFFERENT NUMERICAL SCHEMES: A RANS-BASED SIMULATION (HOEKSTRA, 2000) AND A LES DUE TO DERKSEN AND VAN DEN AKKER (2000)

Numerical scheme Spatial and temporal properties Number of grid nodes Total wall-clock time per simulationa Wall-clock time per grid node per stepd

RANS

LES

Finite volume (FLUENT) 3-D, steady state 1.2 105 117 hb 1.7 10–4 s

Lattice-Boltzmann 3-D, transient 4.9 106 864 hc 6.1 10–6 s

Both simulations were run on an HP Convex S-Class computer. a Based on a simulation on one processor. b Converged after 20,000 iterations. c Required 95,500 time steps for obtaining reliable ﬂow ﬁeld statistics. d One ‘step’ denotes one ‘time step’ or one ‘iteration step.’

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C. LATTICE-BOLTZMANN TECHNIQUES Very promising with a view to simulating turbulent ﬂows is the LB scheme. The method originates from lattice-gas automata, which for ﬂuid ﬂow applications were introduced by Frisch et al. (1986) and by McNamara and Zanetti (1988) and has been studied, reﬁned, and applied ever since. In spite of LB being relatively new and still ‘under construction,’ the method receives an increasing amount of attention among scientists and engineers. Rather than taking the discretized NS equations as a starting point for the numerical analysis, a discrete system that mimics ﬂuid ﬂow is designed directly. Although LB therefore nowadays may be considered as a solver for the NS equations, there is deﬁnitely more behind it. The method originally stems from the lattice gas automaton (LGA), which is a cellular automaton. In a LGA, a ﬂuid can be considered as a collection of discrete particles having interaction with each other via a set of simple collision rules, thereby taking into account that the number of particles and momentum is conserved. In the LB technique, the ﬂuid to be simulated consists of a large set of ﬁctitious particles. Essentially, the LB technique boils down to tracking a collection of these ﬁctitious particles residing on a regular lattice. A typical lattice that is commonly used for the effective simulation of the NS equations (Somers, 1993) is a 3-D projection of a 4-D face-centred hypercube. This projected lattice has 18 velocity directions. Every time step, the particles move synchronously along these directions to neighboring lattice sites where they collide. The collisions at the lattice sites have to conserve mass and momentum and obey the socalled collision operator comprising a set of collision rules. The characteristic features of the LB technique are the distribution of particle densities over the various directions, the lattice velocities, and the collision rules. Such an approach is conceptually different from the continuum description of momentum transport in a ﬂuid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable ‘viscosity,’ since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). Lattice-Boltzmann is an inherently time-dependent approach. Using LB for steady ﬂows, however, and letting the ﬂow develop in time from some starting condition toward a steady-state is not a very good idea, since the LB time steps need to be small (compared to, e.g., FV time steps) in order to meet the incompressibility constraint. The LB method is especially attractive if complexly shaped boundaries are involved; see, e.g., Chen and Doolen (1998). Eggels (1996) was the ﬁrst to study

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the ﬂow near a Rushton turbine in a bafﬂed stirred tank reactor by means of a LES, thereby using LB on a uniform grid. Derksen and Van den Akker (1998, 1999) performed a similar study for a tank reactor where the ﬂow was driven by either a pitched blade turbine or a Rushton turbine. These simulations revealed a range of ﬂow ﬁeld characteristics, such as the turbulence statistics and the trailing vortex structure near the impeller. The computer code they developed is based on a LB scheme proposed by Somers (1993). Boundary conditions can be imposed through locally forcing the ﬂuid to a prescribed velocity, or (in case of no-slip walls) by simple bounce-back rules for the ﬁctitious LB particles. In an agitated vessel, the action of the revolving impeller is described by means of an adaptive force-ﬁeld procedure (Derksen and Van den Akker, 1998, 1999). Lattice-Boltzmann can simply be conceived as an alternative method of ﬁnding a solution for the NS equations and is now being used for a growing number of applications, ranging from laminar blood ﬂow through irregularly shaped veins (Artoli et al., 2003), the swirling ﬂow in a vessel with a revolving bottom (Derksen et al., 1996) and the vortex street behind a cylinder in cross-ﬂow (Derksen et al., 1997) to turbulent ﬂow in industrial devices such as stirred vessels, swirl tubes and reverse ﬂow cyclones (Derksen and Van den Akker, 1998, 1999, 2000; Derksen, 2001, 2002a, b). There is also a commercial LB code available (Power FLOW, EXA Corp., USA) that is used for ﬂow problems related to, e.g., the automotive and aerospace industries. A disadvantage of LB techniques is that they require a (locally) uniform and cubic computational grid. This raises problems with curved boundaries (see Section III). Furthermore, the ﬂow is calculated in great detail everywhere, irrespective of the degree of turbulence; in some parts of the ﬂow domain, particularly in the more quiescent parts of the vessel, the LES may turn into some type of DNS, the SGS contribution to the transport equation becoming quite obsolete. Rohde (2004) developed a technique for local grid reﬁnement which, however, has not yet been applied to the cylindrical geometry of a stirred vessel. Lu et al. (2002) successfully attempted the use of a nonuniform grid in their LB LES of the turbulent ﬂow in a stirred tank. Eggels and Somers (1995) used an LB scheme for simulating species transport in a cavity ﬂow. Such an LB scheme, however, is more memory intensive than a FV formulation of the convective-diffusion equation, as in the LB discretization typically 18 single-precision concentrations (associated with the 18 velocity directions in the usual lattice) need to be stored, while in the FV just 2 or 3 (double-precision) variables are needed. Scalar species transport therefore can better be simulated with an FV solver.

D. A MUTUAL COMPARISON

OF

FINITE VOLUME

AND

LATTICE-BOLTZMANN

Lattice-Boltzmann exhibits some inherent properties that favor the speed of the simulations, and therefore make the method a serious candidate and

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alternative for FV techniques for simulating ﬂows in process industry equipment. These properties are: The hydrodynamic quantities such as the velocities and velocity gradients are

determined locally in a LB code. As a result, the local character of the LB technique is strongly in favor of (massive) parallellization of the computational job via domain decomposition: the communication between CPU’s relates to grid cells very near to the subdomain boundaries only. On the contrary, in solving the NS equations iteratively with a FV solver the ﬁeld properties result in long-range variations propagating across the boundaries of subdomains and, hence, requiring intense communication between CPU’s dealing with neighboring subdomains. Since LB describes the NS equations in the incompressible limit, the local pressures can directly be obtained from the local densities and the speed of sound. Hence, a distinct step for calculating the pressures via a Poisson equation (derived from the continuity equation) as required in incompressible FV schemes, is absent in LB. Implementing complex boundaries in LB simulations is relatively easy compared to doing so for FV techniques (Chen and Doolen, 1998). In view of the usually complex boundaries of process equipment, particularly in the case of stirred vessels with a revolving impeller, this is a distinctive advantage. More basically, LB with its collision rules is intrinsically simpler than most FV schemes, since the LB equation is a fully explicit ﬁrst-order discretized scheme (though second-order accurate in space and time), while temporal discretization in FV often exploits the Crank–Nicolson or some other mixed (i.e., implicit) scheme (see, e.g., Patankar, 1980) and the numerical accuracy in FV provided by ﬁrst-order approximations is usually insufﬁcient (Abbott and Basco, 1989). Note that ‘fully explicit’ means that the value of any variable at a particular moment in time is calculated from the values of variables at the previous moment in time only; this calculation is much simpler than that with any other implicit scheme. A comprehensive and more extensive overview of the pros and cons of LB with respect to applications can be found in Succi (2001). By the way, LB methods are continuously improved to increase speed and accuracy, particularly by introducing grid reﬁnement techniques and advanced techniques for arbitrarily shaped boundaries (e.g., Rohde et al., 2002, 2003, 2006; Rohde, 2004). It makes sense to compare the implications (in terms of simulation times) of using FV vs. LB in simulating turbulent-ﬂow ﬁelds in process devices. Hoekstra (2000) demonstrated the numerical implications of applying different numerical schemes in an industrial application. He compared the outcome of his RANS simulation for a gas cyclone with that of a LES carried out by Derksen and Van den Akker (2000). Table I presents a number of numerical features of the two types of simulations.

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Not only did the results of the LES agree much better with the experimental data than those of the RANS simulation, the LB code was also roughly 30 times faster than the FV code in terms of simulation time per grid node per step. To put it in a more practical sense: performing this LES on, e.g., 18 processors with the FV code would take 2 months, whereas a simulation as such with LB would take 2 days only. Another comparison is due to Van Wageningen et al. (2004) who performed a similar study (in terms of the numerical scheme used) on unsteady laminar ﬂow in a Kenicss static mixer. They found that the LB code was 500–600 times faster than FLUENT in terms of simulation time per grid node per time step and that FLUENT used about 5 times more memory than LB. The difference in speed between a LB code and a FV code in the above studies partly originates from their different character: the FV code in a generalpurpose commercial CFD code should be robust and suited in many applications, while the LB codes used are of a research type and usually strongly dedicated and geared to a speciﬁc job.

IV. Boundary Conditions Solving sets of (partial) differential equations inherently requires the speciﬁcation of boundary conditions and, in case of transient simulations, also initial conditions. This is not as simple as it looks like, especially for turbulent ﬂows in complex process equipment. Whenever a free surface is present at some (mean) ﬁxed position, most CFD codes assume it to be strictly ﬂat, while in the direction normal to the free surface velocities and gradients of most variables are taken zero. Usually, this is accomplished by deﬁning ‘mirror cells’ at the free surface. It is not clear what the effect is of the use of such mirror cells on the ﬂow ﬁeld in the upper part of the vessel in comparison with real life where the surface is not necessarily ﬂat.

A. MOVING BOUNDARIES An aspect of CFD in stirred vessels that needs separate discussion is the issue of the revolving impeller and the way its motion is dealt with in the simulations. In the early days, see, e.g., Bakker and Van den Akker (1994a), a black box representing the impeller swept volume was often used in RANS simulations, with boundary conditions in the outﬂow of the impeller which were derived from experimental data. The idea behind this approach was that such nearimpeller data are hardly affected by the rest of the vessel and therefore could be used throughout. Generally, this is not the case of course. Furthermore, this approach necessitates the availability of accurate experimental data, not only

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with respect to the average velocity components, but also for k and e. The latter variable in particular can hardly be measured directly. Nowadays, this approach is no longer used, also due to the steep increase in computer power which no longer urges for such drastic simpliﬁcations. In those early days, when computer power was limited, often use was made of a symmetry assumption: each quarter of the vessel containing one of the four bafﬂes at the vessel wall was supposed to behave identically; hence, a steady ﬂow in the RANS approach was simulated in just a quarter vessel. Such strong simpliﬁcations are no longer in use. Precessing vortices moving around the vessel centerline contribute to ﬂow unsteadiness and, therefore, exclude models that just assume ﬂow steadiness or allow for domain reductions through geometrical symmetries. The most correct response to this ﬂow unsteadiness is the concept of LES. Later on, in 1994, novel options were introduced such as Sliding Mesh (SM) and Multiple Frames of Reference (MFR) in which the ﬂow domain is divided into two parts, each with their own meshing; one mesh is connected to the stationary vessel wall and the other one to the revolving impeller. The interaction of the ﬂow ﬁelds at either side of the interface between these two meshes requires delicate bookkeeping of ﬂuxes and forces among cells moving with respect to one another at the interface. Of course, these methods are a drastic improvement over the black box description of the impeller swept area applied in the early days of stirred vessel CFD. The mesh associated with the impeller in SM is perfectly capable of simulating the unsteady ﬂow around the impeller blades including the trailing vortices. The MFR technique introduced by Luo et al. (1993, 1994) starts from a steady-state description of the ﬂow ﬁeld and therefore ﬁts in RANS-based simulations only. The SM approach (e.g., Murthy et al., 1994; Bakker et al., 1997) is a fully transient method, also applicable in LES (see, e.g., Bakker et al., 2000; Jahoda et al., 2006; Gao and Min, 2006; Gao et al., 2006). Yeoh et al. (2004a, b) adopted a Sliding and Deforming Mesh (SDM) technique in which the two grids do not only slide with respect to each other but also feel shear resulting in deforming interface cells. Keeping mass and momentum conserved in this technique requires bookkeeping somewhat different from that with the SM technique. Generally, however, it is unclear what (with SM) the effect is of the position of the interface between stationary and moving meshes on the simulated unsteadiness of the overall ﬂow in the vessel. In addition, simulations making use of SM and MFR may suffer from slower or poor convergence. Of course, the transient SM technique is more accurate, though at the cost of larger computer time consumption. Montante et al. (2006) even reported that MFR yielded unphysical ﬁndings, viz. a region of opposite swirl. Harvey et al. (1995) and Harvey and Rogers (1996) proposed a multiblock impeller-ﬁtted grid structure for dealing with the exact geometry of the impeller. The ﬁrst of these two papers introduces an approximate steady-state method

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that solves the viscous ﬂow with the impeller at one position with respect to the bafﬂes ignoring its relative motion, while the second paper is about an unsteady moving grid approach that now takes the relative motion of the impeller with respect to the bafﬂes into account. This approach, however, never made it into to the turbulent-ﬂow regime. Ranade and Van den Akker (1994) and Ranade and Dommeti (1996) introduced a snapshot approach in which the impeller was put in a standstill and the revolving ﬂow in the simulation was obtained by imposing velocity jets at one or more ﬁxed positions of the impeller blades. At a ﬁrst glance, realistic ﬂow ﬁelds were obtained, energy dissipation levels and the total amount of energy dissipated being in the right order of magnitude. A more detailed assessment of this approach, however, reveals that the equations used are not invariant to the type coordinate transformation used. Furthermore, elementary turbulence theory (e.g., Tennekes and Lumley, 1972) indicates that, with a view to the ﬂow ﬁeld in the impeller swept volume, a jet issuing from the face of an impeller blade may not be equivalent to a wake behind an impeller blade, as jets and wakes obey different laws for their expansion in downstream direction. As a result, the shape of the impeller connected zones of high k and e values may not be predicted conﬁdently. Ranade’s snapshot approach (still in use, see, e.g., Khopkar et al., 2006) discussed further on page 207 should therefore be abandoned, in spite of all explanations devoted to it (Ranade, 2002). Applying Immersed or Embedded Boundary Methods (Mittal and Iaccarino, 2005) circumvents the whole issue of the friction between the more or less steady overall ﬂow in the bulk of the vessel and the strongly transient character of the ﬂow in the zone of the impeller. These methods are introduced below. In the context of a LES, Derksen and Van den Akker (1999) introduced a forcing technique for both the stationary vessel wall and the revolving impeller. They imposed no-slip boundary conditions at the revolving impeller and at the stationary tank wall (including bafﬂes). To this purpose, they developed a speciﬁc control algorithm.

B. CURVED BOUNDARIES In stirred vessels and static mixers the ﬂow domain is bounded by complex boundaries due to the curvature of containing walls, the revolving impeller axis and/or static mixing elements. While in the early days a staircase representation of a curved boundary in a cubic grid was quite common, commercial CFD software nowadays exploits boundary ﬁtted meshing. The staircase representation usually was not a problem—in terms of mass conservation or of introducing artefacts such as additional small eddies—as long as the steps did not involve more than a single grid cell. The currently widely adopted boundary ﬁtted meshing, i.e., generating body-conformal either structured or unstructured grids providing adequate

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local resolution with a minimum but large number of grid cells, requires either commercial grid generating software or extensive coding (usually far beyond the reach of academic groups). The impact of a poor grid on accuracy and on convergence properties of the solver may not be underestimated. Whenever a cubic grid is mandatory—either due to coding limitations from the part of academic groups or due to the inherent properties of, e.g., LB techniques—and a staircase approach is to be avoided (e.g., for a revolving impeller axis) one can take refuge to some immersed boundary method (see, e.g., Mittal and Iaccarino, 2005). One may distinguish between embedded boundary methods using cut cells: these methods adjust cell volume

and face areas to the geometry of the body in the ﬂow domain; while in a FV approach this method guarantees global and local conservation of mass and momentum, it creates problems in an explicit formulation of the FV scheme; immersed boundary methods exploiting boundary forcing methods: a boundary is simply treated as a force exerted or felt at the position of the boundary; this position may even be time dependent, e.g., in the case of a revolving impeller; and immersed boundary methods using ghost cells: ghost cells are boundary cells the centers of which are lying outside the ﬂow domain; in this approach, values of variables in these ghost cells are required to satisfy, e.g., a Neumann boundary condition (velocity gradients normal to a wall zero, velocity component along a wall zero at the wall). Which of the various immersed or embedded boundary methods is best— generally or for a particular case—is still an open question. Thornock and Smith (2005) introduced a Cell Adjusted Boundary Force Method for a stirred vessel. All methods proposed so far have their own pros and cons. Immersed boundary methods are also exploited in LB techniques (e.g., Derksen and Van den Akker, 1999). Rohde (2004) investigated the use of triangular facets for representing a spherical particle.

C. THE DOMAIN

AND THE

GRID

With a view to any simulation, a few important items have to be addressed. First of all, it has to be decided whether the ﬂow to be simulated is 2-D, 21/2-D, or 3-D. When the ﬂow is, e.g., axis-symmetrical and steady, a 2-D simulation may sufﬁce. For a ﬂow ﬁeld in which all variables, including the azimuthal velocity component, may not depend on the azimuthal coordinate, a 21/2-D simulation may be most appropriate. Most other cases may require a full 3-D simulation. It is tempting to reduce the computational job by casting the 3-D ﬂow ﬁeld into a 2-D mode. The experience, however, is that in 2-D simulations the turbulent viscosity tends to be overestimated; in this way, the ﬂow

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may become more or less diffusion dominated and less capable of sustaining a transient behavior. Furthermore, between 2-D and 3-D, the wall to volume ratio is different, turbulence intensity may go down, and hydrodynamic stability may be affected. Of course, the above remarks apply to RANS-based simulations only, as LES are inherently transient and 3-D. A second choice to be made relates to the size of the ﬂow domain. It may be worthwhile to limit the calculational job by reducing the size of the ﬂow domain, e.g., by identifying an axis or plane of symmetry, or, in a cylindrical vessel with bafﬂes mounted on the wall, due to periodicity in the azimuthal direction. Commercial software accomplishes these choices by means of ‘symmetry cells’ and ‘cyclic cells,’ respectively; although such choices reduce the size of the simulation, they may eliminate the possibility of ﬁnding the real (asymmetric, unstable, or transient) 3-D ﬂow ﬁeld. The presence of feed pipes or drain or withdrawal pipes may also make the use of symmetry or cyclic cells impossible. Again, this issue only plays a role in RANS-type simulations. A third issue is how many grid cells should be used for the domain size selected. In general, using more computational cells implies more detailed insight in the ﬂow ﬁeld as a result of the simulation, at the cost of longer CPU times (although sometimes convergence rate may increase as a result of increasing the number of grid cells.) This is a trade-off that is to be decided upon, each time a new ﬂow ﬁeld is to be simulated. In some cases, a stepwise approach may be pursued to zoom into a particular zone within a ﬂow device (see, e.g., Stekelenburg et al., 1994). A fourth issue is the use of local grid reﬁnement: many commercial codes offer this modality, often with unstructured grids. The rationale behind the idea of introducing local grid reﬁnement techniques is that the grid is only reﬁned in those parts of the calculation domain in which the ﬂow exhibits strong spatial or temporal gradients. By doing so, one does not waste grid cells in parts of the domain where the ﬂow does not vary signiﬁcantly. Rohde et al. (2006) explored the use of a generic, mass-conservative local grid reﬁnement technique within a LB technique. He successfully attempted this technique for various rather simple cases; so far, it has not been exploited in a cylindrical stirred vessel in which the ﬂow would require a ﬁne grid not only in the impeller swept region but also near to the bafﬂes. Anyhow, the result of any ﬂow simulation should be grid independent, i.e., the ﬂow ﬁeld should not be different if a ﬁner grid is used for the simulation. This test is to be carried out always and everywhere, and may really be described as a conditio sine qua non. Finally, it is good to quote Shaw (1992, p. 227): ‘the most important asset in a CFD analysis process is the analyst, who actually translates the engineering problem into a computational simulation, runs the CFD solver and analyzes the results. It is the skill of this person, or set of persons, that will determine whether all the hardware and software will be utilized in the best possible way and produce good quality results.’ CFD is certainly not a panacea that may solve all

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possible design and optimization problems. It rather is a tool that in the hands of a well-trained professional may provide valuable insights in the local phenomena and processes, which take place in (bio)reactors and all sorts of process equipment including mixing devices. CFD-simulations may provide a good starting point for a mechanistic description of operations such as blending, suspending solids, dispersing gas, and carrying out (bio)chemical reactions.

V. Simulations of Turbulent Flows in Stirred Vessels When exploiting computational techniques for studying the mixing performance of stirred tanks and static mixers, the ﬁrst question is how well the turbulent-ﬂow ﬁeld and the characteristic turbulence properties are predicted by the various forms of CFD discussed above. The different forms of CFD (RANS, LES, DNS) and the various strategies, turbulence models, and submodels used by them may suffer from a different degree of validity in the various mixing devices considered. The impact of a limited validity of the models used may vary from case to case. That is why validation of simulation results is still an urgent issue. Sometimes, the results of computational simulations just look qualitatively correct. It is important to check code performance by means of quantitative experimental data as to average quantities as well as local and transient ﬂuctuations. The next question then is whether the processes taking place inside this turbulent-ﬂow ﬁeld can be modeled with conﬁdence. We will now ﬁrst consider the ﬁrst question.

A. TURBULENCE PROPERTIES It is worthwhile to verify whether or not some basic assumptions of turbulence theory (local equilibrium between production and dissipation of turbulence, isotropy, homogeneous turbulence) which are used in modeling certain aspects of momentum transport via turbulent eddies are met with in stirred tanks. To be honest, at hardly any position in a stirred tank the production rate of turbulent kinetic energy balances the rate of its dissipation. Production of turbulence mainly takes place in the impeller swept volume, while a much larger part of the vessel takes part in dissipating the turbulence. In addition, the action of a revolving impeller and its interaction with nearby bafﬂes turns the ﬂow intrinsically unsteady. As a result, the turbulent ﬂow in a stirred vessel is certainly not an equilibrium ﬂow, as presumed in using, e.g., the Smagorinsky SGS model in LES. Most RANS-based simulations make use of the k–e model for taking into account the momentum transport due to the turbulent eddies. This model is an

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eddy-viscosity model and as such it assumes isotropic turbulent transport. The question is whether everywhere in a stirred tank the turbulent ﬂow is locally isotropic. This issue might have to be explained a bit further as ‘local isotropy’ should be distinguished from just ‘isotropy.’ Local isotropy means that the ﬂuctuations can be modeled with an (isotropic) eddy viscosity, while isotropy has been deﬁned as having the same ﬂuctuation levels in the three coordinate directions. Local isotropy does not imply isotropy of the ﬂuctuations: a k–e model can predict nonisotropic ﬂuctuations. The question whether or not stirred tank ﬂow is locally isotropic, may be investigated with the help of a LES which resolves a great deal of the Reynolds stresses. To this end, the Reynolds stress data are best presented in terms of the so-called anisotropy tensor aij and its invariants A1, A2, and A3. The anisotropy tensor is related to the turbulent stresses, of course, and is deﬁned as aij ¼

ui uj 2 dij 3 k

(15)

Its ﬁrst invariant A1 is equal to zero by deﬁnition. The second and third invariants of this tensor are A2 ¼ aijaji and A3 ¼ aijajkaki, respectively. The range of physically allowed values of A2 and A3 is bounded and represented by the socalled Lumley triangle in the (A3, A2) plane (Lumley, 1978). The distance|A| ¼ O(A22+A32) from the isotropic state, i.e., from the origin (A2 ¼ 0, A3 ¼ 0), is a measure of the degree of anisotropy. See also Escudie´ and Line´ (2006) for a more extensive discussion as to how to quantify and visualize how different from isotropic turbulence a stirred vessel is. Phase-averaged values of|A|in a plane midway between two bafﬂes of a stirred tank have been plotted in Fig. 1 (from Hartmann et al., 2004a) for two different SGS models (Smagorinsky and Voke, respectively) in LES carried out in a LB approach. The highest values, i.e., the strongest deviations from isotropy, occur in the impeller zone, in the boundary layers along wall and bottom of the tank, and at the separation points at the vessel wall from which the anisotropy is advected into the bulk ﬂow. In the recirculation loops, the turbulent ﬂow is more or less isotropic. Fig. 2 (also from Hartmann et al., 2004a) shows how the values of the invariants A2 and A3 found in a LES using a Smagorinsky SGS model are distributed within the Lumley triangle. Most but not all of the points are clustered in the lower part of this triangle. This implies that in a large part of a stirred tank the turbulent ﬂow is more or less isotropic and the use of a turbulent viscosity and a k–e model are permitted. On the whole, however, turbulent ﬂow is not really isotropic. This may also explain why in RANS-based simulations turbulent kinetic energy levels generally are underestimated. As far as e is concerned: it is extremely difﬁcult to measure its spatial distribution and its wide range of values experimentally. LES may provide an

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FIG. 1. Phase-averaged plots of the anisotropy distance|A|in a plane midway between two bafﬂes in a stirred vessel provided with a Rushton turbine, as obtained by means of LES, with two different SGS models: (a) the Smagorinsky model; (b) the Voke model. Reproduced with permission from Hartmann et al. (2004a).

FIG. 2. This plot shows to which degree, according to a LES, the turbulence in a plane midway between two bafﬂes in a stirred vessel provided with a Rushton turbine can be typiﬁed. For clarity, not all grid points in such a plane have been used for this plot. According to Lumley (1978), the borders represent different types of turbulent ﬂows: 3-D isotropic turbulence, 2-d axis-symmetric turbulence, 2-D turbulence, and 1-D turbulence. Most but not all points are concentrated in the lower part of this Lumley triangle. Reproduced with permission from Hartmann et al. (2004a).

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attractive alternative. Micheletti et al. (2004) present a valuable discussion on this issue.

B. VALIDATION

OF

TURBULENT FLOW SIMULATIONS

In view of the different requirements as to computer power, it is very worthwhile to compare the outcome of RANS and LES simulations mutually and/ or with quantitative experimental data. Several authors have done this, e.g., Derksen and Van den Akker (1998, 1999), Derksen (2001), Lu et al. (2002), Ranade (2002), Derksen (2002b), and Yeoh et al. (2004a,b). In most cases, the experimental data have been obtained by means of Laser-Doppler Anemometry (LDA, or LDV): see, e.g., Yianneskis et al. (1987), Wu and Patterson (1989), Scha¨fer et al. (1997, 1998), and Derksen et al. (1999). In this review, we will mainly refer to the validation study due to Hartmann et al. (2004a). In comparing RANS results and LES results with LDA data, the focus is ﬁrst on the global, phase-averaged ﬂow ﬁeld: see Fig. 3 (from Hartmann et al., 2004a) that relates to a plane midway between two bafﬂes. The two types of simulations capture the dominant ﬂow feature, viz. the two large circulation loops, more or less to the same extent. The RANS simulation better predicts the position of the point where the upper loop separates from the vessel wall. As far as turbulent kinetic energy (k) levels are concerned, k-values obtained in a LES relate to the resolved turbulent eddies. Hartmann et al. (2004a) argued that the SGS ﬂuctuations hardly contribute to the k-levels. It is evident from Fig. 4 that the RANS simulation underestimates the kinetic energy levels created by the blades of the Rushton impeller. The same conclusion applies to phase-resolved kinetic energy data (see Fig. 5). The RANS simulation is hardly capable of catching the remainders of the trailing vortices created by the preceding impeller blade, while LES nicely reproduces this succession of vortices. Whenever one is interested in details of the turbulent-ﬂow ﬁeld because they may affect the performance of the mixing device, one should really consider carrying out a LES. The ﬁndings due to Yeoh et al. (2004a) are completely in line with this. As long as the interest is in ﬁelds of averaged velocity components and in overall mixing patterns, RANS-based simulations may sufﬁce. Examples of such satisfactory simulation results are plentiful, e.g., Marshall and Bakker (2004) and Montante et al. (2006). When, however, the interest is in the details of the turbulent-ﬂow ﬁeld and in processes affected by these details, LES is the option to be preferred. From the ﬁndings reproduced above and from the validation studies of Derksen and Van den Akker (1999) and Derksen (2001) the general conclusion is that, as long as the spatial resolution is sufﬁcient, LB LES deliver results in excellent agreement with experimental turbulence data. Large eddy simulations explicitly resolves the inherently unsteady character of the turbulent ﬂow in a stirred tank into account, including the periodic phenomena associated with the motion of the impeller and their interaction with

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FIG. 3. Velocity vector ﬁelds and levels of turbulent kinetic energy in a plane midway between two bafﬂes in a stirred vessel, according to LDA data (a); a RANS-based simulation (b); and two LES (c) and (d). Reproduced with permission from Hartmann et al. (2004a).

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FIG. 4. Phase-averaged plots of turbulent kinetic energy in the vicinity of the Rushton impeller as found in different types of simulations as indicated. Reproduced with permission from Hartmann et al. (2004a).

the bafﬂes. Global as well as subtle ﬂow features are in quantitative agreement with experimental data. Typical examples of such a quantitative agreement are—for six-bladed Rushton turbines—the path along which the trailing vortices developing at the impeller blades are swept into the bulk of the tank as well as the turbulent kinetic energy levels in the wakes of the impeller blades; in a vessel equipped with a pitched-blade turbine, we could mention the primary and secondary recirculations in the phase-averaged ﬂow ﬁeld. Derksen (2001) and Hartmann et al. (2004a) demonstrated that various choices in SGS modeling did not have a big impact on the quality of the ﬂow ﬁeld predictions. In addition, all these studies did not reveal a signiﬁcant effect of the value of the Smagorinsky coefﬁcient within the range 0.08–0.14. One of the complications in stirred tank ﬂows is the presence of macroinstabilities (i.e., low-frequency mean ﬂow variations) that may affect the mixing performance. Various authors have distinguished between various types and investigated their occurrence and their frequencies under varying operating conditions and with several types of vessels and impellers (Yianneskis et al., 1987; Haam et al., 1992; Myers et al., 1997; Hasal et al., 2000; Nikiforaki et al., 2002).

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FIG. 5. Phase resolved plots of velocity vector ﬁeld and turbulent kinetic energy in a plane 15o behind an impeller blade (obtained by sampling data only when the measuring point is at the speciﬁed position with respect to the impeller blades). Not all vectors have been plotted for clarity. Reproduced with permission from Hartmann et al. (2004a).

Roussinova et al. (2003) and Hartmann et al. (2004b) found that the LES methodology is excellently capable of reproducing various types of macroinstabilities. While Nikiforaki et al. (2002) experimentally found precessing frequencies in the range 0.013–0.018 N, the LES of Hartmann et al. (2004b) arrived at 0.0228 and 0.0255 N as the dominant frequencies, where N stands for the impeller speed (in number of revolutions/s). The discrepancy between experimental and numerical frequencies is a challenge for further improving particularly the SGS-model and some numerical settings.

VI. Operations and Processes in Stirred Vessels In simulating physical operations carried out in stirred vessels, generally one has the choice between a Lagrangian approach and a Eulerian description. While the former approach is based on tracking the paths of many individual ﬂuid elements or dispersed-phase particles, the latter exploits the continuum concept. The two approaches offer different vistas on the operations and require different computational capabilities. Which of the two approaches is most

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suited is hard to say: it depends on the details of the issue of interest and on the computational power available. In whichever approach, the common denominator of most operations in stirred vessels is the common notion that the rate e of dissipation of turbulent kinetic energy is a reliable measure for the effect of the turbulent-ﬂow characteristics on the operations of interest such as carrying out chemical reactions, suspending solids, or dispersing bubbles. As this e may be conceived as a concentration of a passive tracer, i.e., in terms of W/kg rather than of m2/s3, the spatial variations in e may be calculated by means of a usual transport equation. In the context of the RANS-methodology, this e is also required for solving the spatial distributions of the velocity components, while in LES e just serves the purpose of providing a basis for modeling the operation(s) of interest. Even when the number of grid cells in a LB LES simulation of a stirred vessel 1.1 m3 in size amounts to some 36 106 grid cells, this implies a cell size, or grid spacing, of 5 mm only. Even a cell size of just a few millimeters makes clear that substantial parts of the transport of heat and species as well as all chemical reactions take place at scales smaller than those resolved by the ﬂow simulation. In other words: concentrations of species and temperature still vary and ﬂuctuate within a cell size. The description of chemical reactions and the transport of heat and species therefore ask for subtle approaches to these SGS ﬂuctuations.

A. MIXING

AND

BLENDING

One of the most crucial (design) parameters in blending two miscible liquids or distributing a particular miscible addition over a heel of liquid is the so-called mixing time, i.e., the time needed to achieve complete homogenization or a predetermined degree of homogeneity (see, e.g., Grenville and Nienow, 2004). In this review, the focus is on blending operations carried out with low viscosity ﬂuids in the turbulent regime. Kramers et al. (1953) were among the ﬁrst to study mixing times as a function of bafﬂe position and impeller rotational speed. Results of several experimental studies have been combined into empirical correlations relevant to industrial applications (Procha´zka and Landau, 1961; Hoogendoorn and Den Hartog, 1967; Sano and Usui, 1985; Grenville, 1992; Ruszkowski, 1994; Nienow, 1997). Grenville and Nienow (2004) present a concise review as to such correlations. Bouwmans (1992; see also Bouwmans et al., 1997) used a particle tracking technique in a RANS ﬂow ﬁeld to estimate trajectories of neutral and buoyant additions, to construct Poincare´ sections of additions crossing speciﬁc horizontal cross-sectional planes, to predict probabilities of surfacing for buoyant additions, and to mimic the temporal response of conductivity probes. Exploiting the Eulerian point of view, Ranade et al. (1991) was one of the ﬁrst to simulate mixing times for a vessel provided with a pitched blade turbine; in

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this early work, the impeller swept area was still modeled as a black box just delivering inlet conditions for the turbulent ﬂow in the remainder of the vessel. Various authors (Osman and Varley, 1999; Jaworski et al., 2000; Bujalski et al., 2002) used SM or MFR techniques in RANS-based simulations for Rushton turbines driven ﬂow combined with species transport equations to predict mixing times, but arrived at values 2–3 times higher than the experimental values. This is in line with the ﬁndings for Rushton turbines that in such RANS-based simulations turbulence levels are underpredicted and that mixing across the central plane of the discharge plane is poorly reproduced. Montante and Magelli (2004) studied a homogenization process in a bafﬂed vessel stirred with various sets of Rushton turbines; while effects of varying impeller number and spacing were correctly forecasted, a very low turbulent Schmidt number had to be adopted for obtaining good quantitative agreement in terms of tracer response curves. RANS-based simulations of mixing in tanks provided with pitched blade turbines prove to underpredict mixing times, probably owing to mesoscale concentration ﬂuctuations not really reproduced by the simulation. All these ﬁndings of disappointing quantitative agreement with experimental data stem from the inherent drawback of the RANS-approach that there is no clear distinction between the turbulent ﬂuctuations modeled by the Reynolds stresses and (mesoscale) ﬂuctuations. In LES, however, the distinction between resolved and unresolved turbulence is clear and relates to the cell size of the computational grid chosen. The LES methodology has recently been applied in a number of studies simulating in a stirred tank the mixing in response to the addition of a tracer. Yeoh et al. (2004a) carried out an FV-simulation that matched the experimental set-up of Lee (1995); the focus of their study is on mixing patterns, on traces of concentrations at certain monitoring points, and on a comparison of predicted mixing time with correlations from literature. Hartmann (2005) and Hartmann et al. (2006) coupled an FV solver for the species transport to an LB ﬂow solver. In his study, different from Yeoh et al. (2005), the passive scalar is injected at zero speed to avoid an effect of jet mixing on mixing times. Hartmann aimed at reproducing the experiments of Distelhoff et al. (1997) and focused on the effect of impeller size and ‘injection’ position on mixing time. A sample of his results is presented in Fig. 6. For the time being, the results due to Hartmann still suffer from an unphysical mass increase caused by the current implementation of his novel immersed boundary method using ghost cells. Comparative studies on simulating mixing times by means of the traditional RANS approach and the more sophisticated LES are due to Gao and Min (2006), Gao et al. (2006), and Jahoda et al. (2006). They all show that RANS-based simulations fail in reproducing the transient responses of probes monitoring the local tracer concentrations, while LES is able to mimic the experimental traces quite accurately (see Fig. 7, from Jahoda et al., 2006). The latter traces strongly resemble those presented by Hartmann (2005) and Hartmann et al. (2006).

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FIG. 6. A sample of Hartmann’s results (2005). The tracer is injected (with zero speed) at the top of the tank in a plane midway between two bafﬂes (black dot in the plots). The concentration ﬁelds shown are also in the midway bafﬂe plane. T/D ¼ 3, cN denotes the ﬁnal uniform concentration.

FIG. 7. Time traces of normalized concentration as ‘seen’ by a probe in the lower part of a vessel in simulations of a mixing time experiment. The vessel is provided with two Pitched Blade Turbines. Three different types of simulations are shown, where ‘ske’ stands for a standard k–e simulation and ‘sm’, ‘mrf,’ and ‘les’ have the usual meaning. Reproduced with permission from Jahoda et al. (2006).

B. SUSPENDING SOLIDS An important class of stirred tank applications involves suspending discrete solid particles in a continuous liquid phase where the turbulent-liquid-ﬂow

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generated by the impeller induces particle motion and should prevent sedimentation. Such suspensions are for instance encountered in industrial crystallization and in catalytic slurry reactors. The usual procedure in designing such processes is making use of the Zwietering correlation (1956). Note that in the NAMF Handbook of Mixing, Chapter 10 on Solid-Liquid Mixing does not refer to any CFD results. Although the Zwietering correlation provides valuable guidance to chemical engineers in, e.g., solving practical engineering problems and dealing with scaling-up issues, at a more fundamental level there are many unresolved issues such as: What is actually going on at the particle scale (in terms of, e.g., heat and mass

transfer, or mechanical load on particles as a result of particle–particle and particle–impeller collisions) and how are these microscale events affected by the larger-scale phenomena? On the reverse, how does the presence of particles affect local and global ﬂow features in the vessel such as the vortex structure in the vicinity of the impeller, power consumption, circulation and mixing times, and the spatial distribution of turbulence quantities; more speciﬁcally: colliding particles have an impact on the liquid’s turbulence (Ten Cate et al., 2004) while local particle concentrations affect the effective (slurry) viscosity which may be useful in the macroﬂow simulations? Most of these issues may best be studied by DNS, while other can better be tackled by LES. Anyhow, RANS-based simulations are not very suited to this purpose as the turbulence in the RANS-approach is not resolved at all but just modeled. Below, typical DNS-, LES-, and RANS-based simulations of solids suspensions will be reviewed in succession. 1. A Direct Numerical Simulations Approach In view of secondary nucleation in crystallizers, Ten Cate et al. (2004) were interested in ﬁnding out locally about the frequencies of particle collisions in a suspension under the action of the turbulence of the liquid. To this end, they performed a DNS of a particle suspension in a periodic box subject to forced turbulent-ﬂow conditions. In their DNS, the ﬂow ﬁeld around and between the interacting and colliding particles is fully resolved, while the particles are allowed to rotate in response to the surrounding turbulent-ﬂow ﬁeld. Ten Cate et al. (2004) were able to learn from their DNS about the mutual effect of microscale (particle scale) events and phenomena at the macroscale: the particle collisions are brought about by the turbulence, and the particles affect the turbulence. Energy spectra conﬁrmed that the particles generate ﬂuid motion at length scales of the order of the particle size. This results in a strong increase in the rate of energy dissipation at these length scales and in a decrease

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of the turbulence at larger length scales. All these details were obtained just by restricting the DNS to a small periodic box. Details of their approach will be dealt with further on; a typical result is Fig. 12. 2. Results of Large Eddy Simulations Derksen (2003), on the contrary, was interested in simulating the process of solids suspension in a stirred tank; to this end, he tracked particles in the whole tank in a Lagrangian sense, considering the particles as point particles and not resolving the detailed ﬂow ﬁeld between the particles. In other words, Derksen applied a more superﬁcial view on the particle suspension by dropping details, and was rewarded with a picture of the full tank: see Fig. 8. Yet, Derksen was able to track just over 6.7 million particles, to include effects such as particle rotation, particle–particle collisions, particle–impeller collisions and even two-way coupling, and to include ﬂuid–particle interaction forces such as the Saffman force, the Magnus force, and forces due to stress gradients. Tracking all these particles was done in a turbulent-ﬂow ﬁeld obtained via an LB LES. Derksen (2006a) continued along this line of approach and—by means of a clever strategy—mimicked the long-time behavior of solids suspension in an unbafﬂed tall stirred tank equipped with four hydrofoil impellers (Lightnins A310). The time span covered by his LES amounted to some 20,000 impeller 5 revolutions (some 20 min). Running a LES for a Reynolds number of 1.6 10 over the entire time span is not an option, and for that reason a particular ﬂow

FIG. 8. This is a snapshot of a spatial particle distribution. The plane shown is the horizontal cross-section just below the disc of a Rushton turbine in a ﬂat-bottomed stirred tank. The impeller revolves in the counter clockwise direction. Particle size is some 0.468 mm; Re ¼ 1.5? 105; volume fraction amounts to 3.6%; number of particles tracked in the simulation just over 6.7 million. Reproduced with permission from Derksen (2003).

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time series of sufﬁcient length is stored and repeatedly played for tracking the paths of 20,000 particles 0.33 mm in diameter and with a density of 2,450 kg/m3. The advantage of this approach is that use is made of the highly resolved ﬂow information of the LES. The simulations of the particle response to the typical ﬂow ﬁeld created by the four impellers showed the counterintuitive behavior observed by Pinelli et al. (2001): almost all solid particles rise to the top of the tank. 3. Results of Reynolds Averaged Navier– Stokes Simulations A very different type of CFD-simulation results are those due to, e.g., Montante and Magelli (2005) who studied solids suspension in stirred tanks by means of two-ﬂuid CFD simulations, i.e., a RANS-type of Euler–Euler simulations. These authors tested two commercial CFD codes, viz. FLUENT 6.0 and CFX4, for various formulations of the two-ﬂuid model, of the ﬂuid–particle interaction force, and of the k–e turbulence model for multiphase ﬂow. Montante and Magelli just focused on predicting axial proﬁles of the solids concentration for three bafﬂed stirred tanks agitated with single and multiple impellers; they evaluated the effect of the various model formulations on these proﬁles and compared their various predicted curves with experimental data. This type of simulation delivers data relevant for engineering purposes of limited scope and depth only. In this context, one component of the phase interaction force may need separate discussion: the drag force. While most authors use the Schiller–Nauman equation CD ¼

24 1:0 þ 0:15Re0:687 p Rep

(16)

for the relation between particle drag coefﬁcient and the particle Reynolds number (see, e.g., Ranade, 2002, or Derksen, 2003). Brucato et al. (1998), however, reported experimental data showing that free stream turbulence may signiﬁcantly increase particle drag coefﬁcients. They proposed a novel correlation for predicting the effect of free stream turbulence on the particle drag coefﬁcient: 3 C Dt C D 4 d p ¼ 8:76 10 CD ZK

(17)

in which C D t denotes the particle drag coefﬁcient in a turbulent ﬂow and CD stands for the usual particle drag coefﬁcient as given by Eq. (16). Equation (17) implies that the drag coefﬁcient for particles the size of which is smaller than or comparable with that of the smallest turbulent eddies (represented by the local Kolmogorov length scale ZK) is hardly affected by free-stream turbulence, while the ﬂuid-mechanical interaction of larger particles with turbulent eddies becomes signiﬁcant and leads to an increase of particle drag.

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Montante et al. (2000), Micale et al. (2000), and Montante and Magelli (2005) incorporated this novel correlation into their two-ﬂuid simulations of solids suspension in stirred vessels and arrived at pretty good agreement with experimental data. These authors claim that this correlation due to Brucato et al. (1998) plays an essential role in arriving at this good agreement: introducing the turbulence effect reduces the tendency of the larger particles to collect in the bottom part of the vessel. One may argue, however, that the introduction of a higher particle drag coefﬁcient is rather artiﬁcial and just compensates for shortcomings in the two-ﬂuid formulation used, such as ignoring all other components of the phase interaction force. By the way, Micale et al. (2004) in simulating particle suspension height ignored this effect of free stream turbulence and identiﬁed it as a second-order effect only. In some way, introducing an increased particle drag by means of Eq. (17) resembles the earlier proposal raised by Bakker and Van den Akker (1994b) to increase viscosity in the particle Reynolds number due to turbulence (in agreement with the very old conclusion due to Boussinesq, see Frisch, 1995) with the view of increasing the particle drag coefﬁcient and eventually the bubble holdup in the vessel. Lane et al. (2000) compared the two approaches for an aerated stirred vessel and found neither proposal to yield a correct spatial gas distribution. In addition, it is dubious whether this new correlation due to Brucato et al. (1998) should be used in any Euler–Lagrangian approach and in LES which take at least part of the effect of the turbulence on the particle motion into account in a different way. So far, the LES due to Derksen (2003, 2006a) did not need a modiﬁed particle drag coefﬁcient to attain agreement with experimental data. Anyhow, the need of modifying particle drag coefﬁcient in some way illustrates the shortcomings of the current RANS-based two-ﬂuid approach of two-phase ﬂow in stirred vessels. The present author wonders whether, due to its very nature (particularly, the various assumptions as to averaging and the various modeling uncertainties), a RANS-based two-ﬂuid approach is suited for reproducing the details of solids suspension in a stirred vessel. May be we should be satisﬁed with the gross predictions of the current RANS methods and turn to LES for the details of those processes which are dominated by the spatially distributed turbulence. It is really a valid question how long we should keep trying and improving the RANS two-ﬂuid approach as now the increased computer power brings the much more sophisticated LES within reach.

C. DISSOLVING SOLIDS Properly simulating a dissolution process of solid particles in a stirred vessel operated in the turbulent-ﬂow regime urges for a very detailed simulation of the turbulent-ﬂow ﬁeld itself. Just reproducing the overall ﬂow pattern by means of

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a RANS-type of simulation is not sufﬁcient! Our estimate is that a two-ﬂuid simulation (on the basis of RANS) yields too rough an approximation of the process. The dissolution process may too strongly depend on the heavily ﬂuctuating ﬂow and concentration ﬁelds around particles which—as the result of the presence and action of eddies of various scales—may move chaotically with respect to one another while being advected through the vessel. One really may need an inherently transient LES to capture all these details. The ﬁner the grid for such a LES, the more reliably the local transient conditions may be taken into account in reproducing this turbulent mass transfer process (while ignoring the issue of supplying the heat for the dissolution which may also depend on a proper representation of the turbulent-ﬂow ﬁeld). An additional important issue is how many particles have to be tracked for a proper representation of the transient spatial distribution of the particles over the vessel. Hartmann et al. (2006) reported very detailed simulation results (see also Hartmann, 2005) (Fig. 9). Their LB simulation was restricted to a lab-scale vessel 10 L in size only for which 2403 lattice cells a bit smaller than 1 mm2 were used; the temporal resolution was 25 ms only. A set of 7 million mono-disperse spherical particles 0.3 mm in size was released in the upper 10% part of the vessel. At the moment of release, the local volume fraction amounted to 10%. The particle properties were those of calcium chloride. The simulation was carried out on 30 parallel processors of an SGI Altrix 3700 system and required for 6 weeks for 100 impeller revolutions. Figure 9 presents some typical results for the spatial distribution of the dissolving particles, their sizes, and their size distribution. Initially, the particles respond to the centrifugal forces in the vortices and collect at the outer regions of the vortices, giving rise to streaky patterns. The dissolution process results in decreasing particle sizes and, hence, in decreasing inertia to the effect that gradually they start behaving like ﬂuid tracers. Overall, the solids and scalar concentrations become more homogeneous in the course of the dissolution process. The simulation even predicts that a plot of the Sauter mean diameter d32 vs. time may exhibit a minimum (not shown here).

D. PRECIPITATION

AND

CRYSTALLIZATION

The counterparts of dissolving particles are the processes of precipitation and crystallization the description and simulation of which involve several additional aspects however. First of all, the interest in commercial operations often relates to the average particle size and the particle size distribution at the completion of the (batch) operation. In precipitation reactors, particle sizes strongly depend on the (variations in the) local concentrations of the reactants, this dependence being quite complicated because of the nonlinear interactions of ﬂuctuations in velocities, reactant concentrations, and temperature.

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FIG. 9. Snapshots of particle sizes and their spatial distribution in a vertical midway bafﬂe plane at two moments in time, along with the pertinent respective overall particle size distributions. The diameter of the particles is enlarged by a factor of 10 for reasons of clarity. Grey colors represent particle size with respect to the original particle size. Reproduced with permission from Hartmann (2005).

The same applies to crystallizers, in which particle sizes and particle number concentrations not only depend on nucleation and growth from supersaturated mother liquid, but are also affected by shear-dominated agglomeration and by secondary nucleation as a result of particle–particle and particle–impeller collisions. Some of the subprocesses involved may be limited to speciﬁc and different parts of the vessel: e.g., nucleation may be restricted to a ﬂame-like region around the outlet of a feed pipe (Van Leeuwen, 1998). In addition, in

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many cases, many more product properties are relevant, such as color, texture, and particle morphology. Gradually, the insight is growing that all these properties are not only affected by the averaged process conditions in reactor or crystallizer, but also by their spatial and temporal variations felt by the particles during their stay in and on their pathways through the vessel (see, e.g., Rielly and Marquis, 2001). The ﬁrst response of modelers was to introduce compartmental modeling (Van Leeuwen, 1998; Bermingham et al., 1998; Ten Cate et al., 2000): the vessel is divided into several compartments, each compartment being considered as (more or less) ideally mixed and described in terms of averaged values of the process parameters such as temperature, concentrations, and rate of turbulent kinetic energy dissipation; these averaged values may vary strongly from compartment to compartment. Although this approach is an improvement over the traditional method of lumping all variations into a single average value, an important problem still is how to decide on the number and size of the compartments. Carrying out RANS-type simulations would be an option for selecting proper compartmentalization on a case-to-case basis. Seckler et al. (1995), Van Leeuwen et al. (1996), and Wei and Garside (1997) were among the ﬁrst to exploit commercial CFD codes (FLUENT and PHOENICS) for simulating precipitation reactors (of a particular simpliﬁed design) by adding to their codes some simple precipitation kinetics, i.e., relations for nucleation and particle growth. Van Leeuwen et al. (1996) included the ﬁrst four moments of the crystal size distribution in their simulations. Van Leeuwen (1998) was the ﬁrst to study precipitation in a stirred vessel; among other things, he explored the option for extending his RANS-type simulations with a routine involving the use of PDFs to account for variations in the species concentrations (see one of the next sections on chemical reactors). He did not arrive at a satisfactory agreement with experimental data. As discussed in several of the above sections, LES is much better suited to represent the spatial and temporal variations in a turbulent-ﬂow ﬁeld. Of course, this is very relevant in precipitation reactors and crystallizers where the particle formation is a highly nonlinear process. Progress is being made in developing LES including detailed models (e.g., PDFs, or methods of moments) for speciﬁc reactive systems (e.g., combustion, polymerization) under speciﬁc conditions in simple geometries (e.g., tubular reactors): see also one of the next sections on chemical reactors. In turbulent agitated precipitation reactors and crystallizers, however, mixing is so intense and complex and so heavily dominated by the revolving impeller that so far no one has succeeded in simulating the full process of nucleation, particle growth, agglomeration, and particle break-up and in arriving at a reasonable prediction of the eventual particle size distribution. Certain isolated aspects of precipitation and crystallization which have successfully been studied by means of LES, are discussed below to illustrate the progress being made in the ﬁeld of precipitation and crystallization. The LB

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technique exploited in these studies is very helpful, since LB easily allows parallel computing and the related computational acceleration. These detailed studies may make a similar simulation possible as presented above for a dissolution process. 1. Agglomeration Hollander (2002) and Hollander et al. (2001a,b, 2003) studied agglomeration in a stirred vessel by adding a single transport equation for the particle number concentration m0 (actually, the ﬁrst moment of the particle size distribution) @m0 1 þ r ðv~ m0 Þ ¼ b0 m20 2 @t

(18)

Note that the particle diffusion term is ignored, just like particle dispersion due to SGS motions (this was found justiﬁed in a separate simulation). The shape of the sink term in the right-hand term of this equation is due to Von Smoluchowski (1917) while the local value of the agglomeration kernel b0 is assumed to depend on the local 3-D shear rate according to a proposition due to Mumtaz et al. (1997). This additional Eq. (18) was discretized at the same resolution as the ﬂow equations, typical grids comprising 1203 and 1803 nodes. At every time step, the local particle concentration is transported within the resolved ﬂow ﬁeld. Furthermore, the local ﬂow conditions yield an effective 3-D shear rate that can be used for estimating the local agglomeration rate constant b0. Fig. 10 (from Hollander et al., 2003) presents both instantaneous and time-averaged spatial distributions of b0 in vessels agitated by two different impellers; color versions of these plots can be found in Hollander (2002) and in Hollander et al. (2003). The results of these simulations conﬁrm the suspicion of a large spread in b0-values across the vessel. Furthermore, the simulations show that agglomeration does not occur in the impeller region, as the hydrodynamic conditions (shear!) are too severe for agglomerates to survive. A bulk region can be deﬁned in which agglomeration conditions are beneﬁcial. Due to the typical structure of the ﬂow (i.e., trailing vortices, large-scale turbulent motion), the vessel contains large gradients in agglomeration rates. It is this effect that causes large differences in the local particle consumption rate and leaves the vessel with a very uneven distribution in particle concentration. Fig. 11 presents the results of some 30 simulations for various conditions and two impeller types in terms of the mean agglomeration rate constant observed in the various simulations vs. the vessel-averaged shear rate found in the simulations. The simulations all started from the dotted curve for relating local agglomeration rate constant to local shear rate. A clear decrease in the maximum of b0 as well as a shift toward higher average shear rates was found which are caused by the local nature of the nonlinear ﬂow interactions only. These

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FIG. 10. Results of LES-based simulations of an agglomeration process in two vessels: one agitated by a Rushton turbine (left) and one agitated by a Pitched Blade Turbine (right). The two plots show the agglomeration rate constant b0 normalized by the maximum value, in a vertical crosssectional plane midway between two bafﬂes and through the center of the vessel. Each of the two plots consists of two parts: the right-hand parts present instantaneous snapshots; the left-hand parts present spatial distributions of time-averaged values after 50 impeller revolutions. Reproduced with permission from Hollander et al. (2003).

FIG. 11. The discrepancy between the original kinetic relation due to Mumtaz et al. (1997) and the observed relation between mean agglomeration rate constant b0 and volume-averaged shear rate. Symbols refer to individual numerical simulations (LES). RT stands for Rushton Turbine, PBT for Pitched Blade Turbine. Reproduced with permission from Hollander et al. (2001b).

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nonlinear effects may play an important role in our mediocre understanding of the scale-up behavior of agglomeration processes in stirred vessels. The computational demands for Hollander’s simulations (in 2000) were typically of the order of 1 CPU week on a PentiumIII/500 MHz at 400 MB of memory per run. The computer code was fully parallelized and was run on a 12 CPU Beowulf cluster.

2. Colliding Particles Finally, Ten Cate et al. (2001) studied secondary nucleation as a result of crystal–crystal collisions by means of a two-step approach. The ﬁrst step involved a LES of the complete crystallizer in which the liquid phase (typically containing 10–20%v of solids) is treated as a single phase with a homogeneous density and viscosity. (In principle, density and viscosity may be allowed to vary locally in the simulation, by invoking the help of a particle dispersion routine (e.g., Liu, 1999) and coupling the resulting particle concentrations back to a SGS model to modify the local values of density and viscosity.) The spatial grid resolution (some 5.0 mm) used by Ten Cate typically was at least 10 times the crystal size; the grid comprised some 36 million cells. The second step in Ten Cate’s two-step approach was to focus on crystal– crystal interaction by means of an explicit two-phase DNS of the turbulent suspension that completely resolves the translational and rotational motions and collisions of the spherical particles plus the turbulence of the liquid between the particles. The particle motions are driven by the turbulent ﬂow and the particles affect the turbulent ﬂow of the liquid in between. When particles approach one another down to a distance smaller than the grid spacing, lubrication theory is exploited to bridge the gap between them. At the start of a simulation, the particles (up to 3,900) are placed randomly, without mutual contact and with zero velocity, into a fully developed turbulent single-phase ﬂow ﬁeld in a periodic box. This starting situation is subjected to a precalculated force ﬁeld; this forcing involves a divergence-free white-noise signal, distributed over the wave number domain as a Gaussian distribution about a desired wave number, with a characteristic root mean square velocity and a characteristic length scale. In this way, turbulent conditions are generated which accurately recover a priori set values such as the Kolmogorov length scale, the integral length scale, and the integral time scale, derived from the LES at some position somewhere in the crystallizer. For details, the reader is referred to Ten Cate (2002) and Ten Cate et al. (2004). The evolution of the two-phase turbulence depends on the initial random position of the particles, the motion of which modiﬁes the turbulent-ﬂow ﬁeld directly. These DNS are therefore a nice example of two-way coupling between the two phases: see Fig. 12. From these DNS, detailed knowledge can be derived as to the frequency of the particle–particle collisions and the forces involved

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1.2·101

0.4

1.2·10-2

FIG. 12. Snapshot from a two-phase DNS of colliding particles in an originally fully developed turbulent ﬂow of liquid in a periodic 3-D box with spectral forcing of the turbulence. The particles (in blue) have been plotted at their position and are intersected by the plane of view. The arrows denote the instantaneous ﬂow ﬁeld, the colors relate to the logarithmic value of the nondimensional rate of energy dissipation.

(as a function of local conditions in a crystallizer) which may yield quantitative data on secondary nucleation (fragmentation of crystals due to collisions).

VII. Stirred Gas–Liquid and Liquid–Liquid Dispersions Similar problems as encountered in simulating agitated suspensions also play a role in agitated gas–liquid and liquid–liquid systems. Again, there is the dilemma whether to use a Lagrangian approach or to apply a two-ﬂuid, Euler–Euler, model. Then, there is again the question of one-way coupling vs. two-way coupling. Thirdly, the issue of taking the particle size distribution into account (yes or no) should be addressed. Further, particle–particle interaction may not be ignored in many cases, and then the additional problem is coalescence of bubbles or droplets, and their break-up. All these issues have not been tackled simultaneously in the past, not in the least because of computational limitations. Furthermore, the physics of the interaction between turbulence and bubbles in the complex ﬂow of a stirred vessel, with its implications for coalescence and break-up of bubbles and drops, is still far from being understood. Up to now, simple correlations are available for scale-up of industrial processes; generally, these correlations have been derived in experimental investigations focusing on the eventual mean drop diameter and the drop size distributions as brought

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about by the power input (mainly) via the impeller given the ﬂuid properties (e.g., Colenbrander, 2000). In the 1980s, Issa and Gosman (1981), Pericleous and Patel (1987), and Tra¨ga˚rdh (1988) made the ﬁrst attempts to simulate aerated stirred vessels computationally. Their results were rather approximate indeed, and were not validated by means of experimental data. A. BAKKER’S GHOST! CODE In the early 1990s, Bakker and Van den Akker (1991, 1994) introduced an approximate but effective Euler–Euler approach (see also A. Bakker’s PhD Thesis, 1992): on the basis of a single-phase RANS ﬂow ﬁeld calculated by FLUENT, a code named GHOST! calculated local and averaged values of bubble size db, gas hold-up a, and speciﬁc mass transfer rate kla. Their GHOST!-code essentially consisted of a mass balance for the gas, a transport equation for the bubble number density nb, and a force balance for a single bubble, respectively, which run as r ða¯ u¯ Þ r ðrðD¯aÞÞ ¼ S¯ g

(19)

S¯ g @nb þ r ðnb uÞ ¼ oðnb1 nb Þ þ @t V b;in

(20)

rl V b

w2 1 p r^ rl gV b z^ ¼ C D rl jus jus d 2b 2 4 r

(21)

respectively. Of course, the variables nb, db, and a are interrelated. The second term in the mass balance, Eq. (19) stands for the extra transport of the bubbles due to the larger turbulent eddies; usually, this turbulent transport is taken into account by a separate term in the momentum balance for the dispersed phase which, however, in Bakker’s approach is replaced by the simple force balance, Eq. (21), for a single bubble. The coefﬁcient o in the transport equation for nb is just a relaxation parameter that stands for the rate at which nb responds to the local turbulence, i.e., adapts—either by coalescence (in the bulk of the stirred vessel) or by bubble break-up (in the impeller swept domain of the vessel)—to the local equilibrium number density nbN The latter variable corresponds to the local maximum stable bubble size dbN which—according to Hinze (1955)— depends on the local value of the speciﬁc rate of energy dissipation e: s 3=5 2=5 d b1 ¼ C b1 12 rl

(22)

Bakker increased the local values of e as obtained with FLUENT by a contribution related to the slip velocity of the bubbles. Working with a single

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relaxation parameter o is a phenomenological description avoiding detailed relations for coalescence and bubble break-up. The Eulerian gas velocity ﬁeld required in both the mass balance and the above transport equation for nb is found by an approximate method: ﬁrst, the complete ﬁeld of liquid velocities obtained with FLUENT is adapted downward because the power draw is smaller under gassed conditions; next, in a very simple way of one-way coupling, the bubble velocity calculated from the above force balance is just added to this adapted liquid velocity ﬁeld. This procedure makes a momentum balance for the bubble phase redundant; this saves a lot of computational effort. Finally, Bakker and Van den Akker calculated local values for the speciﬁc mass transfer rate kla, by estimating local kl-values from local values of the Kolmogorov time scale O(n/e) and by deriving local values of the speciﬁc interfacial area a from local values for bubble size and bubble hold-up. In spite of all the simpliﬁcations Bakker and Van den Akker applied and given the black box approach for the impeller swept domain, their simulations resulted in values for the bubble size just below the liquid surface, overall holdup, and average kla values which are in good agreement with their experimental data (see Table II). The major step forward they made was the acquisition of the different spatial distributions of average bubble size (see Fig. 13), bubble holdup and kla as effected by three common impeller types. As a matter of fact, their approach may be restricted to low values of the gas hold-up. B. VENNEKER’S DAWN CODE Venneker et al. (2002) extended the GHOST! approach due to Bakker and Van den Akker (1994b) by replacing the single transport equation for the TABLE II COMPUTATIONAL AND EXPERIMENTAL RESULTS FOR OVERALL HOLD-UP A, SIZE /DB,OUTS OF THE BUBBLES LEAVING THE LIQUID HEEL, AND OVERALL SPECIFIC MASS TRANSFER RATE KLA (FROM: BAKKER, 1992) S. no.

1 2 3 4 5

/db,outS (mm)

a (%)

DT A315 A315 PBT PBT

Exp.

Sim

4.770.2 4.670.2 4.870.2 4.170.3 1.170.3

4.9 4.2 4.3 4.1 1.0

Exp 3.25 3.76 3.44

Sim 2.91 3.59 3.82 3.39 2.00

kla (l/s) Exp

Sim

0.038 0.035 0.038 0.036 0.011

0.038 0.036 0.036 0.037 0.013

Note: The respective impellers used are a classical Rushton turbine (DT), a hydrofoil impeller (A315) manufactured by Lightnin, and a Pitched Blade impeller (PBT). The cases 1 through 4 all relate to a superﬁcial gas rate of 3.6 mm/s only, with impeller speeds varying between 5 and 10/s (gas ﬂow numbers between 0.01 and 0.02); cases 2 and 3 differ in sparger size and position.

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FIG. 13. Spatial distributions of bubble size in three vessels agitated by different impellers: a classical Rushton turbine (DT), a hydrofoil impeller (A315) manufactured by Lightnin, and a Pitched Blade Impeller (PBT). The gas ﬂow numbers in these simulations are in the range 0.01–0.02. These simulation results have been obtained by using GHOST! Reproduced with permission from Bakker (1992).

bubble number density including the effective relaxation parameter o by a number of population balance equations. Actually, these population balance equations are convective-diffusive transport equations for the bubble number density in a speciﬁc bubble size class and include separate birth and death terms which take coalescence and break-up into account. In addition, the procedure for adapting the liquid ﬂow ﬁeld to the lower power draw under gassed conditions has been improved on the basis of experimental ﬁndings due to Rousˇ ar and Van den Akker (1994). For the rest, the same approximations are made as in GHOST! with respect to gas velocities, rate of energy dissipation, and speciﬁc mass transfer rate. Venneker et al. (2002) used as many as 20 bubble size classes in the bubble size range from 0.25 to some 20 mm. Just like GHOST!, their in-house code named DAWN builds upon a liquid-only velocity ﬁeld obtained with FLUENT, now with an anisotropic Reynolds Stress Model (RSM) for the turbulent momentum transport. To allow for the drastic increase in computational burden associated with using 20 population balance equations, the 3-D FLUENT ﬂow ﬁeld is averaged azimuthally into a 2-D ﬂow ﬁeld (Venneker, 1999, used a less elegant simpliﬁcation!) The agreement between simulation results and experimental data is encouraging (see Fig. 14), although the simulation gives higher hold-up values in the upper part of the vessel while the overall hold-up is lower in the simulation than

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207

alpha (%) 10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0

FIG. 14. Spatial distribution of the gas hold-up in a turbulent stirred vessel ﬁlled with a 0.075% Keltrol solution in water and agitated by a Rushton turbine: (a) experimental data obtained by means of an optical probe; (b) computational result from DAWN. Overall hold-up amounts to some 3.1% in the simulation and 3.7% in the experiment. Reproduced with permission from Venneker (1999), improved.

in the experiment. The latter discrepancy may be due to the use of optical probes overlooking bubbles smaller than, say, 1 mm.Venneker et al. (2002) present some more comparisons between computational and experimental results. One should realize, however, that for validation purposes hardly any detailed experimental data as to bubble size distributions in stirred vessels are available. This same shortage of experimental data hampers the assessment of the so-called MUltiple-SIze Group approach MUSIG due to Lo (2000) as incorporated in the commercial CFD code CFX.

C. FURTHER SIMULATIONS In comparison with Bakker and Van den Akker (1994b) and Venneker et al. (2002), Khopkar et al. (2005) applied a more sophisticated two-ﬂuid approach including a standard k–e turbulence model. Using the incorrect snapshot approach due to Ranade (2002), their simulation results (for gas ﬂow numbers being 4 times higher than those of Bakker and Van den Akker, 1994b) still exhibit major discrepancies with respect to experimental data. One of the

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striking features is that their liquid velocities even in the outﬂow of the Rushton impeller are pretty much overestimated in the simulation, may be due to the use of the snapshot approach (cf. the dicussion on page 180). The spatial distributions of gas hold-up found in the simulations are compared with experimental data obtained by means of computed tomography (CT); there is substantial room for improvement (see Fig. 15) in spite of the much more sophisticated type of simulation. The paper due to Gentric et al. (2005) exploits several features and options of the two-ﬂuid mode of the commercial code STAR-CD and illustrates its capabilities in comparing the mixing performance of two industrial mixing vessels, but does not present validation by means of experimental data. The combination of two-phase ﬂow, turbulence, and a revolving impeller poses tremendous simulation problems and still requires excessive computer time or power. While in bubble columns and gas lift loops population balances are used with some success (see, e.g., Wang et al., 2006), the ﬂow ﬁeld in a stirred tank is so much more complicated and turbulent and dominated by the revolving impeller that implementing them in stirred vessel simulations still causes serious convergence problems. Laakkonen et al. (2006) reduced such problems by restricting his simulation to a multiblock approach subdividing the stirred vessel into just 23 ideally mixed subregions: this approach actually is a kind of network of zone approach (see also, e.g., Hristov et al., 2004, who even used 36,000 zones) extended with population balances and may not be named CFD indeed. The use of the so-called Quadrature Method of Moments (QMOM) has been suggested as a proper tool for combining population balances with CFD (Marchisio et al., 2003), but so far has not been used for simulating gassed stirred tanks.

FIG. 15. Comparison of simulated and experimental gas hold-up distribution in a horizontal plane 10 cm above the bottom of the vessel. The gas ﬂow number amounts to 0.084. Reproduced with permission from Khopkar et al. (2005).

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D. A PROMISING PROSPECT Just like in the context of simulating solids suspension, one may wonder whether much may be expected from just sticking to the two-ﬂuid approach combined with population balances. A better way ahead might rather be to combine population balances with LES, while proper relations for the various kernels used for describing coalescence and break-up processes could be determined from DNS of periodic boxes comprising a certain number of bubbles (or drops). The latter simulations would serve to study the detailed response of bubbles or drops to the ambient turbulent ﬂow. An attractive framework for investigating these phenomena is provided by Derksen (2006b) who carried out DNS of liquid–liquid dispersions in a 3-D periodic box. Derksen investigated the response of a turbulent dispersion of droplets to a history comprising ﬁrst a rapidly increasing turbulent activity, then a quasi-steady situation of high turbulence intensity and ﬁnally a rapid decay in turbulence intensity; this history may be equivalent to what a ﬂuid package experiences during its passage through the impeller stream. Again, Derksen applied a particular LB method for mimicking the two phases. The drop size distribution and the Sauter mean diameter were tracked in time. Furthermore, the presence of the droplets affected the turbulence spectrum because of the small-scale ﬂuid motions induced by the droplets, on the analogy of the interaction between solid particles and turbulence in the work of Portela and Oliemans (2003) and Ten Cate et al. (2004).

VIII. Chemical Reactors So far, most (stirred) chemical reactors have been designed and scaled up by traditional methods exploiting simple concepts—such as ‘continuous stirred tank reactor’ and residence time distribution—and scale-up rules involving usually a single dimensionless number such as the (mixing) Damko¨hler number being the ratio of the turbulent macrotime scale to the characteristic reaction time scale. In addition, various types of local mixing times and their ratios are used to characterize or categorize the interaction of mixing and chemical reactions at scale-up (Patterson et al., 2004). These methods hardly take spatial distributions of velocity ﬁeld and chemical species or transient phenomena into account, although most chemical reactors are operated in the turbulent regime and/or a multiphase ﬂow mode. As a result, yield and selectivity of commercial chemical reactors often deviate from the values at their laboratory or pilot-scale prototypes. Scale-up of many chemical reactors, in particular the multiphase types, is still surrounded by a fame of mystery indeed. Another problem relates to the occurrence of thermal runaways due to hot spots as a result of poor local mixing effects.

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Patterson (1985) presented a concise review of the early developments in computational modeling of second-order chemical reactions and of more complicated and multiple reaction sets which are affected by an intermediate rate of local turbulent mixing. At that moment in time, closing the cross-correlation terms stemming from the turbulent ﬂuctuations by means of micromixing models was still in its infancy. He also just hinted on the use of PDFs. Furthermore, the limited computer power of those days kept detailed simulations and their assessment impossible. Stirred vessels in particular were too difﬁcult a type of ﬂow devices to allow for application of rigorous CFD techniques, although some attempts were made with a very small number of zones or mixing segments only.

A. MECHANISTIC MICROMIXING MODELS In the 1980s, Bourne along with a long series of co-workers at ETH Zurich developed a mechanistic micromixing approach in which lamellar structures were central. His lamellar structures represent the small ﬂow structures of the size of the Kolmogorov length scale within which molecular diffusion is the mechanism bringing the chemical species into the intimate contact required for a chemical reaction. The best reference might be two papers due to Baldyga and Bourne (1984a, b). Such lamellar structures have also been described and modeled by Ranz (1979) and Ottino (1980) in the context of chemical reactions in laminar ﬂows. In Bourne’s micromixing models for chemical reactors operated in the turbulentﬂow regime, various assumptions are raised as to the engulfment, the deformation, and the lifetime of these lamellar structures which, along with the diffusion of the reacting species, all affect the yield of the chemical reactions taking place within these structures. Actually, a CDR equation—see Eq. (13)— is solved explicitly for a chemical species within a single Kolmogorov eddy. The most appealing models Baldyga and Bourne proposed for the evolution of such eddies are the Strain (St) Model, the Shear (Sh) Model, the EngulfmentDeformation-Diffusion (EDD) model, and then the simpler Engulfment (E) model. Bourne applied his technique to various sets of competing parallel or consecutive model reactions each carried out in a fed batch reactor. In the 1990s, Bakker and Van den Akker (1994, 1996)—see also R.A. Bakker’s PhD thesis (1996)—continued this mechanistic modeling approach by attempting a completely deterministic description of the 3-D small-scale ﬂow ﬁeld in which the chemical reactions take place at the pace the various species meet. Starting point is a lamellar structure of layers intermittently containing the species involved in the reaction. These authors conceived such small-scale structures as Cylindrical Stretched Vortex (CSV) tubes being strained in the direction of their axis and—as a result—shrinking in size in a plane normal to

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their axis. Such CSV tubes showed up around 1990 in several studies exploiting DNS of turbulent ﬂows in a periodic box. The evolution of a CSV tube during its lifetime can be found by means of an analytical solution of the vorticity equation. For the parameters typical of the turbulent-ﬂow ﬁeld in a stirred reactor, however, the vorticity distribution does not result in substantial winding of material lines during the pertinent short time scales. As a result, the main effect of the stretching of the CSV is just the exponential shrinking rate of more or less ﬂat, only slightly curling material layers. Consequently, Bakker (1996) described the concentration evolution of a single layer subjected to the vorticity ﬁeld of a single CSV by means of a onedimensional differential equation where both the nondimensional time and the nondimensional spatial coordinate contain the exponential shrinking rate. In this respect, the CSV approach differs from the various Bourne models in which the successive generation of several multiple-layer stacks is required and vortex age is a crucial element.

B. A LAGRANGIAN APPROACH In addition, Bakker and Van den Akker (1994, 1996) were the ﬁrst to track the path such structures follow in the turbulent-ﬂow ﬁeld of a fed batch reactor computationally. This is extremely relevant as both vortex age (in Bourne’s multiple-layer models) and Kolmogorov length scale strongly depend on the spatially strongly varying e. Precisely this latter variable exhibits a very inhomogeneous spatial distribution that only can be estimated by means of CFD. The idea is that during the microscale process of mixing and reaction the macroﬂow ﬁeld advects the reaction zone throughout the reactor, thereby exposing the zone to regions of varying e. The ﬂow ﬁeld and the spatial e-distribution were obtained via a RANS-type of simulation (FLUENT), while the tracking was done by means of a Discrete Random Walk approach. (It should be kept in mind that at the time of their simulations LES was not really an option yet!) In addition to their own CSV model, Bakker and Van den Akker also validated some of Bourne’s micromixing models. Some typical results from their simulations are presented in Fig. 16 in which the yield XQ of the product Q from the slow reaction of a set of two competitive reactions in a fed batch reactor has been plotted vs. impeller speed for two micromixing models, viz. their own CSV model and Bourne’s EDD model; their simulation results are compared with experimental data from Bourne and Yu (1991). For the cases shown, the CSV model may perform better than Bourne’s EDD model, in particular when A is fed near to the impeller where mixing is most intense. An alternative but similar approach (Akiti and Armenante, 2004) is to deﬁne the reaction zones (or blobs) as a separate phase distinct for the ambient ﬂuid

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FIG. 16. The yield XQ of the product Q of the slower reaction of a set of two competitive parallel reactions in a fed batch reactor plotted vs. impeller speed (in /s). The experimental data are due to Bourne and Yu (1991); the crosses refer to feeding reactant A at the top of the vessel, while the diamonds refer to feeding more closely to the impeller. The various types of lines refer to simulations as speciﬁed in the legend. Reproduced with permission from R. A. Bakker (1996).

and to track these reaction zones by means of a Volume-of-Fluid (VOF) technique, which may be conceived being a pseudo-multiphase model, originally designed by Hirt and Nichols (1981). Rather than a k–e model, Akiti and Armenante used a RSM model to reproduce the turbulence characteristics of the ﬂow ﬁeld needed for the tracking procedure. The most important drawback of using a Lagrangian approach for simulating (micro) mixing in chemical reactors is that some model is required for describing the formation of the Kolmogorov-scale ﬂow structures at some (which?) distance from the mouth of the feed tube. This so-called feed discretization, aimed at deﬁning the starting conditions (size, number, concentrations) for the lamellar structures, may have an unknown impact on the eventual yield.

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213

C. A EULERIAN PROBABILISTIC APPROACH An alternative approach (e.g., Patterson, 1985; Ranade, 2002) is the Eulerian type of simulation that makes use of a CDR equation—see Eq. (13)—for each of the chemical species involved. While resolution of the turbulent ﬂow down to the Kolmogorov length scale already is far beyond computational capabilities, one certainly has to revert to modeling the species transport in liquid systems in which the Batchelor length scale is smaller than the Kolmogorov length scale by at least one order of magnitude: see Eq. (14). Hence, both in RANS simulations and in LES, species concentrations and temperature still ﬂuctuate within a computational cell. Consequently, the description of chemical reactions and the transport of heat and species in a chemical reactor ask for subtle approaches as to the SGS ﬂuctuations. In order to obtain a realistic estimate for the reaction rate, the joint distribution of the reactants at the smallest turbulent scales is required. Any model disregarding this joint distribution may lead to an erroneous estimate of the reaction rate. For instance, ﬁltering the reaction term, on the analogy of the LES ﬁlter for the ﬂuid ﬂow, would result in an over-prediction of the reaction rate due to the segregation at the subgrid scales. It may be just due to peculiar operating or mixing conditions when, such as in the FLUENT simulations reported on page 845 of Patterson et al. (2004), CFD simulations ignoring SGS ﬂuctuations result in yield predictions close to experimental data. The value of the Damko¨hler number, denoting the ratio of the turbulent macrotime scale to the characteristic reaction time scale, plays an important role as well. During the years, quite some proposals have been raised as to closure equations in the CDR equations for the spatial species distributions. These closure equations relate to the correlation terms in general and to the SGS ﬂuctuations in species concentrations in particular. These proposals substantially differ in degree of sophistication. Patterson et al. (2004) present several examples of closure models which were reasonably successful in reproducing a particular set of experimental data. This, however, does not necessarily say something about their universal applicability. Bakker and Fasano (1993) applied the so-called Magnussen model and arrived at reasonable yield predictions for a competitiveconsecutive reaction system in a stirred reactor (see also Marshall and Bakker, 2004). This Magnussen model, originally derived for combustion, locally calculates several reaction rates as a function of both mean concentrations and turbulence levels and then selects the lower rate for the source term in the CDR equation. Particular attention is to be paid to closure models exploiting various types of PDFs such as beta, presumed, or full PDFs (e.g., Baldyga, 1994; Fox, 1996, 2003; Ranade, 2002). While PDFs have successfully been exploited for describing chemical reactions in turbulent ﬂames, tubular reactors (Baldyga and Henczka, 1997), and a Taylor-Couette reactor (Marchisio and Barresi, 2003), they have never been used successfully in stirred reactors so far.

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D. A PROMISING PROSPECT At the end of this review on chemical reactors, special room is reserved for a very promising approach, although this approach, too, has not yet been applied for simulating a stirred chemical reactor. Van Vliet et al. (2001, 2005) exploited an elegant probabilistic approach (see also Van Vliet, 2003), where the PDF methodology incorporates the joint scalar information by solving the transport equation for the full joint scalar PDF (Pope, 1985). In this way, the second-order and thus nonlinear reaction terms in the CDR equation are kept in closed form, making further modeling of the chemical reaction term redundant. In order to implement the PDF equations into a LES context, a ﬁltered version of the PDF equation is required, usually denoted as ‘ﬁltered density function’ (FDF). Although the LES ﬁltering operation implies that SGS modeling has to be taken into account in order to capture micromixing effects, the reaction term remains closed in the FDF formulation. Van Vliet et al. (2001) showed that the sensitivity to the Damko¨hler number of the yield of competitive parallel reactions in isotropic homogeneous turbulence is qualitatively well predicted by FDF/LES. They applied the method for calculating the selectivity for a set of competing reactions in a tubular reactor at Re ¼ 4,000. Although this LES/FDF methodology is a promising technique, the (current) drawback is the high computational costs involved to obtain a numerical solution of the FDF transport equation. In the above study due to Van Vliet (2003), the LES ﬂuid transport was computed with the help of an LB solver on a 6 5 10 computational grid. Solving a transport equation for the joint PDF of the chemical species is most effectively done in a Lagrangian Monte-Carlo (MC) manner: the chemical composition of the ﬂow as a function of time and space is represented by a collection of ﬁctitious particles that are randomly released in the ﬂow domain and that carry with them the full chemical composition. The assembly of MC particles is tracked through physical and chemical space by a set of stochastic ordinary differential equations, where the random term represents diffusion. These equations need closure as to the way the particles interact with their direct chemical environment, more speciﬁcally for the scalar energy dissipation rate. The model used is the rather common Interaction by Exchange with the Mean (IEM) model in which a mixing frequency describes the mixing at SGS. 8 Van Vliet et al. (2005) tracked 1 10 computational nodes to obtain a stochastic solution of their FDF equations. In order to deal with the high computational costs, the code was run in parallel on a Linux cluster of 11 dual AMD Athlon (TM) MP 1800+processors. In this way, about one ‘turbulent macro time scale’ (or 8,000 computational steps) per 2 days was computed. Van Vliet et al. (2004, 2006) investigated the formation of hot spots and reactor efﬁciency in various geometrical conﬁgurations of a tubular reactor for manufacturing Low-Density Polyethylene (LDPE) by means of the above

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215

FIG. 17. Three different representations (using increasing temperature thresholds) of hot spots in a tubular LDPE reactor as found by the LES/FDF-methodology due to Van Vliet et al. (2004).

LES/FDF-approach. An ‘In situ Adaptive Tabulation’ (ISAT) technique (due to Pope) was used to greatly reduce (by a factor of 5) the CPU time needed to solve the set of stiff differential equations describing the fast LDPE kinetics. Fig. 17 shows some of the results of interest: the occurrence of hot spots in the tubular LDPE reactor provided with some feed pipe through which the initiator (peroxide) is supplied. The 2004-simulations were carried out on 34 CPU’s (3 GHz) with 34 GB shared memory, but still required 34 h per macroﬂow time scale; they served as a demo of the method. The 2006-simulations then demonstrated the impact of installing mixing promoters and of varying the inlet temperature of the initiator added. The above simulations as to the occurrence of hot spots once more illustrate the power and promises of LES over RANS-type simulations. The hot spots can never be found by means of a RANS-type of simulation. The same technique was used by Van Vliet et al. (2006) to study the inﬂuence of the injector geometry and inlet temperature on product quality and process efﬁciency in the LDPE reactor. On the contrary, the RANS-based simulations due to R. A. Bakker and Van den Akker (1994, 1996) were pretty much suited to arrive at yield predictions for a fed batch reactor as a whole. So far, to the best of our knowledge, the above LES/FDF-approach has not been applied to stirred chemical reactors in which the turbulent-ﬂow ﬁeld is far more complex than in a tubular reactor. This LES/FDF-approach, however, may be the way to go, as it provides highly detailed information on turbulent reactive ﬂows with the usage of a minimum of modeling assumptions. Although the high computational demands make LES/FDF simulations currently accessible to academic research groups only, the continued exponential growth of

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computer resources will make them a versatile tool for process and geometry optimization of turbulent reactive ﬂows in the process industries.

IX. Summary and Outlook The above review has shown that 20 years of developing CFD-techniques has yielded us substantial simulation capabilities for studying and predicting mixing under turbulent conditions. The start in the 1980s was slow and with much trial and error. In the early 1990s, we had to be content with RANS-based simulations on—speaking afterwards—coarse grids of limited size only. With increasing computer power and memory becoming available at lower cost, ﬁner and larger grids offered the potential of getting more detailed pictures, among other things via DNS. LES entered the mixing scene, although their proliferation suffered from slow convergence of the FV solvers. In the late 1990s, however, LB solvers entered the mixing ﬁeld and, owing to being faster and better geared to parallellization, made LES much more attractive and viable.

A. THE VARIOUS COMPUTATIONAL FLUID DYNAMICS OPTIONS Nowadays, it is therefore essential to distinguish between the various main CFD options for dealing with turbulent mixing issues, viz. RANS simulations: usually exploiting some k–e turbulence model, intended

for global information on the average ﬂow ﬁeld and the global transport phenomena in full-scale process equipment, with additional output (of limited conﬁdence level) on spatial distributions of k and e; DNS simulations: delivering fully resolved transient ﬁelds of velocities and other variables in either a ﬂow domain of limited size under laminar or very moderately turbulent ﬂow conditions or in a periodic box with some prescribed turbulence level; LES: with some model of the SGS ﬂow and transport phenomena, suited for reproducing—at the level of the grid cell size—rather detailed transient ﬁelds of velocities and other transport variables in full-scale process equipment operated under turbulent-ﬂow conditions. Commercial CFD software has become a reliable tool for carrying out simulations for laminar ﬂows and—based on RANS—for turbulent ﬂows. Practising engineers gradually have become convinced about the usefulness of RANSbased simulations. This review, however, emphasizes that CFD now has much more to offer. For practicing engineers confronted with mixing problems, it is

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important to realize that CFD is not inherently restricted to just the average single-phase ﬂow ﬁeld, but gradually is becoming more and more capable of dealing with the details of turbulent eddies and two-phase ﬂows. The performance of many physical operations and the yield and selectivity of many chemically reacting systems strongly depend on nonlinear interactions at the small scales of turbulent ﬂows. An example: Hollander et al. (2001a) nicely demonstrated how the strong inhomogeneities in stirred-tank ﬂow result in unpredictable scale-up behaviour and that the impact of the detailed hydrodynamics and of the nonuniform spatial particle distribution on agglomeration rate is larger and more complex than usually assumed; their study once more illustrated the risks of scale-up on the basis of keeping a single non-dimensional number. Sophisticated CFD, especially on the basis of LES, offers an attractive alternative indeed. Compared to RANS simulations, DNS and LES are much better geared to reproducing these small-scale processes. RANS simulations focus on the average ﬂow only and by their nature just model the small scales rather rudimentarily. On the contrary, a DNS resolves all ﬂuid motions and a LES resolves most part of the turbulence spectrum, i.e., all eddies larger than the grid cell size. While DNS nowadays can be used for turbulent ﬂows at Reynolds numbers up to say 10,000 in simple geometries (channels, curved tubes) only, LES are quite feasible for complex geometries, certainly when LB techniques are adopted.

B. THE PROMISES SIMULATIONS

OF

DIRECT NUMERICAL SIMULATIONS

AND

LARGE EDDY

At the moment, DNS and LES for turbulent ﬂows are still the playground of academic research groups. These groups are making substantial progress, however, in developing dedicated software for—and building up competence in—simulating multiphase ﬂows, transport phenomena, many types of physical operations, and chemical reactions. Such dedicated software makes it possible to dig into the details of the mechanisms of a variety of ﬂow and transport phenomena—often beyond the current capabilities of experimental techniques. That is why this review paper is an—admittedly provocative—plea for starting the exploitation of the advantages of DNS and LES. An example: rather than linking average bubble size to just or essentially the (overall) power input of a particular vessel-impeller combination, dedicated CFD (preferably DNS and LES) allows for studying (‘tracking’) the response of bubble size to local and spatial variations in the turbulence levels in a stirred vessel. In this way, the validity of certain modeling assumptions may be afﬁrmed or disproved. Particularly, effects of spatial variations in e which

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remain hidden in the traditional engineering techniques, may surface as a result of such dedicated CFD approaches. This type of dedicated CFD simulations offer a better and closer look into the details of ﬂow and transport phenomena than experimental techniques which, e.g., still are not capable of delivering reliable high-resolution e-values. The advantage of LES over RANS-based simulations is that in the former approach modeling the effect of the unresolved scales of the ﬂow is ‘easier’ and more straightforward, just because the SGS eddies are distinctly separated from the vessel boundary conditions and—as a result—their behavior is closer to the ideal of isotropy rendering universal turbulence modeling feasible. This makes the outcome of simulations less sensitive to deﬁciencies in turbulence modeling. This—along with the inherently transient character and the degree of detail of the simulations—turns LES highly suited as a base for simulating physical operations and chemical reactions carried out in stirred vessels. Whenever the performance of these processes is strongly dependent on turbulent mixing, the degree to which CFD simulations can be trusted depends on the ability to reproduce the complicated nonlinear interactions of ﬂow and transport phenomena across the various turbulence scales. LES is then the CFD option to be recommended. The present author even wonders whether we should not be satisﬁed with the gross predictions of the current RANS methods and turn to LES for the details of those single-phase and multiphase mixing processes which are dominated by the spatially distributed turbulence. It is really a valid question how long we should keep trying and improving the various RANS methods now the increased computer power brings the much more sophisticated LES within reach. The very nature of the RANS approach itself—particularly the basic assumptions as to averaging and the various modeling uncertainties as to turbulence and multiphase ﬂow—may really set limits to its exploitation. The modest demands on computer resources RANS-based simulations require these days are no excuse in this respect. In addition, DNS of turbulent ﬂow in a periodic box offer interesting opportunities for studying in a fully resolved mode the intimate details of the ﬂow ﬁeld, its interaction with particles and the mutual interaction between particles (including particle–particle collisions and coalescence). Such simulations may yield new insights; see, e.g., Ten Cate et al. (2004) and Derksen (2006b). The same can be said about our understanding of particle–turbulence interactions in wallbounded ﬂows: this has increased due to Portela and Oliemans (2003) exploiting both DNS and LES and due to Ten Cate et al. (2004).

C. AN OUTLOOK Nowadays, CFD research at academia is heavily engaged in attempts to include microscale transport phenomena and microscale processes in the dedicated codes under development with a view to reproduce such divergent processes as

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blending, dissolution, crystallization, precipitation, coalescence and redispersion of bubbles and droplets, suspending solids, and chemical reactions. Essential physical challenges are in ﬁnding proper models for the details of the ﬂow. In single-phase ﬂows, we need better models for the unresolved contribution of microscale transport phenomena such as micromixing, while multiphase ﬂow CFD looks for better models for the mutual interaction of turbulence and dispersed phase particles and for the interaction force(s) between the dispersed phase particles and the ambient continuous phase. ‘The devil is in the detail’ here fully applies. In developing multiphase ﬂow CFD and in combining CFD with population balances and various types of PDF approaches, one needs to keep the size of the computational job under control—in spite of the overwhelming growth in computational power (processor speed, memory, communication tools). This requires on the one hand efﬁcient and effective numerical tools and on the other hand clever strategies for handling the enormous amounts of data. Local grid reﬁnement techniques may be of great help in avoiding an unnecessary degree of detail. The development of the above more dedicated LES and DNS is promoted by the introduction of LB techniques into the world of turbulent mixing simulations. LB techniques provide a viable alternative for the more classical FV solvers of the commercial CFD software, in particular in the context of parallel simulations on multiple processors. LB techniques are also inherently faster than FV techniques due to the locality of their operations. In addition, complex boundaries are easier to implement in the LB approach than with FV solvers. Substantial improvements in LB techniques have been effected—in terms of immersed or embedded boundary methods for dealing with moving and curved boundaries (impeller blades, solid particles) and of grid reﬁnement techniques— which have had a positive impact on the fast proliferation of dedicated CFD tools. Here, too, the details of the computational techniques do matter. Finally, the large number of processors used in many of the parallel simulations cited is striking. It illustrates the enormous progress made in the size of the simulations academic groups have realized. The falling prices of such processors and the ease at which these can be combined into platforms for parallel simulations may have the effect that—just like in the past decade with RANSbased simulations—pretty soon industrial users can afford such dedicated and detailed simulations, both LES and DNS, and can beneﬁt from their outcome in dealing with their commercial targets.

NOTATION a aij

speciﬁc interface area anisotropy tensor, comprising, essentially, the turbulent stresses made nondimensional with the turbulent kinetic energy k

220 A A1, A2, A3 c cs C b1 CD CDt Cm db dbN D Dij Dk De g I k kl ksgs m0 nb nb1 N p p~ P Pij Pk Pe q r r^ S~ Sg t T Ui Uj Uk ui uj u Us v

HARRY E. A. VAN DEN AKKER

distance to origin in (A3, A2) plane invariants of anisotropy tensor aij concentration Smagorinsky constant (in SGS modeling) coefﬁcient in relation for local maximum bubble size dbN drag coefﬁcient drag coefﬁcient in a free stream turbulence coefﬁcient in model equation for nt (in RANS models) bubble size local maximum stable bubble size diffusion (or dispersion) coefﬁcient speciﬁc rate of production of turbulent stresses speciﬁc rate of production of turbulent kinetic energy speciﬁc rate of production of e (Kolmogorov eddies) gravitational acceleration constant unity tensor concentration of turbulent kinetic energy mass transfer coefﬁcient turbulent kinetic energy contained in the SGS eddies particle number concentration particle number density local equilibrium number density impeller speed (number of revolutions per unit of time) pressure pressure as resolved in LES average pressure (in RANS context) speciﬁc rate of production of turbulent stresses speciﬁc rate of production of turbulent kinetic energy speciﬁc rate of production of e (Kolmogorov eddies) speciﬁc heat production rate speciﬁc rate of chemical reaction producing or consuming a particular species radial vector component local resolved deformation rate (in LES) source term in mass balance for gas phase (due to gas supply) time temperature components of velocity vector v (in sufﬁx notation) average turbulent, or Reynolds, stresses gas velocity vector slip (or: relative) velocity vector ﬂuid velocity vector

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v~ V Vb Vb,in w xi, xk XQ z^

221

ﬂuid velocity vector as resolved in LES average ﬂuid velocity vector (in RANS context) bubble volume bubble volume at position of gas supply azimuthal component of velocity vector spatial coordinate yield of product Q vertical vector component

GREEK SYMBOLS a b0 dij D e ij ZB ZK k n ne nt Pij Pk r rl s s s0 o o O

volume fraction of gas agglomeration coefﬁcient Kronecker delta grid spacing speciﬁc rate at which turbulent kinetic energy is dissipated (in the Kolmogorov eddies) speciﬁc rate at which turbulent stresses are dissipated Batchelor length scale (proportional to penetration depth for diffusion) Kolmogorov length scale (smallest scale in turbulent ﬂow) thermal conductivity coefﬁcient kinematic viscosity coefﬁcient effective SGS viscosity coefﬁcient (in LES) turbulent viscosity coefﬁcient (in RANS) speciﬁc rate of production of turbulent stresses speciﬁc rate of production of turbulent kinetic energy ﬂuid density liquid density interfacial tension shear stress tensor as resolved in LES part of s, see Eq. (3) effective break-up/agglomeration coefﬁcient (a kind of relaxation parameter) vorticity speciﬁc rate of destruction of e

DIMENSIONLESS NUMBERS Re Sc

Reynolds number Schmidt number

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ABBREVIATIONS ASM CDR CFD CSV CT DNS E EDD FDF FV IEM ISAT LB LDA LDV LDPE LES LGA MC MFR NS PDF QMOM RANS RSM SGS SDM SM Sh St TVD VOF

Algebraic Stress Model Convection-Diffusion-Reaction Computational Fluid Dynamics Cylindrical Stretched Vortex Computed Tomography Direct Numerical Simulation Engulfment Engulfment-Deformation-Diffusion Filtered Density Function Finite Volume Interaction by Exchange with the Mean In situ Adaptive Tabulation Lattice-Boltzmann Laser Doppler Anemometry Laser Doppler Velocimetry Low Density Poly Ethylene Large Eddy Simulation Lattice Gas Automaton Monte Carlo Multiple Frames of Reference Navier–Stokes Probability Density Function Quadrature Method of Moments Reynolds Averaged Navier Stokes Reynolds Stress Model Sub Grid Scale Sliding and Deforming Mesh Sliding Mesh Shear Strain Total Variation Diminishing Volume of Fluid

ACKNOWLEDGEMENTS First of all, Dr. Jos J. Derksen of the Department of Multi-Scale Physics at Delft University of Technology is gratefully acknowledged for a fruitful longtime collaboration and for his critical review of the draft paper. The author is also indebted to all former PhD students of the ‘Kramers Laboratorium voor

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Fysische Technologie’ of Delft University of Technology for contributing through their PhD projects and theses to the development of the views and capabilities described in this chapter.

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CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS Rodney O. Fox Department of Chemical Engineering, Iowa State University, Ames, IA 50011-2230, USA I. Introduction II. Computational Fluid Dynamics for Reacting Systems A. Basic Formulation of CFD Models B. Length and Time Scales in Turbulent Flows C. Models for SubGrid Scale Phenomena D. Reactor Systems Amenable to CFD III. Mixing-Dependent, Single-Phase Reactions A. Acid–Base and Equilibrium Chemistry B. Consecutive-Competitive and Parallel Reactions C. Detailed Chemistry IV. Production of Fine Particles A. Mixing-Dependent Nucleation and Growth B. Brownian and Shear-Induced Aggregation and Breakage C. Multivariate Population Balances V. Multiphase Reacting Systems A. Multiﬂuid CFD Models B. Interphase Mass/Heat-Transfer Models C. Coupling with Chemistry VI. Conclusions and Future Perspectives Acknowledgments References

231 234 234 238 244 250 252 254 257 267 273 275 278 282 287 288 296 299 300 302 302

I. Introduction The ﬁeld of chemical reaction engineering (CRE) is intimately and uniquely connected with the design and scale-up of chemical reacting systems. To achieve the latter, two essential elements must be combined. First, a detailed knowledge of the possible chemical transformations that can occur in the system is required. This information is represented in the form of chemical kinetic schemes, kinetic rate parameters, and thermodynamic databases. In recent years, considerable progress has been made in this area using computational chemistry and carefully Corresponding author. Tel: +1 515 294 9104; Fax: +1 515 294 2689. E-mail: [email protected]

231 Advances in Chemical Engineering, vol. 31 ISSN 0065-2377 DOI 10.1016/S0065-2377(06)31004-6

Copyright r 2006 by Elsevier Inc. All rights reserved.

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controlled experiments to isolate the chemical kinetics in the absence of transport effects. The second essential element is detailed knowledge of the transport phenomena in chemical reacting systems. Indeed, because commercial chemical reactors are almost always operated in a regime where mass and energy transport affect or even control the product distribution from the system, understanding the coupling between transport processes and chemical reactions is an essential step in the design and scale-up of chemical reacting systems. From its beginning, the ‘‘holy grail’’ of CRE has been a computational model that is capable of predicting reactor performance based on the fundamental molecular-scale parameters describing the chemical kinetics and the transport coefﬁcients. In principle, the latter can be measured experimentally in smallscale laboratory experiments (or estimated using computational chemistry). The chemical reaction engineer then incorporates this information into a computational model to predict the behavior of the plant-scale reactor. By avoiding the need for pilot-scale experiments, this ‘‘experiment-free’’ scale-up approach should result in more rapid process development at much lower cost. Admittedly, while CRE has made considerable progress toward this goal, much work remains to be accomplished. Nevertheless, due to the tremendous expansion in computing power over the last 30 years, computational models used in CRE can now account for much more detail than was previously thought possible. This trend is unlikely to abate, and thus, to remain relevant to industry, chemical reaction engineers of the future must become adept at employing detailed ﬂow models for chemical reacting systems. For example, over the past 15 years computational ﬂuid dynamics (CFD) has become an important tool in CRE for understanding the coupling between transport processes and chemical reactions. The deﬁnition of ‘‘CFD’’ has in the process expanded from its original emphasis on ﬂuid dynamics (i.e., momentum transport) to include mass and energy transport, as well as detailed chemical reactions. One might argue that it would, therefore, be more accurate to refer to the ﬁeld as ‘‘computational transport phenomena.’’ On the other hand, because CFD models rarely resolve all of the relevant scales (as described below), one might also argue that ‘‘computational chemical reaction engineering’’ is a more accurate description. However, both of these names are perhaps too broad and lose the essential focus on the fact that CFD always includes a description of momentum transport in the ﬂuid phase(s). Thus, the original name continues to be used to designate all computational approaches that solve for the spatial distribution of the velocity, concentration, and temperature ﬁelds. CFD models have also been developed for multiphase systems, and commercial CFD codes now offer a wide range of options for modeling chemical reactors. Despite the many advances, the users of CFD codes must keep in mind that the underlying transport equations are based on models, which may or may not be valid for a particular application. This fact often escapes the minds of newcomers to the ﬁeld who are typically overwhelmed by the numerical issues associated with convergence, grid-independence, and post-processing. Even

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many CFD experts tend to avoid the issue by working exclusively in an application area where acceptably accurate models are already available. Unfortunately (at least for industrial users!), the application of CFD to chemical reactor analysis introduces new modeling challenges with each new reactor type. In the simplest case of laminar ﬂow with fully resolved concentration and temperature ﬁelds, the models have a strong fundamental basis that typically involves molecular-scale transport coefﬁcients. These CFD models are based on the so-called microscopic balance equations that are taught to undergraduate students in chemical engineering courses on transport phenomena. The extension of the microscopic balance equations to multiphase ﬂow systems is also well understood, but brings with it a wide range of new ﬂow phenomena in even the simplest cases. In reality, most of the applications in CRE for which CFD is used cannot be treated using the microscopic balance equations alone. Instead, CFD models are introduced to describe the effects of unresolved phenomena on the resolved quantities. These models introduce phenomenological transport coefﬁcients, much like the ones developed in CRE models for chemical reactors. In fact, in many cases, spatial transport is dominated by convection and the molecularscale spatial transport can be neglected. Nevertheless, just as in ‘‘classical’’ CRE models for interphase mass/energy transport, the molecular-scale transport coefﬁcients appear in the dimensionless numbers used to formulate the phenomenological coefﬁcients. Indeed, because the accuracy of CFD predictions are strongly dependent on the accuracy of these so-called subgrid-scale (SGS) models, the modeling skills developed in CRE over its long history are a crucial component in the development of CFD for chemical reactor design and analysis. In fact, it would not be pretentious to claim that, due to their considerable abilities to deal with the coupling between chemical reactions and transport phenomena, chemical reaction engineers are uniquely qualiﬁed to develop the SGS models needed for CFD modeling of chemical reacting systems. In the remainder of this work we review the current status of CFD models for chemical reacting systems. In some cases (e.g., single-phase reacting ﬂow) the current models are fully predictive in the limits of high and low Reynolds numbers, and quite accurate in the transition region between these limits. In other cases (e.g., multiphase reacting ﬂow), the predictive abilities of current CFD models are, in general, limited. Nevertheless, for particular multiphase reactors (e.g., gas–solid reacting ﬂow), powerful models exist and are making their way into commercial CFD codes. The goal of the presentation will not be to describe every model in detail, but rather to indicate the current status of models for treating reacting ﬂows and to point out areas where further research is needed. The reader interested in a deeper understanding of the particular aspects a model will be pointed to the appropriate research literature for further reading. Moreover, consistent with our desire to use CFD for chemical reactor analysis and design, we will not discuss models whose primary purpose is to describe nonreacting ﬂows. For single-phase reactors, excellent descriptions of

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such models are available in the literature. Likewise, CFD models for multiphase ﬂows are described by other authors in this issue. The organization of the material is as follows. In Section II we provide a general introduction to CFD models for chemical reacting systems, and to the critical modeling issues that arise in their development and application. In Section III we describe the current state of the art in CFD models for singlephase reacting ﬂows. In Section IV we extend the discussion of single-phase reacting ﬂows to include systems that produce ﬁne particles that follow the continuous-phase ﬂow. In these systems, the principal novelty is the inclusion of a population balance model for the particulate phase. In Section V, we describe the current state of the art in CFD models for multiphase reacting ﬂows. Because this last area is the least developed, but most rapidly advancing, we will limit our discussion to CFD models that can potentially be used to describe plant-scale reactors (i.e., multiﬂuid models and related mean-ﬁeld descriptions). Even for these models, we will not cover the details on how momentum exchange is treated between phases. Rather, we will focus our attention on factors that affect directly the chemical reactions. Finally, in Section VI conclusions are drawn concerning the current status of CFD models for chemical reactor analysis, and an attempt is made to point out the research directions where progress can be expected in the near future.

II. Computational Fluid Dynamics for Reacting Systems In this Section we give an overview of the formulation of CFD models for reacting systems, with particular emphasis on systems requiring SGS models. For the reader to understand the procedure followed to create a CFD model for a chemical reactor, we cover ﬁrst the basic formulation. Then, because the SGS models are often needed due to the ﬂow being turbulent, we next review the principal length and time scales present in turbulent transport. We then give examples of SGS phenomena and their corresponding models in turbulent reacting ﬂows. Finally, we end the section with a brief discussion of the types of reactor systems that can currently be treated using CFD.

A. BASIC FORMULATION

OF

CFD MODELS

When applying CFD to model a chemical reactor, we are interested in knowing how the basic quantities (density, velocity, concentrations, etc.) vary with the spatial location in the reactor at a given time instant. The starting point for developing a CFD model is the microscopic balance equation, described in detail in standard textbooks on transport phenomena (Bird et al., 2002). Letting F denote a quantity of interest, the general form of its microscopic balance

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equation is @F þ = ðUFÞ þ = Jf ¼ Sf @t

(1)

where U is the convective velocity, Jf is a molecular-scale model for the diffusive ﬂux, and Sf is a molecular-scale source term. Typical examples of quantities of interest are ﬂuid density r, species mass fractions rYa, and the ﬂuid momentum rU. Likewise, for multiphase systems similar quantities are of interest, but for each individual phase present in the reactor. The generalized source term Sf will then include mass/momentum/heat-transfer between phases. For complex ﬂuids (e.g., non-Newtonian ﬂows), molecular-scale models for Jf and Sf can be quite complicated and can lead to numerical difﬁculties, requiring specially developed CFD solvers. As mentioned in Section I.A, CFD codes were originally developed to solve for the ﬂuid momentum for Newtonian ﬂuids, for which the right-hand side of Eq. (1) is well understood (Bird et al., 2002). However, even for such ﬂuids, it is not possible to accurately solve the microscopic balance equation for Reynolds numbers commonly observed in chemical reactors. It is thus necessary to distinguish between direct-numerical simulations (DNS) and CFD models using phenomenological descriptions of the turbulence. The two most widely used CFD approaches for describing turbulent ﬂow are large-eddy simulations (LES) and Reynolds-averaged Navier–Stokes (RANS) models (Pope, 2000). In both approaches, it is no longer possible to solve Eq. (1) directly for F due to the enormous computational cost. Instead, Eq. (1) is ﬁltered (LES) or ensembleaveraged (RANS), yielding ~ @F f þ = Jf ¼ S~ f ~ FÞ ~ þ = ðufÞ þ = ðU @t

(2)

This transport equation cannot be solved directly because it involves several f represents the spatial transport of F by the unclosed terms. The SGS ﬂux uf unresolved velocity ﬂuctuations. Models for this term can generally be written in the form of a generalized transport equation: ~ @F ~ FÞ ~ þ = J~ f þ J~ Tf ¼ S~ f þ S~ Tf þ = ðU @t

(3)

where the SGS diffusive ﬂux is denoted by J~ Tf and SGS source term by S~ Tf . To distinguish this expression from Eq. (1), we will refer to Eq. (3) as the CFD transport equation. Thus, only in the (rare) case of DNS will the CFD transport equation correspond to the microscopic balance equation. In chemical reacting systems, the Reynolds number of the ﬂow is not the only source of computational challenges. Indeed, even for laminar reacting ﬂows the chemical source term can be extremely stiff and tightly coupled to the diffusive transport terms. Averaging, as done above to treat turbulent ﬂows, does not

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alleviate this difﬁculty. Thus, turbulent reacting ﬂows offer many difﬁcult challenges and require specialized models to describe the coupling between molecular diffusion and chemical reactions (Fox, 2003). We will look at some of the more widely applicable models in Section III. Keeping in mind the discussion leading to Eq. (3), the formulation of a CFD model for a chemical reactor can be broken down into the following broad steps: (i) First we must identify the set of state variables needed to completely describe the reactor. Typical examples are ~ 2 r; ~ r~ Y~ a ; r~ T; ~ r~ f~ b ~ r~ U; F where, in addition to the quantities introduced earlier, T~ is the ﬂuid e is a set of scalar quantities. The latter are introduced to temperature and f b deﬁne, for example, the closure for the chemical source term (Fox, 2003) and the turbulence model (Pope, 2000). Note that the identiﬁcation of the state variables is analogous to what is done in ‘‘classical’’ CRE models. Thus, chemical reaction engineers are generally well acquainted with the methodology needed to complete this step. The only new quantity that does not appear in ‘‘lumped’’ CRE models is the ﬂuid velocity. However, chemical engineers are typically introduced to momentum balances in courses on transport phenomena, and thus understand its signiﬁcance. (ii) The next and arguably the most difﬁcult step is to ﬁnd closures for the CFD transport equation, expressed in terms of the state variables. For example, in turbulent ﬂows the diffusive-ﬂux terms can often be modeled successfully as gradient-diffusion terms: ~ J~ f þ J~ Tf ¼ Df þ DTf =F (4) where Df is a molecular-diffusion coefﬁcient and DTf is a turbulent-diffusion coefﬁcient. In high-Reynolds-number ﬂows, Df is negligible compared to e apDTf. Note that in general DTf will depend on the scalar quantities f b pearing in the turbulence model. Closure of the source terms in Eq. (3) is much more difﬁcult, and requires fundamental knowledge about how the local ﬂow ﬁeld interacts with the quantity of interest (e.g., how the local turbulence level affects the rates of diffusive mixing and chemical reactions at the subgrid scale). Nevertheless, the ﬁnal closures must be expressed as follows in terms of the state variables: ~ ~ ~ Y~ a ; T; ~ Y~ a ; T; ~ f ~ f ~ U; ~ U; and S~ Tf r; S~ f r; b

b

Note that these closures describe SGS phenomena and hence are essentially local in space (i.e., interior to a computational grid cell). For this reason, it is

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often possible to use DNS of statistically homogeneous systems (i.e., for which ~ do not depend on x) to develop closure models the [ﬁltered] state variables F for S~ f and S~ Tf . This procedure has been widely used in single-phase turbulence modeling (Pope, 2000; Fox, 2003), and more recently in multiphase ﬂow systems (e.g., Bunner and Tryggvason, 2003; Nguyen and Ladd, 2005). For the latter, the generalized source terms include mass/momentum/heattransfer between phases, and as discussed in Section V the closure models involve dimensionless parameters such as the particle Reynolds number. (iii) The coupled system of CFD transport equations now appears as follows in closed form: ~ @F ~ FÞ ~ ¼ = Df þ DTf =F ~ þ S~ f þ S~ Tf þ = ðU @t

(5)

and the remaining task is to ﬁnd a suitable numerical algorithm to solve them. Fortunately, CFD experts have developed powerful and robust algorithms for solving equations in the form of Eq. (5), and these are now available in commercial CFD codes. Thus, from the perspective of the chemical reaction engineer working in industry, the efﬁcient application of CFD to chemical reactor analysis and design will inevitably involve the use of a commercial CFD code. The next step in the CFD model formulation will thus be to introduce the closure models developed in the previous step into the CFD code. This is facilitated in most commercial CFD codes by the availability of so-called ‘‘user-deﬁned scalars.’’ In many cases, the basic turbulence and multiphase models will already be available in a commercial code. The chemical reaction engineer will thus only need to add the specialized closure models (in terms of f~ b ) needed to describe the state variables in a particular application. (iv) Once the CFD model equations have been implemented in the code, the next step is to create a computational grid to capture the speciﬁc geometry of the chemical reactor. The qualities of the grid strongly affect the accuracy and the speed of convergence of the numerical algorithm. Thus, for complex reactor geometries, it may make sense to hire a specialist in grid generator to carry out this step. (v) The remaining steps involve solving the CFD model and postprocessing of the results. The latter is greatly facilitated by the built-in functions available in most commercial CFD codes. It is at this point that reactor analysis and design actually come into full play. By experimenting with variations in the operating conditions and reactor geometry, the CFD model can be used to enhance product selectivity and reactor performance. When applying the steps outlines above, the prudent engineer will start by modeling an existing reactor for which plant-scale data are available for validation of the CFD results. If the agreement is poor, usually this will be due to

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inadequate choices for the state variables and/or closure models. Nevertheless, one should also examine the computational results to see if there are numerical errors leading, for example, to inconsistencies in the mass, species, or energy balances. Getting acceptable agreement may take several iterations of changes in the closures. For many cases, this process can be facilitated by breaking it down into independent steps (e.g., ﬂow-ﬁeld predictions can be validated before adding the chemistry). After reasonable agreement between the model and data is obtained, the CFD model can be safely used to explore alternative design scenarios.

B. LENGTH

AND

TIME SCALES

IN

TURBULENT FLOWS

As mentioned before in Eq. (3), the most common source of SGS phenomena is turbulence due to the Reynolds number of the ﬂow. It is thus important to understand what the principal length and time scales in turbulent ﬂow are, and how they depend on Reynolds number. In a CFD code, a turbulence model will provide the local values of the turbulent kinetic energy k and the turbulent dissipation rate e. These quantities, combined with the kinematic viscosity of the ﬂuid n, deﬁne the length and time scales given in Table I. Moreover, they deﬁne the local turbulent Reynolds number ReL also given in the table. The integral scale of a turbulent ﬂow characterizes the largest and most energetic ﬂow structures. In a CFD simulation, the local grid size will be proportional to the integral length scale Lu. Likewise, the characteristic lifetime of the largest eddies is proportional to the integral time scale tu. The Kolmogorov scale characterizes the smallest ﬂow structures and is resolved by neither LES nor RANS simulations (only in DNS). Note that the ratios of the length and time scales are as follows: Lu 3=4 ¼ ReL Z

and

tu 1=2 ¼ ReL tZ

(6)

Thus, as the local turbulent Reynolds number increases, the separation between the scales will increase. As a general rule, ReL will be proportional to the

TABLE I THE PRINCIPAL LENGTH AND TIME SCALES, AND REYNOLDS NUMBERS CHARACTERIZING A TURBULENT FLOW DEFINED IN TERMS OF THE TURBULENT KINETIC ENERGY k, AND TURBULENT DISSIPATION RATE , AND THE KINEMATIC VISCOSITY m Quantity

Integral scale

Kolmogorov scale

Length Time Reynolds number

Lu ¼ k3/2/e tu ¼ k/e ReL ¼ k2/en

Z ¼ (n3/e)1/4 tZ ¼ (n/e)1/2 ReZ ¼ 1

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239

macroscopic Reynolds number for the ﬂow (i.e., Re deﬁned in terms of a characteristic ﬂow velocity and length scale.) In general, for a ﬁxed-ﬂow geometry, the integral length scale will remain approximately constant (e.g., in a turbulent jet the integral length scale is proportional to the jet diameter). Likewise, the integral scale velocity, deﬁned by Lu/tu, will be proportional to the characteristic velocity of the ﬂow (e.g., the jet velocity). Thus, as the Reynolds number increases (e.g., to enhance turbulent mixing), Z, tu, and tZ will all decrease. In the CFD simulation, the grid will remain approximately the same and the time step must decrease to follow tu. This implies that at high Reynolds numbers less and less of the small-scale ﬂow structures are captured by the CFD simulation. To estimate the amount of turbulent kinetic energy lost when ﬁltering at a given grid size, it is useful to introduce a normalized model energy spectrum (Pope, 2000) as follows: E u ðkÞ ¼ Ck5=3 f L ðkÞf Z ðkÞ

(7)

where k is the dimensionless wavenumber (inverse length), and the Kolmogorov constant is C ¼ 1.61 (based on the most recent DNS (Watanabe and Gotoh, 2004)). The nondimensional cut-off functions are deﬁned by f L ðkÞ

k

!5=3þp0

ðk2 þ cL Þ1=2

(8)

and h i f Z ðkÞ exp b ½k4 =Re3L þ c4Z 1=4 cZ

(9)

wherein p0 ¼ 2 and b ¼ 5.2. The parameters cL and cZ are found by applying two integral constraints as follows: Z 1 1¼ E u ðkÞ dk (10) 0

and Z

1

2k2 E u ðkÞdk

ReL ¼

(11)

0

Note that the ﬁnal form of the energy spectrum depends only on the local turbulent Reynolds number. As an example, spectra found with different ReL are shown in Fig. 1. In the normalized energy spectrum, k ¼ 1 corresponds to the inverse of the 3=4 local integral length scale and k ¼ ReL to the inverse of the local Kolmogorov length scale. The range of wavenumbers in Fig. 1 over which the slope is 5/3 is

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RODNEY O. FOX

ReL = 100 ReL = 102 ReL = 104 ReL = 106 ReL = 108

100 10-2

Eu

10-4 10-6 10-8 10-10 10-12

10-1

100

101

102

103 κ

104

105

106

107

FIG. 1. The normalized model turbulent energy spectrum for a range of Reynolds numbers.

called the inertial range. Thus, for the ﬂow to be considered turbulent (as opposed to transitional ﬂow), ReL must be larger than approximately 20. In contrast, high-Reynolds-number turbulence (i.e., with a signiﬁcant inertial range) does not begin until ReL is larger than 450. In RANS turbulence models designed for low-Reynolds-number turbulent ﬂows, the model parameters are functions of ReL, and as the local turbulent Reynolds number approaches zero, the microscopic balance equation (Eq. 1) is recovered. In contrast, in LES turbulence models the ﬁlter size is typically ﬁxed at some Reynolds-numberindependent wavenumber kc410. Thus, the fraction of turbulent kinetic energy captured by LES can be found from Z

kc

fc ¼

E u ðkÞdk

(12)

0

and varies from fc ¼ 1 for small ReL up to a constant value less than one for very large ReL (Pope, 2000). In the discussion above, we have considered only the velocity ﬁeld in a turbulent ﬂow. What about the length and time scales for turbulent mixing of a scalar ﬁeld? The general answer to this question is discussed in detail in Fox (2003). Here, we will only consider the simplest case where the scalar ﬁeld f is inert and initially nonpremixed with a scalar integral length scale Lf that is approximately equal to Lu. If we denote the molecular diffusivity of the scalar by G, we can use the kinematic viscosity to deﬁne a dimensionless number in the following way: Sc

n G

(13)

called the Schmidt number. In gases, typical values of the Schmidt number are near unity, while in liquids values near 1,000 are quite common. The Schmidt

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241

number and the Kolmogorov length scale can be used to deﬁne the Batchelor length scale as follows: 3=4

lB Sc1=2 Z ¼ Sc1=2 ReL

Lu

(14)

which is the length scale where molecular diffusion occurs. In a nonpremixed turbulent ﬂow seen under magniﬁcation (e.g., using planar laser-induced ﬂuorescence), the smallest observable structures over which concentration gradients are seen have characteristic size lB. Note that for large Sc, the Batchelor scale can be very small even at low Reynolds numbers. The degree of local mixing in a RANS simulation is measured by the scalar variance hf02 i, which ranges from zero for complete mixing (i.e., f ¼ hfi is uniform at the SGS) up to ðfmax hfiÞðhfi fmin Þ where hfi is the mean concentration and fmax and fmin are the maximum and minimum values, respectively. The rate of local mixing is controlled by the scalar dissipation rate ef (Fox, 2003). The scalar time scale analogous to the turbulence integral time scale is (Fox, 2003) as follows: tf

2hf02 i f

(15)

In a RANS simulation of scalar mixing, a model for ef must be supplied to compute hf02 i. In fully developed turbulence, tf can be related to tu by considering the scalar energy spectrum, as ﬁrst done by Corrsin (1964). To determine how the scalar time scale deﬁned in Eq. (15) is related to the turbulence integral time scale given in Table I, we can introduce a normalized model scalar energy spectrum (Fox, 2003) as follows: ð3bðkÞ5Þ=4 bðkÞ

E f ðkÞ ¼ C OC ReL

k

f L ðkÞf B ðkÞ

(16)

where the scaling exponent is deﬁned by bðkÞ 1 þ

2 7 6f D ðkÞ f Z ðkÞ 3

(17)

and Obukhov–Corrsin constant is COC ¼ 0.67–0.68 (Sreenivasan, 1996; Watanabe, and Gotoh, 2004; Yeung et al., 2005). In the model spectrum, the diffusive-scale cut-off function is deﬁned by 3=4 3=4 f D ðkÞ 1 þ cD ScdðkÞ=2 k=ReL exp cD ScdðkÞ=2 k=ReL

(18)

with cD ¼ 2.59, and the diffusive exponent is dðkÞ

1 1 þ f ðkÞ 2 4 Z

(19)

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RODNEY O. FOX

The Batchelor-scale cut-off function is deﬁned by 3=4 3=4 f B ðkÞ 1 þ cd ScdðkÞ k=ReL exp cd ScdðkÞ k=ReL

(20)

wherein the scalar-dissipation constant cd is found by applying an integral constraint as follows: Z

1

2k2 E f ðkÞdk

ReL Sc ¼

(21)

0

Note that the scalar-dissipation constant computed from Eq. (21) depends only on ReL and Sc. In Fig. 2, the normalized model scalar energy spectrum is plotted for a ﬁxed Reynolds number (ReL ¼ 104) and a range of Schmidt numbers. In Fig. 3, it is shown for Sc ¼ 1000 and a range of Reynolds numbers. The reader interested in the meaning of the different slopes observed in the scalar spectrum can consult Fox (2003). By deﬁnition, the ratio of the time scales is equal to the area under the normalized scalar energy spectrum as follows: Z 1 tf ¼ E f ðkÞdk (22) 2tu 0 Thus, from Figs. 2 and 3 we can conclude that the time-scale ratio will depend on ReL and Sc. In the literature on turbulent mixing, the mechanical-to-scalar time-scale ratio is deﬁned by R

2tu tf

(23)

10-1

Sc = 0.05 Sc = 0.5 Sc = 5 Sc = 50 Sc = 500 Sc = 5000

E

10-3

10-5

10-7

10-9

10-1

100

101

102 κ

103

104

105

106

FIG. 2. Normalized model scalar energy spectra for a range of Schmidt numbers and ReL ¼ 104.

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

100

ReL = 100 ReL = 102 ReL = 104 ReL = 106 ReL = 108

10-2 10-4 10-6 E

243

10-8 10-10 10-12 10-14

100

102 κ

104

106

108

FIG. 3. Normalized model scalar energy spectra for a range of Reynolds numbers and Sc ¼ 103.

4.0 3.5 3.0 R

2.5 Sc = 0.05 Sc = 0.5 Sc = 5 Sc = 50 Sc = 500 Sc = 5000

2.0 1.5 1.0 0.5 0.0 100

101

102

103 104 ReL

105

106

107

FIG. 4. Mechanical-to-scalar time-scale ratio found from the normalized model scalar energy spectra.

and is plotted based on Eq. (16) for a range of Schmidt numbers as a function of Reynolds number in Fig. 4. Note that for very large Reynolds numbers, R is independent of Sc and approaches a constant value of R ¼ C/COC ¼ 2.37, i.e., the ratio of the Kolmogorov and the Obukhov–Corrsin constants. In contrast, for ReL less than 106, R is strongly dependent on both the Reynolds and Schmidt numbers. The dependence on Reynolds number is especially signiﬁcant for Schmidt numbers far from unity. For example, liquid-phase reactors used for material processing (Mahajan and Kirwan, 1993, 1996; Johnson and Prud’homme, 2003a,b) have high Schmidt numbers and operate at low to moderate Reynolds numbers (Liu and Fox, 2006). In a CFD simulation, tu can be found from the turbulence model and tf from the data in Fig. 4. Thus, the curves in Fig. 4 deﬁne a SGS model for tf parameterized in terms of ReL and Sc.

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RODNEY O. FOX

We will revisit this topic in Section III when discussing CFD models for mixingsensitive reactions. Note that while the discussion above applies to RANS turbulence models, the method can be extended to LES by integrating over the SGS wavenumbers (i.e., starting at kc). In summary, we have seen that the principal length and time scales in a singlephase turbulent ﬂow depend on the local turbulent Reynolds number ReL. In a CFD code, standard turbulence models will provide the local values of k and e. Given the ﬂuid viscosity, it will thus be possible to compute the local turbulent Reynolds number and related integral-scale quantities such as Lu and tu. Using the model energy spectra, we have also shown how the scalar mixing time tf depends on the Schmidt number and local turbulent Reynolds number. In principle, a similar analysis could be carried out for multiphase turbulent ﬂows to understand the scaling laws for the length and time scales and their dependence on the relevant dimensionless numbers. Unfortunately, DNS of multiphase ﬂow is still in its infancy and experimental measurements of energy spectra are difﬁcult to obtain. Nevertheless, we can expect signiﬁcant progress in our understanding of turbulent multiphase ﬂows using DNS for particular systems (e.g., gas–solid ﬂows) in the coming years. C. MODELS

FOR

SUBGRID SCALE PHENOMENA

As noted earlier, in most applications of CFD to chemical reactor design and analysis the CFD transport equation (Eq. 3) will require SGS closures. (Here we use ‘‘subgrid scale’’ to refer to LES and RANS models for terms not fully resolved by the computational grid.) Thus, one of the principal tasks of a chemical reaction engineer is to develop the SGS models that accurately describe the chemistry and physics occuring at the unresolved scales of the ﬂow. As discussed above, for a single-phase turbulent ﬂow, the unresolved scales are those smaller than the integral length and time scales (Lu and tu, respectively). Obviously, the SGS models will be highly dependent on the type of ﬂow under consideration (e.g., single-phase vs. multiphase, nonreacting vs. reacting, etc.), and a complete listing of all such models would be lengthy and uninformative to the general reader. Thus, instead of giving general examples, in this section we will demonstrate how the SGS model is developed for a particular example (single-phase turbulent mixing). In the subsequent sections we will extend this model to reacting scalars of various types. Readers interested in more details on the models can consult Fox (2003). One of the ﬁrst questions that arises when considering a chemical reactor is ‘‘Can the reactor be considered perfectly mixed?’’. In CRE, this question implies at least two physical situations: 1. The reactor is perfectly macromixed if the mean concentration at every point in the reactor is equal to the volume average. 2. The reactor is perfectly micromixed if the instantaneous, local concentration at every point in the reactor is equal to the local mean concentration.

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245

The ‘‘classical’’ CRE model for a perfectly macromixed reactor is the continuous stirred tank reactor (CSTR). Thus, to ﬁx our ideas, let us consider a stirred tank with two inlet streams and one outlet stream. The CFD model for this system would compute the ﬂow ﬁeld inside of the stirred tank given the inlet ﬂow velocities and concentrations, the geometry of the reactor (including bafﬂes and impellers), and the angular velocity of the stirrer. For liquid-phase ﬂow with uniform density, the CFD model for the ﬂow ﬁeld can be developed independently from the mixing model. For simplicity, we will consider this case. Nevertheless, the SGS models are easily extendable to ﬂows with variable density. Following the steps for formulation of a CFD model introduced earlier, we begin by determining the set of state variables needed to describe the ﬂow. Because the density is constant and we are only interested in the mixing properties of the ﬂow, we can replace the chemical species and temperature by a single inert scalar ﬁeld x(x, t), known as the mixture fraction (Fox, 2003). If we take x ¼ 0 everywhere in the reactor at time t ¼ 0 and set x ¼ 1 in the ﬁrst inlet stream, then the value of x(x, t) tells us what fraction of the ﬂuid located at point x at time t originated at the ﬁrst inlet stream. If we denote the inlet volumetric ﬂow rates by q1 and q2, respectively, for the two inlets, at steady state the volume-average mixture fraction in the reactor will be x¯

q1 q1 þ q2

(24)

Thus, the reactor will be perfectly mixed if and only if x ¼ x¯ at every spatial location in the reactor. As noted earlier, unless we conduct a DNS, we will not compute the instantaneous mixture fraction in the CFD simulation. Instead, if we use a RANS model, we will compute the ensemble- or Reynolds-average mixture fraction, denoted by hxi. Thus, the ﬁrst state variable needed to describe macromixing in this system is hxi. If the system is perfectly macromixed, hxi ¼ x¯ at every point in the reactor. The second state variable will be used to describe the degree of local micromixing, and is the mixture-fraction variance hx02 i. When the variance is zero, the ﬂuid is perfectly micromixed so that x ¼ hxi. The maximum value of the variance at any point in the reactor is hxið1 hxiÞ, and varies from zero in the feed streams to a maximum of 1/4 when hxi ¼ 1=2. At this point, we should clarify an all-to-common misconception in the chemical-engineering literature concerning the meaning of ‘‘Reynolds average.’’ Unfortunately, many textbooks and journal articles still deﬁne it as a time average or a volume average over an interval that ‘‘is not too long, but not too short.’’ This deﬁnition confuses methods for estimating the expected value from experimental data for a single realization (i.e., time and volume averages), which are statistics, with the underlying expected values with respect to all possible realizations. In general, a statistic will be different every time it is computed, while an expected value is constant at a given point in space and time. Thus, when deriving closures for expected values such as hxi and hx02 i, we start with a general transport theory based on the joint probability density function (PDF)

246

RODNEY O. FOX

as described in Pope (2000) and Fox (2003). Space or time averages only come into the picture when we must validate the predictions of the CFD model against experimental data. For example, if the ﬂow is statistically stationary, then the time average of x(x, t) can be used to estimate hxi(x). (Note that by deﬁnition of ‘‘statistically stationary’’ the expected values will not depend on time.) Likewise, if the ﬂow is statistically homogeneous, then the volume average of x(x, t) can be used to estimate hxi(t). In chemical reactors, the ﬂow is almost never homogeneous (if it were, CFD would not be needed). Nevertheless, one still ﬁnds authors who confuse micromixing, rigorously deﬁned in terms of the variance hx02 i, with deviations of the mean hxi from its volume-average. In reality, such ﬂuctuations correspond to poor macromixing and are a mathematical artifact caused by ‘‘lumping’’ inhomogeneous ﬂow into a homogeneous model (e.g., by modeling laminar ﬂow in a tubular reactor using a plugﬂow model). Finally, we should note that identical statistical concepts can be used to derive CFD models for scalar mixing in low-Reynolds-number chaotic ﬂows encountered in high-viscosity mixing. The principal difﬁculty in these ﬂows is ﬁnding general state variables to describe the length and time scales of the ﬂow. The remaining state variables in our CFD model for turbulent mixing are needed to describe the ﬂow ﬁeld in the reactor. In a RANS model for turbulent ﬂow, the mean velocity hUi appears in the Reynolds-average momentum balances. The latter is closed by providing a turbulence model for the Reynolds stresses (Pope, 2000). If a turbulent-viscosity-based model is used, two state variables are introduced to describe the local turbulent integral time scale and length scale (see Table I). Common choices are the turbulent kinetic energy k, and the turbulent dissipation rate, e. The set of state variables used to described turbulent mixing in the reactor are thus ~ 2 hUi; k; ; hxi; hx02 i F Note that when solving the CFD transport equations, the mean velocity and turbulence state variables can be found independently from the mixture-fraction state variables. Likewise, when validating the CFD model predictions, the velocity and turbulence predictions can be measured in separate experiments (e.g., using particle-image velocimetry [PIV]) from the scalar ﬁeld (e.g., using planar laser-induced ﬂuorescence [PLIF]). Now that the state variables have been determined, we can go to steps (ii) and (iii), which involve ﬁnding closed CFD transport equations. The derivation of the RANS equations is described in detail in Fox (2003), and will not be repeated here. Instead, we will simply give the CFD transport equations and discuss the closures appearing in the equations. The ﬁve transport equations are @rhUi þ = ðrhUihUiÞ ¼ = m þ mT =hUi =p þ rg @t

(25)

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247

@rk þ = ðrhUikÞ ¼ = m þ mT =sk =k þ Pk r @t

(26)

@r þ = ðrhUiÞ ¼ = m þ mT =s = þ ðC 1 Pk C 2 rÞ @t k

(27)

@rhxi þ = ðrhUihxiÞ ¼ = ðD þ DT Þ=hxi @t

(28)

and @rhx02 i þ = rhUihx02 i ¼ = ðD þ DT =sx Þ=hx02 i @t þ 2DT j=hxij2 rx

ð29Þ

Note that although the density is constant, we have included it in the transport equations to be consistent with the formulation used in commercial CFD codes. The left-hand sides of Eqs. (25)–(29) have the same form as Eq. (5) and represent accumulation and convection. The terms on the right-hand side can be divided into spatial transport due to ‘‘diffusion’’ and source terms. The diffusion terms have a molecular component (i.e., m and D), and turbulent components. We should note here that the turbulence models used in Eqs. (26) and (27) do not contain corrections for low Reynolds numbers and, hence, the molecular-diffusion components will be negligible when the model is applied to high-Reynoldsnumber ﬂows. The turbulent viscosity is deﬁned using a closure such as mT ¼ rC m k2 =

(30)

The turbulent diffusivity is deﬁned by introducing a so-called ‘‘turbulent’’ Schmidt number ScT: DT ¼ mT =ScT

(31)

which should not be confused with the molecular Schmidt number Sc. Like the other diffusion-model constants (i.e., sk, se, and sx), ScT has been determined using canonical turbulent mixing experiments (see Pope (2000) and Fox (2003) for details). We should note, however, that these ‘‘constants’’ must sometimes be adjusted for noncanonical ﬂows. The source terms on the right-hand sides of Eqs. (25)–(29) are deﬁned as follows. In the momentum balance, g represents gravity and p is the modiﬁed pressure. The latter is found by forcing the mean velocity ﬁeld to be solenoidal ð= hUi ¼ 0Þ. In the turbulent-kinetic-energy equation (Eq. 26), Pk is the source term due to mean shear and the ﬁnal term is dissipation. In the dissipation equation (Eq. 27), the source terms are closures developed on the basis of the form of the turbulent energy spectrum (Pope, 2000). Finally, the source terms

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RODNEY O. FOX

for the mixture-fraction variance (Eq. 29) are due to production by mean mixture-fraction gradients and dissipation by micromixing. As written, Eq. (29) is not yet closed: we need to add a model for the mixture-fraction dissipation rate ex. Using Eq. (23), the latter can be modeled by x ¼ R hx02 i k

(32)

where R depends on ReL and Sc as shown in Fig. 4. The CFD model for turbulent mixing is now complete, and can be solved to investigate the degree of macro- and micromixing in a chemical reactor. The next step in the CFD model formulation involves adding Eqs. (25)–(29) to a CFD code. For this particular example, this step is facilitated in some commercial CFD codes that have the model already included in the standard release of the code. The ﬁnal step is to solve the model and to compare with experimental data when available. In this step, it may be useful to deﬁne new variables during postprocessing to quantify the degree of mixing, mixing zones or the characteristic times for macro- and micromixing. See Liu and Fox (2006) for examples of how this can be done using output from a CFD mixing model. The CFD model developed above is an example of a ‘‘moment closure.’’ Unfortunately, when applied to reacting scalars such as those considered in Section III, moment closures for the chemical source term are not usually accurate (Fox, 2003). An alternative approach that yields the same moments can be formulated in terms of a presumed PDF method (Fox, 1998). Here we will consider only the simplest version of a multi-environment micromixing model. Readers interested in further details on other versions of the model can consult Wang and Fox (2004). The basis idea behind multi-environment models is that the mixture fraction at any location in the reactor can be approximated by a distribution function in the form of a sum of delta functions as follows: f x ðz; x; tÞ ¼

N X

pn ðx; tÞdðz xn ðx; tÞÞ

(33)

n¼1

where pn is the mass fraction of environment n, and xn the mixture fraction in environment n. Using the deﬁnition of mixture-fraction moments, we have hx i ¼

N X

pn xn

(34)

pn x2n hxi2

(35)

n¼1

and hx02 i ¼

N X n¼1

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

249

In other words, if we know pn and xn at every point in the reactor, then we can compute up to 2N1 independent mixture-fraction moments. The simplest model of this type is the two-environment model (N ¼ 2) for which the independent state variables in the CFD model are ~ 2 hUi; k; ; p1 ; x1 ; x2 F In theory, this model can be used to ﬁx up to three moments of the mixture fraction ðe:g:; hxi; hx2 i; and hx3 iÞ. In practice, we want to choose the CFD transport equations such that the moments computed from Eqs. (34) and (35) are exactly the same as those found by solving Eqs. (28) and (29). An elegant mathematical procedure for forcing the moments to agree is the direct quadrature method of moments (DQMOM), and is described in detail in Fox (2003). For the two-environment model, the transport equations are @rp1 þ = rhUip1 ¼ = ðD þ DT Þ=p1 @t @rp1 x1 þ = rhUip1 x1 ¼ = ðD þ DT Þ=p1 x1 þ rgp1 p2 ðx2 x1 Þ @t 2 2 DT þ p1 =x1 þ p2 =x2 x1 x2

(36)

ð37Þ

and @rp2 x2 þ = rhUip2 x2 ¼ = ðD þ DT Þ=p2 x2 þ rgp1 p2 ðx1 x2 Þ @t 2 2 DT þ p1 =x1 þ p2 =x2 x2 x1

ð38Þ

where p2 ¼ 1p1. Summing together Eqs. (37) and (38) and using Eq. (34), the reader can easily show that Eq. (28) is recovered. With a little algebra (Fox, 2003), one can also show using all three equations and Eq. (34) that Eq. (29) will be recovered if we let the micromixing parameter be g¼

R 2k

(39)

Although we will not do so here, with a little more work one can use Eqs. (36)–(38) to ﬁnd the transport equation for hx3 i. The two-environment model thus provides an extra piece of information that can be compared to experimental data. The next step would be to implement the CFD transport equation for the state variables in a CFD code. This is a little more difﬁcult for the two-environment model (due to the gradient terms on the right-hand sides of Eqs. 37 and 38) than for the moment closure. Nevertheless, if done correctly both models will

250

RODNEY O. FOX

predict exactly the same values for the mean and variance of the mixture fraction. (See Wang and Fox (2004) and Liu and Fox (2006) for speciﬁc examples.) The real advantage of the two-environment model comes when dealing with reacting scalars. Unlike the moment method, multi-environment models can easily be extended to multiple reacting scalars with virtually no changes in the model formulation and, more importantly and surprisingly, are often nearly as accurate as more sophisticated closures (Wang and Fox, 2004). We will look at examples of how this is done in Sections III and IV. In summary, we have presented two different SGS models for single-phase turbulent mixing of an inert scalar. The goal of this presentation was not to show the reader the speciﬁc details of how models are derived and tested, but rather to show how a rather complicated physical process can be modeled by adding additional scalars to a CFD model in the form of Eq. (5). Once the equations are in this form, they can be solved in a commercial CFD code for arbitrarily complex reactor geometries. The primary task faced when developing a CFD model for a new reacting system is to develop closures in terms of an appropriate set of state variables. For chemical reaction engineers, the usual starting point will be an existing CFD model for the ﬂuid phase(s), which has been developed and (hopefully!) validated by experts in ﬂuid dynamics. Given such a model for momentum transport, the chemical reaction engineer can focus on the signiﬁcant task of describing local mass/heat-transfer and chemical reactions. Thus, the availability of accurate models for momentum transport is the baseline requirement when faced with a new reactor system, and essentially determines which systems are amenable to CFD.

D. REACTOR SYSTEMS AMENABLE

TO

CFD

It would be difﬁcult to construct an exhaustive list of reactor systems that can be treated to some degree using CFD. However, we can give a partial list with a few examples to illustrate the technical issues. First, the simplest systems to treat with CFD are laminar-ﬂow reactors for which the microscopic transport equation can be solved directly (i.e., no SGS modeling is required.) For such reactors, it is possible to use detailed chemical kinetics in complex ﬂow geometries involving heat and mass-transfer and catalytic surfaces. Nevertheless, even for laminar systems computational difﬁculties can arise, for example, when the working with liquid systems wherein the Schmidt (Prandtl) number is much larger than unity. For such cases the scalar ﬁeld will require a much ﬁner grid than the velocity ﬁeld to fully resolve all of the chemistry and physics (i.e., reaction-diffusion layers). One might therefore consider using a micromixing model for the scalar ﬁelds to describe the molecular mixing below the grid resolution of the velocity ﬁeld. In principle, such a model would have the same form as those used for turbulent reacting ﬂows (see Section III), but with a micromixing rate (or local scalar dissipation rate) found from the local strain-rate tensor of the velocity ﬁeld.

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251

Finally, we can also mention that laminar-ﬂow systems with non-Newtonian ﬂuids often require special numerical algorithms that are usually not available in CFD codes designed mainly for turbulent ﬂows. Turbulent single-phase ﬂow reactors can also be treated quite accurately with current CFD technology. The key issues in this case are the SGS models and the modeling of heat/mass transfer and reactions at ﬂow boundaries. These issues arise in the CFD transport equation due to the inability to resolve the smallest scales of the ﬂow or boundary layers. For turbulent reacting ﬂows, it is now possible to handle relatively complex chemistry. Nevertheless, due to the computational cost, the total number of chemical species that can be transported by a CFD code for a large computational grid is on the order of 10–100. Furthermore, due to numerical stiffness of many kinetic schemes, simply adding a large number of scalar equations coupled through the chemical source terms leads to unrealistically long computing times. It is thus still very much of interest to ﬁnd ‘‘smart’’ methods for reducing the number of transported scalars needed to describe complex chemistry. Several useful methods have been proposed to do this (e.g., using the quasi steady state for free radicals (Kolhapure et al., 2005), but methods based on tabulation in terms of a set of ‘‘progress variables’’ (e.g., Fiorina et al., 2005) appear to be promising for complex gas-phase reaction systems. Difﬁcult complications arise, however, if the chemical reactor has multiple inlets or recycled product streams. Such reactors cannot be classiﬁed as either nonpremixed or premixed (which are the types that can be most easily handled using tabulation), and the number of degrees of freedom in scalar phase space is large enough that it is very difﬁcult to determine a priori an appropriate set of progress variables to describe the ﬂow. It, thus, may be necessary to carry a large number of scalar ﬁelds to describe such reactors, but one can still use tabulation schemes (Raman et al., 2004) to handle the numerical stiffness of the chemical source term. Multiphase reactor systems offer many challenges to CFD modelers. In terms of complexity, ﬂuid–solid systems are more amenable to CFD modeling than gas–liquid systems. Nevertheless, progress has been made in both ﬁelds. For ﬂuid–solid systems, we can distinguish (see Section IV) between reactor systems with ﬁne particles that follow exactly the ﬂuid and systems with solid-particle velocities different than the ﬂuid’s velocity. In the ﬁrst case, the solids can be treated as a scalar ﬁeld advected by the (single-phase) ﬂuid. In the second case, the solid phase must have its own momentum equation that is coupled to the momentum equation for the ﬂuid. The momentum equation for the solid phase requires many modeling assumptions to describe all possible ﬂow regimes. From the point of view of the ﬂuid dynamics, in general, dilute ﬂuid–solid systems (e.g., circulating ﬂuidized beds), dominated by ﬂuid-phase turbulence, are easier to deal with than dense systems (e.g., bubbling ﬂuidized beds). However, the addition of chemical species and reactions is challenging in both cases. In theory, a general CFD code for ﬂuid–solid ﬂows must account for homogeneous reactions occuring in the ﬂuid (or solid) phase and heterogeneous reactions occuring at the interface between phases. Given that the solid phase is very often a complex

252

RODNEY O. FOX

porous material with microchannels that cannot be resolved by the CFD code, it will be necessary to develop subgrid-scale models to describe mass/heat transfer to the particle surface and through the pores. Indeed, in many cases, the solidphase and surface reactions will be mass-transfer limited and the overall conversion predicted by the CFD code will be determined by the model used to describe mass/heat transfer between the ﬂuid and solid phases. Additional complications arise for ﬂuid–solid systems when the solid particles change in size due, for example, to surface growth, breakage, or agglomeration. It is then necessary to include a description of the particle size distribution (Fan et al., 2004). Finally, we should note that although turbulence models have made considerable progress for dilute gas–solid systems (Minier and Peirano, 2001), the same cannot be said for dense systems. As a result, CFD simulations of bubbling ﬂuidized beds are usually done without turbulence models (see Section V) and require relatively ﬁne computational grids to capture integral-scale properties of the ﬂow (e.g., total pressure drop). The high computational cost of such models makes them intractable for analyzing plant-scale ﬂuidized-bed reactors. From the perspective of CFD, the most difﬁcult reactor systems are gas–liquid and liquid–liquid ﬂows. Using the denser phase as the reference phase, such systems range from dilute (e.g., liquid sprays) to dense (e.g., bubbly ﬂow). From the point of view of the ﬂuid dynamics, these systems are challenging because the interface between the phases is deformed by (and deforms) the ﬂow. A completely general CFD model would need to keep track of the ﬂuid velocity in each phase and the location and velocity of the interface. Although it is possible to use this approach for speciﬁc model problems, it is intractable for actual reactor systems where a less-detailed approach must be applied. For example, continuum model can be used that describes gas and liquid as interpenetrating ﬂuids. However, it is then necessary to introduce models for momentum, mass, and energy exchange between the phases that describe the unresolved processes occuring at the phase interphase. Unlike in ﬂuid–solid ﬂows where the interfaces are rigid, in gas–liquid ﬂows the interface can change due to the ﬂuid dynamics and chemical/physical processes occuring at the interface. Moreover, under industrial conditions where the volume fraction of the gas phase is often very high, turbulence and bubble coalescence and breakage must be accounted for in the CFD model (Sanyal et al., 2005). Unfortunately, it is very difﬁcult to validate (and thus to improve) multiphase turbulence models using modern laser-based measurement techniques. CFD models for gas–liquid chemical reactors remain, therefore, the least developed and should be applied with caution for reactor design and analysis.

III. Mixing-Dependent, Single-Phase Reactions CFD models for single-phase chemical reactors are by far the most advanced and widely used in industry. The number of different chemical reactors that can be

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253

modeled by CFD is very broad and ranges for laminar ﬂow reactors with detailed gas-phase chemistry coupled with catalytic-surface chemistry to complex turbulent hydrocarbon ﬂames. For many of the more complicated ﬂow conﬁgurations, specialized CFD codes have been developed to take advantage of the particular characteristics of the ﬂow. Thus, we will not attempt to describe the entire range of ﬂow phenomena that can be predicted using CFD in any detail. For turbulent reacting ﬂows, a recent monograph (Fox, 2003) covering speciﬁc SGS models with abundant references to the literature is recommended as a starting point for anyone wanting to know more about the subject. Here, we will conﬁne our attention to a few speciﬁc examples of single-phase turbulent reacting ﬂows to give the reader a general idea of the key modeling issues in the context of CFD. The topics in this section are arranged in the order of the computational difﬁculty faced when treating the chemical source term. For completeness, we should note that the simplest case, which simply neglects SGS ﬂuctuations of the scalar ﬁelds, will not be discussed for two reasons. First, its implementation in a CFD code is trivial (at least for cases where it is accurate!) and it is the usual default model in a commercial CFD code. Second, since it is accurate in the limit where the reaction rates are slow compared to the ﬂow time scales, CFD is often not required to understand how to scale up chemical reactors with slow chemistry. The discussion here will thus proceed in the opposite direction: starting with very fast chemistry and progressing toward so-called ﬁnite-rate chemistry. As discussed in Fox (2003), the speed of the reactions is taken with respect to the resolved scales of the ﬂuid ﬂow. Thus, in a DNS, the smallest characteristic time scale is the Kolmogorov time scale tZ (see Table I), and a fast reaction occurs on time scales shorter than tZ. In contrast, in RANS simulations the ﬂow time scale is given by the local integral time scale tu. In comparison, for a ‘‘classical’’ CRE model the ﬂow time scale is the residence time, which is typically much larger than tu. Therefore, we can conclude that CFD models will start to have utility for reactor analysis whenever the reaction time scales are smaller than the residence time of the reactor. Before looking at speciﬁc SGS models, we should highlight highly exothermic chemical reactions (e.g., combustion). The CFD models for these systems are complicated by the fact that the reaction rates change dramatically across the ﬂow domain depending on the local temperature. Thus, these systems can behave as not only nonreacting ﬂows under ambient conditions but also inﬁnitely fast reactions once ignited. For this reason, combustion models for premixed and nonpremixed systems are usually formulated very differently (Peters, 2000; Poinsot and Veynante, 2001; Veynante and Vervisch, 2002). In contrast, if we consider fast, nearly isothermal reactions in the liquid phase the range of behaviors is more limited in terms of the observed reaction rates. For example, it does not make sense to discuss CFD models for a premixed acid–base reaction, because once mixed the reaction occurs instantaneously. For this reason, liquid-phase reactions that are sensitive to mixing are almost always operated under nonpremixed conditions. We will thus limit our attention to these cases in the following discussion.

254 A. ACID– BASE

RODNEY O. FOX AND

EQUILIBRIUM CHEMISTRY

Acid–base reactions are the archetypical instantaneous reactions. If we let A denote the acid concentration and B the base concentration, the chemical source term for both the acid and base can be expressed as SðA; BÞ ¼ kAB

(40)

where the rate constant k is extremely large. In essence, acid and base cannot coexist at the same spatial location so that either A ¼ 0 when B40, or B ¼ 0 when A40. These zones with excess acid or base are separated by stoichiometric surfaces whereon A ¼ B ¼ 0. In a CFD simulation of an acid–base reaction it makes no sense to try to solve the problem directly using the chemical source term. Indeed, even if k were only moderately large, including the source term will lead (at best) to very slow convergence. To overcome this difﬁculty, we can introduce a new variable x deﬁned in terms of a linear combination of A and B such that the chemical source term for x is null. Consider an acid–base reaction of the form A þ rB ! P

(41)

The microscopic transport equations for A and B are, respectively, @A þ U =A ¼ = GA =A kAB @t

(42)

@B þ U =B ¼ = GB =B rkAB @t

(43)

and

where G denotes the molecular-diffusion coefﬁcient. Note that if we multiply Eq. (42) by r and then subtract Eq. (43), we can eliminate the chemical source term as follows: @ðrA BÞ þ U =ðrA BÞ ¼ r2 ðGA rA GB BÞ @t

(44)

However, the diffusion term is now more complicated and cannot be closed unless A and B are known separately. As discussed for the general case in Fox (2003), to proceed further we must assume that GAEGB so that Eq. (44) can be written as @ðrA BÞ þ U =ðrA BÞ ¼ Gr2 ðrA BÞ @t

(45)

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255

The applicability of this approximation depends on the relative importance of the convection and the diffusion terms, and it becomes more accurate for cases dominated by convection (i.e., at large Reynolds numbers). As discussed earlier, acid–base reactions are always nonpremixed. For example, a semi-batch reactor could initially be ﬁlled with base at concentration B0 and acid is added with concentration A0. Likewise, a continuous reactor could be run with two feed streams: one for acid and one for base. For both of these examples, the degree of mixing between the acid stream and the rest of the reactor contents can be quantiﬁed by introducing the mixture fraction x, which obeys @x þ U =x ¼ Gr2 x @t

(46)

with boundary conditions x ¼ 1 in the acid inlet stream and x ¼ 0 in the base inlet stream (and inside the reactor at t ¼ 0). For reactors with more than two inlet streams, it is possible to deﬁne a mixture-fraction vector n (Fox, 2003), which obeys Eq. (46) and has N components with the properties 0 xn 1 and SN n¼1 xn ¼ 1. The modeling approaches discussed below for a single mixture fraction component x can thus be extended to n to treat more complex ﬂow conﬁgurations (Fox, 2003). The mixture fraction as deﬁned above describes turbulent mixing in the reactor and does not depend on the chemistry. However, by comparing Eqs. (45) and (46), we can note that they have exactly the same form. Thus, for the acid–base reaction, the mixture fraction is related to rAB by x¼

rA B þ B0 rA0 B0

(47)

and the stoichiometric mixture fraction is given by xst ¼

B0 rA0 þ B0

(48)

Using the fact that A and B cannot coexist at the same spatial location, we then ﬁnd ( ðx xst Þ A0 þ B0 =r if x4xst A¼ (49) 0 otherwise and

( B¼

0 if x4xst ðxst xÞðrA0 þ B0 Þ otherwise

(50)

Thus, the CFD simulation need to only treat the turbulent mixing problem for the mixture fraction. Once x (or its statistics) are known, the acid and base concentrations can be found from Eqs. (49) and (50), respectively.

256

RODNEY O. FOX

The acid–base reaction is a simple example of using the mixture fraction to express the reactant concentrations in the limit where the chemistry is much faster than the mixing time scales. This idea can be easily generalized to the case of multiple fast reactions, which is known as the equilibrium-chemistry limit. If we denote the vector of reactant concentrations by / and assume that it obeys a transport equation of the form @/ þ U =/ ¼ Gr2 / þ Sð/Þ @t

(51)

then in the equilibrium limit we need to only consider the solution to a simpler equation that includes only the chemical source term as follows: d/ ¼ Sð/Þ dt

(52)

In fact, we are only interested in the value of / for t ¼ N subject to initial conditions that depend on the mixture fraction as follows: /0 ðxÞ ¼ x/1 þ ð1 xÞ/2

(53)

where /1 is the reactant concentration vector in the ﬁrst inlet stream (deﬁned by x ¼ 1) and /2 is the reactant concentration vector in the second inlet stream. When formulating the equilibrium-chemistry approximation we implicitly assume that the solution to Eq. (52) for large t depends only on the mixture fraction, and not on the mixing history of the ﬂuid element. For some mixingsensitive reactions (see Section III.B below), this assumption does not hold and the equilibrium-chemistry approximation is not applicable. These reactions are typically irreversible and the ﬁnal product distribution depends on the mixing path in concentration phase space traversed by the ﬂuid particle. In general, the equilibrium-chemistry approximation should only be used for systems of fast reversible reactions. For this case, /N(x) found from solving Eq. (52) will depend only on x. Note that adding a reverse reaction to the acid–base reaction (Eq. 41) discussed above will not change the basic conclusion that A and B can be determined from x. Only the ﬁnal formulae (Eqs. 49 and 50) will change, and these can be found using the methods described in Fox (2003). In a turbulent ﬂow, the local value (i.e., at a point in space) of the mixture fraction will behave as a random variable. If we denote the probability density function (PDF) of x by fx(z) where 0rzr1, the integer moments of the mixture fraction can be found by integration: n

Z

hx i ¼

1

zn f x ðzÞ dz

(54)

0

In most applications, the moments of principal interest are the mean hxi and variance hx02 i ¼ hx2 i hxi2 . The most widely used approach for approximating

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

257

fx is the presumed PDF method wherein the PDF depends only on a small set of moments. For example, the beta PDF can be used and has the functional form f x ð zÞ ¼

ða þ b 1Þ! a1 z ð1 zÞb1 ða 1Þ!ðb 1Þ!

where a and b depend on the mean and variance as follows: hxið1 hxiÞ 1 hx i a 1 and b ¼ a ¼ h xi 02 hx i hx i

(55)

(56)

The spatial dependencies of hxi and hx02 i are found by solving Eqs. (28) and (29), respectively. A typical CFD model for acid–base and equilibrium chemistry solves Eqs. (25)–(29), and then uses Eq. (55) to approximate fx. Once fx is known, the expected values of the reactant concentrations are computed by numerical quadrature from the formula Z 1 h/i ¼ /1 ðzÞf x ðzÞ dz (57) 0

For example, hAi and hBi can be computed using Eqs. (49) and (50), respectively. Note that instead of Eq. (55), we could use the simpler expression for fx given by Eq. (33), which avoids the need for numerical quadrature. In both cases, the mean and variance of the mixture fraction are identical (and thus both models account for ﬁnite-rate mixing effects.) In practical applications, the differences in the predicted values of h/i can often be small (Wang and Fox, 2004). B. CONSECUTIVE-COMPETITIVE

AND

PARALLEL REACTIONS

To go beyond the equilibrium-chemistry limit to consider cases where some of the reaction rates are ﬁnite compared to the ﬂow time scales, we need an efﬁcient method to solve for chemical species with chemical-source terms. The straightforward approach for doing this is to simply solve a transport equation for each chemical species with its corresponding chemical-source term. However, it is often the case that one or more of the reaction steps is very fast compared to the ﬂow time scales, leading to numerical difﬁculties or poor convergence. An elegant method for avoiding this problem is to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables (Fox, 2003). Some typical examples of the reactions that can be treated in this manner are consecutive–competitive reactions: k1

A þ B!R k2

A þ R!S

ð58Þ

258

RODNEY O. FOX

and parallel reactions as follows: k1

A þ B ! P1 k2

A þ C ! P2

ð59Þ

In most applications, the ﬁrst reaction in each set is an acid–base reaction so that k1 is very large. For Eq. (59), B and C are premixed and added to A under conditions such that B is in stoichiometric excess to A. Likewise, for Eq. (58), B is reacted in stoichiometric excess with A to produce the desired product R. Under these conditions, the ﬁrst reaction in each set is favored. However, if mixing occurs with the same time scale as the second reaction, the undesired byproduct (S in Eq. (58) and P2 in Eq. (59)) will be produced. Thus, the amount of by-product produced is a sensitive measure of the quality of mixing in the chemical reactor. The description of Eqs. (58) and (59) in terms of the mixture fraction and reaction-progress variables is described in detail by Fox (2003). Here we will consider a variation of Eq. (59) wherein the acid acts as a catalyst in the second reaction (Baldyga et al., 1998): k1

A þ B ! P1 k2

A þ C ! A þ P2

ð60Þ

This parallel reaction set was used, for example, by Johnson and Prud’homme (2003a) to investigate the quality of mixing in a conﬁned impinging-jets reactor. Following the steps outline in Fox (2003), the reactant concentrations in Eq. (60) can be written in terms of the mixture fraction x and two reaction-progress variables Y1 and Y2 as cA ¼ A0 ½1 x ð1 xs1 ÞY 1

(61)

cB ¼ B0 ðx xs1 Y 1 Þ

(62)

cC ¼ C0 ðx xs2 Y 2 Þ

(63)

and

where the two stoichiometric mixture fractions are xs1 ¼

A0 A0 þ B0

and xs2 ¼

A0 A0 þ C0

(64)

and A0, B0, and C0 are the inlet concentrations of reactants A, B, and C, respectively. Note that in the absence of chemical reactions, the reactionprogress variables are deﬁned such that Y1 ¼ Y2 ¼ 0.

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259

The microscopic transport equations for the reaction-progress variables can be found from the chemical species transport equations by generalizing the procedure used above for the acid–base reactions (Fox, 2003). If we assume that GAEGBEGC, then the transport equations are given by @Y a þ U =Y a ¼ Gr2 Y a þ S a ðx; Y 1 ; Y 2 Þ @t

for a ¼ 1; 2

(65)

where the chemical-source terms are k1 1x x cA cB ¼ B0 xs1 k1 Y1 Y1 S 1 ðx; Y 1 ; Y 2 Þ ¼ 1 xs1 xs1 B0 xs1

(66)

k2 1x x cA cC ¼ B0 xs1 k2 Y1 Y2 S 2 ðx; Y 1 ; Y 2 Þ ¼ 1 xs1 xs2 C 0 xs2

(67)

and

Note that since the reaction rates must always be nonnegative, the chemically accessible values of the reaction-progress variables will depend on the value of the mixture fraction. We will discuss this point further by looking next at the limiting case where the rate constant k1 is very large and k2 is ﬁnite. In many applications, due to the large value of k1, the ﬁrst reaction is essentially instantaneous compared to the characteristic ﬂow time scales. Thus, if the transport equation is used to solve for Y1, the chemical-source term S1 will make the CFD code converge slowly. To avoid this problem, Y1 can be written in terms of x by setting the corresponding reaction-rate expression (S1) equal to zero as follows:

Y 11

x 1x ¼ min ; xs1 1 xs1

(68)

It is then no longer necessary to solve a transport equation for Y1 and the numerical difﬁculties associated with treating the ﬁrst reaction with a ﬁnite-rate chemistry solver are thereby avoided. As with the acid–base reaction, Eq. (68) implies that A and B cannot coexist at any point in the ﬂow. Using this inﬁnite-rate approximation, we need only solve transport equations for x and Y2, where the source term for Y2 is now S21 ðx; Y 2 Þ ¼ B0 xs1 k2

1x Y 11 1 xs1

x Y2 xs2

(69)

Note that S2N must be nonnegative, and thus the expression above only holds for x and Y2 values that satisfy this condition. For all other values, S2N is null.

260

RODNEY O. FOX

Inaccessible region

Y2max

'mixing' line

Y2

non-zero source term 0

ξ s1

0

1

ξ FIG. 5. Region in x–Y2 phase space with non-zero chemical source term and the mixing line.

Applying Eq. (68), we ﬁnd that when S2N is nonzero, it equals x x Y2 S 21 ðx; Y 2 Þ ¼ A0 k2 1 xs1 xs2 if 0 x xs1 and 0 Y 2 x=xs2

ð70Þ

The region in x–Y2 composition space where this chemical source term is nonzero is shown in Fig. 5. Note that the maximum conversion of C occurs when x ¼ xs1 and corresponds to Y2max ¼ xsl/xs2 or (using Eq. 63) to cC ¼ 0 (i.e., complete conversion). As mentioned earlier, in applications of Eq. (60) the reactor is operated with excess B so that xs1o1/2. If the mixing in the reactor is good, the mixture fraction in all ﬂuid particles at the exit of the reactor will be equal to the mean ðx ¼ hxi41=2Þ. Thus, if mixing were much faster than the characteristic reaction time of Eq. (70) ((A0k2)1), then the chemical-source term in Eq. (70) would be zero and no reaction would occur so that Y2 ¼ 0 at the exit. In contrast, any local deviations from perfect mixing can lead to zones in the reactor where xrxs1 and hence to the production of the by-product Y2. In the opposite limit where k2 is large compared to the mixing rates, the maximum attainable value for Y2 when xs1rxr1 is the mixing line (Fox, 2003), deﬁned by Y 2mix ðxÞ ¼ Y 2 max

1x 1 xs1

for xs1 x 1

(71)

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

261

and shown as a dashed line in Fig. 5. Using this expression, we ﬁnd that the maximum attainable conversion is

X max ¼

¯ xs1 ð1 xÞ for xs1 x¯ 1 ¯ xs1 Þ xð1

(72)

The accessible region in xY2 phase space for the reaction given in Eq. (60) is represented by the triangular region in Fig. 5 found by connecting the feed points and the maximum conversion point. Phase-space trajectories begin at the two feed streams [stream 1: (0; 0) and stream 2: (1; 0)]. If xs1 is less than the outlet value of the mixture fraction, then the amount of by-product formed is determined by the amount of time spent in the region with nonzero source term (tmix) and the characteristic time of the second reaction (tr). If tr is large compared to tmix, then the by-product concentration will be near zero. If the inverse is true, then the by-product concentration will be near Xmax. The CFD model for the reaction given in Eq. (60) in the limit where the ﬁrst reaction is very fast must account for ﬂuctuations in x and Y2 due to turbulent mixing. In general, this is done by solving for their joint PDF (Fox, 2003), denoted here by f (z, y). There are several ways this can be accomplished: 1. Solve the joint PDF transport equation. 2. Assume a functional form for the joint PDF. 3. Assume a functional form for the conditional PDF of Y2 given x and use a presumed PDF for fx. Method 1 will be discussed in Section III.C. Method 3 can be implemented in several different forms (Baldyga, 1994; Klimenko and Bilger, 1999; Fox and Raman, 2004), but the lowest order approximation requires a model for the conditional expected value of Y2 given that x ¼ z (denoted by hY 2 jzi) where hY 2 j0i ¼ hY 2 j1i ¼ 0. By deﬁnition, hY 2 jzi will be a single-valued function of mixture fraction and will lie in the triangular region in Fig. 5. The simplest such model is the one proposed by Baldyga (1994), which uses a linear-interpolation procedure to ﬁnd the conditional moment from the unconditional moment hY 2 i (Fox, 2003). Here we will look at a multi-environment model that is based on method 2. The multi-environment model for the joint PDF generalizes Eq. (33) by writing f ðz; y; x; tÞ ¼

N X

pn ðx; tÞ dðz xn ðx; tÞÞ dðy Y 2n ðx; tÞÞ

(73)

n¼1

where Y2n is the value of Y2 corresponding to environment n. Here we will consider only the two-environment model (N ¼ 2) where the CFD models for

262

RODNEY O. FOX

p1, x1 and x2 are given by Eqs. (36), (37), and (38), respectively. Similarly, the CFD models for the reaction-progress variable in the two environments are @rp1 Y 21 þ = rhUip1 Y 21 ¼ = ðD þ DT Þ=p1 Y 21 þ rp1 S21 ðx1 ; Y 21 Þ @t DT þrgp1 p2 ðY 22 Y 21 Þ þ p1 j=Y 21 j2 þ p2 j=Y 22 j2 Y 21 Y 22

ð74Þ

and @rp2 Y 22 þ = rhUip2 Y 22 ¼ = ðD þ DT Þ=p2 Y 22 þ rp2 S21 ðx2 ; Y 22 Þ @t DT þrgp1 p2 ðY 21 Y 22 Þ þ p j=Y 21 j2 þ p2 j=Y 22 j2 Y 22 Y 21 1

ð75Þ

Except for the chemical source term, these equations have the same form as those used for the mixture fraction. Note that the chemical source term (S2N) is evaluated using the mixture fraction and reaction-progress variable in the particular environment. The average chemical source term hS 21 ðx; Y 2 Þi will thus not be equal to S 21 ðhxi; hY 2 iÞ unless micromixing occurs much faster than the second reaction. The CFD model for the reaction given in Eq. (60) with k1 ¼ N has state variables ~ 2 hUi; k; ; p1 ; x1 ; x2 ; Y 21 ; Y 22 F By deﬁnition of the reaction-progress variables, Y21 and Y22 are zero for the inlet streams, and nonnegative inside the reactor due to the chemical source term. Once the CFD model has been solved, the reactant concentrations in each environment n are found from cAn ¼ A0 ½1 xn ð1 xs1 ÞY 1n

(76)

cBn ¼ B0 ½xn xs1 Y 1n

(77)

cCn ¼ C0 ðxn xs2 Y 2n Þ

(78)

and

where

Y 1n

x 1 xn ¼ min n ; xs1 1 xs1

(79)

The mean reactant concentrations are then deﬁned by (p2 ¼ 1p1) hcA i ¼ p1 cA1 þ p2 cA2

(80)

hcB i ¼ p1 cB1 þ p2 cB2

(81)

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

263

hcC i ¼ p1 cC1 þ p2 cC2

(82)

and

The overall conversion of C, denoted by X, is computed using X ¼1

hc C i C 0 h xi

(83)

5

5

4

4

3

3

2

2 Z(mm)

Z(mm)

The value of X at the reactor outlet is a sensitive measure of the degree of mixing in the reactor. If X51, then mixing in the reactor is rapid compared to the second reaction in Eq. (60). In contrast, if XE1, then mixing is slow. The CFD model described above has been used by Liu and Fox (2006) to simulate the experiments of Johnson and Prud’homme (2003a) in a conﬁned impinging-jets reactor. In these experiments, two coaxial impinging jets with equal ﬂow rates are used to introduce the two reactant-streams. The jet Reynolds number Rej determines the ﬂuid dynamics in the reactor. Typical CFD results are shown in Fig. 6–9 for a jet Reynolds number of Rej ¼ 400 and a reaction time of tr ¼ 4.8 msec. The latter is controlled by ﬁxing the inlet concentrations of the reactants. Further, details on the reactor geometry and the CFD model can be found in Liu and Fox (2006).

1 p1

0

-2 -2

-1

0 X(mm)

1

p2

0

1.0 0.9 0.8 0.7 0.6 0.5 0.5 0.4 0.3 0.2 0.1 0.0

-1

1

1.0 0.9 0.8 0.7 0.6 0.5 0.5 0.4 0.3 0.2 0.1 0.0

-1 -2 2

-2

-1

0

1

2

X(mm)

FIG. 6. Volume fractions p1 and p2 in the cross-section of the conﬁned impinging-jets reactor.

264 5

5

4

4

3

3

2

2 Z(mm)

Z(mm)

RODNEY O. FOX

1 ξ1

0

-2 -2

-1

0

1

ξ2

0

1.00 0.91 0.82 0.73 0.64 0.55 0.45 0.36 0.27 0.18 0.09 0.00

-1

1

1.00 0.91 0.82 0.73 0.64 0.55 0.45 0.36 0.27 0.18 0.09 0.00

-1 -2

2

-2

-1

X(mm)

0

1

2

X(mm)

5

5

4

4

3

3

2

2 Z(mm)

Z(mm)

FIG. 7. Mixture fractions x1 and x2 in the cross-section of the conﬁned impinging-jets reactor.

1 Y21

0

0.67 0.61 0.55 0.49 0.43 0.37 0.30 0.24 0.18 0.12 0.06 0.00

-1 -2 -2

-1

0 X(mm)

1

2

1 Y22

0

0.67 0.61 0.55 0.49 0.43 0.37 0.30 0.24 0.18 0.12 0.06 0.00

-1 -2 -2

-1

0

1

2

X(mm)

FIG. 8. Reaction-progress variables Y21 and Y22 in the cross-section of the conﬁned impinging-jets reactor.

5

5

4

4

3

3

2

2 Z(mm)

Z(mm)

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

1 cC1

0 -1 -2 -2

-1

0 X(mm)

1

1 cC2

0

619 563 506 450 394 338 281 225 169 113 56 0

619 563 506 450 394 338 281 225 169 113 56 0

-1 -2 2

265

-2

-1

0

1

2

X(mm)

FIG. 9. Reactant concentrations cC1 and cC2 in the cross-section of the conﬁned impinging-jets reactor.

In Fig. 6, the volume fractions for each environment p1 and p2 are shown in a cross-section of the reactor, which includes the inlet jets and the outlet tube. Note that because p2 ¼ 1–p1 and the inlet ﬂow rates are equal, the contour plots are symmetric about the vertical axis. For the same reason, p1 ¼ p2 ¼ 1/2 on the vertical axis. In the left-hand inlet stream p1 ¼ 1, corresponding to reactant A. In the right-hand inlet stream p2 ¼ 1, corresponding to reactants B and C. On the outlet cross-section mixing is nearly complete so that p1Ep2. Finally, note that since Eq. (36) does not contain a term for micromixing, the distribution of p1 and its deviation from 1/2 measures the degree of macromixing at any point in the reactor. We can therefore conclude the reactor is fairly well macromixed everywhere except in the region near the inlet jets. In Fig. 7, the mixture fractions in each environment x1 and x2 are shown. By deﬁnition of the inlet conditions, in the inlet tubes x1 ¼ 0 and x2 ¼ 1. The variations away from the inlet values represent the effect of micromixing. For example, if we set g ¼ 0 in Eqs. (36) and (37) to eliminate micromixing, then x1 and x2 would remain at their inlet values at all points in the reactor. Note that the spatial distributions of x1 and x2 are antisymmetric with respect to the vertical axis (as would be expected from the initial conditions.) In the outlet tube, x1 and x2 are very near the perfectly micromixed value of 1/2. Finally, by comparing Fig. 6 and Fig. 7, we can observe that macromixing occurs slightly faster than micromixing in this reactor (i.e., pn are closer to their outlet values than are xn.)

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RODNEY O. FOX

The results shown in Figs. 6 and 7 can be combined to compute the mean mixture fraction hxi and its variance hx02 i from Eqs. (34) and (35), respectively. Example plots are shown in Liu and Fox (2006) and, as expected, they agree with the solution found by solving the moment transport equations directly (Eqs. 28 and 29). In Fig. 8, the reaction-progress variables in each environment Y21 and Y22 are shown. By deﬁnition of the inlet conditions, in the inlet tubes Y21 ¼ Y22 ¼ 0. Recall that Y2 is produced by the second (ﬁnite-rate) reaction in Eq. (60). Thus, as observed in the plots, it is largest in zones in the reactor where A is in excess. The largest values are found near the left wall of the reactor where the convective velocity is low and A is in slight stoichiometric excess. The residence time for ﬂuid particles in this region is relatively long compared to the reaction time. In general, Y21 is larger than Y22, which is easily explained by the fact that A enters the reactor in environment one. Finally, note that at the reactor outlet Y2 is not uniformly mixed across the tube. Thus, despite the high energy dissipation in the reactor (as measured by the pressure drop), the macromixing at the outlet is not complete (Liu and Fox, 2006). In Fig. 9, the distribution of reactant C is shown in each environment. As cC is a linear combination of x and Y2 (Eq. 78), we can distinguish features of both Fig. 7 and Fig. 8 in the plots in Fig. 9. In particular, because C is injected in the right-hand inlet stream, cC2 and x2 appear to be quite similar. Finally, as shown in Liu and Fox (2006), the CFD predictions for the outlet conversion X are in excellent agreement with the experimental data of Johnson and Prud’homme (2003a). For this reactor, the local turbulent Reynolds number ReL is relatively small. The good agreement with experiment is thus only possible if the effects of the Reynolds and Schmidt numbers are accounted for using the correlation for R shown in Fig. 4. Further details on the simulations and analysis of the CFD results can be found in Liu and Fox (2006). The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called ‘‘simple chemistry’’ (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reactionprogress variablereaction-progress variable –4 depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces: (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of ‘‘elementary’’ reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003).

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C. DETAILED CHEMISTRY In a CFD model with detailed chemistry, the user must provide a chemical mechanism involving K chemical species Ab of the form (Fox, 2003) K X b¼1

kfi

K X

ki

b¼1

ufbi Ab Ðr

urbi Ab

for i 2 1; . . . ; I reactions

(84)

The rate constants (kfi and kri ) and the stoichiometric coefﬁcients (ufbi and urbi ) are all assumed to be known. Likewise, the reaction rate functions Ri for each reaction step, the equation of state for the density r, the speciﬁc enthalpies for the chemical species Hk, as well as the expression for the speciﬁc heat of the ﬂuid cp must be provided. In most commercial CFD codes, user interfaces are available to simplify the input of these data. For example, for a combusting system with gas-phase chemistry, chemical databases such as Chemkin-II greatly simplify the process of supplying the detailed chemistry to a CFD code. The reaction rates Ri will be functions of the state variables deﬁning the chemical system. While several choices are available, the most common choice of state variables is the set of species mass fractions Yb and the temperature T. In the literature on reacting ﬂows, the set of state variables is referred to as the composition vector /: / ¼ ½Y 1 ; . . . Y K ; T T where the mass fractions sum to unity: Y1+?+YK ¼ 1. The microscopic balance equation for the composition vector has the form of Eq. (1) (Bird et al., 2002). For a turbulent reacting ﬂow, the CFD transport equation will thus have the form of Eq. (3) after averaging. With detailed chemistry, the most difﬁcult term to close in the CFD transport g As described in detail equation will be the averaged chemical source term Sð/Þ: in Fox (2003), the chemical source term depends on the reaction rates Ri, which can be highly nonlinear in the composition vector /. For this reason, simple closures that neglect correlations between components of the composition vector are usually inaccurate. Nevertheless, the default closure for detailed chemistry in most commercial CFD is the so-called laminar-chemistry g ¼ Sð/Þ. ~ In words, this closure approximates the average approximation: Sð/Þ chemical source term by its value evaluated at the average composition vector. In general, the laminar-chemistry approximation overpredicts the reaction rate of the principal reactants, which in reality will be reduced by ﬁnite-rate turbulent mixing (Fox, 2003). The simplest example is the reaction rate for the g ¼ 0, which is null because acid and base do not acid–base reaction Se ¼ kAB ~B ~ which coexist. The laminar-chemistry approximation for this case is Se ¼ kA e e forces either A or B to be null for large k. In reality, this is usually not the case in a turbulent ﬂow unless hx02 i ¼ 0.

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RODNEY O. FOX

To represent correlations between the components of /, we can introduce the joint composition PDF denoted by f/(w), where w is the sample-space composition vector (Fox, 2003). Starting from the microscopic transport equation for /, it is possible to derive a CFD transport equation for f/ (w; x, t), in which the chemical source term appears in closed form. This CFD transport equation is the starting point for transported PDF methods. In transported PDF methods, closures are required for turbulent dispersion and molecular mixing. However, once such closures are introduced into the CFD transport equation, we are still faced with the computational challenge of dealing with a large number of independent variables (w; x, t). One method for overcoming this difﬁculty is to use Monte–Carlo simulation techniques wherein f/ is represented by a ﬁnite number of so-called notional particles (Fox, 2003). Typically, 50–100 notional particles are needed for each CFD grid cell to accurately capture the correlations between the components of / and to control statistical errors. For comparison, the laminar-chemistry approximation uses in effect one notional particle per grid cell. The transported PDF method will therefore be more computationally demanding, but represents the state of the art for treating detailed chemistry coupled to turbulent mixing. Reader interested in further details on PDF methods and the corresponding simulation codes are advised to consult Fox (2003). There are many reported applications of transported PDF methods to combusting systems (see, for example, results for a turbulent diffusion ﬂame in Raman et al. (2004), and Fox (2003) for other references). Applications to chemical process systems are rarer, but some recent examples are Raman et al. (2003) and Liu et al. (2004). The relatively high cost of transported PDF methods has led us to explore lower-cost methods for approximating the CFD transport equation for f/ (Wang and Fox, 2004). In principle, transported PDF simulations can be made cheaper by using only a small number of notional particles N. However, the random ﬂuctuations p introduced by the simulation method lead to statistical errors that ﬃﬃﬃﬃﬃ scale like 1= N : Thus, when N is small the statistical error is quite signiﬁcant. Ideally, we would like to have an acceptably accurate method that works for N as small as one, and whose accuracy increases for larger N in a well-deﬁned deterministic manner. Our method for accomplishing this task is called the direct quadrature method of moments (DQMOM) (Fox, 2003; Wang and Fox, 2004; Marchisio and Fox, 2005). We brieﬂy describe the resulting CFD model below. Readers interested in more details should consult the references given above. The application of DQMOM to the closed composition PDF transport equation is described in detail by Fox (2003). If the IEM model is used to describe micromixing and a gradient-diffusivity model is used to describe the turbulent ﬂuxes, the CFD model will have the form @rpn þ = rhUipn ¼ = ðD þ DT Þ=pn @t for n ¼ 1; . . . ; N

ð85Þ

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

269

@rpn fan þ = rhUipn fan ¼ = ðD þ DT Þ=pn fan þ rpn Sa /n @t þ rpn g hfa i fan þ rban

ð86Þ

and

where the subscript n denotes the environment and the subscript a denotes the component of the composition vector. Thus, pn is the mass fraction of environment n, and /n is the composition vector in environment n. The reader will recognize these equations as an N-environment generalization of the two-environment model introduced earlier. Note that, as in transported PDF simulations, the chemical source term Sa(/n) appears in closed form in Eq. (86). By deﬁnition, the sum of the mass fractions is unity: p1+?+pN ¼ 1. Thus, one of the equations for pn in Eq. (85) is redundant. In Eq. (86), fan is one of the K+1 components of the composition vector in environment n. The mean composition hfa i appearing in the micromixing model is deﬁned by hfa i ¼

N X

pn fan

(87)

n¼1

Note that if the mass fractions are used to deﬁne the composition vector, then by deﬁnition K X

fan ¼ 1

(88)

a¼1

where the sum is over all the K chemical species. This implies that the sum of Eq. (86) over all species must yield Eq. (85), and thus that one of the chemical species equations is redundant. Redundancy can be avoided in the CFD model by solving Eq. (85) only for n ¼ 1,y, N1, and Eq. (86) for n ¼ 1,y, N but with only K1 mass fractions. To avoid numerical errors, the mass fraction of the species not solved for should be relatively large and, preferably, correspond to a chemically inert species. The composition vector will then have K (including temperature) components, and a total of N(K+1)1 transport equations must be solved to represent the model in the CFD code. Alternatively, since the sum of the chemical source term in Eq. (86) over all chemical species is null, the CFD model can solve only Eq. (86) for a ¼ 1,y, K+1 and not use Eq. (85). This leads to N(K+1) equations (i.e., one more than required1), but has the beneﬁt that all of the transport equations have the same form. The ﬁnal term in Eq. (86) is the correction term ban ; which comes from applying DQMOM to the transport equation for the composition PDF (Fox, 1

The ‘‘extra’’ equation is the mass continuity equation for r.

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RODNEY O. FOX

2003). This term is computed by solving a linear system of equations deﬁned for each a ¼ 1,y, K+1 by N X

fm1 an ban ¼

n¼1

N X 2 ðm 1Þpn fm2 an DT j=fan j n¼1

for m ¼ 1; . . . ; N

ð89Þ

This expression results from the transport equations for the ﬁrst N forcing moments of fa, denoted by fm a ; to agree with the composition PDF transport equation (Fox, 2003). For example, with N ¼ 2 the linear system in Eq. (89) can be written in matrix form as " #" # " # 0 ba1 1 1 2 2 ¼ (90) p1 DT =fa1 þ p2 DT =fa2 ba2 fa1 fa2 Solving for ban yields ba1 ¼

2 2 DT p1 =fa1 þ p2 =fa2 fa2 fa1

(91)

ba2 ¼

2 2 DT p1 =fa1 þ p2 =fa2 fa2 fa1

(92)

and

The reader will recognize these terms as having of the same form as the correction terms in the two-environment model discussed earlier. With N ¼ 1, ba1 ¼ 0 and the model reduces to the laminar-chemistry approximation. With N ¼ 2, additional information is obtained concerning the second-order moments of the composition vector. Likewise, by using a larger N, the Nth-order moments are controlled by the DQMOM correction terms found from Eq. (89). As noted earlier, the sum of the mass fractions is unity and thus Eq. (86) will be consistent with Eq. (85) only if the sum of the correction term ban over all chemical species a ¼ 1,y, K is null. In general, this will not be the case if Eq. (89) is used. Another difﬁculty that can arise is that the mass fractions in two environments may be equal, e.g., fa1 ¼ fa2, and thus the coefﬁcient matrix in Eq. (89) will be singular. This can occur, for example, in the equilibrium-chemistry limit where the compositions depend only on the mixture fraction, i.e., / ¼ /N(x). For chemical species that are not present in the feed streams, the equilibrium values for x ¼ 0 and x ¼ 1 are zero, but for intermediate values of the mixture fraction, the equilibrium values are positive. This implies that the equilibrium values will be the same for at least two values of the mixture fraction in the range 0oxo1. Thus, in the equilibrium limit it is inevitable that two environments will have equal mass fractions for certain species at some point in the ﬂow ﬁeld. Since singularity implies an underlying correlation between

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

271

components of the composition vector, in most cases these difﬁculties can be overcome by computing the correction terms using a set of moments that is mp p different than hfm fb i. For example, a i (i.e., one can include cross moments hfa one possible choice of moments that also ensures that the sum of the correction terms is null is to use the cumulative mass fractions. We will describe this choice next, but the reader should keep in mind that other choices may be required if the coefﬁcient matrix in Eq. (89) becomes singular. Let Ya denote the mass P fractions of the K chemical species describing the reacting ﬂow. By deﬁnition, K a¼1 K Y a ¼ 1: Assuming that the chemical species are numbered such that the major species (e.g., reactants) appear ﬁrst,2 followed by the minor species (e.g., products), we can deﬁne a linear transformation by Xb ¼

b X

Ya

for b ¼ 1; . . . ; K

(93)

a¼1

Note that by deﬁnition XK ¼ 1 and thus Xb is the cumulative mass fraction of the ﬁrst b species. The inverse transformation corresponding to Eq. (93) is Y1 ¼ X1

and

Y a ¼ X a X a1

for a 2

(94)

The correction terms can be computed using either Ya or Xa. However, it is clear that the correction terms will depend on which choice is used since the moments m controlled by DQMOM are different (i.e., hY m a i vs. hX a i). Thus, for example, with ﬁxed N one set of moments may lead to singular correction terms, but not the other. In general, we can continue to solve for the mass fractions in the CFD model, but with the correction terms computed using the cumulative mass fractions as follows. 1. Given the mass fractions in each environment Yan, use Eq. (93) to compute the cumulative mass fractions Xan. 2. Use the integer moments of Xa to compute the correction terms as follows: N X

þ X m1 an ban ¼

n¼1

N X

2 ðm 1Þpn X m2 an DT jrX an j

n¼1

for m ¼ 1; . . . ; N

ð95Þ

3. Compute the correction terms for the mass fractions ban using Eq. (94) as follows: b1n ¼ bþ 1n

and

þ ban ¼ bþ an ba1n

for a 2

(96)

2 In the computer code, a sorting algorithm can be used to put the mean mass fractions hY a i in descending order before deﬁning Xb. By keeping track of the order of the indices, one can easily deﬁne the inverse transformation needed to compute Ya from Xb.

272

RODNEY O. FOX

Note that since XKn ¼ 1, Eq. (95) leads to the degenerate case where bþ Kn ¼ 0, and thus the sum of the correction terms over all species is null. Also, if one of the mass fractions is null3 (say Yg ¼ 0) then Xg ¼ Xg1 and thus the correction term for such mass fractions will be null due to Eq. (96). As discussed in detail in Wang and Fox (2004), on the one hand, for nonreacting systems the DQMOM approach with N environments will exactly reproduce the moment equations for each chemical species up to order N. However, N cannot be chosen to be too large because the coefﬁcient matrix in Eq. (89) can become poorly conditioned. In contrast, the moments estimated from the transported PDF simulations will have statistical ﬂuctuations that can only be reduced by time/ensemble averaging. In this sense, DQMOM is preferable. On the other hand, for reacting systems the moments of chemical species obey transport equations that contain unclosed averages involving the nonlinear chemical source term. In both approaches, these averages are approximated by

fm1 Sa ð/Þ a

Z ¼

cm1 Sa ðwÞf / dw a

N X

pn fm1 an S a /n

(97)

n¼1

where in a transported PDF code pn is the particle weight and /n is the particle composition vector (Fox, 2003). The integral in Eq. (97) is just the deﬁnition of the expected value. It can be seen that the integral is replaced by a ﬁnite sum over a set of N ‘‘quadrature’’ points. Thus, the accuracy of the moments will depend on the degree of nonlinearity of S(/). For a weakly nonlinear source term, a low-order quadrature method may be adequate and DQMOM would be preferred. Such cases have been successfully investigated by Wang and Fox (2004). In contrast, for strong nonlinearities (e.g., typical of combusting systems), a large value of N may be needed to approximate the integral in Eq. (97) and transported PDF simulations would be preferred. In practice, it may be difﬁcult to determine in advance which method is best to use for a particular application. For example, the CFD results may be more sensitive to large-scale inhomogeneities in the ﬂow ﬁeld than to the chemical source term closure. A rational approach to determine whether a more detailed SGS model is needed might be to start with N ¼ 1 (laminar-chemistry approximation) and compare the predicted mean chemical species ﬁelds to the twoenvironment model (N ¼ 2). If the differences are small, then the simpler model is adequate. However, if the differences are large, then the CFD simulation can be repeated with N ¼ 3 and the results compared to N ¼ 2. Naturally, once this procedure has ‘‘converged,’’ it will still be necessary to validate the CFD results with experimental data whenever possible. Indeed, it may be necessary to 3

If the mass fractions are sorted in descending order, then all of the null mass fractions will be grouped together at the end of the array. The procedure can thus be simpliﬁed by using only the nonzero mass fractions to deﬁne Xb. In practice, one can deﬁne a lower limit for Ya and set the correction term to zero for mass fractions below this limit.

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273

improve the turbulent transport and micromixing models before satisfactory agreement is obtained even with the state-of-the-art transported PDF method.

IV. Production of Fine Particles The CFD models discussed in the previous section considered mass balances for a ﬁnite number of chemical species. In this section, we will extend these models to include systems wherein a second phase is produced. Such systems include aerosols (Friedlander, 2000; Wright et al., 2001), reactive precipitation (Pohorecki and Baldyga, 1983; Garside and Tavare, 1985; Pohorecki and Baldyga, 1988; Villermaux and David, 1988; Marcant and David, 1991; Mahajan and Kirwan, 1993; David and Marcant, 1994; Seckler et al., 1995; Mahajan and Kirwan, 1996; Aoun et al., 1999; Johnson and Prud’homme, 2003b,c), colloids (Oles, 1992; Sandku¨hler et al., 2003, 2005; Waldner et al., 2005), and ﬂame synthesis of nanoparticles (Kodas et al., 1987; Akhtar et al., 1991; Xiong and Pratsinis, 1991; Kruis et al., 1993; Pratsinis et al., 1996; Zhu and Pratsinis, 1997; Briesen et al., 1998; Pratsinis, 1998; Kammler and Pratsinis, 1999, 2000; Kammler et al., 2001; Mueller et al., 2004a,b; Tani et al., 2004a,b) (to name just a few examples). We will limit the discussion to systems wherein the particles are ‘‘not too large,’’ in other words, to particles that on average follow the local ﬂuid velocity (Davies, 1949). Systems with larger particles will be considered in Section V. In terms of the CFD model, the primary difference between ‘‘ﬁne’’ and ‘‘large’’ particles is that the former can be treated as pseudo species by extending the mass balances and chemical kinetic expressions that we have already considered (Piton et al., 2000; Johannessen et al., 2000, 2001). In contrast, for larger particles, a separate momentum balance is required for each phase (Fan et al., 2004), which signiﬁcantly complicates the solution procedure used to solve the CFD model. The deﬁnition of a ‘‘ﬁne’’ particle can be made more quantitative by introducing the article Stokes number St, which measures the particle-to-ﬂuid response-time ratio to changes in the velocity (Fuchs, 1964): St ¼

grs d 2s 12rf nf

(98)

In this deﬁnition, rs and rf are the solid and ﬂuid densities, respectively. The characteristic diameter of the particles is ds (which is used in calculating the projected cross-sectional area of particle in the direction of the ﬂow in the drag law). The kinematic viscosity of the ﬂuid is nf and g is a characteristic strain rate for the ﬂow. In a turbulent ﬂow, g can be approximated by 1/tZ when ds is smaller than the Kolmogorov length scale Z. (Unless the turbulence is extremely intense, this will usually be the case for ﬁne particles.) Based on the Stokes

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RODNEY O. FOX

number, we can neglect the momentum equation for the solid phase when Sto0.14 (Dring, 1982). Note that for systems with growing or aggregating particles, the characteristic diameter of the largest particles should be used to compute the Stokes number. Thus, as a general rule of thumb, in a liquid-phase turbulent ﬂow solid particles (with rsErf) will follow the ﬂuid if dsrZ. Since typical minimum values for the Kolmogorov length scale in practical systems are in the range 10–100 mm, nucleation and growth of nanoparticles and crystals, and colloidal aggregation can all be treated as systems involving ﬁne particles. A fundamental modeling challenge that arises when dealing with particulate systems is the need to describe the particle size distribution (PSD). Depending on the application (Randolph and Larson, 1988; Ramkrishna, 2000), this is done by deﬁning a population balance equation (PBE) governing a number density function (NDF) n(l; x, t). The latter is deﬁned by an ‘‘internal coordinate’’ l, which may correspond to mass, volume, or (as done here) length.4 In words, n(l; x; t, is the number concentration of particles with lengths in the range [l, l+dl]) and thus it has units of (number)/(volume length). The NDF is very similar to the PDFs introduced in the previous section to describe turbulent reacting ﬂows. However, the reader should not confuse them and must keep in mind that they are introduced for very different reasons. The NDF is in fact an extension of the ﬁnite-dimensional composition vector / to the case of an inﬁnite number of scalars (parameterized by 0ploN). Thus, even in the case of laminar ﬂow where the PDFs are not needed, the NDF still introduces an extra dimension (l) to the problem description. The choice of the state variables in the CFD model used to solve the PBE will depend on how the internal coordinate is ‘‘discretized.’’ Roughly speaking (see Ramkrishna (2000) for a more complete discussion), there are two approaches that can be employed: 1. Sectional methods that represent the NDF by a ‘‘histogram’’ (Kumar and Ramkrischna, 1996). 2. Quadrature methods that approximate integral constraints (e.g., moments) of the NDF (McGraw, 1997). For cases with only one internal coordinate, either approach can be implemented in a CFD code (but the computational cost for the same accuracy can be very different). However, for cases with more than one internal coordinate, only the quadrature methods are computationally tractable on current computers. Thus, in the examples below, we will describe only CFD models based on the 4

The preferred choice of internal coordinates is discipline dependent. Nevertheless, the conservation of solid mass will imply constraints on particular moments of the PSD. In general, given the relationship between the various choices of coordinates, it is possible (although not always practical) to rewrite the PBE in terms of any choice of internal coordinate.

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275

quadrature approach. Details on particular applications of this approach can be found in our recent publications (Marchisio et al., 2001a,b, 2003a,b; Wang and Fox, 2003; Wang et al., 2005a).

A. MIXING-DEPENDENT NUCLEATION

AND

GROWTH

As a ﬁrst example of a CFD model for ﬁne-particle production, we will consider a turbulent reacting ﬂow that can be described by a species concentration vector c. The microscopic transport equation for the concentrations is assumed to have the ‘‘standard’’ form as follows: @c þ = ðUcÞ ¼ = ðG=cÞ þ SðcÞ @t

(99)

All of the chemical species, except one, will be assumed to be completely soluble. The one partially insoluble species will nucleate and grow a solid phase. A typical example is A+B-P where P is a sparingly soluble compound. The rates of nucleation J and molecular surface growth G can be functions of the local concentration vector c, the particle size l, and the local turbulence properties. Neglecting aggregation and breakage processes, a microscopic PBE for this system can be written as follows: @n @ þ = ðUnÞ ¼ = ðGn =nÞ þ Jðl; cÞ ½Gðl; cÞn @t @l

(100)

Note that we have used the ﬂuid velocity U to describe convection of particles, which is valid for small Stokes number. In most practical applications, J is a highly nonlinear function of c. Thus, in a turbulent ﬂow the average nucleation rate will depend strongly on the local micromixing conditions. In contrast, the growth rate G is often weakly nonlinear and therefore less inﬂuenced by turbulent mixing. The quadrature approach for treating Eq. (100) introduces a moment transformation deﬁned by Z 1 mk ¼ l k nðlÞdl for k ¼ 0; 1; . . . ; 1 (101) 0

Applying this transformation to Eq. (100) yields @mk þ = ðUmk Þ ¼ = ðG=mk Þ þ J k ðcÞ þ kGk1 ðcÞ @t where the moments of the nucleation function are deﬁned by Z 1 J k ðcÞ ¼ l k Jðl; cÞ dl 0

(102)

(103)

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RODNEY O. FOX

and the growth rates for the moments are deﬁned by Z

1

G k ðcÞ ¼

l k Gðl; cÞnðlÞ dl

(104)

0

Note that in the special case of size-independent growth, this term can be expressed as a closed function of the moments, i.e., Gk ðcÞ ¼ GðcÞmk . Note also that when deriving Eq. (102) we have neglected the size-dependence of Gn. This is justiﬁed in turbulent ﬂows and, in any case, to do otherwise would require a micromixing model that accounts for differential diffusion (Fox, 2003). In principle, any functional form could be used for the nucleation rate. However, to simplify the discussion, we will assume that only particles of zero size are formed by nucleation so that Eq. (103) becomes J k ðcÞ ¼ d0;k JðcÞ

(105)

where dj,k is the Kronecker delta and J(c) contains the dependence of the nucleation rate on the local composition vector. Note that under this assumption nucleation only appears in the equation for the moment of order k ¼ 0. Due to the presence of the unknown NDF inside the integral in Eq. (104), the growth term is not closed (i.e., it cannot be computed exactly in terms of the moments unless G is independent of l). In the quadrature method of moments (QMOM), the integral is approximated by a sum over a set of M weights (wm) and M abscissas (lm): G k ðcÞ ¼

M X

wm l km Gðl m ; cÞ

(106)

m¼1

The weights and abscissas are determined in QMOM by forcing them to agree with the quadrature approximation of the ﬁrst 2 M moments: mk ¼

M X

wm l km

for k ¼ 0; 1; . . . ; 2M 1

(107)

m¼1

This system of 2 M nonlinear equations is ill-conditioned for large M, but can be efﬁciently solved using the product-difference (PD) algorithm introduced by McGraw (1997). Thus, given the set of 2 M moments on the left-hand side of Eq. (107), the PD algorithm returns wm and lm for m ¼ 1,y, M. The closed microscopic transport equation for the moments can then be written for k ¼ 0,y, 2 M1 as M X @mk þ = ðUmk Þ ¼ = ðG=mk Þ þ d0;k JðcÞ þ k wm l k1 m Gðl m ; cÞ @t m¼1

(108)

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277

To proceed further, this expression and Eq. (99) must be averaged to ﬁnd the CFD transport equations for the species concentrations and the moments. By deﬁning the composition vector to include the species concentrations and the moments as follows: / ¼ ½c1 ; . . . ; cK ; m0 ; . . . ; m2M1 T we can observe that the microscopic transport equations have the same form as those used for detailed chemistry in Section III.C. Thus, any turbulent reacting ﬂow model that can be used for detailed chemistry can also be used as a CFD model for ﬁne-particle production. For example, using the DQMOM approach to treat the composition PDF leads to Eqs. (85) and (86) with source terms for the moments given by the last two terms in Eq. (108) as follows:

S k ð/Þ ¼ d0;k JðcÞ þ k

M X

wm l k1 m Gðl m ; cÞ

(109)

m¼1

Note that each environment in the micromixing model will have its own set of concentrations can and moments mkn, reﬂecting the fact that the PSD is coupled to the chemistry and will thus be different at every SGS point in the ﬂow. The PD algorithm is applied separately in each environment to compute the weights (wmn) and abscissa (lmn) from the quadrature formula as follows: mkn ¼

M X

wmn l kmn

for k ¼ 0; 1; . . . ; 2M 1

(110)

m¼1

Thus, the source terms for each environment S(c) and Sk (/) will be closed. Of particular interest are the local nucleation rates J(cn). As discussed in Wang and Fox (2004), due to poor micromixing the local nucleation rates can be much larger than those predicted by the average concentrations J(hci). This results in a rapid increase in the local particle number density m0n due to the creation of a very large number of nuclei. As discussed below, this will have signiﬁcant consequences on the local rate of aggregation. The CFD model for nucleation and growth can now be solved to determine the average species concentrations hci and the average moments of the NDF hmk i. However, to properly interpret the computational results, care must be taken in deﬁning the averages for terms involving the moments. Starting from the deﬁnition of the moments from the microscopic NDF (Eq. 101), the average moments are deﬁned by Z

1

l k hnðlÞi dl

hm k i ¼ 0

(111)

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RODNEY O. FOX

where hnðlÞi is the Reynolds-average NDF. For the multi-environment model, the Reynolds-average quantities are deﬁned by hm k i ¼

N X

pn mkn and hnðlÞi ¼

n¼1

N X

pn nn ðlÞ

(112)

n¼1

where nn (l) is the NDF in environment n. Using the deﬁnition of the weights and abscissas (Eq. 110), the average moments can be expressed as follows: hm k i ¼

N X n¼1

pn

M X

wmn l kmn ¼

m¼1

M X

hwm l km ia

m¼1

M X

hwm ihl m ik

(113)

m¼1

Thus the weights and abscissas for the average NDF cannot be found to be averaging those for the NDF in each environment. Due to the nonlinear relationship between the moments and weights and abscissas, this result is not surprising.5 However, it does illustrate that hwm i and hl m i are not the relevant quantities needed to reconstruct hmk i.

B. BROWNIAN

AND

SHEAR-INDUCED AGGREGATION

AND

BREAKAGE

As mentioned above, when local nucleation rates are high, the local number density of particles will be large and aggregation will be favored. For very small particles (e.g., submicron), the dominant aggregation mechanism is Brownian motion (Einstein, 1905). Physically, the particles move by random walks driven by momentum exchange with solvent molecules. When two particles collide, they will stick together with a probability pB to form a doublet. By this mechanism, larger clusters are eventually formed with a fractal dimension df (Meakin, 1988; Sorensen, 2001; Lattuada et al., 2003a,b). If the volume fraction of particles is high enough, the clusters can reach ‘‘inﬁnite’’ size in a ﬁnite time 5

If we interpret the weights and abscissas as a delta-function representation of the NDF:

nn ðlÞ ¼

M X

wmn dðl l mn Þ

m¼1

then the average NDF is represented by N+M delta functions:

hnðl Þi ¼

N X M X

pn wmn dðl l mn Þ

n¼1 m¼1

By using the PD algorithm, this set of N M delta functions is reduced to a set of only M delta functions, but with the same values for the ﬁrst 2 M moments hmk i as the original set. It should be obvious that this cannot be accomplished by simply averaging the weights and abscissas.

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279

span through a process called gelation (Sandku¨hler et al., 2005). However, if the clusters are growing in a turbulent ﬂow ﬁeld, once their characteristic size is greater than approximately one micron other shear-driven processes become dominant (Oles, 1992). These are typically classiﬁed as shear-induced growth, breakage and restructuring. As in the case of chemical kinetics, aggregation and breakage models are required to describe these phenomena. From the perspective of developing CFD models, we can assume that such ‘‘kinetic expressions’’ are available. Thus, our focus here will be on how to implement the PBE in a CFD simulation to predict the moments of the PSD. The simplest aggregation and breakage models can be formulated in terms of the NDF n(u), which uses volume as the independent variable.6 The microscopic transport equation for the NDF has the form (Wang et al., 2005a,b) @n @ þ = ðUnÞ ¼ = ðGn =nÞ þ Jðu; cÞ ½Gðu; cÞn @t @u þ AðuÞ þ BðuÞ

ð114Þ

where A(u) and B(u) are the aggregation and breakage terms, respectively. Although we do not do so here, these terms can be assumed to be dependent on the species concentrations c without changing the form of the CFD model. For binary aggregation and breakage, the aggregation term can be expressed as follows (Ramkrishna, 2000): AðuÞ ¼

1 2

Z

u

bðu s; sÞnðu sÞnðsÞ ds Z 1 nðuÞ bðu; sÞnðsÞds 0

ð115Þ

0

where b is the aggregation kernel. A typical breakage term has the form Z1 BðuÞ ¼

bðujsÞaðsÞnðsÞds aðuÞnðuÞ

(116)

u

where b is the daughter-size distribution and a is the breakage kernel. For Brownian aggregation, the aggregation kernel can be written as follows (Elimelech et al., 1995; Sandku¨hler et al., 2003): bðu; sÞ ¼ 6

2pB kB T 1=d f u þ s1=d f u1=d f þ s1=d f 3m

(117)

It is also possible to use length as the independent variable as described in Wang et al. (2005b).

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RODNEY O. FOX

while for shear-induced aggregation it has the form (Oles, 1992; Elimelech et al., 1995) as follows: 3 bðu; sÞ ¼ gaðu; sÞ u1=d f þ s1=d f

(118)

A general expression can be found by combining these two cases (Melis et al., 1999). In these expressions, kB is the Boltzmann constant, T is the ﬂuid temperature (Kelvin), m is the ﬂuid viscosity, g is the local shear rate, and a is an efﬁciency factor. For shear-induced breakage, the kernel is usually ﬁt to experimental data (Wang et al., 2005a,b). A typical form is (Pandya and Spielman, 1983) as follows: aðuÞ ¼ c1 gc2 uc3 =d f

(119)

where c1c3 are empirical ﬁtting parameters. A typical daughter-size distribution is bðujsÞ ¼ dðu fsÞ þ dðu ð1 f ÞsÞ

(120)

where f ¼ 1/2 corresponds to equal-size daughters and f51 corresponds to erosion (i.e., a very small and a very large daughter). At this point, we should step back and make a few comments concerning Eq. (114). First, it should be obvious to the reader that many modeling assumptions have already been invoked to arrive at forms for the aggregation and breakage terms and the rate functions needed to deﬁne them. Therefore, we should keep in mind that, just as when working with chemical kinetics, the CFD predictions can be no better than the basic physical/chemical models used to describe the source terms. By their nature, aggregation and breakage terms are much more difﬁcult to formulate accurately than gas-phase chemical mechanisms. Thus, we should expect a greater degree of mismatch with experimental data for CFD solutions for aggregation-breakage systems than we are accustomed to with, for example, combustion systems (Raman et al., 2004). Due to the uncertainties in the ‘‘kinetics’’ for aggregation and (especially) breakage, the use of a highly sophisticated SGS model for turbulent mixing (e.g., transported PDF or CMC models) is likely unwarranted for most systems of interest. In fact, at present the greatest need in this domain is carefully designed experiments to accurately measure the rate constants appearing in the aggregation and breakage expressions. A second general observation can be made by comparing the aggregation terms (Eq. 115) to the breakage terms (Eq. 116): The former is second order in n(u) and the latter is ﬁrst order. This implies that when n(u) is very large (e.g., due to high local nucleation), aggregation will be favored. In fact, in sheardominated systems gelation is prevented only by virtue of the fact that the aggregation efﬁciency a drops off rapidly for large clusters. Thus, in systems

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

281

with nucleation, growth, aggregation, and breakage (e.g., aggregating nanoparticles), the PSD can be nonzero over a very wide range of cluster volumes (i.e., 3–4 orders of magnitude is not uncommon). If sectional methods are used to approximate n(u), such systems typically require a relatively large number of sections (e.g., 25–100) for reasonable accuracy (Marchisio et al., 2003b). For this reason, when used in a CFD model, sectional methods require an unfavorable trade-off between reasonable computational cost and accuracy. Quadrature methods, which have lower cost for equivalent accuracy (Marchisio et al., 2003b), are thus a better choice for combining with CFD to describe these complex systems. A ﬁnal observation concerns the shear rate g appearing in both the aggregation and breakage kernels. Since from the outset we have assumed that the clusters are small compared to the Kolmogorov length scale, the local shear rate seen by a cluster is the instantaneous shear (e/u)1/2, where e is the ﬂuctuating dissipation rate (Fox, 2003). The ﬂuctuating dissipation is very different than the average dissipation e computed by the turbulence model (Fox and Yeung, 2003). On average hi ¼ , but e(t) ﬂuctuates strongly on a characteristic time scale proportional to the Kolmogorov time scale tZ. Whether these ﬂuctuations must be accounted for in the CFD model depends on the characteristic aggregation time. Marchisio et al. (2006) estimate that when the local solid volume fraction exceeds 103, the ﬂuctuations must be included in the CFD model. Otherwise, g can be set equal to (e/u)1/2. In any case, we see from this discussion that g scales with the local turbulent Reynolds number like g Re1/2 L . Thus, shear-induced aggregation and breakage will be important phenomena in turbulent ﬂows and their importance will increase with increasing Reynolds number. We now turn to the question of developing a CFD model for ﬁne-particle production that includes nucleation, growth, aggregation, and breakage. Applying QMOM to Eq. (114) leads to a closed set of moment equations as follows: @mk þ = ðUmk Þ ¼ = ðG=mk Þ þ S k ð/Þ @t

(121)

where the moment source term is given by (Marchisio et al., 2003a) S k ð/Þ ¼ d0;k JðcÞ þ k

M X

wm uk1 m Gðum ; cÞ þ

m¼1

þ

M X

k wm aðum Þ bðkÞ m um

m¼1

h

i 1 wm wp ðum þ up Þk ukm ukp bðum ; up Þ 2 m¼1 p¼1 M X M X

ð122Þ

and bðkÞ m ¼

Z

1

uk bðujum Þdu 0

(123)

282

RODNEY O. FOX

Thus, just as we saw with Eq. (109), the moment source term has the form found with detailed chemistry (i.e., the right-hand side of Eq. (122) depends only on /).7 The CFD transport equation can therefore be developed along the same lines that we discussed earlier for nucleation and growth. In other words, the DQMOM model can be used to describe micromixing of the moments of the PSD at the sub-grid scale, along with turbulent transport models to describe macromixing. One new factor that can arise when dealing with aggregation is that the moment source terms can be ‘‘stiff.’’ To handle this problem in CFD simulations, Wang and Fox (2003) successfully implemented a tabulation method originally designed for combustion chemistry. C. MULTIVARIATE POPULATION BALANCES The CFD model described above is adequate for particle clusters with a constant fractal dimension. In most systems with ﬂuid ﬂow, clusters exposed to shear will restructure without changing their mass (or volume). Typically restructuring will reduce the surface area of the cluster and the fractal dimension will grow toward df ¼ 3, corresponding to a sphere. To describe restructuring, the NDF must be extended to (at least) two internal coordinates (Selomulya et al., 2003; Zucca et al., 2006). For example, the joint surface, volume NDF can be denoted by n(s, u; x, t) and obeys a bivariate PBE. With two (or more) internal coordinates, numerical approaches for the PBE using sectional methods become intractable in the context of CFD. A practical alternative is to use a ﬁnite number of samples to approximate the NDF in terms of delta functions as follows: nðs; uÞ ¼

M X

wm dðs sm Þdðu um Þ

(124)

m¼1

The weights wm and abscissas sm and um are related to the bivariate moments by ZZ mk 1 k 2 ¼

sk1 uk2 nðs; uÞdsdu ¼

M X

wm skm1 ukm2

(125)

m¼1

Thus, it would be natural to attempt to extend the QMOM approach to handle a bivariate NDF. Unfortunately, the PD algorithm needed to solve the weights and abscissas given the moments cannot be extended to more than one variable. Other methods for inverting Eq. (125) such as nonlinear equation solvers can be used (Wright et al., 2001; Rosner and Pykkonen, 2002), but in practice are computationally expensive and can suffer from problems due to ill-conditioning. 7 There are, however, important differences. For example, in detailed chemistry the source terms do not depend on the ﬂow quantities. In contrast, all of the rate functions for particulate systems can potentially depend on the local ﬂow quantities such as the instantaneous shear rate.

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

283

To overcome the difﬁculty of inverting the moment equations, Marchisio and Fox (2005) introduced the direct quadrature method of moments (DQMOM). With this approach, transport equations are derived for the weights and abscissas directly, thereby avoiding the need to invert the moment equations during the course of the CFD simulation. As shown in Marchisio and Fox (2005), the NDF for one variable with moment equations given by Eq. (121) yields two microscopic transport equations of the form @wm þ = ðUwm Þ ¼ = ðG=wm Þ þ am þ am @t

(126)

@wm um þ = ðUwm um Þ ¼ = ðG=wm um Þ þ bm þ bm @t

(127)

and

where the source terms am and bm are found from the moment source terms Sk, and am and bm are ‘‘correction’’ terms that depend on G9rum92. The role of the correction terms is to ensure that Eqs. (126) and (127) are mathematically equivalent to Eq. (121), and they result from the diffusion terms during the nonlinear change of variables (Eq. 107). Thus, they will be null in the absence of molecular diffusion and their forms do not depend on the moment source terms. The extension of DQMOM to bivariate systems is straightforward and, for the surface, volume NDF, simply adds another microscopic transport equation as follows: @wm sm þ = ðUwm sm Þ ¼ = ðG=wm sm Þ þ cm þ cm @t

(128)

Example calculations for a bivariate system can be found in Marchisio and Fox (2006) and Zucca et al. (2006). We should note that for multivariate systems the choice of the moments used to compute the source terms is more problematic than in the univariate case. For example, in the bivariate case a total of 3 M moments must be chosen to determine am, bm and cm. In most applications, acceptable accuracy can be obtained with 3rMr5. Thus, the maximum number of moments that will be required is 15, and one might decide to use bivariate moments up to order four as shown below: m0,0, m1,0, m2,0, m3,0, m4,0,

m 0,1, m1,1, m 0,2, m2,1, m1,2, m0,3, m3,1, m2,2, m1,3, m0,4.

However, there is no guarantee that this set of moments will be linearly independent. Even if they do work, we can note that compared to the univariate case where the ten moments m0–m9 would be employed, the accuracy (as measured

284

RODNEY O. FOX

by the highest-order moment used) decreases as the number of internal coordinates increases. Thus, the only way to obtain equivalent accuracy would be to increase M. Leaving aside the question of determining which moments to use, we also need a consistent method for deriving a CFD transport equation for turbulent reacting ﬂows from the microscopic transport equations for the weights and abscissas. In particular, since the moments have a nonlinear dependence on the weights and abscissas, the deﬁnition of the micromixing model in terms of the weights and abscissas must be consistent with that used for the moments.8 The development of a consistent model is most easily done by proceeding in three independent steps as follows: 1. Find the multi-environment model for the case where the moment source terms are null and there is no micromixing: am ¼ am ¼ bm ¼ bm ¼ cm ¼ cm ¼ 0: 2. Extend the model to include micromixing, but no moment source terms: am ¼ bm ¼ cm ¼ 0. 3. Extend the model to include the moment source terms due to nucleation, growth, etc. We will now brieﬂy illustrate how these steps are carried out. The multi-environment model for turbulent transport of the bi-variate moments in the absence of moment source terms has the form @pn þ = ðUpn Þ ¼ = ðGT =pn Þ @t

(129)

and, for arbitrary values of k1 and k2 @pn mk1 k2 n þ = ðUpn mk1 k2 n Þ ¼ = ðGT =pn mk1 k2 n Þ þ bk1 k2 n @t

(130)

where the bi-variate moments in environment n are given in terms of the weights and abscissas by mk1 k2 n ¼

M X

1 k2 wmn vkmn smn

(131)

m¼1

As discussed earlier with other transported scalars, the correction term appearing on the right-hand side of Eqs. (130) is found by solving a linear system deﬁned by N X n¼1 8

n mk1 k 1 k 2 n bk 1 k 2 n ¼

N X 2 ðk 1Þpn mk2 k1 k2 n GT =mk1 k2 n

for k ¼ 1; . . . ; N

(132)

n¼1

We develop the CFD equations using the DQMOM model for micromixing. Nevertheless, care must also be taken when using other micromixing models, including transported PDF methods.

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285

Note that in this expression k is unrelated to k1 and k2, which are determined by the choice of moment mk1 k2 n . Equation (132) deﬁnes the correction term bnk1 k2 n for a ﬁxed set of indices (k1, k2, n). In other words, given the weights and abscissas we can use Eq. (131) to compute the bi-variate moments, which can then be used in Eq. (132) to solve for bnk1 k2 n . The multi-environment model for the weights and abscissas has the form @pn wmn þ = ðUpn wmn Þ ¼ = ðGT =pn wmn Þ þ Amn þ Amn @t

(133)

@pn wmn umn þ = ðUpn wmn umn Þ ¼ = ðGT =pn wmn umn Þ @t þ Bmn þ Bmn

ð134Þ

@pn wmn smn þ = ðUpn wmn smn Þ ¼ = ðGT =pn wmn smn Þ @t þ C mn þ C mn

ð135Þ

and

where the ﬁnal two terms on the right-hand sides are null in the absence of micromixing and moment source terms. The correction terms Anmn ; Bnmn and C nmn are determined such that they are equivalent to the correction term bnk1 k2 n in Eq. (130). In other words, if we use Eq. (131) to replace mk1 k2 n in Eq. (130), then Eqs. (133)–(135) will be equivalent to Eq. (130) when the correction terms satisfy the following linear system: ð1 k1 k2 Þ

M X

1 k2 ukmn smn Anmn þ k1

m¼1

M X

1 1 k 2 ukmn smn Bnmn þ k2

m¼1

¼ bnk1 k2 n þ k1 ðk1 1Þpn

M X

M X

1 k 2 1 ukmn smn C nmn

m¼1

1 2 k 2 wmn vkmn smn GT j=vmn j2

m¼1

þ k 1 k 2 pn

M X

1 1 k 2 1 wmn vkmn smn GT ð=vmn =smn Þ2

m¼1

þ k2 ðk2 1Þpn

M X

1 k 2 2 wmn vkmn smn GT j=smn j2

ð136Þ

m¼1

Note that the correction terms are proportional to GT and result from turbulent velocity ﬂuctuations (represented by a gradient-diffusion model). For the multi-environment model the composition vector is deﬁned by / ¼ ½w1 ; . . . ; wM ; w1 v1 ; . . . ; wM vM ; w1 s1 ; . . . ; wM sM T

286

RODNEY O. FOX

Thus, a total of 3 M variables are needed to describe the evolution of the bi-variate NDF in a turbulent ﬂow. The ﬁrst step in the construction of the CFD model is now complete. In the second step we must add the micromixing terms from the DQMOM model to Eqs. (133)–(135). However, as we discussed earlier, we need to keep in mind that micromixing conserves the moments of the NDF, and not the weights and abscissas (see Eq. 113). The micromixing model in environment n for the bivariate moments has the form M k1 k2 n ¼ g hmk1 k2 i mk1 k2 n (137) where the moments in environment n are given by Eq. (131) and the average moments are found by averaging over all N environments: hmk1 k2 i ¼

N X

pn mk1 k2 n

(138)

n¼1

Thus, given the weights and abscissas, the micromixing term for the moments is closed. Applying DQMOM, the micromixing source terms (which are added to the right-hand sides of Eqs. (133)–(135)) can be shown to obey for each n ¼ 1,y, N the linear system deﬁned by ð1 k1 k2 Þ

M X

1 k2 ukmn smn Amn þ k1

m¼1

M X

1 1 k 2 ukmn smn Bmn

m¼1

þk2

M X

1 k 2 1 ukmn smn C mn ¼ pn M k1 k2 n

ð139Þ

m¼1

Note that the right-hand side of this expression contains the closed micromixing term for the moments (Eq. 137). To ﬁnd the 3 M micromixing source terms (Amn, Bmn, Cmn) from this expression, we must choose a set of 3 M bi-variate moments. Note that because the moment equations are closed when only micromixing is considered, the chosen moments will be reproduced exactly. A convenient choice is to use the uncoupled moments mk0 and m0k. (Note that this same choice should then be used in Eq. (136).) This yields the linear system ð1 kÞ

M X m¼1

ukmn Amn þ k

M X

uk1 mn Bmn ¼ pn M k0n

m¼1

for k ¼ 0; . . . ; 2M 1

ð140Þ

which can be solved to ﬁnd Amn and Bmn for each n ¼ 1,y, N, and the linear system k

M X m¼1

sk1 mn C mn ¼ pn M 0kn þ ðk 1Þ

M X

skmn Amn

m¼1

for k ¼ 1; . . . ; M

ð141Þ

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

287

which can be solved to ﬁnd Cmn. The second step in the construction of the CFD model is now complete. The third step is to add the moment source terms due to nucleation, growth, aggregation, and restructuring. The exact form of these terms will depend on the models used to describe these processes (see for example Eq. 122). However, if we denote the source term for the bi-variate moments as S k1 k2 ð/Þ, where / is the composition vector including all variables needed to describe the kinetics, then the source terms (including micromixing) in the CFD model (Amn, Bmn, Cmn) can be found by simply adding pn S k1 k2 ð/n Þ to the right-hand side of Eq. (139). The extension of DQMOM to multi-variate systems with more than two internal coordinates is discussed in Marchisio and Fox (2005). As far as the CFD model equations are concerned, no additional complications arise for higherdimensional systems. Nevertheless, from a practical point of view the reader should keep in mind that going to higher dimensions will necessarily diminish the accuracy of the method as compared to the uni-variate case. We end here with our discussion of CFD models for ﬁne-particle production. The reader hopefully has a good feel for the issues involved and at least a cursory understanding of the available models. The current status of the ﬁeld is such that the CFD tools available for the analysis of ﬁne-particle systems are adequate for most applications. Currently, the weakest link in the modeling process is description of physical processes such as aggregation and breakage, and their coupling to the ﬂow ﬁeld. As mentioned earlier, there is a need for carefully designed experiments with local measurements of the PSD and turbulence ﬁelds that can be used for CFD validation. As in the ﬁeld of turbulent mixing, the ideal experiment would be spatially homogeneous, or a most one-dimensional, to focus on the source terms for the chemical reactions and the moments. It would also be useful to conduct a few ‘‘idealized experiments’’ using direct numerical simulations (DNS) for homogeneous ﬁneparticle systems undergoing nucleation, growth, aggregation, and breakage. Although such ‘‘experiments’’ cannot be directly compared to real systems, they would still be useful for calibrating CFD models used for micromixing and shear-induced aggregation and breakage.

V. Multiphase Reacting Systems The CFD models considered up to this point are, as far as the momentum equation is concerned, designed for single-phase ﬂows. In practice, many of the chemical reactors used in industry are truly multiphase, and must be described in the context of CFD by multiple momentum equations. There are, in fact, several levels of description that might be attempted. At the most detailed level, direct numerical simulation of the transport equations for all phases with fully resolved interfaces between phases is possible for only the simplest systems. For

288

RODNEY O. FOX

example, for ﬂuid-particle systems the Navier–Stokes equation must be solved for the ﬂuid phase between the particles with enough detail to capture momentum transport at the particle surfaces (Nguyen and Ladd, 2005). At the same time, Newtonian equations for the particle positions and momenta must be solved simultaneously to account for ﬂuid surface forces and particle–particle collisions. Obviously, such a detailed model could not be used to describe a large chemical reactor such as a ﬂuidized bed. Less costly methods have thus been developed and are described in other chapters of this issue. In general, the methods available in commercial CFD codes are based on the so-called ‘‘multiﬂuid’’ model (Drew and Passman, 1999) that makes no attempt to capture the details at the interfaces between phases. Instead, the ﬂuid at the subgrid scale is described by the volume or mass fractions for each phase much in the same way that environments are used to describe micromixing in single-phase ﬂows). We will thus look brieﬂy at the structure of multiﬂuid models and describe some of the modeling assumptions that are required for multiphase reacting ﬂows.

A. MULTIFLUID CFD MODELS In this section, we will look brieﬂy at multiﬂuid CFD models. Our primary objective is to understand the modeling issues that arise and how they are dealt with in CFD codes. To ﬁx ideas, we will look at a gas–solid system (e.g., a ﬂuidized bed) wherein the solid particles undergo growth, aggregation, and breakage. Unlike in Section IV, we will assume here that the particle Stokes number (St) can be larger than 0.14. Thus, it will be necessary to account for particle momentum separately from the gas phase (or at least to account for the effect of the second phase in the momentum balance). Nevertheless, the particle size distribution (PSD) can be accounted for using DQMOM (Fan et al., 2004). For simplicity, we will assume that the particle density rs is constant and independent of particle size, and that the gas density rg is constant. At the subgrid scale, the two phases are described by the volume fractions ag and as for the gas and solid phases, respectively. By deﬁnition, ag+as ¼ 1. In addition, the DQMOM representation of the solid phase will introduce a volume fraction for each abscissa asm for m ¼ 1,y, M. By deﬁnition, as1 þ þ asM ¼ as . (See Fan et al. (2004) for details.) The multiﬂuid CFD model at its most basic level consists of mass and momentum balances for each ‘‘phase.’’ For the present example, the mass balance for the gas phase can be written as follows: M X @rg ag þ = ðrg ag Ug Þ ¼ M gm @t m¼1

(142)

where Mgm is the mass-transfer rate from the gas to the solid phases. As we will discuss later (Section V.B), a model must be provided to close the mass-transfer

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term. In words, rgag is the mass of gas per unit volume of the multiphase mixture (i.e., gas and solid phases), whereas rg is the mass of gas per unit volume of gas. The gas velocity Ug appears in the convection term for the gas phase, and is normally not equal to the solids velocities. The mass balance for the solid phases (m+1,y, M) can be written as follows: @rs asm þ = ðrs asm Usm Þ ¼ M gm þ 3kv rs l 2m bm 2kv rs am @t

(143)

The solid velocities Usm (one for each abscissa lm) appear in the convection term. The source terms am and bm are found from the DQMOM representation of aggregation and breakage of the solid particles (Fan et al., 2004). Each particle phase is represented by its volume fraction (instead of its weight wm) and its characteristic length lm. Note that growth of solid particles (with constant density rs) requires mass transfer from the gas phase, represented by Mgm. In the absence of growth the total solids volume fraction as does not change. Thus, the aggregation and breakage terms will cancel as follows: M X

3l 2m bm 2am ¼ 0

(144)

m¼1

This property will result from applying DQMOM to the aggregation and breakage terms in the PBE (Fan et al., 2004). While the mass balances given above are relatively straightforward (assuming that a suitable closure can be derived for the mass-transfer terms), the momentum balances are signiﬁcantly more complicated. In their simplest forms, they can be written as follows: @rg ag Ug þ = ðrg ag Ug Ug Þ ¼ ag =p þ = rg @t M X þ f gm þ rg ag g

ð145Þ

m¼1

and @rs asm Usm þ = ðrs asm Usm Usm Þ ¼ asm =p þ = rsm f gm @t M X þ f smn þ rs asm g

ð146Þ

n¼1

for the gas and solid phases, respectively. The two terms of the left-hand sides of the momentum balances correspond to accumulation and convection. Note that the conserved quantities (for example rgagUg) are the momentum of a given phase per unit volume of the mixture (hence the appearance of ag, etc.), and that

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each phase has its own characteristic convection velocity (e.g., Ug). The terms on the left-hand side of Eqs. (145) and (146) account for changes in momentum of each phase. Obviously, since momentum is exchanged between phases at the interfaces and we are not resolving the interfaces, such phase-interaction terms will require models. In contrast, body forces (i.e., gravity g in this example) appear in closed form. The ﬁrst term on the right-hand side of Eqs. (145) and (146) is a pressure term shared by both phases. The purpose of this term (when rs and rg are constant) is to ensure that the volume-average velocity, deﬁned by Uvol ¼ ag Ug þ

M X

asm Usm

(147)

m¼1

is solenoidal in the absence of mass transfer. Indeed, dividing the mass balances (Eqs. 142 and 143) by the densities and neglecting the mass-transfer terms lead to = Uvol ¼ 0

(148)

Thus, just as for incompressible single-phase ﬂow, the pressure p constrains the velocity ﬁelds to ensure (in the case of multiphase ﬂows) that the sum of the phase volume fractions equals unity. In the presence of mass transfer, the righthand side of Eq. (148) is nonzero; nevertheless, the role of the pressure is still the same. Finally, we should note that in gas–solid ﬂows the maximum volume fraction of the solid phase is less than unity due to physical constraints (i.e., when particles are close packed there is still room for the gas phase so that 0oag). To accommodate this constraint, it is common to introduce a ‘‘solid-pressure’’ term ps that becomes extremely large when ag approaches its minimum value (e.g., ag ¼ 0.4). The second term on the right-hand side of Eqs. (145) and (146) contains the viscous-stress models sg and ssm. Even for laminar ﬂow, suitable forms for these models are difﬁcult to determine a priori. Typical models used in CFD introduce an effective viscosity meff for each phase, and describe the viscous stresses as follows. rx ¼ meff;x rUx þ ðrUx ÞT x ¼ g; sm (149) Leaving aside the difﬁcult question of whether this model holds for multiphase ﬂows, we still have the problem of determining meff,x in terms of the computed properties of the ﬂow. The reader should appreciate that choosing an effective viscosity for a multiphase ﬂow is much more complicated than just adding a turbulence model as done in single-phase turbulent ﬂows. Indeed, even for a case involving two ﬂuids (e.g., two immiscible liquids) for which the molecular viscosities are constant, the choice of the effective viscosities is not obvious. For example, even if the mass-average velocity deﬁned by P rg ag Ug þ M m¼1 rs asm Usm Umass ¼ (150) rg ag þ rs as

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were laminar, the ﬂow around individual particles could be turbulent (as measured by the particle Reynolds number deﬁned below) and the effective viscosity should reﬂect this fact. The simplest models account for ‘‘particle-scale’’ turbulence using an expression of the form meff;g ¼ mg ð1 þ C s as Res Þ

(151)

where mg is the molecular viscosity of the gas phase and Res is a particle Reynolds number deﬁned by Res ¼

rg d s jUs Ug j mg

(152)

and ds is the characteristic diameter of the particles. The model constant Cs is order unity, but must be ﬁt to experimental data. The effective viscosity in Eq. (151) has the desired behavior in the limits where as or Res are very small; however, it is unlikely to be accurate when the product of these terms is large. The situation for cases where Umass is also turbulent is even more complicated. First, such ‘‘large-scale’’ turbulence can be due to a variety of physical phenomena, and thus have different characteristics. For example, large-scale turbulence can (as in single-phase ﬂows) be introduced through the boundary conditions (e.g., turbulent jets) or by using mixing devices (e.g., stirred tanks). For such cases, it may be possible to make suitable modiﬁcations to single-phase turbulence models to arrive at useful expressions for the effective viscosity. In contrast, large-scale turbulence that arises due to internal properties of a multiphase ﬂow (e.g., density differences between phases) is more difﬁcult to describe by simple modiﬁcations of standard turbulence models. Second, due to the difﬁculty of accessing multiphase ﬂows with laser-based ﬂow diagnostics, there is very little experimental data available for validating multiphase turbulence models to the same degree as done in single-phase turbulent ﬂows. For example, thanks to detailed experimental measurements of turbulence statistics, there are many cases for which the single-phase k-e model is known to yield poor predictions. Nevertheless, in many CFD codes a multiphase k-e model is used to supply multiphase turbulence statistics that cannot be measured experimentally. Thus, even if a particular multiphase turbulent ﬂow could be adequately described using an effective viscosity, in most cases it is impossible to know whether the multiphase turbulence model predicts reasonable values for meff. Third, many of the multiphase ﬂows of interest to chemical engineers are in regimes where both particle-scale and large-scale turbulence are signiﬁcant. For example, in gas–liquid bubble columns the particle Reynolds number (based on the bubble rise velocity) is typically large. Thus, even for low gas-ﬂow rates, particle-scale turbulence will be signiﬁcant. However, at low gas-ﬂow rates and with uniform sparging, a bubble column will have no large-scale turbulence (i.e., the ﬂow regime will be ‘‘homogeneous’’) (Garnier et al., 2002; Harteveld et al., 2003), and thus only the effective viscosity of individual ‘‘particles’’ should be

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included in the model. As the gas ﬂow rate is increased to a critical value of the gas holdup (which can be as high as ag ¼ 0.55 (Mudde, 2005)), the ﬂow will become unstable and large-scale turbulence will be generated. Although it has been attempted in the literature (Thorat and Joshi, 2004), it is unlikely that a twoﬂuid k-e turbulence model has the necessary mathematical structure to correctly predict ﬂow transitions or even the ‘‘turbulence’’ levels observed in homogeneous bubbly ﬂows. In contrast, a two-ﬂuid model using only a ‘‘particle-scale’’ effective viscosity and appropriate force models can predict ﬂow transitions (Monahan et al., 2005) in reasonable agreement with experiments (Harteveld et al., 2003). Nevertheless, it is likely that different CFD models will be required for different ﬂow regimes (e.g., homogeneous vs. churn turbulent) and the user must be careful not to extend a particular model beyond its range of applicability. Returning to the momentum equations, the third term on the right-hand side of Eqs. (145) and (146) contains the gas–solid momentum-exchange models fgm. Likewise, the fourth term on the right-hand side of Eqs. (146) contains the solid–solid momentum-exchange model fsmn. Note that because solid–solid interactions conserve momentum, the latter must be deﬁned such that M X M X

f smn ¼ 0

(153)

m¼1 n¼1

Determination of accurate models for fgm and fsmn is nontrivial (Drew and Passman, 1999), and no consensus exists on the exact forms needed to describe particular ﬂows. Nevertheless, it is generally acknowledged that the momentumexchange model must include ‘‘drag’’ terms with forms similar to f gm ¼ ag asm C D ðRes Þ

3mg Res 4d 2s

ðUsm Ug Þ

(154)

where CD(Res) is a drag coefﬁcient, and f smn ¼ asm asn C mn ðUsn Usm Þ

(155)

where Cmn depends on the properties of the solid phases (Gao et al., 2006). Note that the drag models depend on the velocity difference between two phases, and thus can be nonzero even for cases where the velocity ﬁelds are uniform in time and space. Other forces that can be included depend on gradients (temporal or spatial) of the velocities or volume fractions (Drew and Passman, 1999), and thus are only signiﬁcant for inhomogeneous ﬂows. However, as can be shown using linear stability analysis (Batchelor, 1988; Lammers and Biesheuvel, 1996; Minev et al., 1999; Jackson, 2000; Johri and Glasser, 2002; Sankaranarayanan and Sundaresan, 2002), spatially uniform solutions to the multiﬂuid model are usually unstable, implying that the stationary, homogeneous solution to the multiﬂuid model is not representative of the ﬂow. Thus, even when simulating ‘‘homogeneous’’ ﬂows, it is important to include all relevant forces when

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293

comparing numerical simulations with experiments (Monahan et al., 2005). Moreover, the computational requirements in terms of gird size needed to attain grid-independent solutions are relatively high for the ‘‘laminar’’ two-ﬂuid model (Monahan et al., 2005). In the context of modeling chemical reactors, it will be necessary to develop CFD models for the unresolved scales when applying multiﬂuid models to real reactors. Although procedures for developing such models are still being actively investigated and no clear consensus has yet to emerge (Sundaresan, 2000), here we will limit ourselves to a brief discussion of the relevant issues. To simplify the presentation, let us consider the transport equation for Umass found by summing together Eqs. (145) and (146): ^ mass @rU ^ mass Umass Þ þ = ðr^ ud ^ ^ þ rg þ = ðrU d ud Þ ¼ =p þ = r @t

(156)

where the phase-average density, deﬁned by r^ ¼ rg ag þ rs as ; obeys @r^ ^ mass Þ ¼ 0 þ = ðrU @t

(157)

Note that these expressions (Eqs. 156 and 157) appear deceptively simple (i.e., as if the problem can be reduced to modeling a variable-density, single-phase ﬂow) because we have hidden the ‘‘difﬁcult’’ terms in the deﬁnition of some new symbols! First, the phase-average stress r is deﬁned by r^ ¼ rg þ

M X

(158)

rsm

m¼1

and (based on the model in Eq. (149)) it is not a simple function of Umass. Second, a new ‘‘multiphase’’ stress term ud d ud has be introduced and is deﬁned by r^ ud d ud ¼ rg ag ug ug þ rsm asm

M X

usm usm

(159)

m¼1

where ug ¼ UgUmass and usm ¼ UsmUmass are the differences between the phase velocities and the mass-average velocity. We can note that for a constant-density, two-phase system with rg¼ 6 rs, Ug and Us are related to Umass and Uvol by rs r^ Uvol Umass ag ðrs rg Þ ag ðrs rg Þ rg r^ Uvol þ Umass Us ¼ as ðrs rg Þ as ðrs rg Þ

Ug ¼

ð160Þ

Thus the two-ﬂuid model can be formulated in terms of any two velocities chosen from the set Ug, Us, Umass and Uvol, which might be useful, for example, to

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RODNEY O. FOX

examine the limiting case where rg rs . The closures for shear stresses r^ and convection due to differences in the phase velocities ud d ud in Eq. (156) are necessarily ﬂow dependent. Nevertheless, the simplest closures might use Eq. (149) with Umass and an effective viscosity depending on r^ and assume that the slip velocity between phases ud is a known constant (e.g., depending on density difference and bubble size). We should also note that it is possible to write the model in terms of the volume-average velocity Uvol (Eq. 147). However, the resulting expressions are more complicated than Eqs. (156) and (157). Up to this point we have not introduced any modeling concepts to deal with ‘‘large-scale’’ turbulence. However, if the Reynolds number corresponding to Eq. (156) is large enough, the velocity ﬁeld Umass will become turbulent. In this case, the computational resources needed to resolve all of the relevant ﬂow scales will increase drastically, and the multiﬂuid CFD model will no longer be tractable for analyzing chemical reactors. To deal with this difﬁculty, we can introduce a multiphase turbulence model based on Reynolds averaging Eq. (156). Because r^ is not constant, we will in fact use the Favre average. For example, if the Reynolds-average velocity is denoted by hUmass i, then the Favreaverage velocity is deﬁned by ^ mass i e mass ¼ hrU U ^ hri

(161)

Note that we have also introduced the Reynolds-average, phase-average density ^ Applying the Favre average to Eq. (157) yields a closed expression for the hri. mass balance as follows: ^ @hri e mass Þ ¼ 0 ^ U þ = ðhri @t

(162)

Applying the same process to Eq. (156) yields e mass ^ U @hri e mass U e mass Þ þ = ðhri g umass Þ ^ U ^ umass þ = ðhri @t d ^ ug ^ þ = ðhri u Þ ¼ =hpi þ = hr^ i þ hrig d d

ð163Þ

where the velocity ﬂuctuations due to large-scale turbulence are denoted by ~ mass . The most important unclosed terms in Eq. (163) are the umass ¼ Umass U g umass and the Favre-average multiphase stresses ug d turbulence stresses umass d ud . The ﬁrst of these is usually closed by introducing a multiphase turbulence model with appropriate modiﬁcations to include the effect of interfacial momentum exchange on production (dissipation) of large-scale turbulence. The second term d ug d ud is not directly related to turbulent velocity ﬂuctuations. Instead, it will depend on correlations between the phase-average density r^ and the velocity difference ud. For example, if the bubble rise velocity Ur were constant and 2 d independent of the gas volume fraction, then ug d ud U r ev ev where ev ¼ g/|g|is

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295

the unit vector in the vertical direction. More generally, the turbulent two-ﬂuid model for gas–liquid ﬂow should have properties similar to turbulence generated by buoyancy in single-phase ﬂows (Riley and DeBruynKops, 2003). Likewise, in gas–solid ﬂows ud will depend on the Favre-average drag term, and thus on the particle Stokes number through correlations between gas- and solidphase velocity ﬂuctuations (Fan and Zhu, 1998). Although computationally expensive using present-day computers (Agrawal et al., 2001; Zhang and VanderHeyden, 2002), it might be instructive to use direct simulations of the laminar two-ﬂuid model (i.e., before Favre averaging) to parameterize multiphase turbulence models as has been done for stably stratiﬁed ﬂows (Shih et al., 2005). This possibility is especially attractive because it offers access to ﬂow statistics that cannot be measured experimentally. At the very least, it might allow us to distinguish between the adequacy of the various multiphase turbulence models available in the literature (Mudde, 2005). The goal of the discussion above was obviously not to describe multiphase turbulence models, but rather to point out the difﬁculties encountered when trying to derive a consistent set of transport equations. Although it is usually not done, it is worthwhile to think of the averaging process used to arrive at Eqs. (162) and (163) in two distinct steps: (1) ensemble averages over different phase conﬁgurations to derive the ‘‘laminar’’ multiﬂuid model (i.e., Eq. 146) that can be used to describe multiphase ﬂows without large-scale turbulence, and (2) Reynolds or Favre averages (or even LES) to describe turbulent multiphase ﬂows. Ideally, direct numerical simulations (DNS) of two-phase ﬂows with resolved interfaces could be used to develop two-ﬂuid models for laminar multiphase ﬂows. For example, the recent work of Nguyen and Ladd (2005) uses DNS to understand the sedimentation of mono- and poly-disperse hard-sphere suspensions when the large-scale ﬂow is laminar, and the work of Bunner and Tryggvason (2003) uses DNS to investigate bubbly ﬂows. If a two-ﬂuid model (which does not resolve the interfaces) could be derived that adequately reproduces these DNS data, then it could be used to investigate the effects of large-scale turbulence that arises, for example, when the system is subjected to shear (Lakehal et al., 2002). The results from direct simulations of the two-ﬂuid model in the turbulent regime could then be used to develop and validate multiphase turbulence models along the lines suggested by Sundaresan (2000). Fortunately, with the continuing advances in computer power, steady advances in DNS of two-phase systems can be expected. There is thus reason to be optimistic that more powerful multiphase turbulence models will eventually be available for modeling practical systems such as chemical reactors. In the current state of the art, almost all multiphase CFD models available in commercial codes use some type of turbulence model based on extending models originally developed for single-phase ﬂows. Such CFD models are thus meant to describe fully turbulent ﬂows (as opposed to laminar or transitional ﬂows). Nevertheless, many of these models have not been validated

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experimentally in the context of dense multiphase ﬂows for the reasons discussed earlier, and thus should be used with caution even for turbulent multiphase ﬂows. In any case, there is still considerable room for improvement of multiphase CFD models through comparison with carefully designed experiments for canonical ﬂows. Even more so than for single-phase turbulence, it can be expected that particular models will have limited ranges of applicability and will have to be ‘‘tuned’’ for multiphase ﬂows with different physics. For example, a two-ﬂuid model for a solid–liquid slurry in an agitated reactor will require different physical models than a churn-turbulent bubble column. In the ﬁrst case, large-scale turbulence is generated by the agitation system, while in the second case, it is generated by buoyancy and interfacial dynamics. However, for speciﬁc ﬂows of interest to the chemical industry, it should be possible to develop reliable CFD models for multiphase ﬂow dynamics that can be used to investigate scalar transport and chemical reactions needed to model chemical reactors. In the next section, we will thus look brieﬂy at the additional models needed to describe the transport and production of chemical species and thermal energy.

B. INTERPHASE MASS/HEAT-TRANSFER MODELS Our discussion of multiphase CFD models has thus far focused on describing the mass and momentum balances for each phase. In applications to chemical reactors, we will frequently need to include chemical species and enthalpy balances. As mentioned previously, the multiﬂuid models do not resolve the interfaces between phases and models based on correlations will be needed to close the interphase mass- and heat-transfer terms. To keep the notation simple, we will consider only a two-phase gas–solid system with ag +as ¼ 1. If we denote the mass fractions of Nsp chemical species in each phase by Yga and Ysa, respectively, we can write the species balance equations as @rg ag Y ga þ = ðrg ag Y ga Ug Þ ¼ = Jga M a þ Rga , @t

(164)

@rs as Y sa þ = ðrs as Y sa Us Þ ¼ = Jsa M a þ Rsa . @t

(165)

and

The terms Jga and Jsa are the diffusive ﬂuxes of species a in the gas and solid phases, respectively. Note that in addition to molecular-scale diffusion, these terms include dispersion due to ‘‘particle-scale’’ turbulence. The latter is usually modeled by introducing a gradient-diffusion model with an effective diffusivity along the lines of Eqs. (149) and (151). Thus, for large particle Reynolds numbers the molecular-scale contribution will be negligible. The term Ma is the

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297

mass-transfer rate from the gas to the solid phase for species a. By deﬁnition, the N sp N sp N sp mass fractions sum to unity so that Sa¼1 Jga ¼ Sa¼1 Jsa ¼ 0 and M g ¼ Sa¼1 Ma (see Eq. 142). The terms Rga and Rsa are the reaction rates for species a in each N sp N sp phase. By deﬁnition, mass is conserved so that Sa¼1 Rga ¼ 0 and Sa¼1 Rsa ¼ 0. Although we do not write them explicitly here, the reader can appreciate that the enthalpy balances for each phase will have a form similar to Eqs. (164) and (165), and can be used to determine the temperatures Tg and Ts of each phase. Likewise, mass transfer will lead to corresponding terms in the momentum balances (Eqs. 145 and 146) (Bird et al., 2002). CFD models for turbulent multiphase reacting ﬂows do not solve the ‘‘laminar’’ two-ﬂuid balances (Eqs. 164 and 165) directly. First, Reynolds averaging is applied to eliminate the large-scale turbulent ﬂuctuations. Using Eq. (164) as an example, we can apply Reynolds averaging to ﬁnd (with rg constant) @rg hag i Ye ga 00 00 e g þ = rg hag i Yg þ = rg hag i Ye ga U ga ug @t ¼ = hJga i hM a i þ hRga i

ð166Þ

The Reynolds-average gas volume fraction hag i is found from @rg hag i e g ¼ hM g i þ = rg hag iU @t

(167)

wherein the Reynolds-average mass-transfer term hM g i is unclosed. The Favree g is deﬁned by average gas velocity U e g ¼ hag Ug i U hag i

(168)

and its transport equation is found by Reynolds averaging Eq. (145). Although we do not write it out explicitly here, the reader should appreciate that the Reynolds-average gas-phase momentum equation has a number of unclosed terms that require models. Returning to Eq. (166), the third term on the left-hand side involves the turbulent scalar ﬂuxes, deﬁned by 00 Y 00g ga u g ¼

hag Y 00 ga u00 g i hag i

(169)

where the scalar and velocity ﬂuctuations are deﬁned by Y 00 ga ¼ Y ga Ye ga and e g ; and respectively. The usual model for the scalar ﬂuxes is grau00 g ¼ Uga U dient diffusion as follows: 00 e Y 00g ga u g ¼ GT =Y ga

(170)

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where GT is a turbulent diffusivity that is computed from the multiphase turbulence model. The CFD model can then be written as @rg hag iYe ga eg þ = rg hag iYe ga U @t ¼ = Geff =Y~ ga hM a i þ hRga i

ð171Þ

where we have combined the (usually negligible) particle-scale diffusive ﬂux and the turbulent ﬂuxes into an effective-diffusion coefﬁcient (Geff). As mentioned earlier, since the interfaces between phases are not resolved in the CFD model, the Reynolds-average mass-transfer terms (hM a i), and the Reynolds-average reaction rates (hRga i) in Eq. (171) must be modeled in terms of known quantities. This situation is very much like classical reaction engineering models for multiphase reactors with the important difference that all quantities are known locally. Such quantities include e g; U es ag ; Ye ga ; Ye sa ; Teg ; Tes ; U and local multiphase turbulence statistics. Note that these variables, although local, tell us nothing about the internal structure of the phases (i.e., subgridfs represents the Favre-average temperature scale information). For example, T of a solid particle consistent with the Favre-average enthalpy of a single particle.9 If, as is often the case, the temperature varies strongly between the center and outer surface of a particle, a SGS model will be required to account for this effect on, for example, the chemical reactions. The principal advantage of using the CFD model over a classical CRE model is thus the ability to eg U e s j) on the mass/heataccount for the effect of local ﬂuid dynamics (e.g., jU transfer rate between phases. In CFD codes, this is typically done by using correlations for hM a i written in terms of the local Sherwood (or Nusselt) number and particle Reynolds number (modiﬁed perhaps by a function of hag i to account for particle–particle interactions). In the case where large-scale mixing is inﬁnitely rapid, these correlations will reduce to the classical CRE models for homogeneous multiphase reactors. However, such cases are rare (and need not be modeled using CFD), and it is more likely that large-scale mixing will be rate limiting at certain locations within the reactor. Indeed, it is exactly in such cases that CFD modeling will be of most beneﬁt for reactor design and analysis. 9 This discussion also applies to the original variable Ts, which represents the ensemble-average temperature of particles located at a particular point at a given time. Basically, we know the total enthalpy of each particle, but we do not know how it is distributed inside any given particle. Since the reaction rate can be very sensitive to the local temperature, we will need a SGS model to describe the coupling between intraparticle transport processes and chemical reactions.

CFD MODELS FOR ANALYSIS AND DESIGN OF CHEMICAL REACTORS

C. COUPLING

WITH

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The species balances given above (Eqs. 164 and 165) include the reaction source terms on the right-hand sides. However, for these expressions to be useful in CFD modeling, the user must supply the reaction rate functions and the kinetic parameters. In addition, just as in single-phase turbulent reacting ﬂows (Fox, 2003), it may be necessary to account for micromixing effects on the chemical kinetics by using SGS models in the Reynolds-average transport equations (e.g., for hRga i in Eq. (171)). For example, consider the parallel reactions AþB!R AþC!S with A in the gas phase and excess B and C in the liquid phase. If the ﬁrst reaction is very rapid, then in the absence of micromixing effects in the liquid phase only R would be produced. However, if A cannot be rapidly mixed into the liquid phase (after mass transfer from the gas phase), then some S will be produced as an unwanted by-product. In many ways, this situation is analogous to the reaction systems discussed in Section III.B. If we deﬁne a mixture fraction x that is unity in the gas phase (i.e., for pure A) and initially zero in the liquid phase, then the degree of conversion of the ﬁrst reaction (Y1N) can be parameterized by the value of x in the liquid phase (see Eq. 68). However, the rate of the second reaction will depend on the local mixture fraction in a nontrivial manner (see Eq. 70). Thus, if we simply ignore SGS ﬂuctuations in the liquid mixture fraction we will likely severely underestimate the extent of the second reaction. In theory, it is possible to write a presumed PDF transport model for the mixture fraction in the liquid phase. However, unlike in single-phase turbulence, the source term for the mixture fraction in the liquid phase is the masstransfer term and the sink term for mixture-fraction ﬂuctuations will depend on the rate of molecular mixing in different regions around the interphase (e.g., boundary layer, wake, far ﬁeld). A similar situation is encountered in spray combustion (Reve´illon et al., 2004) where evaporating liquid droplets act as source terms for reactants in the gas phase. It thus may be useful to adapt SGS models for spray combustion (at least the parts modeling the mixture fraction mean and variance) to describe SGS mixing in more general settings such as gas–liquid ﬂows. One can also use direct simulations of bubbly reacting ﬂows (Khinast, 2001; Khinast et al., 2003; Koynov and Khinast, 2004; Raffensberger et al., 2005) to explore the validity of SGS models developed for two-phase reacting ﬂows. From the discussion above, we should keep in mind that even if no SGS micromixing model is used to describe the multiphase ﬂow, it may often be the case that chemical reactions (and indeed micromixing) will be limited by mass/ heat transfer between the phases. Because the multiﬂuid model (see Eqs. 164 and

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165) includes mass-transfer terms, the reaction rates will usually be mass transfer limited in cases where the chemical kinetics are fast. Thus, since we are already relying on correlations to calculate the mass/heat-transfer rates, it may not be fruitful to try to include a detailed description of micromixing inside each phase. Indeed, it will more likely be the case that improving the correlations (e.g., to include the effect of chemical reactions on local mass transfer) will have a greater impact on the accuracy of CFD model predictions. Finally, to conclude our discussion on coupling with chemistry, we should note that in principle fairly complex reaction schemes can be used to deﬁne the reaction source terms. However, as in single-phase ﬂows, adding many fast chemical reactions can lead to slow convergence in CFD simulations, and the user is advised to attempt to eliminate instantaneous reaction steps whenever possible. The question of determining the rate constants (and their dependence on temperature) is also an important consideration. Ideally, this should be done under laboratory conditions for which the mass/heat-transfer rates are all faster than those likely to occur in the production-scale reactor. Note that it is not necessary to completely eliminate mass/heat-transfer limitations to determine usable rate parameters. Indeed, as long as the rate parameters found in the lab are reliable under well-mixed (vs. perfect-mixed) conditions, the actual mass/ heat-transfer rates in the reactor will be lower, leading to accurate predictions of chemical species under mass/heat-transfer-limited conditions.

VI. Conclusions and Future Perspectives From the brief overview of CFD models presented in this work, the reader will hopefully have gained an initial appreciation of the utility and power of CFD tools for chemical reactor analysis and design. In Section II we have discussed the basic formulation and speciﬁc steps needed to set up a CFD model. We have also introduced the key concept of subgrid-scale (SGS) modeling and its importance in describing unresolved phenomena in reactor-scale CFD models. However, it is worth repeating here that the development of SGS models for chemical reactions and molecular transport is a natural extension of ‘‘traditional’’ chemical reaction engineering modeling activities, and thus one of the key areas where chemical engineers can have a large impact on the ﬁeld. In Section III we gave an example of an SGS model developed for mixing-sensitive chemical reactions. This simple model for mass-transfer-limited chemical reactions can be easily extended to other applications such as high-Schmidt-number laminar ﬂows. In Section IV we discussed efﬁcient methods for adding a population balance equation to a CFD code to model the production of ﬁne particles. More generally, the ability to represent the evolution of a population of entities (e.g., particles, bubbles, drops) in the context of CFD results in a very powerful tool for describing complex reacting ﬂows. At present, methods based on the direct quadrature method of moments

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(DQMOM) are still in their initial stages of development. Nevertheless, they have already been applied with great success by a growing number of researchers to a wide variety of problems. The versatility of DQMOM was demonstrated in Section IV by applying it twice to the one modeling problem: ﬁrst to model micromixing between ﬂuid elements containing a bivariate number density function (NDF), and then to represent the bivariate NDF itself. Finally, in Section V we discussed the challenges associated with CFD models for multiphase reacting ﬂows. Although there are still a number of open problems to be solved in multiphase ﬂow CFD, when used with caution existing CFD models can be used for at least qualitative analysis of chemical reactors. In my opinion, the perspective for future developments in the ﬁeld of CFD modeling of chemical reactors is quite strong. On the one hand, the continued growth of computational power both through faster computers and better algorithms will make it possible to solve more and more complex problems. On the other hand, we are fortunate in this ﬁeld that the basic microscopic balance equations are known, even though they lead to complex multiscale, multiphysics phenomena. The accessibility of large-scale computing facilities will allow us to explore this complexity using direct simulations for speciﬁc ‘‘academic’’ problems that can be used to test the SGS models needed for CFD simulations of industrial reactors. In general, advances in the development of SGS models will require collaboration between computational physicist/chemist working on direct simulation of academic problems and chemical reaction engineers developing multiphysics models. Indeed, the reader should appreciate that it is almost never the case that a reliable SGS model can be developed by simply analyzing the results from a large-scale direct simulation. Inversely, SGS model developed without validation against detailed experimental or direct simulation data are usually of limited value. Instead, the more fruitful approach is to ﬁrst develop a tentative SGS model based on a preliminary understanding of the physics/ chemistry of the problem, and then to design a large-scale direct simulation to test key assumptions/predictions of the model. This dialogue between model development and model validation is continued until a suitably reliable SGS model is found. SGS models developed in this manner have a strong fundamental underpinning and have a much greater chance of being applicable to a wide range of operating conditions. In summary, my recommendation for future progress in the ﬁeld is not to follow the deceptively simple path of rushing toward the application of largescale CFD simulations to complex industrial reactor systems if the basic SGS models have not ﬁrst been shown to be reliable on ‘‘academic’’ problems.10 Rather, I would recommend that we proceed more cautiously with adequate attention given to the development of the fundamental physical understanding required to develop reliable CFD models. While this path will obviously require 10

Despite this word of caution, one should not lose site of the fact that there are many industrial reactor systems that can be accurately simulated with existing SGS models!

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patience and perseverance, in the long run it will undoubtedly be the surest way to attain the chemical reaction engineer’s long-sought goal of ‘‘experiment-free’’ scale-up of chemical reactors.

ACKNOWLEDGMENTS The author’s work in the area of CFD analysis of chemical reactors has been supported nearly continuously for the last 15 years by the U.S. National Science Foundation. The work on gas–solid multiphase ﬂows and population balances was funded by the U.S. Department of Energy. The author would also like to acknowledge support from several companies, including Air Products and Chemicals, BASF, BASELL, BP Chemicals, Dow Chemical, DuPont Engineering, and Univation Technologies. Last, but not least, the author wishes to acknowledge his many collaborators over the years who are many in number to name them individually.

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PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN USING COMPUTATIONAL FLUID DYNAMICS Anthony G. Dixon1, Michiel Nijemeisland2 and E. Hugh Stitt2 1

Worcester Polytechnic Institute, Department of Chemical Engineering, Worcester, MA 01609, USA 2 Johnson Matthey Catalysts, Billingham, Cleveland, UK I. Introduction A. CFD and Packed Reactor Tube Modeling B. CFD Approaches to Interstitial Flow in Fixed Beds II. Principles of CFD for Packed-Tube Flow Simulation A. CFD Basics and Turbulence Modeling B. Packed Bed CFD Model Development C. Packed Bed CFD Simulation Issues D. Validation of CFD Simulations for Packed Beds III. Low-N Packed Tube Transport and Reaction Using CFD A. Hydrodynamics and Pressure Drop B. Mass Transfer, Dispersion, and Reaction C. Heat Transfer IV. Catalyst Design for Steam Reforming Using CFD A. Steam Reforming and Principles of Catalyst Design B. CFD Simulation of Reformer Tube Heat Transfer with Different Catalyst Particles C. Reaction Thermal Effects in Spheres Using CFD D. Reaction Thermal Effects in Cylinders Using CFD V. Future Prospects References

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Abstract Computational ﬂuid dynamics (CFD) is rapidly becoming a standard tool for the analysis of chemically reacting ﬂows. For single-phase reactors, such as stirred tanks and ‘‘empty’’ tubes, it is already wellestablished. For multiphase reactors such as ﬁxed beds, bubble columns, trickle beds and ﬂuidized beds, its use is relatively new, and methods are still under development. The aim of this chapter is to present the application of CFD to the simulation of three-dimensional interstitial ﬂow in packed tubes, with and without catalytic reaction. Although the use of 307 Advances in Chemical Engineering, vol. 31 ISSN 0065-2377 DOI: 10.1016/S0065-2377(06)31005-8

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CFD to simulate such geometrically complex ﬂows is too expensive and impractical currently for routine design and control of ﬁxed-bed reactors, the real contribution of CFD in this area is to provide a more fundamental understanding of the transport and reaction phenomena in such reactors. CFD can supply the detailed three-dimensional velocity, species and temperature ﬁelds that are needed to improve engineering approaches. In particular, this chapter considers the development of CFD methods for packed tube simulation by ﬁnite element or ﬁnite volume solution of the governing partial differential equations. It discusses speciﬁc implementation problems of special relevance to packed tubes, presents the validation by experiment of CFD results, and reviews recent advances in the ﬁeld in transport and reaction. Extended discussion is given of two topics: heat transfer in packed tubes and the design of catalyst particles for steam reforming.

I. Introduction A. CFD

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The design of chemical reactors to make useful chemical products must be able to accommodate new catalysts, new feedstocks, and new product speciﬁcations, while facing ever-tighter economic, environmental, safety, and social acceptability constraints. The reactor designer must strive for higher yields, less energy use, smaller reactor capital costs, and other aspects of sustainable processing. To accomplish these objectives, the reactor designer must be able to understand, quantify, and control the individual chemical and physical phenomena present in reactors. An essential part of such a task is the development of predictive models of reactor behavior, based on the true representation of the physical and chemical processes that occur, on different length scales. Capturing the real physics and chemistry is especially important for heterogeneous reactors, such as gas–solid or liquid–solid ﬁxed beds, and gas–liquid–solid trickle beds. It is necessary to know the spatial distribution of reactants, catalysts, inerts, and products in detail (Lerou and Ng, 1996). A catalytic ﬁxed bed reactor is a (usually) cylindrical tube that is randomly ﬁlled with porous catalyst particles. These are frequently spheres or cylindrical pellets, but other shapes are also possible. The use of rings or other forms of particles with internal voids or external shaping is on the increase. During single-phase operation, a gas or liquid ﬂows through the tube and over the catalyst particles, and reactions take place on the surfaces, both interior and exterior, of the particles. Single-phase catalytic ﬁxed bed reactors are the main reactor type used for largescale heterogeneously catalyzed gas-phase reactions. Frequently, multitubular

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ﬁxed beds with low tube-to-particle diameter ratio (N) are used for strongly exothermic reactions such as partial oxidations and selective hydrogenations as well as strongly endothermic reactions such as steam reforming of methane. In these processes heat must be rapidly transferred into or out of a narrow reactor tube, while the need to reduce compressor costs dictates a low pressure drop along the tube, and so the particle size cannot be too small. These constraints combine to give tubes with low values of N. The presence of the tube wall has a strong inﬂuence on heat transfer, reaction rates, and selectivity for these reactor tubes, which have proved to be exceptionally difﬁcult to model. Current ﬁxed bed reactor models have been based on fairly strong simplifying assumptions, such as pseudo homogeneity, unidirectional plug ﬂow, effective transport parameters, and uniform catalyst pellet surroundings. These simpliﬁcations have been motivated in the past by the need for computational savings, which continues to become less of an issue. The complex structure of randompacked tubes has also prompted a simpliﬁed approach. These idealized models have led, however, to problems. Even the most advanced models today cannot quantitatively predict reactor behavior if independently determined kinetics and transport parameters are used (Schouten et al., 1994; Landon et al., 1996). The effects of tube and catalyst pellet design changes are masked by the use of effective parameters and simpliﬁed models. Reaction engineers are in agreement that the entire ﬁeld has neglected the role of ﬂuid ﬂow in reactor modeling. For ﬁxed beds, a better understanding of ﬂuid ﬂow through arrays of realistic catalyst particle shapes would be of great help, with special attention to the problematical wall region. The presence of the tube wall causes changes in bed structure, ﬂow patterns, transport rates, and the amount of catalyst per unit volume, and is usually the location of the limiting heat transfer resistance. New techniques in experimentation and computation that allow us to understand and model ﬁxed-bed phenomena at the particle or subparticle level are needed. The desire to measure ﬂuid ﬂow inside the bed has led several researchers to use noninvasive experimental methods. McGreavy et al. (1984, 1986) used laser Doppler velocimetry (LDV) in low-N packed beds, for both liquid and gas experiments, although only results using liquids were presented. Particle tracking methods were used by Rashidi et al. (1996) and also by Stephenson and Stewart (1986). The latter authors used marker bubbles as a noninvasive method to measure the radial distribution of ﬂow of a matchedrefractive index ﬂuid in transparent packed beds of equilateral cylinders with N ¼ 10.7. They found that the local superﬁcial velocity attained its global maximum at 0.2dp from the wall and its global minimum at 0.5dp from the wall. Both studies found an oscillatory radial velocity proﬁle. This type of proﬁle was conﬁrmed recently, again using LDV (Giese et al., 1998), for a column with a tube-to-particle diameter ratio of approximately 9. Comparisons were made with the extended Brinkman model, and good agreement was obtained when an adjustable effective viscosity was introduced into the term for wall effects.

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A noninvasive experimental method that has been used to obtain local ﬂow patterns in ﬁxed beds is magnetic resonance imaging (MRI). This method can show ﬂow patterns in complicated geometries. The method has so far been restricted to relatively low ﬂow rates, and to ﬂuids that can produce a suitable signal for measurement, such as water. Gas ﬂow has rarely been investigated by MRI techniques. Generally the packed beds used for MRI have had a considerably higher tube-to-particle diameter ratio, which will result in less pronounced wall effects. Qualitatively the MRI results show generally accepted ﬂow concepts such as ﬂow increase in bed voids, as well as inhomogeneous velocity distribution in different pores (Sederman et al., 1997, 1998). The larger tube-to-particle diameter ratio also allows for a statistical view of the velocity distribution over the column cross section. When averaged over a long evolution time, the data approached Gaussian behavior (Park and Gibbs, 1999). With a tube-to-particle diameter ratio of 6.7 and relatively low ﬂow rates, the velocity proﬁle was roughly parabolic with the maximum being near the center of the tube. Also, negative velocities or reversed ﬂow within the bed were shown (Kutsovsky et al., 1996). Recent applications of the noninvasive MRI technique have been made by Suekane et al. (2003) to a cubic-packing ﬁxed bed unit cell and by Yuen et al. (2002) to isothermal reacting ﬂows. Ren et al. (2005) used NMR methods to obtain proﬁles of velocity in a narrow tube. All these methods allow observations of ﬂow inside the ﬁxed bed without disturbing the bed structure and can thus further our understanding, but each is subject to severe limitations. LDV requires windows for optical access and is restricted to beds of very low N where such voids occur naturally. It also requires the ﬂuid to be refractive-index-matched with the transparent material of the column. MR methods are mainly used for liquids, and techniques to allow imaging of fast ﬂows are starting to appear (Gladden et al., 2005). Particle tracking methods require observation and counting of the markers, and problems with choice of ﬂuid similar to those with LDV are found. Experimental methods do not yet let us get to conditions of interest for ﬁxed bed reactor design, i.e., gas-phase high ﬂow rates at elevated temperatures with conduction and reaction in catalyst particles. While experimental techniques continue to improve, a complementary approach is to take advantage of the recent advances in scientiﬁc computing to simulate the ﬂow ﬁelds. Computational ﬂuid dynamics (CFD) has become a standard tool in the ﬁeld of chemical engineering. The general setup of most CFD programs allows for a wide range of applications, and several commercial packages have introduced chemical reactions into the CFD code allowing rapid progress in the use of CFD within the ﬁeld of chemical reaction engineering (Bode, 1994; Harris et al., 1996; Kuipers and van Swaaij, 1998; Ranade, 2002). The application of CFD to packed bed reactor modeling has usually involved the replacement of the actual packing structure with an effective continuum (Kvamsdal et al., 1999; Pedernera et al., 2003). Transport processes are then represented by lumped parameters for dispersion and heat transfer (Jakobsen

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et al., 2002). The reactions that take place in the porous catalyst particles are represented by source or sink terms in the conservation equations (Ranade, 2002) and corrected for volume fraction and particle transport limitations. The velocity ﬁeld can be obtained from a modiﬁed momentum balance (Bey and Eigenberger, 1997) or a form of the Brinkman-Forcheimer-extended Darcy (BFD) equation (Giese et al., 1998). These approaches provide an averaged superﬁcial velocity ﬁeld, usually in the form of a radially varying axial component of velocity, which is an improvement over the classical assumption of plug ﬂow (constant unidirectional ﬂow). These velocity ﬁelds have been used in improved models of ﬁxed bed transport and reaction (Winterberg et al., 2000). The disadvantages of the BFD approach have been the continued lumping of transport processes, thus obscuring the physical basis of the model, and the necessity to introduce an effective viscosity for the bed to bring computed and experimental velocity proﬁles into agreement (Bey and Eigenberger, 1997; Giese et al., 1998). This form of CFD in ﬁxed beds is an extension of the classical pseudo-continuum approach, in which both ﬂuid and solid phases are modeled as inter-penetrating continua, i.e., as if they coexisted at every point in the tube (Fig. 1a). An alternative and complementary use of CFD in ﬁxed bed simulation has been to solve the actual ﬂow ﬁeld between the particles (Fig. 1b). This approach does not simplify the geometrical complexities of the packing, or replace them by the pseudo-continuum that is used in the ﬁrst approach. The governing equations for the interstitial ﬂuid ﬂow itself are solved directly. The contrast is thus between the interstitial ﬂow ﬁeld type of simulation and the superﬁcial ﬂow

(a) pseudocontinuum CFD

(b) interstitial CFD FIG. 1. Comparison of (a) pseudo-continuum and (b) interstitial CFD approaches to packed-tube simulation.

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ﬁeld models of the BFD approach (Tobis´ , 2000). The equations of the interstitial approach are well established and relatively straightforward; however, the geometric modeling and grid generation become complicated and the computational demands rise signiﬁcantly (Ranade, 2002). Owing to the computational requirements, the approach can so far be applied only to small, periodic regions of the reactor. It is therefore useful mainly as a learning tool, from which we can develop detailed insight into ﬁxed bed ﬂow structures and understand how they inﬂuence transport and reaction. The understanding from this more rigorous approach to CFD can then be used to inform the simplifying decisions made in the development of the more computationally tractable pseudo-continuum ﬁxed bed models.

B. CFD APPROACHES

TO

INTERSTITIAL FLOW

IN

FIXED BEDS

The development of CFD calculations of interstitial ﬂow in ﬁxed beds has increased and become more realistic, as greater computational power has become more available over the recent years. Calculations of velocity and pressure proﬁles for creeping ﬂow between spheres were done by Snyder and Stewart (1966) using Galerkin’s method, followed by Sørensen and Stewart (1974) using specially designed collocation methods. They were able to obtain the velocity and temperature proﬁles in cubic arrays of spheres, a highly symmetric arrangement. Their calculations yielded insight into the behavior of the heat transfer coefﬁcient for particle-to-ﬂuid heat and mass transfer, over a wide range of values of the Peclet number Pe ¼ RePr, where Re is the particle Reynolds number, based on superﬁcial velocity (rv0dp/m) and Pr is the Prandtl number (mcp/kf). Flow through cubic arrays of particles was also studied by Lahbabi and Chang (1985) by analytical methods, with focus on ﬂow transitions. Dalman et al. (1986) investigated ﬂow around two spheres near a wall using two-dimensional (2D) ﬁnite element models in an axisymmetric radial plane. This study showed that eddies formed between the spheres, which led to regions of poor heat transfer. Lloyd and Boehm (1994) also did a 2D study, with eight spheres in line, to determine the inﬂuence of the sphere spacing on the drag coefﬁcients and the particle-ﬂuid heat transfer coefﬁcient. A 3D ﬁnite element method was used by Mansoorzadeh et al. (1998) to simulate ﬂow past a heated/ cooled sphere at moderate Re. They found good agreement between calculated drag coefﬁcients and a literature correlation, and that the axisymmetry of the wake ﬂow broke down at Re ¼ 400. Heat transfer from or to the sphere increased the drag coefﬁcient at higher Re. It was found that heat transfer from the spheres decreased with decreased sphere spacing. McKenna et al. (1999) used discrete particle CFD to obtain valuable insight into the effect of particle size on particle-ﬂuid heat transfer during oleﬁn polymerization in a ﬂuidized bed. They used a commercial code, Fluent, to conduct a 2D CFD study of small clusters of catalyst particles and a 3D study of a single catalyst sphere close to a

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wall. They explored two ways to include the local energy effects of reaction. The ﬁrst was by specifying a surface heat ﬂux for systems of two touching particles or three particles with one small hot particle touching two larger, colder ones. Their second approach was to utilize a constant volumetric heat source in particles with solid conduction included, to allow for dilution of active sites and heat generation as the particle grows. Their main ﬁnding was that signiﬁcant heat is removed from these particles by conduction, as well as the usually assumed path of convection. Three-dimensional (3D) models have been developed more recently. A simple three-sphere model (Derkx and Dixon, 1996) focused on obtaining wall heat transfer coefﬁcients. An eight-sphere model followed (Logtenberg and Dixon, 1998a, b) in which the packing was modeled as two layers of four spheres, perpendicular to the ﬂow in a tube with a tube-to-particle diameter ratio of N ¼ 2.43. This study was limited by the absence of contact points between the spheres and the wall and between the spheres themselves. Subsequently, a 3D 10-sphere model was developed, with N ¼ 2.68, incorporating contact points between the particles and between the particles and the wall (Logtenberg et al., 1999), which used spherical dead volumes with estimated diameters, around the contact points. These studies focused on using CFD to obtain the traditional radial heat transfer modeling parameters such as the wall heat-transfer coefﬁcient (hw) and the effective radial thermal conductivity (kr), and gave reasonable qualitative agreement with experimental estimates. Other heat transfer work in ﬁxed beds has explored the use of CFD to simulate ﬂow and transport in structured packings (Von Scala et al., 1999) and to investigate the effects of roughness gaps in particle–particle heat conduction (Lund et al., 1999). So far, there have only been a few modeling studies to try to link local ﬂuid ﬂow to bed structure. Chu and Ng (1989) and later Bryant et al. (1993) and Thompson and Fogler (1997) used network models for ﬂow in packed beds. Different beds were established using a computer simulation method for creating a random bed. The model beds were then reduced to a network of pores, and either ﬂow/pressure drop relations or Stokes’ law was used to obtain a ﬂow distribution. Several groups have studied the connection between ﬂuid ﬂow and bed structure in complete particle beds. Esterl et al. (1998) and Debus et al. (1998) applied a computational code by Nirschl et al. (1995) to ﬁnd ﬂow proﬁles in a square channel, using an adapted chimera grid. This grid consisted of a structured grid, based on the ﬂowing medium, which was overlaid by a separate structured grid, based on the packing particles. Calculated pressure drops were compared against predicted pressure drops using, amongst others, Ergun’s relation for a bed with the same porosity; the simulation data gave the same order of magnitude. Simulations were performed in beds with up to 300 spheres, although the bed for which results were discussed consisted of 120 spheres. One of the aspects that may have affected the accuracy of the simulations is that the

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bed was only approximately ﬁve layers deep, resulting in a ﬂow that mainly consisted of inlet and outlet effects. Still other research groups (Georgiadis et al., 1996; Maier et al., 1998; Manz et al., 1999; Zeiser et al., 2001) have used the lattice Boltzmann method (LBM) for simulation of ﬂow in a ﬁxed bed of spheres. A dense packing of spheres in a cylindrical column was created from experimental observations, such as MRI, or by computer simulation, for example by using a raining and compression algorithm. The created packing geometry was then divided into an equidistant Cartesian grid, where individual elements were labeled as solid or ﬂuid regions as in a marker-and-cell approach. A high resolution of the grid made it possible to obtain accurate ﬂow proﬁles. Recently (Zeiser et al., 2002; Freund et al., 2003; Yuen et al., 2003) simple reactions have been added to the simulation, showing species and conversion proﬁles inside the bed. Limitations of the LBM are that simulations of turbulence are expensive, as the method corresponds to a direct numerical simulation (DNS) approach to the Navier-Stokes equations, and that it is difﬁcult to include heat transfer. Incorporating heat transfer into ﬁxed bed simulations is extremely challenging, due to the need to mesh and solve both convection in the voids and conduction in the particles. For LBM thermal models in the ﬂuid, a multispeed approach must be used, usually from two to four speeds is feasible. Due to the small number of speeds, the variation in temperature is restricted. In addition, LBM methods are inherently transient and thus more computationally expensive than steady-state differential equation-based approaches, and they suffer from instability, which is worse for multispeed methods (Chen and Doolen, 1998). A growing number of studies are appearing in which CFD methods are being used to simulate multi-physics ﬂow and heat transfer at higher ﬂow rates in ﬁxed beds. Calis et al. (2001) applied the commercial code CFX-5.3 to a structured packing of spheres. They simulated ﬂow in a number of channels of square cross-section ﬁlled with spherical particles. Several different types of structured packing were investigated, all based on structured packing of spheres. The repetitive sections had varying N, from 1 to 4. Values for pressure drop obtained from the simulations were validated against experimental values. The turbulence models used (k–e and RSM) showed similar results, with an average error from the experimental values of about 10%. Pressure drop in a structured packing has also been studied by Larachi et al. (2003) and Petre et al. (2003), who used the Fluent CFD code to obtain ﬂow ﬁelds that allowed them to construct submodels of different contributions to the overall pressure drop. Pressure drop and dispersion were the focus of work by Magnico (2003) who simulated ﬂow at lower Re by direct numerical simulation (DNS) in beds of spheres with an in-house code. Tobis´ (2000) simulated a small cluster of four spheres with inserts between them to compare to his experimental measurements of pressure drop. Gunjal et al. (2005) also focused on ﬂow and pressure drop through a small cell of spheres, in order to validate the CFD approach by comparison to the MRI measurements in the same geometry made by Suekane

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et al. (2003). Romkes et al. (2003) extended the study of Calis et al. to include calculations for the heat transfer coefﬁcient for a single sphere in an inﬁnite medium, the wall heat transfer coefﬁcient for laminar and turbulent ﬂow in an empty tube, and the particle-to-ﬂuid heat transfer coefﬁcient in a packed tube. Heat transfer in full beds and bed segments has been the main focus of our own work (Dixon and Nijemeisland, 2001; Nijemeisland and Dixon, 2001, 2004) with special emphasis on ﬂow patterns and heat transfer near the tube wall. Guardo et al. (2004, 2005) have performed similar calculations for pressure drop and wall heat transfer rates. Our most recent work has extended ﬂow and heat transfer simulations to include source or sink terms in the solid particles, to emulate the energetic effects of reaction (Dixon et al., 2003; Nijemeisland et al., 2004). Several of these contributions will be discussed in more detail below. The focus of the remainder of this chapter is on interstitial ﬂow simulation by ﬁnite volume or ﬁnite element methods. These allow simulations at higher ﬂow rates through turbulence models, and the inclusion of chemical reactions and heat transfer. In particular, the conjugate heat transfer problem of conduction inside the catalyst particles can be addressed with this method.

II. Principles of CFD for Packed-Tube Flow Simulation A. CFD BASICS

AND

TURBULENCE MODELING

CFD may be loosely thought of as computational methods applied to the study of quantities that ﬂow. This would include both methods that solve differential equations and ﬁnite automata methods that simulate the motion of ﬂuid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier–Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire ﬂow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will inﬂuence the accuracy of the solutions. After boundary conditions have been implemented, the ﬂow and energy balances are solved numerically; an iteration process decreases the error in the solution until a satisfactory result has been reached. Commercially available CFD codes use one of the three basic spatial discretization methods: ﬁnite differences (FD), ﬁnite volumes (FV), or ﬁnite elements (FE). Earlier CFD codes used FD or FV methods and have been used in stress and ﬂow problems. The major disadvantage of the FD method is that it is limited to structured grids, which are hard to apply to complex geometries and

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are mostly used for stress calculations in beams, etc. A 3D structured grid results in a grid with all rectangular elements. The rectangular elements can undergo limited deformation to ﬁt the geometry, but the adaptability of the grid is limited. The FV and FE methods support both structured and unstructured grids, and therefore can be applied to a more complex geometry. An unstructured grid is a 2D structure of triangular cells or a 3D structure of tetrahedral cells, which is interpolated from user-deﬁned node distributions on the surface edges or a triangular surface mesh, respectively. The interpolation part of the creation process is less directly inﬂuenced by the user in an unstructured mesh than in a structured mesh because of the random nature of the unstructured interpolation process. This aspect does however allow the mesh to adapt more easily to a complex geometry. The FE method is in general more accurate than the FV method, but the FV method uses a continuity balance per control volume, resulting in a more accurate mass balance. FV methods are more appropriate for ﬂow situation, whereas FE methods are used more in stress and conduction calculations, where satisfying the local continuity is of less importance. The equations for both laminar and turbulent ﬂows, and the ﬁnite volume methods used to solve them, have been presented extensively in the literature (Patankar, 1980; Mathur and Murthy, 1997; Ranade, 2002; Fluent, 2003). The following summary focuses on aspects of particular concern for simulation of packed tubes and also those options chosen for our own work. 1. Navier– Stokes Equations The general equation used for conservation of mass (the continuity equation) may be written as follows: @r @ðrui Þ þ ¼ Sm @t @xi

(1)

The source term Sm contains the mass added through phase changes or userdeﬁned sources. In general, and in the simulations described here, the source term was equal to zero. The equation for conservation of momentum in direction i and in a nonaccelerating reference frame is given by @ðrui Þ @ðrui uj Þ @p @tij þ ¼ þ þ rgi þ F i @t @xi @xj @xj

(2)

In this balance p is the static pressure, tij is the stress tensor, and rgi is the gravitational body force. Fi is an external body forces component; it can include forces from interaction between phases, centrifugal forces, Coriolis forces, and

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user-deﬁned sources. For single-phase ﬂow through packed tubes it is usually zero. The stress tensor tij for a Newtonian ﬂuid is deﬁned by @ui @uj 2 @ul tij ¼ m þ dij m 3 @xl @xj @xi

(3)

Here m is the molecular viscosity; the second term on the right-hand side of the equation is the effect of volume dilation. 2. RANS and the Standard k– e Turbulence Model A time-dependent solution of the Navier–Stokes equations for high-Reynoldsnumber turbulent ﬂows in complex geometries is currently beyond our computational capabilities. Two methods have been developed to transform the Navier–Stokes equations so that the small-scale turbulent ﬂuctuations do not have to be directly simulated. These are Reynolds averaging (RANS) and ﬁltering or Large-Eddy simulation (LES). Both methods introduce additional terms in the governing equations that need to be modeled in order to achieve closure. LES has not yet been applied to packed-tube modeling to any signiﬁcant extent. With RANS the solution variables in the Navier–Stokes equations are decomposed into mean, u¯ i , and ﬂuctuating u0i components, and integrated over an interval of time large compared to the small-scale ﬂuctuations. When this is applied to the standard Navier–Stokes equations (Eqs. (1)–(3)), the result is @ðrui Þ @ðrui uj Þ þ ¼ @t @xj

@ðru0i u0j Þ @p @ @ui @uj 2 @ul m þ m þ þ @xi @xj 3 @xl @xj @xj @xi

ð4Þ

The velocities and other solution variables are now represented by Reynoldsaveraged values, and the effects of turbulence are represented by the ‘‘Reynolds stresses,’’ ðru0i u0j Þ that are modeled by the Boussinesq hypothesis: 0

0

rui uj ¼ mt

@ui @uj þ @xj @xi

2 @ui rk þ mt dij 3 @xi

(5)

The k– e turbulence model was developed and described by Launder and Spalding (1972). The turbulent viscosity, mt, is deﬁned in terms of the turbulent kinetic energy, k, and its rate of dissipation, e. mt ¼ rC m

k2

(6)

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ANTHONY G. DIXON ET AL.

The turbulent kinetic energy and its dissipation rate are obtained from the adapted transport equations. @ðrkÞ @ðrui kÞ @ m @k þ ¼ mþ t (7) þ G k þ G b r @t @xi @xi sk @xi @ðrÞ @ðrui Þ @ þ ¼ @t @xi @xi

mt @ 2 Gk þ ð1 C 3 ÞG b C 2 r mþ þ C 1 k s @xi k (8)

In these equations, Gk is the generation of turbulent kinetic energy, k, due to turbulent stress, and is deﬁned by 0

0

Gk ¼ rui uj

@uj @xi

(9)

Gb is the generation of turbulent kinetic energy, k, due to buoyancy, G b ¼ bgi

mt @T Prt @xi

(10)

Here, Prt is the turbulent Prandtl number for temperature or enthalpy and b is the thermal expansion coefﬁcient, 1 @r b¼ (11) r @T p The default values of the constants C1e ¼ 1.44, C2e ¼ 1.92, Cm ¼ 0.09, sk ¼ 1.0, se ¼ 1.3, and Prt ¼ 0.85 have been established from experimental work with air and water, and have been found to work well for a wide range of wall-bounded and free shear ﬂows (Launder and Spalding, 1972). In a system with both heat and mass transfer, an extra turbulent factor, kt, is included which is derived from an adapted energy equation, as were e and k. The turbulent heat transfer is dictated by turbulent viscosity, mt, and the turbulent Prandtl number, Prt. Other effects that can be included in the turbulent model are buoyancy and compressibility. The energy equation is solved in the form of a transport equation for static temperature. The temperature equation is obtained from the enthalpy equation, by taking the temperature as a dependent variable. The enthalpy equation is deﬁned as, P @ hj J j @ðrhÞ @ðrui hÞ @ @T Dp @ui j þ þ ðtik Þeff ¼ ðl þ lt Þ þ þ Sh (12) @t @xi @xi @xi Dt @xi @xk

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319

In this equation Sh includes heat of chemical reaction, any interphase exchange of heat, and any other user-deﬁned volumetric heat sources. lt is deﬁned as the thermal conductivity due to turbulent transport, and is obtained from the turbulent Prandtl number lt ¼

cp mt Prt

(13)

The enthalpy h is deﬁned as h¼

X

Y j hj

(14)

j

where Yj is the mass fraction of species j and, Z

T

hj ¼

cp;j dT

(15)

T ref

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 3. Alternative Turbulence Models The Reynolds-averaged approach is widely used for engineering calculations, and typically includes models such as Spalart–Allmaras, k– e and its variants, k–o, and the Reynolds stress model (RSM). The Boussinesq hypothesis, which assumes mt to be an isotropic scalar quantity, is used in the Spalart–Allmaras model, the k– e models, and the k– o models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, mt. For the Spalart–Allmaras model, one additional transport equation representing turbulent viscosity is solved. In the case of the k– e and k–o models, two additional transport equations for the turbulence kinetic energy, k, and either the turbulence dissipation rate, e, or the speciﬁc dissipation rate, o, are solved, and mt is computed as a function of k and either e or o. Alternatively, in the RSM approach, transport equations can be solved for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (usually for e) is also required. This means that seven additional transport equations must be solved in 3D ﬂows. Our group has made extensive use of the RNG k– e model (Nijemeisland and Dixon, 2004), which is derived from the instantaneous Navier–Stokes equations using the Renormalization Group method (Yakhot and Orszag, 1986) as opposed to the standard k– e model, which is based on Reynolds averaging. The

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major differences, in application, from the standard k– e model are different empirical constants in the k and e balances and extra terms in the turbulent dissipation balance (e). The Renormalization group methods are a general methodology of model building based on the stepwise coarsening of a problem. The main idea is that the RNG theory is applicable to scale-invariant phenomena that do not have externally imposed characteristic length and time scales. In the case of turbulence, the RNG theory is applicable to the small-scale eddies, which are independent of the larger scale phenomena that create them. The RNG theory as applied to turbulence reduces the Reynolds number to an effective Reynolds number (Reeff) by increasing an effective viscosity (meff). Through this process the small-scale eddies are eliminated, which reduces computational demand considerably. The new equation for the variation of the effective viscosity is as follows: meff ð‘Þ ¼ mmol

3A 1 þ 3 ð‘4 ‘4d Þ 4mmol

1=3 ð‘ ‘d Þ

(16)

where A is a constant derived by the RNG theory, ‘ is the eddy length scale, and ‘d is the Kolmogorov dissipation scale. So in this case when the eddy length scale is the Kolmogorov scale, the effective viscosity is the molecular viscosity. This equation then gives the interpolation formula for meff(‘) between the molecular viscosity mmol valid at dissipation scales and the high Reynolds number limit L ‘d . Using the deﬁnition for the turbulent viscosity (mt ¼ meff – mmol), which gives a result similar to the standard k– e model with only a small difference in the modeling constant, the effective viscosity is now deﬁned as a function of k and e in Eq. (16) in algebraic form. " meff ¼ mmol

sﬃﬃﬃﬃﬃﬃﬃﬃﬃ #2 Cm k pﬃﬃ 1þ mmol

(17)

The differential form of this equation is used in calculating the effective viscosity in the RNG k– e model. This method allows varying the effective viscosity with the effective Reynolds number to accurately extend the model to low-Reynolds-number and near-wall ﬂows. The transport equations for the turbulent kinetic energy, k, and the turbulence dissipation, e, in the RNG k– e model are again deﬁned similar to the standard k– e model, now utilizing the effective viscosity deﬁned through the RNG theory. The major difference in the RNG k– e model from the standard k– e model can be found in the e balance where a new source term appears, which is a function of both k and e. The new term in the RNG k– e model makes the turbulence in this model sensitive to the mean rate of strain. The result is a model that responds to the effect of strain and the effect of streamline curvature,

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a feature that is nonexistent in the standard k– e model. The inclusion of this effect makes the RNG k– e model more suitable for complex ﬂows. The RNG model provides its own energy balance, which is based on the energy balance of the standard k– e model with similar changes as for the k and e balances. The RNG k– e model energy balance is deﬁned as a transport equation for enthalpy. There are four contributions to the total change in enthalpy: the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is ﬁxed and user-deﬁned; the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). Guardo et al. (2004, 2005) have compared various turbulence models for a four-layer packed tube with N ¼ 3.92 containing 44 spheres. In their ﬁrst report (Guardo et al., 2004) they compared laminar, standard k– e, and Spalart– Allmaras models, and in the follow-up work (Guardo et al., 2005) they extended this to cover the Spalart–Allmaras, standard k– e, RNG k– e, and realizable k– e and k– o models. Based on comparisons of computed pressure drop to the predictions of the Ergun equation, they conclude that the Spalart–Allmaras equation is preferable. This is explained by the fact that it is formulated to be valid all the way to solid surfaces, thus avoiding problems with wall functions (discussed below). Our comparisons of the standard k– e, RNG k– e, and RSM models (Nijemeisland and Dixon, 2001) showed that there were no signiﬁcant differences in the results. Similar results for their geometry were found by Calis et al. (2001). The RNG k– e model was chosen in our work because it deals better with ﬂow with high streamline curvature and high strain rates, such as would be expected in packed tubes. 4. Wall Functions Turbulent ﬂows in packed tubes are strongly inﬂuenced by the solid surfaces, both the tube wall and the surfaces of the packing. Collectively, solid surfaces are referred to as ‘‘walls’’ in the CFD literature, and in this section we will continue that tradition. Besides the no-slip boundary condition on the velocity components that has to be satisﬁed, the turbulence is also changed by the presence of the wall. Very close to the wall, the tangential velocity ﬂuctuations are reduced by viscous damping and the normal ﬂuctuations are reduced by kinematic blocking. In the outer part of the near-wall region, in contrast, turbulence is increased by the production of turbulence kinetic energy due to the large gradients in mean velocity. The near-wall region is conceptually subdivided into three layers, based on experimental evidence. The innermost layer is the viscous sublayer in which the ﬂow is almost laminar, and the molecular viscosity plays a dominant role. The outer layer is considered to be fully turbulent. The buffer layer lies between

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the viscous sublayer and the fully turbulent layer, and the effects of molecular viscosity and turbulence are equally important. To numerically resolve a solution in the sublayer requires a very ﬁne mesh, since the sublayer is thin and gradients there are large. Models that are modiﬁed to enable the viscosityaffected region to be resolved with a mesh all the way to the wall, including the viscous sublayer such as the Spalart–Allmaras and k– o models, are termed lowReynolds number models. To save computational effort, high-Reynolds number models, such as k– e and its variants, are coupled with an approach in which the viscosity-affected inner region (viscous sublayer and buffer layer) are not resolved. Instead, semiempirical formulas called ‘‘wall functions’’ are used to bridge the viscosity-affected region between the wall and the fully turbulent region. The two approaches to the sublayer problem are depicted schematically in Fig. 2 (Fluent, 2003). The standard wall function (Launder and Spalding, 1974) has been widely used for industrial ﬂows. The wall function is based on the assumption that the velocity obeys the log law-of-the-wall 1 U ¼ lnðEy Þ k

(18)

where 1=2

U P C 1=4 m kP tw =r

(19)

turbulent core

U

?

buffer & sublayer

Wall Function Approach

Near-Wall Model Approach

The viscosity-affected region is not resolved, instead is bridged by the wall functions.

The near-wall region is resolved all the way down to the wall.

High-Re turbulence models can be used.

The turbulence models ought to be valid throughout the near-wall region.

FIG. 2. Near-wall treatments (reproduced from Fluent Inc., Version 6.1 Manual, 2003, by permission).

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1=2

y

C 1=4 m k P yP n

(20)

and k and E are universal constants, and UP is the mean velocity at P, the centroid of the cell next to the wall, and yP is the distance of point P from the wall. We shall follow the original reference and present the wall functions in terms of y and U, although the usual notation in the turbulence ﬁeld is to use pﬃﬃﬃﬃﬃﬃﬃ t =ry P yþ wn P and U þ pUﬃﬃﬃﬃﬃﬃﬃ : tw =r

It is important to place the ﬁrst near-wall grid node far enough away from the wall at yP to be in the fully turbulent inner region, where the log law-of-the-wall is valid. This usually means that we need y 430–60 for the wall-adjacent cells, for the use of wall functions to be valid. If the ﬁrst mesh point is unavoidably located in the viscous sublayer, then one simple approach (Fluent, 2003) is to extend the log-law region down to y ¼ 11.225 and to apply the laminar stress–strain relationship: U ¼ y for yo11.225. Results from near-wall meshes that are very ﬁne using wall functions are not reliable. The heat ﬂux to the wall and the wall temperature are related through a wall function 0:5 ðT w T P Þrcp C 0:25 C 0:25 k0:5 m kP ¼ Prt k1 lnðEy Þ þ P þ 12rPr m q_00 P Prt U 2P þ ðPr Prt ÞU 2c 00 w q_ w

(21) where P can be computed using (Launder and Spalding, 1974) 0:24 0:5 p=4 A Pr Prt P¼ 1 sinðp=4Þ k Prt Pr

(22)

where TP is the temperature at the cell adjacent to the wall, Tw is the temperature at the wall, Prt is the turbulent Prandtl number, Uc is the mean velocity magnitude at the edge of the thermal conduction layer, and A, k and E are universal constants. An analogous approach is used for species transport. The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the ﬂow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects.

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Apart from reﬁnements to the standard wall function approach, such as the nonequilibrium model, there has been little progress in the 30 years since its ﬁrst development. Craft et al. (2002) have suggested that this is because those who work on fundamental phenomena in the near-wall region regard any wall functions as a basically inadequate approach, whereas for industrial applications for which resolution of the boundary layer is usually impractical, the more advanced wall functions have not given clearly better results than the simpler original formulation. More recently, there has been a renewed interest in developing improved wall functions, based on sub-grid models of the near-wall cell (Craft et al., 2002) or on analytical integration of simpliﬁed boundary-layer equations to get proﬁles to be used in the near-wall cell (Utyuzhnikov, 2005). The development of improved wall function treatments, especially for situations where the near-wall node must be located in the ‘‘buffer’’ region, will be of great importance to the application of CFD to packed tubes, as will be further discussed below in the context of mesh generation. An ‘‘enhanced wall function’’ option that uses blending functions to obtain a single equation valid for all three near-wall layers has been developed in Fluent, but has yet to be extensively tested for packed-tube ﬂows. 5. Finite Volume Methods and Codes There is considerable difference of opinion in the CFD research community regarding the use of commercial CFD codes versus the development of in-house code. The main advantage of the latter is that the user has complete control of the code, and can modify it as he/she sees ﬁt. This approach is therefore essential for those whose research is directed toward the development of improved turbulence models, wall functions, or numerical algorithms. The downside of the in-house code is that it is usually restricted to specialized, if not simple, computational domains, and that post-processing and visualization facilities are usually primitive, in the absence of a third-party post-processing package. When the research is directed more toward the use of CFD methods to obtain insight or detailed information about a complex geometry such as a packed tube, the mesh generation and post-processing facilities available in commercial codes can be invaluable. Some of the commercial codes that have been used for packed-tube simulations are those by Fluent Inc. (FLUENT, FIDAP), ANSYS, Inc. (CFX), and Computational dynamics, Ltd. (Star-CD). Other codes that are often used originate in the governmental or academic sectors, e.g., CFDLIB from Sandia National Labs and PHOENICS, which is available as both shareware and commercially from Simuserve Ltd. CFD codes, can be modiﬁed to include extra terms, such as the Ergun terms for porous regions (Tierney et al., 1998). Some have interfaces for user-deﬁned code (e.g., FLUENT) while others make the original source code available (e.g., CFDLIB). The two main approaches to CFD for packed-tube simulations have been LBM and FV methods. The LBM codes are in-house codes; the methods behind

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them are well described in Chen and Doolen (1998) and are beyond the scope of this chapter. FV methods that directly discretized the Navier–Stokes equations have been extensively described by Patankar (1980), Mathur and Murthy (1997), and Ranade (2002). Brieﬂy, the governing equations are integrated over control volumes to obtain discrete algebraic equations that conserve the quantity on the control volume. Variables are stored on a collocated grid with a local pressure gradient correction. The pressure–velocity coupling is achieved by the SIMPLE or the SIMPLEC scheme, and the equations are stabilized by underrelaxation. Second-order centered differences are used for spatial discretization of the viscous terms, while interpolation of face values from cell center values is done by ﬁrst- or second-order upwind differences for the convective terms. The equations are solved iteratively, and either the energy equation can be a part of the main iteration in a coupled calculation or the ﬂow ﬁeld can be calculated ﬁrst and then substituted into the energy equation in a segregated calculation. The solution of the large sets of algebraic equations is carried out by multigrid schemes, to accelerate convergence by computing corrections on a series of coarse grid levels. This is particularly important for unstructured meshes, as the line iterative methods used for structured meshes are unavailable. Multigrid methods use the property that on coarse meshes the global error can be reduced quickly, while on ﬁne meshes the local error can be reduced by ﬁne-grid relaxation schemes. Further details of the computational methods may be found in standard references, and are not repeated here as our focus is on the use and specialization of CFD methods for packed tubes, and the issues arising from this application area.

B. PACKED BED CFD MODEL DEVELOPMENT To create a useful CFD simulation the model geometry needs to be deﬁned and the proper boundary conditions applied. Deﬁning the geometry for a CFD simulation of a packed tube implies being able to specify the exact position and, for nonspherical particles, orientation of every particle in the bed. This is not an easy task. Our experience with different types of experimental approaches has convinced us that they are all too inaccurate for use with CFD models. This leads to the conclusion that the tube packing must either be computergenerated or be highly structured so that the particle positions can be calculated analytically. Some groups have worked with models of unit cells of periodic repeating packing, such as cubical or rhombohedral, which are representative of the bed far from the containing walls (Tobis´ , 2000; Gunjal et al., 2005). Others have developed models of complete tube cross-sections in relatively short-axial segments of tubes with low enough N to be completely structured, for example N ¼ 2 (Nijemeisland and Dixon, 2001) and 1 rN r2 in a square cross-section channel (Calis et al., 2001). Several groups used computer-generated packings,

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at N ¼ 7 (Esterl et al., 1998) and at N ¼ 2.42 and N ¼ 4 in square channels (Calis et al., 2001) and at N ¼ 5.96 and N ¼ 7.8 in circular tubes (Magnico, 2003). Others used various ad hoc methods based on a combination of structure and experimental observation to guide model development for N ¼ 4 (Dixon and Nijemeisland, 2001) and N ¼ 3.92 (Guardo et al., 2004, 2005). In this section we brieﬂy review and illustrate some of these approaches, including our wall-segment (WS) approach that aims to retain the essential near-wall phenomena in a reduced-size model. 1. Computer-generated Packed Tubes The geometric complexity of random-ﬁxed bed structures has usually been handled by a statistical description of the geometry, in terms of void fraction proﬁles and pore size distributions. This approach gives a practical way to implement bed structure into the ﬂow model by retaining the statistical characteristics of the void space without having to introduce the real void structure, since that is complicated and its 3D structure will vary with repacking (Jiang et al., 2002). For CFD simulation, however, the exact structure of a particular instance of the bed must be known. For truly random large-N beds, the CFD results would then have value only if averaged over an ensemble of realizations of the possible bed structures. For low-N packed tubes, a different point of view can be taken. The packing of such tubes is quite far from random, as the ordering of the particles imposed by the tube walls can extend from two- to ﬁve-particle diameters into the packing. In addition, the packing of the particles against the tube wall results in a small number of repeatable structures. Thus, if the focus of the research is the phenomena in the near-wall region, then CFD simulations in quite-structured arrays of particles can be regarded as a good representation of the full-size tube. The advantage of computer generation of such arrays is that we get complete information on the position and orientation of the particles, with high accuracy. The drawbacks of the routine use of computer-generated packs are as follows: several algorithms do not allow for the inﬂuence of containing walls or require such walls to be planar (rectangular ducts); few algorithms exist for nonspherical particles; it is difﬁcult to generate dense or loose packing as required, and a particular algorithm usually gives one or the other case; and validation of the computer results still requires some development. Despite these problems, packing algorithms are being used to provide geometric models for CFD simulations. Some of these instances are reviewed in this section, along with some of the more recent advances in computer generation algorithms. The approaches to computer-generated packings have been classiﬁed into two main classes (Liu and Thompson, 2000): sequential deposition (SD) and collective rearrangement (CR). Typical of SD are the drop-and-roll algorithms, in which particles are added one by one to the packing and allowed to move until they ﬁnd a gravitationally stable position. These algorithms usually result

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in loose packings of porosity e ¼ 0.42 or higher. The CR algorithms (sometimes called Monte Carlo methods) typically begin with a prescribed number of particles that are moved randomly to either eliminate overlaps if the initial arrangement allowed them, or decrease voidage if the initial arrangement was nonoverlapping. A prescribed voidage can often be achieved, but it may be at some considerable computational cost. It is also possible to classify the algorithms by whether the container is rectangular or cylindrical, and by whether the particles are spherical or nonspherical. One of the earliest algorithms that has been used for CFD geometry generation is an SD method for spheres in rectangular ducts (Chan and Ng, 1986), which was used by Esterl et al. (1998) and Debus et al. (1998). Soppe (1990) used a hybrid method, combining the raining technique and Monte Carlo rearrangement, again for spheres in a rectangular duct. This method was adapted by Freund et al. (2003) on a 3D equidistant orthogonal lattice to give packings of spheres with 1.1 oN o20.3. Reyes and Iglesia (1991) also used an SD algorithm for slow settling of hard spheres, but extended the methodology to cylindrical container walls and arbitrary-size distributions of spheres. Nolan and Kavanagh (1992) developed a CR algorithm for spheres in a cylinder, and then extended it to nonspherical particles (Nolan and Kavanagh, 1995) by representing arbitrary shapes as assemblies of component spheres with varying size distributions. Spedding and Spencer (1995) used SD for spherical packings in cylinders with no edge effects, since N was equal to 50 in their simulations of liquid rivulet ﬂow over spheres. Periodic boundary conditions for the walls were used by Yang et al. (1996) who wanted an efﬁcient method for large numbers of random-size spheres packed into a hexahedral domain. They used a hybrid algorithm with random placement to generate the initial distribution of spheres and rearrangement to stable positions via a drop-and-roll procedure to eliminate overlaps. Liu and Thompson (2000) studied the extent to which the choice of boundary conditions affected internal-packing structure for spheres in rectangular ducts in CR algorithms. They found that even with periodic boundaries, intended to remove structure from the packings, CR algorithms could induce a self-assembly process that resulted in packings with ordered structures. Mueller (1997) paid particular attention to wall effects in his use of an SD method for spheres in cylindrical tubes. His algorithm starts with a base layer of spheres, and then adds each new sphere to one of the two different types of positions, wall sphere positions and inner sphere positions, maintaining stability under gravity. He compared different procedures for determining the next site for addition of a new sphere. Best results were obtained when spheres were added to the lowest vertical positions of the two types of positions according to a set percentage. A more recent reﬁnement of the method (Mueller, 2005) uses a packing parameter for the layers above the base layer, and gave good agreement with experimental voidage proﬁles for 2rNr20. Mueller’s approach seems to be the most useful for generating packed tubes of spheres at low N for CFD

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simulations, and it was adopted by Magnico (2003) for his simulations at N ¼ 5.96 and N ¼ 7.8. Maier et al. (1998) used a CR technique to generate spherical bead packs in cylinders with nonpermeable walls and rectilinear domains with periodic boundaries, which they then used in their own, lattice Boltzmann simulations. Nonspherical particle packings present more difﬁculties than spherical ones and have thus received less attention. Notable efforts have been made by Nandakumar et al. (1999) and Jia and Williams (2001). Nandakumar et al. (1999) represented arbitrary packing objects by a 3D polygonal model with a set of triangular surfaces. Their method did not simulate the dynamics of the packing process, but rather sought stable equilibrium positions for the objects using a collision detection algorithm. They obtained good agreement with experimental data for packings of spheres, Raschig rings and Pall rings. Jia and Williams (2001) took a completely different, digital approach, in which both particle shapes and packing space were digitized. Shapes could be represented by a collection of pixels (in 2D) and particles moved in random directions on a square lattice, with higher probability of moving ‘‘down’’ to simulate the gravity-induced packing process. Extensions to 3D and mixed particle sizes have been reported by Caulkin et al. (2006). Finally, Theuerkauf et al. (2006) reported on the use of the discrete element method (DEM) to generate packing structures in packed beds of spheres for use in CFD modeling, for low-N beds. DEM is a well-established method that uses particle mechanical properties to inﬂuence a ﬁnal packing structure during a settling/deposition type process. The particles can be of arbitrary shape, and a soft contact approach is used, which allows particles to overlap during the process. The authors used a commercial DEM code, and have presented results on packings of cylinders by DEM in conferences. This appears to be a very promising method, which was validated by comparison to experimental radial voidage proﬁles. 2. Complete Wall Models of Packed Tubes In packed beds with regular particles, such as spheres, several packing structures can be identiﬁed. First, a wall-induced structure is regularly found, in which the spheres arrange themselves along the wall in staggered rings (Mueller, 1997). A dense sphere packing can be identiﬁed in the center of the bed, when this is located far enough from the wall. The third structure is a more random transition between the very regular wall and center structures. In the literature, the majority of research is directed toward determining where the transition from the wall-induced structure to the central structure takes place. This is important when a radial porosity proﬁle is used in modeling, in order to determine when to use the bed average porosity value. Generally, it is concluded that the effect of the wall has dissipated at about four particle diameters from the wall (Benenati and Brosilow, 1962) with only minimal contribution at about two particle diameters from the wall (Schuster and Vortmeyer, 1981).

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In beds with very high N, the central structure will dominate throughout the bed. In low-N beds the wall-induced structure will dominate as the inﬂuence of the wall on the structure penetrates relatively deep into the bed. As an example, consider a tube with a diameter of 100, in arbitrary units, and two different spherical packing materials with diameters of 10 (N ¼ 10) and 1 (N ¼ 100); further assume that the wall-induced structure is recognizable as such for four layers of spheres from the wall. In the N ¼ 10 bed the ﬁrst four layers at the wall occupy 96 volume % of the bed. In the N ¼ 100 bed, the wall-inﬂuenced region only makes up 15.6 volume % of the bed. Once the structure of the bed is understood, it must be implemented as a computational geometry. For this the exact positions of the spheres must be established so that they can be placed in the CFD simulation geometry. For the N ¼ 4 bed that we studied (Dixon and Nijemeisland, 2001) there was a regular structure in the bed, which made it possible to obtain the sphere locations using geometric relations. The geometry layout was divided into a nine-sphere wall induced structure and a three-sphere central structure. The nine-sphere wall layer was redistributed regularly along the wall. The speciﬁc tube-to-particle diameter ratio allowed for an almost exact ﬁt of nine spheres along the tube wall, as shown in Fig. 3. All the spheres in a layer were supported by two spheres of the layer below and the column wall, creating a stable packing structure. As the tube-to-particle diameter ratio of the bed was only four, the entire packing structure was controlled by the inﬂuence of the wall. Nevertheless, the packing was divided into an immediate wall layer and a central section, but this should not be taken to imply that the central structure was not wall inﬂuenced. Although a threesphere planar structure would almost ﬁt within the nine-sphere wall layer, there was just not enough room at the same axial coordinate. When, however, the

1-1 1-9

1-2

1-3

1-8

1-7

1-4 1-6

(a)

1-5 (b)

FIG. 3. Complete wall N ¼ 4 geometries: (a) top view of a layer of spheres, with nine wall spheres marked, showing, three spheres in the center region; (b) perspective view of the two-layer full bed of cylinders.

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axial coordinate of this three-sphere structure was located between two ninesphere wall layers, there was enough room. It was eventually found that the three-sphere central stacking was supported by the nine-sphere wall layers. The additional spacing in the central structure results in identical layer spacing of both the nine-sphere wall layers and the three-sphere center layers, creating a stable overall structure. The three-sphere central structure was a spiral repetitive structure in which spheres were supported by only one sphere from the central structure and two spheres from the wall structure. The spiral nature of the structure necessitated a larger overall bed, to be able to accommodate periodic boundaries. The spiral needed six layers for the central layer spheres to return to their original positions; therefore, the full-bed model had to be made over six layers, or 72 spheres in total. More recently, we have created full-bed periodic two-layer models for packings of cylindrical particles (Taskin et al., 2006). The geometry shown in Fig. 3b was created speciﬁcally for comparison of the WS model with cylindrical particles described in the following section, with the structures identical within a 1201 segment of the bed. 3. Wall Segments Experience in CFD modeling in packed tubes has taught us that the number of control volumes increases rapidly with the increasing size of the geometry. To be able to solve for certain details in the model, such as areas where particles in the packing touch each other or the tube wall, a high level of detail is necessary for the required accuracy of the simulation. More particles in a model simulation will lead to more high-detail areas, increasing the computational size of a complete-wall, full-bed simulation very quickly. This led us to the conclusion that we needed to formulate a model that focused on a small number of catalyst particles and their direct neighbors, for an accurate description of the heat transfer and ﬂow processes taking place on the local scale. The near-wall region is the most interesting area since this is where the largest heat transfer gradient occurs. A smaller model was needed with representative particle and periodic boundaries so that it could be compared with any near-wall position in the full-bed geometry. The cylindrical shape of the geometry pointed to the use of a wedge-shaped segment. A WS consisting of a third of the tube circumference (1201 segment) and two axial layers of particles would allow for appropriate detail throughout the simulation geometry. The relationship of the WS to the complete wall geometry is shown in Fig. 3. Within the WS geometry an appropriately large buffer zone around the area of interest was used to limit the effects of the boundary conditions on the ﬂow properties in that area. The smaller WS geometry also gave us the freedom to adjust the mesh density to the requirements of the different packing situations. The major added difﬁculty of this approach was the addition of several new boundaries on which appropriate boundary conditions had to be implemented.

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The most important parts of creating a segment model are the application of the physical boundary conditions and the positioning of the internals to allow for the symmetry and periodic boundary conditions. Without properly applying boundary conditions the simulation results cannot be compared to full-bed results, both as a concept and as a validation, since the segment now is not really a part of a continuous geometry. Our approach was to apply symmetry boundaries on the side planes parallel to the main ﬂow direction, thereby mimicking the circumferential continuation of the bed, and translational periodic boundaries on the axial planes, as was done in the full-bed model. For a WS model, periodic boundary conditions on the top and bottom boundaries are necessary, and a layout that allows for these conditions as well as the symmetry conditions is needed. The N ¼ 4 full-bed model had ninesphere wall layers and three-sphere center layers, as described in the previous section. The 1201 section would contain three wall-layer spheres and one center-layer sphere, for a fully symmetrical layout. In the full-bed model six layers were necessary to create periodic conditions, because of the spiral structure in the three-sphere center layers. Using six layers in the segment model, however, would defeat the purpose of creating a segment model (reduction of size), so it was chosen to slightly adjust the central layer positions to allow for periodic boundary conditions over two layers of spheres, as shown in Fig. 4. There are several sphere segments from the three-sphere central layer that can be seen in the top view. One of the spheres from the central layer structure is completely enclosed in the segment model. The two other spheres, toward the symmetry walls of the segment, are truncated both by the symmetry wall and the top and bottom of the model. The smaller dashed semicircles

FIG. 4. The 1201 wall-segment (WS) model: top (spheres) and front (cylinders).

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within the sphere-outlines depict the intersection of the spheres with the top and bottom planes. To simulate industrial packed tubes, we wanted to extend our WS approach to cylindrical particles, based on the model created for the N ¼ 4 spheres WS. A 1201 segment with two axial layers of particles and translational periodic boundaries on the column inlet and outlet was used for the cylindrical particle beds. An experimental study was performed by packing cylindrical particles with a 1:1 ratio in a bed with a tube-to-cylinder diameter ratio of 4. Dense particle packing structures were created using the Unidense method. From a large selection of packing structures, it could be seen that there were distinct common particle situations near the tube wall, such as the 451 angled particle, or axially aligned particles. The most common arrangement of cylindrical particles near the wall of the column proved to be the one with the round surface of the cylinder against the wall of the column with the particle axis at an angle with the axis of the column (see Fig. 4). To represent this most common orientation the base geometry was created with the main particle axis at a 451 angle with the column axis, as shown. The peripheral particles were then placed in the geometry, based on situations encountered in the physical examples, allowing for the periodic boundary conditions.

4. Contact Points In a packed tube there are a large number of contact points where the particles touch each other and the tube wall. This results in very narrow regions in the computational domain, which creates problems in meshing the packed tube geometry. Automatic mesh generation can result in volumes that are extremely skewed around the contact points, meaning that some of their surfaces can be much larger than others within the one tetrahedron. Manual mesh generation at contact points would involve an unrealistic amount of work as the number of contact points increases rapidly as the number of particles in a tube increase to more realistic levels. Magnico (2003) included contact points with a ﬁne structured mesh and successfully carried out DNS simulations of laminar ﬂow at low to moderate Re. However, Esterl et al. (1998) and Calis et al. (2001) both reported that gaps were introduced between the spheres in their geometries. Tobis´ (2000) was able to avoid the problem, as his experimental set-up had particles glued together which he modelled in his CFD simulation. Gunjal et al. (2005) did not report their treatment of the problem, but from their ﬁgures it seems that they, too, introduced gaps between the spheres. A different approach seems to have been used by Guardo et al. (2004), who actually increased the sizes of their spheres by 1% so that they overlapped slightly. They encountered no convergence difﬁculties, but the impact of this change on heat transfer and velocity distributions has not yet been reported. For our simulations, it was not possible to obtain

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convergence under turbulent ﬂow conditions when actual contact points of the spheres with each other or the wall were incorporated. Our approach (Nijemeisland and Dixon, 2001) was to create a gap between the different entities in the geometry, and then verify that the ﬂow ﬁeld was not disturbed by the gap. To choose the correct gap size a number of different models were created with a small number of spheres, and differing diameters, to allow for different gap sizes. The largest sphere size that would permit a turbulent model to be solved was 99.5% of the original sphere size. Other models were created with sphere sizes of 99%, 97% and 95%. These models were compared using velocity distribution histograms of ﬂuid elements near the contact points. The ﬂuid elements for comparison were selected by limiting the ﬂuid zone to a small area approximately 0.5 cm square around the contact point (Fig. 5a). In the ﬁve different geometries air was ﬂowed through the bed at a Reynolds number of about 20. Velocity magnitude data were taken from the different geometries and compared. It was shown that when the gaps were larger (the 95% and 97% sphere sizes) the velocity distribution tended to move to higher velocities. Both the 99.5% and 99% sphere size models showed negligible difference from the touching model’s velocity distribution (see Fig. 5b). For our heat transfer studies at moderate Re it was decided to create models with 99% spheres. This was chosen because this gap size allowed for easier construction and faster convergence than the 99.5% spheres model, without any loss in accuracy. Calis et al. (2001) also chose to reduce their spheres to 99% of the full size. For our later work under more extreme steam reforming simulation conditions, higher pressure and higher ﬂow rates, it was necessary to re-evaluate the particle size reduction. It was found that at these high ﬂow rates a 99% reduction of the spheres would 1 touching spheres(100%)

fraction

0.8

99.5% sphere diameter 99% sphere diameter

0.6

97% sphere diameter 95% sphere diameter

0.4

0.2

0 0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Flow velocity (m/s)

FIG. 5. (a) Selected elements around the sphere contact points for mesh comparison; (b) Velocity histograms for comparison of the different gap sizes, vin ¼ 0.01 m/s. Reprinted from Chemical Engineering Journal, Vol. 82, Nijemeisland and Dixon, Comparison of CFD simulations to experiment for convective heat transfer in a gas-solid ﬁxed bed, pp. 231–246, Copyright (2001), with permission from Elsevier.

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result in considerable amounts of ﬂow in the gaps, creating an unrealistic ﬂow ﬁeld. The gap size was reduced until the no-ﬂow area around the contact point was re-established. The resulting particle size reduction was 99.5%. These results demonstrate the necessity to evaluate the approach to contact points on small reduced models before implementing it on the simulation model.

C. PACKED BED CFD SIMULATION ISSUES 1. Packed Bed Flow Regimes When running a CFD simulation, a decision must be made as to whether to use a laminar-ﬂow or a turbulent-ﬂow model. For many ﬂow situations, the transition from laminar to turbulent ﬂow with increasing ﬂow rate is quite sharp, for example, at Re ¼ 2100 for ﬂow in an empty tube. For ﬂow in a ﬁxed bed, the situation is more complicated, with the laminar to turbulent transition taking place over a range of Re, which is dependent on the type of packing and on the position within the bed. Several studies have looked at questions such as the transition to turbulence, the level of turbulence intensity in the void space, and the delineation of ﬂow regimes in ﬁxed bed ﬂow. Some of the earliest works investigated turbulence intensity for gas ﬂow in beds of spheres. Mickley et al. (1965) used hot-wire anemometry for air ﬂow in a rhombohedral packing at Re ¼ 4,780 and 7,010. They concluded that the dispersion coefﬁcients for the bed were determined by ‘‘side-stepping’’ of the ﬂuid stream as it passes between particles, as eddy diffusivity in the voids was much smaller than the bed dispersion. They found that eddy shedding did not occur in the packing voids, and that high local heat transfer coefﬁcients in spherical packings must be due to turbulence intensity in the voids, which was as high as 50%, measured behind the seventh layer. Van der Merwe and Gauvin (1971) also found no eddy shedding except on the ﬁrst bank of spheres in their regular packings over the range 2,500oReo27,000, and turbulence intensity values around 25%. Turbulence measurements at much lower Re (50–1470) were made by Kingston and Nunge (1973) in a rhombohedral array, who found that particle geometry and packing conﬁguration had a large effect on maximum intensity and transition Re. Early studies of the transition to turbulence relied on ﬂow visualization techniques for liquid ﬂow through arrays of spheres. Jolls and Hanratty (1966) found a transition from steady to unsteady ﬂow in the range 110oReo150 for ﬂow in a dumped bed of spheres at N ¼ 12, and they observed a vigorous eddying motion that they took to indicate turbulence at Re ¼ 300. In regular beds of spheres, Wegner et al. (1971) found completely steady ﬂow with nine regions of reverse ﬂow on the surface of the sphere for Re ¼ 82, and similar ﬂow elements but with different sizes in an unsteady ﬂow at Re ¼ 200. Dybbs and Edwards (1984) used laser anemometry and ﬂow visualization to study ﬂow

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regimes of liquids in hexagonal packings of spheres and rods. They determined that there are four ﬂow regimes for different ranges of Reynolds number, based on interstitial or pore velocity Rei ¼ Re/e: (1) Reio1: Viscous or creeping ﬂow in which pressure drop is linearly proportional to interstitial velocity and ﬂow is dominated by viscous forces; (2) 10 r Rei r 150: Steady laminar inertial ﬂow in which pressure drop depends nonlinearly on interstitial velocity and and boundary layers in the pores become pronounced with an ‘‘inertial core’’ appearing in the pores; (3) 150 r Rei r 300: Unsteady laminar inertial ﬂow in which laminar wake oscillations appear in the pores and vortices form at around Rei ¼ 250; (4) Rei4300: Highly unsteady ﬂow, chaotic and qualitatively resembling turbulent ﬂow. These results are in reasonable agreement with the earlier work in terms of Re, and also the authors make the point that some workers used the term ‘‘turbulent’’ to cover the unsteady range. This general classiﬁcation of ﬂow regimes appears to have been generally accepted, with some reservations about the transitions. Latiﬁ et al. (1989) sought to remove the inﬂuence of the observer by using microelectrodes as electrochemical sensors, to get more precise regime transitions. They later corrected their results (Rode et al., 1994) to include the transfer function of the electrochemical probe and gave the transition to time-dependent chaotic ﬂow as 110oReo150, but noted that this was not necessarily fully developed turbulence. They made measurements on the tube wall and found that ﬂow was extremely nonhomogeneous at different spatial locations in a packed tube. This result was conﬁrmed by Seguin et al. (1998a) in their study of the end of the stable, laminar regime, which they found to occur at Re ¼ 113 inside the bed, but at Re ¼ 135 at the wall. A similar study from the same group (Seguin et al., 1998b) to determine the transition to the turbulent regime found that the transition was gradual and not at the same Re at all locations. They obtained stabilization of the ﬂuctuation rate, corresponding to local turbulence, at 90% of the electrodes for Re4600. The effects of inertia on ﬂows in both ordered and random arrays of spheres have been approached by lattice-Boltzmann simulations for small Re (Hill et al., 2001a) and for moderate Re (Hill et al., 2001b). These studies illustrate the scaling of drag force with Re, and especially extend the experimental results to beds of more dilute solid fraction. An interesting point of view was provided by Niven (2002), who claimed that the transition to nonlinear behavior was not due to turbulence, but to expansion, contraction, and changes in direction of the ﬂow, i.e., an increase in inertial forces relative to viscous ones. This viewpoint has been challenged by Stevenson (2003), who suggests that the transition from laminar ﬂow to turbulence may occur at much lower Re in a packed tube than

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an empty one, due to the reduced viscous damping of radial velocity components caused by ﬂow instabilities. The detail of the ﬂuid mechanical studies of ﬂow in particle arrays stands in sharp contrast to the highly simpliﬁed pictures used for reaction engineering. From the above survey, it appears that it would be safe to set up CFD simulations with a steady laminar ﬂow model for Reo100, and to use RANS turbulence models for Re4600, and simply to avoid the unsteady intermediate range. A similar point of view was taken by Gunjal et al. (2005), who used a laminar model up to Rei ¼ 204.74 and turbulent models for Rei ¼ 1,000 and 2,000. They also argued that the onset of unsteady ﬂow is delayed by particles packing closely together, and that the ﬂuid is conﬁned and stabilized by neighboring spheres. Magnico (2003) suggests that the stationary hypothesis can be used up to larger Re, and regarded as a coarse approximation. At present, the use of CFD in the transition range can be informed by the intended use of the simulations. If details of the ﬂow environment of the particles are essential, then a very ﬁne mesh should be used and unsteady laminar calculations must be performed. So far this approach has not been taken, to our knowledge. Alternatively, if improved bed-scale modeling is sought, which incorporates more realistic ﬂuid ﬂow on the scale of particles or larger, then an approximate steady ﬂow can be obtained by either turbulent or laminar models, and often both are used and compared to see if the bed scale ﬂow features are the same. 2. Mesh Generation An important part of CFD modeling is the construction of the mesh, especially in complex geometries such as ﬁxed beds. The mesh strongly affects the accuracy of the simulation. It has to be chosen with enough detail to describe the processes accurately and with a degree of coarseness that enables solution within an acceptable amount of time. When an optimal density has been found, reﬁning the mesh will increase the model size without displaying more ﬂow detail. When it is coarsened, the mesh may obscure possibly essential parts of the ﬂow detail. The mesh determines a large part of creating an acceptable simulation. The two main types of mesh are structured and unstructured. In a structured mesh there are families of grid lines, and grid lines of the same family do not cross each other and cross each member of the other families only once (Ranade, 2002). Block-structured grids allow local reﬁnement of structured grids. Unstructured grids are typically made up of tetrahedral cells in 3D, and can be locally reﬁned anywhere. They are very suitable for complex geometries such as those found in packed tubes. The Chimera grid in Fig. 6a was used by Agarwal (1999) to illustrate types of grids for aircraft wings. Similar Chimera grids were used by Nirschl et al. in simulations of a single sphere between moving walls, and by Esterl et al. (1998) and Debus et al. (1998) for packed tube simulations. Each sphere had its own prismatic ﬁtted mesh to resolve the

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FIG. 6. Examples of types of meshes developed to resolve laminar ﬂow around particles: (a) Chimera grid. Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 31 r 1999 by Annual Reviews www.annualreviews.org; (b) Unstructured grid with layers of prismatic cells on particle surfaces. Reprinted from Chemical Engineering Science, Vol. 56, Calis et al., CFD Modeling and Experimental Validation of Pressure Drop and Flow Proﬁle in a Novel Structured Catalytic Reactor Packing, pp. 1713–1720, Copyright (2001), with permission from Elsevier.

boundary layers, which was overlaid on a coarser main mesh for the tube. They had to work with a very ﬁne mesh to capture the gradients in the laminar ﬂow in the narrow gaps between the particles. The grid shown in Fig. 6b was developed by Calis et al. (2001) and consisted of ﬁve layers of prismatic cells on the walls of the spheres and tube, and unstructured tetrahedral cells in between. To obtain grid-independent pressure drops under laminar ﬂow they had to restrict the ﬁrst layer of prismatic cells to be 0.052 mm thick. The thickness then increased for the following four layers. The tetrahedral cells were 0.4 mm in size. In their later work (Romkes et al., 2003), which included heat transfer, they had to reduce the size of the ﬁrst layer of prismatic cells by a factor of three under laminar ﬂow. Other workers have also commented on the need for ﬁne meshes on the wall surface in laminar ﬂow. Magnico (2003) claimed that the spatial resolution or cell size must be less than dp/40, while for LBM simulations, Zeiser et al. (2002) recommended cell size between dp/30 and dp/20 and Freund et al. (2003) used a grid of size dp/30. These cell sizes are in line with the experiences of Tobis´ (2000) who used mesh sizes of 1 mm and 2 mm with particles of diameter 38 mm, and our own experience in which we used average cell sizes in the range 0.5–1 mm depending on the simulation conditions, for particles of 25.4 mm diameter. The preferred range for the thickness of the near-wall cell layer is y+430. However, this is difﬁcult to achieve in packed tubes. The cells sizes are constrained by the need to ﬁt in between the gaps and/or narrow spaces between particles, so they cannot be too large. This can result in the y+ values being too small for proper application of wall functions. The alternative to use small enough cells to resolve the boundary layer (y+o1) increases the computational

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cost unacceptably, at present. For turbulent ﬂow, Calis et al. (2001) employed the k– e model with wall functions on a tetrahedral mesh of size 1.0 mm.They noted that the velocity varies widely near particle surfaces, so that y+ can also vary by a factor of over 40. Their average y+ was in the range 4.4–440 for their range of Re. Guardo et al. (2004) report 4oy+o12, and our simulations result in similar values for y+. Romkes et al. (2003) said that the y+ criterion for the use of wall functions was met in their work only at the highest Re, and for their lowest Re they obtained y+o1. The effects on the computed ﬂow of applying the wall functions outside their preferred range are not yet clear. Guardo et al. (2005) claim that the Spalart–Allmaras model performs well under such conditions, due to the coupling of wall functions with damping functions. They found good predictions of pressure drop despite the range of y+, and that this quantity was not sensitive to near-wall cell size. The heat transfer coefﬁcient, in contrast, was very sensitive. Romkes et al. (2003) found that heat transfer under turbulent ﬂow was only weakly dependent on y+. Clearly this area requires further investigation. We determined an appropriate mesh density for our simulations by comparing results from several different mesh sizes (Derkx and Dixon, 1996; Logtenberg and Dixon, 1998a, 1998b; Logtenberg et al., 1999). An optimal mesh density was chosen from these previous studies on small particle clusters and additional studies for the WS simulation geometries used later (Nijemeisland and Dixon, 2004). Typical examples are shown in Fig. 7 for the unstructured grids. It was shown that there were no differences between the ﬂow solutions whether a completely ﬁne mesh was used or a locally reﬁned mesh. This mesh had a node spacing equal to the size of the gap at the sphere contact points at these locations, gradually grading toward a four times coarser node spacing near the voids in the geometry. The node distributions on the sphere surface were also graded from ﬁne near the contact point to coarser away from the

FIG. 7. Typical examples of unstructured meshes: (a) the surface mesh on a number of spheres and a section of the cylinder and (b) a section of the interior mesh in a plane indicated in part (a).

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contact points, resulting in a graded mesh both in the ﬂuid region as well as in the solid region, the inside of the particles. The results of the graded meshing can be seen in Fig. 7b. Mesh gradation was deﬁned using a ﬁrst–last-ratio principle, in the pre-processing GAMBIT package. In this method a changing node density on an edge in the geometry is created by two user-speciﬁed parameters, an average node spacing, and the size ratio between the ﬁrst and the last node on the edge. The average node spacing in the graded mesh was set to 0.03 inch, and the ﬁrst–last-ratio was set to 0.25, resulting in a factor 4 difference in node spacing between the ﬁne and coarse regions. The indicated settings create a mesh where the node spacing is 0.015 inch near the contact points (the ﬁnely meshed regions) and 0.06 inch at the coarsely meshed regions. 3. Conjugate Heat Transfer When conjugate heat transfer through solid particles in the tube is to be included, the energy balance must be solved in the solid particles, in addition to the ﬂuid ﬂow regions. The energy balance for a solid region is deﬁned by: @ðrhÞ @ @T ¼ l (23) þ Sh @t @xi @xi The last term Sh is the volumetric heat source, which may include user-deﬁned energy source terms. The sensible enthalpy, h, is deﬁned as Z T h¼ cp dT (24) T ref

P which is consistent with Eq. (14) since cp ¼ Y j cp;j . The inclusion of heat transfer in the particlesj means that the solid regions must also be meshed, in addition to the ﬂuid regions. This can considerably increase the mesh size. If the solid particles are only subjected to simple conduction, steady-state gradients within them are not likely to be large, and it may be expected that a coarse grid would sufﬁce. This may not be true if heat effects of reaction are to be included via the source terms. The use of a ﬁne mesh in the ﬂuid region, especially in the vicinity of the contact points, requires the use of a matching ﬁne mesh in the solid particles. Even with mesh coarsening toward the center of the particles, the increase in mesh size may be substantial. One way to deal with this may be to use nonconformal grids, in which the grid points in the solid do not have to coincide with the grid points in the ﬂuid at the ﬂuid–solid interface. This may be a limited remedy, as curvature in the surfaces of the particles will limit the extent to which there can be a mismatch between the grids. 4. Boundary Conditions A no-slip boundary condition is used on all impermeable solid surfaces, but the choice of boundary conditions for the inlet and outlet of the model is not so

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straightforward. It is generally accepted that when ﬂow enters a structured or randomly packed ﬁxed bed, an area near the entrance of the bed has an undeveloped ﬂow proﬁle as the ﬂuid is adjusting from one ﬂow environment to another. A similar situation is found near the exit of the bed, where a sudden change in pressure drop is experienced and the ﬂow ‘‘relaxes’’ before it actually exits the bed. In a full model, in which an actual bed inlet and outlet were modeled, a large portion of the bed would need to be modeled just to eliminate entrance and exit effects from the central portion of the bed that was the main focus of the study. In a real industrial packed tube, in contrast, the entrance and exit of the bed are usually small compared to the length, and an active bed may be preceded or followed by inert packed sections that condition the ﬂow. In our N ¼ 4 models we use translational periodic ﬂow boundaries at both the ﬂow inlet and ﬂow outlet of the column, in which all variables except pressure are identical at the periodic planes. The total mass ﬂow is supplied to the model at the inlet boundary, to give the desired Reynolds number. By imposing the translational periodic boundaries a generic, developed ﬂow solution is obtained. Since the translational periodic boundary deﬁnes the column inlet to be identical to the column outlet, there is no ﬂow development in the bed; a steady-state ﬂow situation is obtained. The periodic boundaries remove the effects of an entrance or exit effect in the bed. The overall size of the model can now be greatly reduced. Other groups have used periodic conditions at the inlet and outlet of a packed tube (Magnico, 2003) or on the ﬂow surfaces of unit cells (Tobis´ , 2000; Gunjal et al., 2005), whereas Guardo et al. (2004) set a constant velocity at their tube inlet and a constant pressure at the outlet. Calis et al. (2001) and Nijemeisland and Dixon (2001) simulated lengths of empty tube before and after their packed regions. Clearly, the pressure itself cannot be periodic even if the geometry of the model is periodic. Instead, the pressure drop is periodic. The local pressure gradient is decomposed into two parts: the gradient of a periodic component, superimposed on the gradient of a linearly varying component. The linearly varying component of the pressure corresponds to the familiar packed-tube pressure drop. Its value is not known before the simulation; it must be iterated on until the mass ﬂow rate that was speciﬁed is achieved in the computational model. When heat transfer is included in the CFD model for a packed tube, the tube wall is heated (or cooled for some applications). Either temperature or heat ﬂux is given at the wall. The gas heats up as it passes through the tube, and the temperature ﬁeld cannot be treated as periodic. Since we want to investigate the energy penetration into the bed from the wall, it is necessary to reinstate the generic bed section as a section in a larger bed, deﬁning different inlet and outlet conditions. For the ﬂuid, the inlet temperature should be speciﬁed. This could also be done for the solid particles; however, an alternative is to choose the inlet heat ﬂux to be zero for the solid regions. Outlet conditions are required for the solid regions only, and again a zero heat ﬂux condition can be used. For full-bed

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models this has the disadvantage that a considerable part of the simulated tube may be atypical of the bed as a whole, due to the uniform inlet temperature proﬁle. In addition, for short-bed segments, the temperature proﬁle may not develop enough for the effects of ﬂow throughout the entire model to be felt. To calculate the development of the temperature proﬁle into the bed, a series of simulations have to be performed. To overcome thermal entry effects, the segments may be virtually ‘‘stacked’’ with the outlet conditions from one segment that becomes the inlet conditions for the next downstream section. In this approach, axial conduction cannot be included, as there is no mechanism for energy to transport from a downstream section back to an upstream section. Thus, this method is limited to reasonably high ﬂow rates for which axial conduction is negligible compared to the convective ﬂow of enthalpy. At the industrial ﬂow rates simulated, it is a common practice to neglect axial conduction entirely. The objective, however, is not to simulate a longer section of bed, but to provide a developed inlet temperature proﬁle to the test section. 5. Convergence The discretized equations of the ﬁnite volume method are solved through an iterative process. This can sometimes have difﬁculty converging, especially when the nonlinear terms play a strong role or when turbulence-related quantities such as k and e are changing rapidly, such as near a solid surface. To assist in convergence a relaxation factor can be introduced: fnew ¼ af fnew þ ð1 af Þfold p p p

(25)

The relaxation factor (af) is a multiplier for the change in the solution variable. When this factor is less than unity, the process is called under-relaxed. When under-relaxed, the iteration process is slower, since the step change is small, but less likely to diverge. Commercial codes, such as Fluent, will typically recommend values for the under-relaxation factors that work well with a wide range of ﬂows. For packed-tube simulations, we have usually needed to reduce the default values by a small amount, usually 0.2. Some simulations, however, did not converge until very small values, of the order of 0.1 or lower, were used and values in this range for pressure and velocity under-relaxation factors have also been suggested by Gunjal et al. (2005). To determine when a solution is converged usually involves examining the residual values. The residual value is a measure of the imbalance in the discretized equation, summed over all the computational cells in the domain. Residuals can be obtained for continuity, velocity components, and turbulence variables. Again, it is common practice to set a ‘‘cut-off’’ value for the normalized residual values. When the set value is reached, the iteration process is stopped. Our experience with packed-tube simulations, especially if low

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under-relaxation factors are used, is that the cut-off values should be set very low, and the iteration continued until the residuals have leveled out. This will normally be a good deal higher for turbulent ﬂows than for laminar ones, and ﬂow residuals are higher than those for energy. In addition, it is a good idea to monitor other measures of convergence besides the residuals, such as pressure drop and/or an averaged wall shear stress or exit temperature (Guardo et al., 2004; Gunjal et al., 2005). We have seen apparently level residuals, while the pressure drop slowly changed, and a substantially different ﬁnal ﬂow ﬁeld was eventually obtained, often after several thousand iterations. Following apparent convergence, it is essential to check both mass and energy balances, as well as performing grid independence studies and comparing to experimental results, to have conﬁdence in the solution. Just because a simulation has converged, does not mean that it is necessarily reliable. Convergence of the discretized system can sometimes be problematic, especially for some of the turbulence models. In such cases, using a laminar ﬂow solution as an initial guess for the turbulent ﬂow ﬁeld can be helpful. Similarly, when using options such as nonequilibrium wall functions or enhanced wall treatment, convergence can be facilitated by ﬁrst obtaining a solution using the k– e model with standard wall functions, and then switching to the desired model.

D. VALIDATION

OF

CFD SIMULATIONS

FOR

PACKED BEDS

1. Flow Field Validation Since the CFD methodology is not speciﬁcally designed for application in constrained geometries, such as particle packed beds, it is necessary to verify if the simulated results are valid. Although the CFD code is based on fundamental principles of ﬂow and heat transfer, some of the boundary issues are modeled using empirical data not necessarily appropriate for ﬁxed bed applications. Validation of CFD ﬂow calculations has generally taken one of the two forms. In the ﬁrst, noninvasive velocity measurements inside the packed bed have been made, and compared to velocities computed from a model of either the entire experimental bed or a representative part of it. In the second form, computed pressure drops have been compared to either measured values or established correlations for pressure drop in ﬁxed beds, such as the Ergun equation (Ergun, 1952). The direct experimental veriﬁcation of the computed ﬂow ﬁeld requires noninvasive measurements of velocity components inside the packed tube. One very promising technique for this is magnetic resonance (MR) as described in a recent review by Gladden (2003). She showed pictures of 3D MR visualization of axial velocity for ﬂow of water in packings of spheres, and her group has used MR to connect the 3D structure of a packed bed to the transport phenomena in

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it (Sederman et al., 1998) and to validate their lattice Boltzmann code for lowRe simulations of water ﬂowing through arrays of spheres (Manz et al., 1999). They have also combined MR spectroscopy with imaging to get noninvasive measurements of chemical conversion inside the bed (Yuen et al., 2002). Maier et al. (1998) also used LBMs to simulate ﬂow through a column of beads for N ¼ 10, which they then compared to the NMR data of Lebon et al. (1996). They obtained encouraging agreement for the qualitative features such as negative velocities; however, they saw differences in peak values of normalized velocity, some of which they attributed to longitudinal dispersion. Most recently, Gunjal et al. (2005) conducted CFD simulations of laminar ﬂow through a simple cubic unit cell corresponding to the set-up used by Suekane et al. (2003) in their MRI studies. Comparisons were made over the range 12.17 rRei r204.74 and showed good agreement for axial components of velocity. At the highest ﬂow rate inertial ﬂow was dominant, with jet-like ﬂow being observed in both experiments and simulations. Some discrepancies were seen at the lowest ﬂow rate, which may have been attributable to experimental difﬁculties in maintaining a constant low ﬂow rate. Overall, the CFD simulations correctly captured the inertial ﬂow structures, including vortices. Comparisons were also made between the ﬁnite-volume CFD calculations and the LBM simulations of Hill et al. (2001a), with good results. Freund et al. (2003) chose to validate their LBM simulations using laser Doppler anemometry (LDA) measurements. Experiments for two packings of spheres at N ¼ 4 and N ¼ 6.15 were used, at a ﬂow rate corresponding to Re ¼ 50. Good agreement was found for the radial proﬁles of axial velocity, with the simulations reproducing the typical oscillations in velocity in the wall region. Some discrepancies were seen near the bed center, but the authors noted that while packing structure of experiment and model were identical near the wall due to the ordering there, some differences existed in the bed center. No direct validations have yet been reported for the unsteady laminar or turbulent ﬂow regimes, due to lack of experimental data at the higher ﬂow rates. Measurements of pressure drop, however, can provide an indirect means of checking on the computations at higher ﬂows, although most comparisons have, in fact, been made at relatively low ﬂow rates. Many groups compare their results to the original Ergun (1952) equation for pressure drop (Gunjal et al., 2005, Guardo et al., 2004; 2005), or to later modiﬁcations or different correlations that take wall effects into account (Esterl et al., 1998; Freund et al., 2003; Magnico, 2003). In general, good agreement is claimed, especially for the lower Re range; however, many comparisons are presented on log-log graphs where it is harder to see differences. There are few reported comparisons to experimental pressure drop data taken by the same workers. An exception is Calis et al. (2001) who compared CFD, the Ergun correlation and experimental data for N ¼ 1–2. They found 10% error between CFD and experimental friction factors, but the Ergun equation

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over-predicted by 80%, which they attributed to a poor prediction of the turbulent contribution. Their results extended to Re ¼ 40,000. The validation of CFD codes using pressure drop is most reliable when actual experimental data are taken in equipment identical to the situation that is being simulated. Existing literature correlations such as the Ergun equation are known to have shortcomings with respect to wall effects, particle shape effects, application to ordered beds and validity at high Re. The applicability of literature correlations to typical CFD simulation geometries needs to be examined critically before fruitful comparisons can be made.

2. Heat Transfer Validation In this section a short description of a comparison between experimental and simulation results for heat transfer is illustrated (Nijemeisland and Dixon, 2001). The experimental set-up used was a single packed tube with a heated wall as shown in Fig. 8. The packed bed consisted of 44 one-inch diameter spheres. The column (single tube) in which they were packed had an inner diameter of two inches. The column consisted of two main parts. The bottom part was an unheated 6-inch packed nylon tube as a calming section, and the top part of the column was an 18-inch steam-heated section maintained at a constant wall temperature. The 44-sphere packed bed ﬁlls the entire calming section and part of the heated section leaving room above the packing for the thermocouple cross (Fig. 8) for measuring gas temperatures above the bed. A radial temperature measurement consisted of establishing and recording a steady-state temperature proﬁle for a combination of a speciﬁc bed length, Reynolds number, and angle of thermocouple cross. A total of four thermocouple cross insulation steam in

P heated wall

r/rt = 0.91 r/rt = 0.70 r/rt = 0.46

rotameter packing

steam out

dryer

r/rt = 0.80 r/rt = 0.56 r/rt = 0.30

P air in

TC

FIG. 8. Experimental setup and detail of the thermocouple cross.

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN 1

z=0.276m CFD z=0.420m CFD

z=0.276m Exp. z=0.420m Exp.

z=0.132m CFD

345

z=0.132m Exp.

0.8

θ

0.6

0.4

0.2

Re = 986

0 0

0.2

0.4

0.6

0.8

1

r/R

FIG. 9. Simulated and experimental radial temperature proﬁles at Re ¼ 986.

thermocouple-cross positions were used for a measurement by rotating the cross 151, 301, and 451 from the initial orientation. By rotating the thermocouple cross a good spread of data points was ensured, giving a full picture of the angular spread of the radial temperature proﬁle. Particle Reynolds numbers were varied from 373 to 1922; bed lengths were varied from 0.132 through 0.42 m. Fig. 9 shows comparisons of CFD results with experimental data at a Reynolds number of 986 at three of the different bed depths at which experiments were conducted. The proﬁles are plotted as dimensionless temperature versus dimensionless radial position. The open symbols represent points from CFD simulation; the closed symbols represent the points obtained from experiment. It can be seen that the CFD simulation reproduces the magnitude and trend of the experimental data very well. There is some under-prediction in the center of the bed; however, the shapes of the proﬁles and the temperature drops in the vicinity of the wall are very similar to the experimental case. More extensive comparisons at different Reynolds numbers may be found in the original reference. This comparison gives conﬁdence in interstitial CFD as a tool for studying heat transfer in packed tubes. 3. Wall-Segment vs. Full-Bed Validation When simulations are done in a WS model, the results need to be validated against a full-bed model. The main reason for this is not only to see if the WS model results are representative for a full bed but also to check that the symmetry boundaries, which are relatively close to all parts of the segment model, do not inﬂuence the solution.

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a. Comparing segment results to full-bed results. A ﬁne mesh was created for the WS and the full-bed models. For the full bed, the mesh was too large for the 32-bit computers so it was reduced in size by excluding the solid parts (the particle internals) from the mesh. This mesh is referred to here as the ‘‘nosphere’’ mesh. The ﬁnest possible mesh for the full bed that was created including all bed internals will be referred to as the ‘‘re-mesh’’. The ‘‘no-sphere’’ full-bed ﬁne mesh was 2.6 million control volumes (cv), while the WS reduced this to only 756,700 cv. The full bed with the ﬁne mesh was over 6 million cv, which was not implemented at the time (but has since become possible with a 64-bit machine), while the full bed with the coarse mesh was 1.97 million cv. In the initial comparisons velocity proﬁles were compared for both the full-bed meshes mentioned above (the no-sphere mesh and the re-mesh) as well as the WS model. Only velocity proﬁle comparisons were eligible, since the no-sphere mesh could not give comparable energy solutions. A section in the full-bed models was isolated that was comparable to the WS model. The layout of these different sections was identical, except that the WS model had a two-layer periodicity and the full-bed models had a six-layer periodicity. To be able to make direct comparisons of velocity proﬁles, several ‘‘sample-points’’ needed to be deﬁned. In the three different models seven tangential planes were deﬁned and on each plane three axial positions were deﬁned. This reduced the data to single radial velocity proﬁles at corresponding positions in all three models, as shown in Fig. 10, for the WS model. Identical planes were deﬁned in the full-bed models. Some spheres and sample planes 4 and 5 are not displayed to improve the visibility of the sample planes and lines. In the right-hand part of the ﬁgure, plane 4 is shown with the axial positions at which data were taken and compared. Simulation data were collected and plotted from the three models. Flow magnitudes were plotted separately for the three different components of the ﬂow, the axial velocity, vz, the radial velocity, vr, and the tangential velocity, vy for seven planes with three data-lines each. Selected results are shown in Fig. 11

FIG. 10. Comparison section with seven tangential planes and axial proﬁle lines indicated.

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN 2.5

2.5

wall segment

wall segment

2

2

full bed, fine mesh, no spheres

full bed, fine mesh, no spheres full bed, coarse mesh

full bed, coarse mesh

1.5

1.5 1

vr/vin

vz/vin

347

0.5

1 0.5

0

0

-0.5

-0.5

-1

-1 0

0.2

0.4

r/rt

0.6

0.8

1

0

0.2

0.4

r/rt

0.6

0.8

1

FIG. 11. Full-bed and WS mesh comparisons of axial and radial velocity components (at Z3).

for the axial and radial velocity components. All simulations were performed at a particle Reynolds number of 420, under atmospheric conditions with no temperature gradients. The superﬁcial velocity in the simulations with a Reynolds number of 420 is 0.58 m/s, and this was used to normalize the different velocity components. The velocity proﬁle plots show interruptions in the velocity proﬁle, where the solid packing was located. In general, the data of the three different cases agreed very well qualitatively; velocity highs and lows are shown at the same points in the bed. Quantitatively, the data of the two full-bed models are practically identical, indicating that the solutions were completely mesh independent. The data from the WS model in some cases deviated slightly from the full-bed models. This could be explained by the slightly different layout of the WS model. Some spheres had to be relocated in the WS model to create a two-layer periodicity from the six-layer periodicity in the full-bed models. The differences in velocity magnitudes were mainly found in the transition area between the wall layers and the center layers. The effect of slightly larger gaps between spheres from the nine-sphere wall layers and the three-sphere central layers, due to the sphere relocations, had a noticeable effect on the velocity proﬁle. Differences were also found in the central layer area where the sphere positions were not identical.

b. Wall-segment mesh independence. In the previous section, two meshes were compared for the full-bed models. Similarly for the WS model, two separate meshes were created: a ﬁne mesh, as reported above, and a coarse mesh with the same mesh density as the full-bed coarse mesh. In Fig. 12, ﬂow proﬁles of the ﬁne mesh (labeled WS) are compared to the results from the coarse mesh. Also included in this comparison are the results from a simulation performed in a scaled-down version of the ﬁne mesh model. A simulation geometry was created at 1/8th the size of the original model, to see if the absolute size of the model had a signiﬁcant inﬂuence on the solution of the physical models, including the wall

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2.5

2.5 wall segment

2

wall segment, fine mesh

2

wall segment, coarse mesh

wall segment, coarse mesh

1/8th size wall segment

1/8th size wall segment

1.5

1

vr/vin

vz/vin

1.5

0.5

1 0.5

0

0

-0.5

-0.5

-1

-1 0

0.2

0.4

r/rt

0.6

0.8

1

0

0.2

0.4

r/rt

0.6

0.8

1

FIG. 12. Wall-segment mesh comparisons for axial and radial velocity components.

functions, which resolve the wall boundary layers. The results in Fig. 12 show complete mesh independence of the WS geometry and no effect of the scaling.

III. Low-N Packed Tube Transport and Reaction Using CFD The broad objective of using CFD to simulate interstitial ﬂow in packed tubes is to obtain a more fundamental understanding of the phenomena, taking place within them. Some more speciﬁc uses of this approach are (i) to obtain information (data) for conditions under which experiments are difﬁcult or impractical, such as high temperatures or pressures; (ii) to obtain simulated measurements where sensors cannot be located without disturbing the bed packing, which would invalidate the measurements being sought; and (iii) to examine the individual contributions to transport and reaction in isolation, or to break down complex transport processes into their contributing mechanisms, in order to establish more fundamental correlations for them. In this section we want to examine the progress so far toward (iii) above, as CFD offers a very promising means of making further progress on problems that have resisted traditional approaches.

A. HYDRODYNAMICS

AND

PRESSURE DROP

As we have seen earlier, it is quite common for CFD pressure drop results to be compared to well-established empirical correlations of experimental data as part of the validation of the simulations. Some studies, in contrast, have used CFD to obtain the contributing mechanisms to pressure drop. One area in particular where this has been fruitful is structured packings. Corrugated-sheet structured packings have been studied by Petre et al. (2003) and Larachi et al. (2003). A layer of this type of structured packing is made of an ensemble of a large number of ‘‘Toblerone-like’’ triangular ﬂow channels having identical

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cross-sections. These channels are the result of the voids between two adjacent corrugated sheets that are placed together at opposite orientations. Structured packing is expensive, so good geometric design of the elements is important, to optimize performance. Design is still empirical, however, and the limits of the correlations and design data are uncertain, since the constants for describing hydrodynamics and mass transfer have been obtained by calibrating models using laboratory experiments. CFD allows the rigorous ﬂuid transport equations to be solved locally, to better understand the phenomena. Petre et al. (2003) have presented an approach that combines a mesoscale and a microscale predictive approach. They identify recurrent mesoscale patterns in the packing, termed representative elementary units (REU), such as criss-crossing junctions, two-layer transitions, entrance regions, etc. They choose these REUs by identifying important dissipative phenomena at different parts of the packing. They then use CFD to simulate each REU over a wide range of Reynolds number from creeping to turbulent regimes. The next step is the conjecture that total resistance to ﬂow can be obtained from adding the resistances of individual REUs. The simulations for each REU gave pressure drop at the microscale, from which they extracted a correlation for its loss coefﬁcient. These coefﬁcients were then used to predict the additive contribution to the total bed pressure drop. The results were checked to see that they were not dependent on the sizes of the REUs, and the bed pressure drops were compared to literature experimental data with good agreement, to within 10–20%. A follow-up paper (Larachi et al., 2003) used this approach together with CFD to evaluate proposed geometric design changes of structured packings in terms of energy loss and pressure drop. Fig. 13 shows typical results in a plot of pressure drop per unit length against the gas ﬂow factor Fs ( ¼ vr0.5). Four types of REUs were identiﬁed and simulated, and correlations were developed. The pressure drop arising from criss-crossing elements was the largest contribution to pressure drop of the four. The total pressure drop predicted from the individual contributions was in excellent agreement with literature data. Returning to more conventional tube packings, several investigators have looked at the contributions to pressure drop in arrays of spheres. Esterl et al. (1998) suggested that CFD could be used to numerically test the assumptions made in deriving the analytical result that dissipation of energy is 75% due to elongational effects and 25% due to shear effects. Dhole et al. (2004) investigated the contributions to drag of power-law ﬂuids in distended beds of spheres at intermediate Re. Since their packings had high void fractions, they could use an idealized cell model where each sphere was surrounded by an envelope of ﬂuid. Their main focus was to write the drag coefﬁcient as the sum of a pressure component and a frictional component, and determine how these behaved as functions of the power law ﬂuid index. Gunjal et al. (2005) analyzed the relative contributions of shear drag and form drag for their unit cell packings of cubic and rhombohedral spheres. They found that the viscous

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FIG. 13. Breakdown into the contributing dissipation mechanisms of dry pressure drops in vessels containing Montz B1-250.45 structured packings. Reprinted from Chemical Engineering Science, Vol. 58, Petre et al., Pressure Drop Through Structured Packings: Breakdown Into the Contributing Mechanisms by CFD Modeling, pp. 163–177, Copyright (2003), with permission from Elsevier.

contribution was between 21% and 27% of the total, and was relatively constant in the laminar regime of 12oReo200. The inﬂuence of bed geometry on frictional resistance has been studied systematically (Tobis´ , 2000, 2001, 2002) using turbulence promoters placed between spheres in a cubic model packing. The experimental and CFD study began with turbulent air ﬂow through six model packings (Tobis´ , 2000), which consisted of a base cubic arrangement of spheres with different turbulence promoters— rectangular bands, triangular prisms, or cylindrical tubes—inserted between them. Wall effects were partially reduced by experimental design, and the remaining effects were eliminated by analysis and did not play a part in this work. The CFD calculations were checked against experiment and different mesh sizes, and turbulence models were used and gave good agreement with each other. The author tested the Ergun equation and concluded that the friction factor could not be correlated using bed porosity e and hydraulic diameter dh as the only measures of bed structure, which goes to explain why the constants in the Ergun equation scatter between the various experimental investigations in the literature. A second study (Tobis´ , 2001) used only the thin bands as inserts, but covered more arrangements, and different numbers of bands were used in the packing. The tested packings were identiﬁed as being made up of three arrangements of representative elementary units (REUs). CFD was used to predict the friction factors for the elementary REUs, and then these micro results were combined to get the macro pressure drop. It was concluded that local bed anisotropy and/or clear passages through the bed affect the pressure drop. Complex bed structures can be handled by the macro-correlation/micro-CFD approach, where local bed anisotropy is taken into account by micro-CFD, and channeling by formulas to

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FIG. 14. CFD predictions of turbulent ﬂow (meshed REU geometry, air pathlines with spheres hidden for clarity, contours of turbulent kinetic energy k) for conﬁguration B2. Copyright 2001 From Turbulent Resistance of Complex Bed Structures by J. Tobis´ . Reproduced by permission of Taylor and Francis, Inc., http://www.taylorandfrancis.com.

combine the friction factors for the REUs. Typical results are shown in Fig. 14, for a horizontal band REU, conﬁguration B2. The ﬁrst part of this ﬁgure shows the triangular surface mesh with periodic conditions on the sides. The middle picture shows a view slightly tilted forward to show path lines of ﬂow accelerating around the sides of the insert, and relatively slow ﬂow in the wake region. Corresponding to these the third picture presents the high amounts of turbulence kinetic energy in the ﬂow trailing from the sides of the insert. The author noted that the turbulence kinetic energy was not homogeneous in these complex packings. There was ﬂow circulation in regions much smaller than the inter-particle space, and ﬂow channeling through clear passages in the packing that provided evidence that there was not complete mixing in the packing interstitial space. In the most recent paper in the series (Tobis´ , 2002), the method was extended to chessboard periodic structures to look at different spatial arrangements of the same packing elements. Experiments showed that pressure drop depended not only on the resistance of the individual REUs but also on their spatial arrangement. The experiments were compared to interstitial ﬂow simulations by CFD, superﬁcial ﬂow modeling by the modiﬁed Forchheimer equation, and a simpliﬁed structural method involving macrocorrelations. Although limiting cases of the macrocorrelations were useful, only the CFD simulations could reproduce observed experimental results, and further development of the structural approach to provide less expensive models is needed. Contributions to pressure drop have also been studied by lattice Boltzmann simulations. Zeiser et al. (2002) postulated that dissipation of energy was due to shear forces and deformational strain. The latter mechanism is usually missed by capillary-based models of pressure drop, such as the Ergun equation, but may be signiﬁcant in packed beds at low Re. For a bed of spheres with N ¼ 3, they found that the dissipation caused by deformation was about 50% of that

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caused by shear, over the range 0.1oReo20. The deformation contribution decreased as Re increased. Later work by the same group (Freund et al., 2003) over a higher range of Re concluded that correlations for pressure drop cannot account for the inﬂuences of local structure on global pressure drop. The authors saw small changes in bed void fraction causing much larger changes in pressure drop than could be explained by the nonlinear terms, which pointed to local effects. An extremely thorough investigation of drag force on spheres, and its dependence on Re, has been performed by Hill et al. (2001a, b), also using LBMs. These studies examined the effects of ﬂuid inertia at small to moderate Re, on ﬂow in ordered and random arrays of spheres over a wide range of void fractions. Although most CFD work has focused on pressure drop, several studies have also reported radial proﬁles of axial velocity, which are also of interest for simpliﬁed 1D models of ﬁxed bed ﬂuid ﬂow. Zeiser et al. (2002) found an oscillating proﬁle, with two peaks at dp/8 and dp from the wall, for a tube with N ¼ 3. Magnico (2003) found a qualitatively similar result for N ¼ 5.96 and N ¼ 7.8, but in those results the near-wall peak was lower, which differs from all other workers. Zeiser et al. (2001) gave fairly qualitative pictures for N ¼ 5 and N ¼ 6, while Freund et al. (2003) also found a maximum close to dp/ 8 for N ¼ 4 and N ¼ 6.15. Overall, most studies show the near-wall peak at dp/8 with a magnitude of about 2.5–3 times the superﬁcial average velocity, which is lower than BFD-type models unless an effective viscosity is introduced. Good agreement has been demonstrated in a few studies with LDA experiments, when they have been available for the same geometry. The similarity of the velocity proﬁles near the wall for beds of spheres is not surprising, given the analogous result for porosity. Toward the bed center, where the wall ordering decreases, the agreement is not so good. Also, beds of cylinders and other shapes of particles might be expected to give interesting results that may not be the same as for spheres.

B. MASS TRANSFER, DISPERSION,

AND

REACTION

A number of studies have used CFD results to obtain information on dispersion or mass transfer in packed tubes, in addition to the usual hydrodynamic results. Some relatively recent work has also included reaction. This area is so far not as well developed as the pressure drop and ﬂow ﬁelds results of the previous section; however, some promising ﬁrst steps have been taken. Structured packings were studied using CFD by Van Baten et al. (2001), who wished to determine the radial liquid residence time distribution of a KATAPAK-S-like structure for comparison to conventional ﬁxed beds for use in heterogeneously catalyzed reactive distillation processes. Such structured packings consist of catalyst particles sandwiched between corrugated sheets of wire gauze that are sealed around the edge. A group of these sandwiches is then

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FIG. 15. The KATAPAK-S structure and its computational grid. Reprinted from Chemical Engineering Science, Vol. 56, van Baten et al., Radial and Axial Dispersion of the Liquid Phase within a KATAPAK-Ss Structure: Experiments vs. CFD Simulations, pp. 813–821, Copyright (2001), with permission from Elsevier.

bound together. An illustration of the ‘‘criss-cross’’ structure is given in Fig. 15 below. The ‘‘Toblerone’’ structure consisting of intersecting and connecting triangular tubes has many similarities with the packings investigated by Petre et al. (2003) and Larachi et al. (2003) that are discussed above. The computational grid used for the simulations is also shown. In these simulations, the authors used the pseudo-continuum approach for the catalyst packing inside the criss-cross structure. The body force describing the resistance to ﬂow offered by the packing was modeled by the Ergun equation. The equation of continuity for a tracer was included with the equations of conservation for total mass and momentum, so their simulation tracked unsteady liquid ﬂow through the illustrated assemblage of triangular tubes. Liquid entered the system through the top eight tubes and exited through the bottom eight tubes. At steady state, a pulse tracer was injected and tracer outlet concentrations were monitored to obtain the RTD curves at different horizontal positions. From these the axial and radial dispersion coefﬁcients were estimated. The authors found good agreement between the coefﬁcients estimated from CFD simulations and those estimated from experiments, and determined that the radial dispersion coefﬁcient for the KATAPAK-S was about an order of

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magnitude higher than that in co-current down-ﬂow trickle beds, which represented a desirable feature. Dispersion in conventional packed beds of spheres has also been simulated by CFD. Manz et al. (1999) compared lattice Boltzmann simulations to NMR velocimetry and propagator measurements. Their ﬂows were at 0.4oReo0.77 with 182oPeo350. Over these ranges, they evaluated contributions to the dispersion tensor: mechanical dispersion, i.e., dispersion arising from the stochastic variations of the velocity ﬁeld due to the structure of the packing; Taylor dispersion, i.e., diffusion of ﬂuid molecules across streamlines; and holdup dispersion, caused by blocked regions of ﬂow in the packing. At larger length scales, dispersion was dominated by mechanical dispersion, while at smaller length scales dispersion was determined by Taylor dispersion. At the largest length scales, they suggested that holdup could play a signiﬁcant role. Zeiser et al. (2001) also used LBM to examine dispersion. They simulated a simple cubic packing only one sphere wide, with periodic boundary conditions perpendicular to the main ﬂow direction to emulate an inﬁnite array, and 40 spheres in the main ﬂow direction to allow dispersion to develop. A pulse was simulated across the transverse plane, which eventually became Gaussian and allowed extraction of the longitudinal dispersion coefﬁcient. They obtained a correlation in terms of Pe1.83, whereas Taylor dispersion would predict Pe2. This may mean that similarly to Manz et al. (1999), mechanical dispersion begins to play a role even at quite low Re. In a more recent paper (Freund et al., 2005) the same group used LBM to obtain ﬂow ﬁelds in a simple cubic packing of spheres, and in a random packing of N ¼ 5. They then used a particle tracking algorithm, together with the ﬂow ﬁelds, to obtain 3D concentration ﬁelds from which they obtained axial and radial dispersion coefﬁcients. They found good agreement with experiment for dispersion in the cubic array, for which ﬂow was relatively simple, but for the more complex ﬂows in the random bed the agreement was less good, although still reasonable compared to the scatter of the data. Freund et al. (2005) also performed simulations that gave residence time behaviour, and warned that the residence time and dispersion coefﬁcients obtained from the popular tracer injection technique could depend on the tracer injection location. Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N ¼ 5.96 and N ¼ 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state ﬂow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between ﬂow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefﬁcients to be smaller than in the literature, which may have been connected to the wall effects. The ﬂow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about dp/4 (at Rei ¼ 7) in which there was no radial velocity component, so that mass transfer across the layer

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would be by molecular transport only. At higher ﬂow rates the layer was too thin to be resolved by the author’s structured mesh, but even at Rei ¼ 200 it was noted that the tracer particles were not easily transported across the region one sphere diameter thick next to the wall. Maier et al. (2003) also investigated wall effects on hydrodynamic dispersion using LBM for tubes at higher N, in the range 10–50, but at relatively low Re. They also found that the presence of the tube wall enhances dispersion, due to the effects on bed structure and velocity proﬁle. The inclusion of chemical reaction into CFD packed-tube simulations is a relatively new development. Thus far, it has been reported only by groups using LBM approaches; however, there is no reason not to expect similar advances from groups using ﬁnite volume or ﬁnite element CFD methods. The study by Zeiser et al. (2001) also included a simpliﬁed geometry for reaction. They simulated the reaction A+B - C on the outer surface of a single square particle on the axis of a 2D channel (Fig. 16). The chemical species were treated as passive scalar tracers in the unsteady LBM equations. The reaction was simulated as being mass-transfer limited at low Re ¼ 166, with diffusivities in the ratios DA : DB : DC ¼ 1 : 3 : 2. The concentration ﬁelds shown in Fig. 16 are different for each species due to the different diffusivities. The slow-diffusing species A is transported mainly by convection and regions of high or low concentration correspond to features of the ﬂow ﬁeld. A more uniform ﬁeld is seen for the concentration of faster

FIG. 16. Snapshot of the 2D concentration ﬁeld for reactive ﬂuids. Reprinted from Chemical Engineering Science, Vol. 56, Zeiser et al., CFD-Calculation of Flow, Dispersion and Reaction in a Catalyst Filled Tube by the Lattice Boltzmann Method, pp. 1697–1704, Copyright (2001), with permission from Elsevier.

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diffusing species B, while species C with intermediate diffusivity has a concentration ﬁeld between the other two. The main conclusion was that the inhomogeneous ﬂow ﬁeld led to a nonuniform concentration distribution around the particle, in contrast to the usual assumption of uniform surroundings for catalyst particles in standard reactor models. A sequel to this study was presented later by the same group (Freund et al., 2003), this time for the simple ﬁrst-order reaction A-B in a cylindrical bed of spheres with N ¼ 5. The reaction was again taken to be mass-transfer limited and to occur on the surfaces of the catalyst particles, but at a very low ﬂow rate at Re ¼ 6.5. It was found that concentration peaks occurred near the wall at values close to the inlet value of species A, indicating that channeling was taking place. There were also local peaks of product concentration that indicated areas of high reactivity that could give rise to hotspots in practice. Yuen et al. (2003) used MR visualization techniques to obtain information on local chemical conversion for the liquid-phase esteriﬁcation of methanol and acetic acid in an optically opaque 3D ﬁxed bed of catalyst spheres. They used LBM simulations to obtain a ﬂow ﬁeld within the ﬁxed bed of catalyst, which was validated by comparison to ﬂow data also obtained from MR visualization. Their experimental data showed signiﬁcant fractional variations in conversion in the tube, at the same axial planes, which they were able to relate to heterogeneities in the ﬂow ﬁeld. They compared local conversion and locally averaged ﬂuid velocity over various length scales. When correlating over the length scale of the entire bed, the effects of local ﬂuid mixing were lost in comparison to heterogeneities in the macroscopic ﬂow ﬁeld. Correlations at the smallest length scale, the size of the packing (about 600–850 mm) showed only a weak link between ﬂow and conversion, as local mixing caused by the bed structure was incomplete at this scale. An intermediate scale of 1.5 mm 1.5 mm 500 mm gave a correlation between the conversion and velocity data that was well ﬁtted by a model based on the assumption of a kinetically controlled esteriﬁcation reaction. A suggested explanation for this result was that over these volumes effective mixing between ﬂuid streamlines had occurred.

C. HEAT TRANSFER There are only a few research groups that investigate heat transfer in packed tubes using CFD, as it involves larger models, and the LBMs cannot, at present, accommodate the energy balance. The added complication of meshing the solid particles for the conjugate heat transfer problem makes heat transfer CFD studies more computationally expensive. Most heat transfer CFD work is done in small sections of packing, representing REUs or periodic elements of the bed. Lund et al. (1999) reported ﬁnite element simulations of conduction between two contacting spheres without ﬂuid ﬂow. They modeled a small separation between the spheres to allow for interparticle micro-asperity gaps, and then the

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thermal conductivity of some of the center-most gap gas elements was increased to that of the solid particles, to include the effect of deformation contact area. From the CFD simulations they derived a correlation for effective bed thermal conductivity that included the effects of the micro-asperity ﬂuid gap, the deformation contact diameter, and the thermal conductivities of solid and ﬂuid. They also considered different packing arrangements to obtain the dependence on bed voidage. One of the earliest heat transfer problems addressed by CFD is the particleto-ﬂuid heat transfer coefﬁcient. Many experimental studies had disagreed on the limiting value of the particle to ﬂuid Nusselt number (Nup ¼ hpdp/kf) as Re decreased to zero. The work of Sørensen and Stewart (1974) provided deﬁnitive calculations for creeping ﬂow that gave the limiting value for spheres in a cubic packing, and showed that it should be nonzero. CFD simulations at higher Re for simple cubic and face-centered cubic arrangements were performed by Gunjal et al. (2005). Their results for the FCC packing were in reasonable agreement with literature correlations as far as the magnitude of Nup, but the trend with Re was somewhat different. The simulations for the SC packing gave Nup values well below the predicted ones. The authors explained these differences by the different ﬂow structures in the interstitial space for the different packing arrangements. Some investigators have considered the particle-ﬂuid heat transfer for a single particle in an inﬁnite medium or in a channel. This is a useful case for the testing of CFD models and grids, as analytical solutions are available for the limiting case Re - 0. Romkes et al. (2003) simulated particle–ﬂuid heat transfer for a single particle in tubes of various sizes intended to approach an inﬁnite domain, and also for composite structured packings (CSP) made up of square channels packed with spheres to give 1 oN o2.05. They also carried out mass transfer experiments with CSP using sublimation of naphthalene from test particles. The CSP structures are designed to reduce pressure drop for packed bed reactor operations, but this also has the effect of reducing heat and mass transfer rates between particle and ﬂuid. Therefore, it is important to test existing correlations for these transport coefﬁcients, and develop new ones if needed. For the single sphere, Romkes et al. (2003) found that over the range 1oReo105 the average heat transfer rate computed by CFD was within 10% of that predicted by well-established literature correlations. They also found that the local heat transfer rate varied along the sphere surface and depended on the angle relative to the stagnation point, in qualitative agreement with prior work. They presented this dependence for high and low Re and showed ﬂow and heat transfer maps for various Re. For the CSP simulations, ‘‘test’’ particles with higher surface temperatures were simulated, analogously to the mass transfer experiments. The authors found that consistent results were obtained for different locations and numbers of active particles, as long as they were not in the ﬁrst two layers, for which the ﬂow ﬁeld had not established periodic behavior, or

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the last layer in the tube, for which exit effects had an inﬂuence. Good agreement with experiment was obtained when simulated Nusselt numbers were compared to experimental Sherwood numbers, to about 15%. Correlations were obtained for each packing (N value) separately, and an overall correlation for N o2.00. The main lesson learned was that low-N beds are special cases, the usual packed-tube particle-ﬂuid transport correlations deviated considerably from both experiment and CFD simulation. Our own group’s work has focused on wall-bed heat transfer in low-N ﬁxed bed tubes. At the high ﬂow rates of industrial practice in such applications as partial oxidation and steam reforming, the bulk of the resistance to radial heat transfer lies in the vicinity of the tube wall; hence, this is a very important problem. The standard approach to modeling radial heat transfer has been to conduct experiments in heated or cooled tubes, measure radial temperature proﬁles at various axial positions in the tube, and ﬁt a 2D pseudo-homogeneous model for temperature, under the assumption of plug ﬂow. The ﬁtting parameters have been the effective radial thermal conductivity, kr, and the apparent wall heat transfer coefﬁcient, hw. The model equations and a discussion of the limitations of this effective or lumped approach have been presented recently (Dixon and Nijemeisland, 2001), although there are literally hundreds of papers presenting variations on the experimental techniques, the types of model ﬁtted, and the theoretical justiﬁcations. Despite this effort, it is generally accepted that approaches to wall heat transfer are empirical and in strong disagreement with each other. Our initial hopes for the use of CFD for packed-tube simulation were to generate more accurate and consistent radial temperature proﬁles, free of experimental artifacts, from which better estimates of kr and hw could be obtained. In our earliest work we estimated these parameters in dimensionless form as kr/ kf and Nuw ( ¼ hwdp/kf), from temperature ﬁelds for ﬂow around small clusters of three particles (Derkx and Dixon, 1996) and eight particles (Logtenberg and Dixon, 1998a). This work was later extended to include the effects of temperature-dependent ﬂuid properties (Logtenberg and Dixon, 1998b) and to a cluster of ten contacting particles (Logtenberg et al., 1999). The results from these studies showed reasonable agreement in magnitude between the parameters estimated from the CFD simulations, and values predicted from commonly used correlations. Trends with Re were followed reasonably well for kr/kf but were off for Nuw, which could be attributed to the use of small clusters of particles that could not fully represent the ﬂow patterns of full beds. We would now regard these efforts as early attempts to check if CFD gave reasonable results for the bed-scale temperature ﬁeld, by comparing the derived heat transfer parameters to literature values. With increased computer power, our next step was to construct full beds of particles: N ¼ 2 for validation studies as described above in Section II.D.2, and N ¼ 4 for further investigation of the temperature ﬁelds and near-wall transport processes as described above in Section II.B.2. Some early ﬂow maps and path

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lines were given in Dixon and Nijemeisland (2001) while temperature ﬁelds were reported in Nijemeisland and Dixon (2004). The intention was to extract radial temperature proﬁles and ﬁt the pseudo-homogeneous model to estimate the heat transfer parameters as previously. Similar work has been published recently by Guardo et al. (2004, 2005). However, we soon faced a dilemma—How should the CFD-generated radial temperature proﬁle be sampled? The steep increase in temperature near the wall, which had been far less apparent in cluster geometries, could not be ﬁtted by the standard model for the case of a complete N ¼ 4 bed. This problem was well illustrated by Von Scala et al. (1999) for the structured packing KATAPAK-Ms. Fig. 17 shows an example of this difﬁculty. In this ﬁgure, dimensionless temperature is plotted against dimensionless radial coordinate. Twelve experimental data points are shown, and curve (a) is ﬁtted by the usual methods to these points. It clearly does not ﬁt as well as we would like either in the bed center or near the wall, due to the limitations of the underlying model. The intercept at r/R ¼ 1 provides the temperature difference used to deﬁne the wall coefﬁcient hw. If an extra point were available closer to the wall (marked as ‘‘speculative’’ in Fig. 17), such as might be provided by CFD simulations, then the ﬁt to all points results in curve (b). This clearly ﬁts the data set even worse than curve (a) and gives a signiﬁcantly different intercept and resulting hw. So, the parameter values that are estimated depend on the location of the measured values. A more consistent approach might be to drop the measurements closest to the wall, as for curve (c) shown, in which this has been done for the three experimental points closest to r/R ¼ 1. This would have the merit of at least ﬁtting the bed center temperatures well, and the parameter estimates would have a greatly reduced dependence on the measurement locations. The ﬁtted proﬁle would be quite inaccurate near the wall,

FIG. 17. Least squares ﬁts of the radial temperature proﬁle in KATAPAK-M. Reprinted from Chemical Engineering Science, Vol. 54, von Scala et al., Heat Transfer Measurements and Simulation of KATAPAK-M Catalyst Supports, pp. 1375–1381, Copyright (1999), with permission from Elsevier.

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however. A decision would also have to be made as to which measurements to accept for ﬁtting. From this illustration we can see that the added detail of the radial temperature proﬁle near the wall that could be provided by CFD simulations does not help in obtaining better estimates for the standard heat transfer parameters. It also implies that experimental efforts to measure temperatures closer to the wall are, in fact, counter-productive. Finally, it is clear that the standard model with plug ﬂow and constant effective transport parameters does not ﬁt satisfactorily to temperature proﬁles in low-N beds. These considerations have led us to look for improved approaches to near-wall heat transfer. At this stage it seemed clear that to improve near-wall heat transfer modeling would require better representation of the near-wall ﬂow ﬁeld, and how it was connected to bed structure and wall heat transfer rates. Our early models of full beds of spheres at N ¼ 4 were too large for our computational capacity when meshed at the reﬁnement that we anticipated to be necessary for the detailed ﬂow ﬁelds that we wanted. We therefore developed the WS approach described above in Section II.B.3. In Fig. 18, ﬂow path lines are shown in a perspective view of the 3D WS. By displaying the path lines in a perspective view, the 3D structure of the ﬁeld, and of the path lines, becomes more apparent. To create a better view of the ﬂow ﬁeld, some particles were removed. For Fig. 18, the particles were released in the bottom plane of the geometry, and the ﬂow paths are calculated from the release point. From the path line plot, we see that the diverging ﬂow around the particle-wall contact points is part of a larger undulating ﬂow through the pores in the near-wall bed structure. Another ﬂow feature is the wake ﬂow behind the middle particle in the bottom near-wall layer. It can also be seen that the ﬂuid is transported radially toward the wall in this wake ﬂow. The second picture in Fig. 18 shows a temperature map for a vertical plane in the middle of the WS. The tube wall is to the right of the picture, and the scale has been chosen to emphasize the temperature gradients in the near-wall region.

FIG. 18. Path lines and a temperature map in the WS geometry.

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Outlines of some of the particles are shown for reference. The boundary layer next to the wall is clearly visible (lightest shade), while the progression of the temperature ﬁeld through the solid particles can also be seen. The temperature ﬁeld is observed to be 3D, with features that do not depend on the radial coordinate alone. When we want to look at the connection between the ﬂow behavior and the amount of heat that is transferred into the ﬁxed bed, the 3D temperature ﬁeld is not the ideal tool. We can look at a contour map of the heat ﬂux through the wall of the reactor tube. Fig. 19 actually displays a contour map of the global wall heat transfer coefﬁcient, h0, which is deﬁned by qw ¼ h0(Tw–T0) where T0 is a global reference temperature. So, for constant wall temperature, qw and h0 are proportional, and their contour maps are similar. The map in Fig. 19 shows the local heat transfer coefﬁcient at the tube wall and displays a level of detail that would be hard to obtain from experiment. The features found in the map are the result of the ﬂow features in the bed and the packing structure of the particles. From Fig. 19, it is clear that the structured packing near the wall causes a pattern in the wall heat ﬂux. To indicate the repeating sections lines were added in Fig. 19. The dotted lines connect the particle–wall contact points, the horizontal line connects these contact points of spheres in the same layer, and the vertical dotted lines connect particle–wall contact points from spheres of alternating layers. The section indicated by the solid box is the repeating section selected for further investigation. The right-hand side of the box is centered at a sphere-wall contact point and the left-hand side of the box is in between contact points. The height of the box is identical to one packing-layer height. For relating the wall heat ﬂux and the near-wall ﬂow patterns quantitatively the separate pieces of information had to be linked. Detailed information on the

FIG. 19. Map of wall heat transfer coefﬁcient for N ¼ 4 bed of spheres.

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near-wall ﬂow ﬁeld was available in conjunction with an equally detailed map of the wall heat ﬂux, similar to Fig. 19. Each surface cell in a repetitive section of the wall was linked directly to a number of ﬂuid volumes with similar tangential and axial coordinates, up to about 5 mm into the ﬂuid. In this way we could probe on a small scale whether the wall heat ﬂux from each surface mesh element (about 1.3 mm in size) could be linked to the corresponding local ﬂow features. Unfortunately, it was found that the quantitative comparison of nearwall ﬂow features with the local wall heat ﬂux by itself did not show enough distinct features to identify any trends. A number of features of the ﬂow ﬁeld were tried, such as velocity components, vorticity components, helicity, and velocity component derivatives. No correlations were found for any of these quantities, suggesting that we cannot relate wall heat ﬂux to local ﬂow ﬁeld on this length scale. By reducing the ﬂow features to simple component magnitudes, larger scale patterns such as the ﬂow path were lost. To test whether wall heat ﬂux could be related to ﬂow patterns on a larger length scale, a conceptual comparison was made using the near-wall repetitive section with the wall heat ﬂux map, as shown in Fig. 19. To capture the 3D volume of the near-wall ﬂow features, the ﬂow ﬁeld was simpliﬁed to a cartoon and the foremost features of the ﬂow were identiﬁed. When the ﬂow features were compared to the wall heat ﬂux, it could be seen that the areas of low wall heat ﬂux were located in the parts where the main through-ﬂow and the wake ﬂow met. The high wall heat ﬂux areas were mostly located near the sphere-wall contact point and just upstream from that area. The area just upstream from the contact point had a diverging ﬂow, consisting of strong axial and tangential components; the wake ﬂow, however, was also characterized by strong tangential ﬂow, but combined with radial ﬂow. When the ﬂow features were separated by component, as was done in the quantitative analysis, these connections were lost. So, analysis at the length scale of a particle was able to reveal connections between ﬂow and wall heat ﬂux (Nijemeisland and Dixon, 2004). The results of the analysis described above have suggested that to lump all the near-wall heat transfer mechanisms into a wall heat transfer coefﬁcient idealized at the wall is too simple an approach for low-N beds. One of our approaches will be based on separating the three main contributions to the extra resistance in the near-wall region: the viscous boundary layer, the decrease in stagnant conductivity of the bed due to decreasing solid fraction, and the reduction in radial convective transport due to increased channeling and changed ﬂow patterns near the wall. We intend to use full-bed simulations to generate temperature ﬁelds such as those shown in Fig. 20, for N ¼ 4. The temperature maps shown in Fig. 20 illustrate the development of the temperature ﬁeld as the ﬂow enters a tube heated at the wall. The ﬁrst (lefthand) map shows the initial heating of the gas at the tube entrance. The development of the boundary layer near the walls is clear and represents one contribution to the heat transfer resistance in the wall region. The more rapid

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN

363

FIG. 20. Temperature contour maps of respectively the x ¼ 0 plane of the ﬁrst stage in the N ¼ 4 geometry stacking, and the y ¼ 0 plane of the fourth stage. Main axial ﬂow direction moves from left to right in these pictures.

penetration of higher temperatures into the bed through the solid particles due to their higher thermal conductivity is also well shown. In the second (righthand) temperature map the arrangement of the spherical particles around the tube wall is illustrated, and the more developed temperature ﬁeld further down the tube. The more uniform temperatures in the bed center, due to convective heat transfer, are evident, as are the temperature gradients across the ﬁrst layer of particles next to the wall. The proper choice of simulation conditions will allow us to separate the individual contributions due to the separate heat transfer mechanisms, and our early results in this direction are promising (Leising, 2005).

IV. Catalyst Design for Steam Reforming Using CFD A. STEAM REFORMING

AND

PRINCIPLES

OF

CATALYST DESIGN

The steam reforming of methane to produce synthesis gas is becoming a particularly important reaction recently, due to the increased availability of natural gas as a feedstock. The use of reformed gas has historically been dominated by hydrogen manufacture for commercial use, or for ammonia and methanol synthesis. Of increasing importance, however, is the manufacture of liquid fuels from remote or stranded natural gas using Fischer Tropsch chemistry, and the generation of hydrogen from natural gas liquid fuels to power mobile and stationary fuel cells. The syngas generation section of such plants comprises over 50% of the capital cost (Abbott et al., 1989). There is thus a strong economic incentive to develop more efﬁcient steam reforming technology, and a major step in this effort is the optimal design of catalyst particles (Stitt, 2005).

364

ANTHONY G. DIXON ET AL.

Steam reforming is traditionally carried out in large ﬁred furnaces containing many catalyst-containing tubes. There are several requirements in reforming that might normally be considered mutually incompatible:

Low pressure drop and thus high voidage and large particles High surface area for high activity which normally requires small particles Good radial mixing for heat transfer that requires a large particle High strength to avoid breakage during handling and charging (Note that no catalyst is strong enough to resist the stresses of tube contraction during cooling but that the fracture patterns are important to preventing pressure drop build up).

The general outcome of this is that the reforming industry uniformly uses pellets of a length to diameter ratio in the range 0.8–1.2 with internal holes. Catalysts ranging from Raschig rings to pellets with 4 to 10 axial holes are now commonly used. Tube-to-particle diameter ratios also vary but at the entry region where good heat transfer is essential they will normally be 5–10. Smaller catalyst particles tend to be used in the lower portion of the tube where the reaction activity becomes a factor next to the heat transfer. This is especially true of top ﬁred reformers. A randomly packed tube, see Fig. 21 for example, is geometrically extremely complex and thus hard to represent. The randomness of the packing makes it hard to construct mappings with periodicity. The impact of the tube-side heat transfer coefﬁcients on the tube wall temperature is shown in Fig. 22: a two-fold improvement in the coefﬁcients facilitates approximately a 401C lowering of the tube wall temperature. Tube wall temperature is an important parameter in the design and operation of steam reformers. The tubes are exposed to an extreme thermal environment. Creep of the tube material is inevitable, leading to failure of the tubes, which is exacerbated if the tube temperature is not adequately controlled. The effects of tube temperature on the strength of a tube are considered by use of the LarsonMiller parameter, P (Ridler and Twigg, 1996): P¼T

logðtÞ þ K 1000

(26)

where T ¼ material temperature [K], t ¼ time [h] and K is a material-dependent constant. The value of this parameter is plotted against the rupture stress of the material. Fig. 23 shows the Larson-Miller diagram for a typical cast high temperature alloy (K ¼ 15), validated by the results of rupture tests at standard conditions. A high tube wall temperature also affects the performance of the catalyst. Higher temperatures lead to increased carbon lay-down on the catalyst and a resultant loss of catalytic activity, as well as potential catalyst breakage. Both

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN

365

FIG. 21. Randomly packed tube packed with reforming catalyst.

Top Fired Reformer

Tube Wall Temp [°C]

860 820 780

Base case

740

Base case with 2x heat transfer

700 0

1

2

3

4

5

6

7

8

9

10

11

12

Distance Down Tube [m] FIG. 22. Impact of tube-side heat transfer coefﬁcients.

lead to a decrease in the local reaction rate. The effective reduction in the reaction-originated heat sink may cause the tube to overheat locally or globally. The effects of excessive temperatures on reformer tubes are in fact quite dramatic. Fig. 24 shows photographs of reformer tube banks with poor performance and tube over-heating. In Fig. 24a, the ﬂame from the burner is visible in the top of the photograph. On several of the tubes clear evidence of hot bands

366

ANTHONY G. DIXON ET AL.

Rupture Stress [MPa]

500 ± 30%

100 50

10 16

18

20 22 24 P (Larson-Miller Parameter)

26

FIG. 23. Larson-Miller diagram summarizing the results of 170 rupture tests on cast 25/20 chrome nickel alloy (adapted from Ridler and Twigg, 1996).

FIG. 24. Photographs of primary steam reformer tube banks showing high tube wall temperature features, (a) showing bands and hot patches and (b) showing an entire tube that has overheated.

and hot patches can be seen. These may be the result of local deactivation, or catalyst voids (settling) which lead to loss of the local endotherm and tube failure. In Fig. 24b, a tube can be seen to have overheated over its entire length. This emphasizes the thermally aggressive environment to which the reformer tube is exposed. From the Larson-Miller analysis, it is possible to derive more easily interpreted information relating to the effects of sustained high temperature on the life of a tube. A common rule of thumb is that a tube wall temperature increase of 201C will shorten a tube life by over 50%: from its design period of 10 years to less than 5 years. The cost of a typical reformer tube is USD 6 000–7 000. With typical reformer sizes in the order of 300–400 tubes and taking on-site expenditure into account, this puts the cost of a complete re-tube in the range

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN

BENEFITS

OF

TABLE 1 MODERN CATALYST PELLET DESIGN 17 mm Raschig Rings

Plant rate (relative) Maximum TWT (1C) CH4 slip (mol% dry) Approach to Equilibrium. (1C) Pressure Drop (kg/cm2)

ON

367

REFORMER PERFORMANCE

17 mm Raschig Rings

LxD 19 14 mm 4-Hole

100 921 4.4

112 940 4.8

112 921 4.3

3 2.3

6 3.1

2 2.8

USD 5–8 million. Avoidance of high tube temperatures, both globally and locally is therefore at a premium. The overall effect of catalyst pellet geometry on heat transfer and reformer performance is shown in the simulation results presented in Table 1. The performance of the traditional Raschig ring (now infrequently used) and a modern 4-hole geometry is compared. The beneﬁts of improved catalyst design in terms of tube wall temperature, methane conversion and pressure drop are self-evident.

B. CFD SIMULATION OF REFORMER TUBE HEAT TRANSFER CATALYST PARTICLES

WITH

DIFFERENT

For the steam reforming reaction, catalyst particles have been developed with internal voids or holes, so as to increase both bed porosity and particle geometric surface area. This results in a lower bed pressure drop and in increased activity for the reforming reaction, respectively. It is not well established, however, what the effect of these features of the catalyst particle would be on wall heat transfer in the tube. Standard heat transfer models characterize the actual particle by using either the diameter of a sphere of equivalent volume to surface area ratio or the diameter of a sphere of equivalent volume, depending on the quantity being correlated. This is a fairly crude approach that can miss the sometimes subtle effects of the particle design on near-wall ﬂuid ﬂow, and thus on convective heat transfer. Our objective in this work was to use CFD to perform a more detailed assessment of the particle design for heat transfer. Simulations were run in the WS conﬁguration described earlier. Cylindrical particles with length ¼ diameter ¼ 0.0254 m were placed in the segment in an arrangement that approximated the most common situation seen from a series of experimental packings in a transparent tube, with N ¼ 4. The arrangement was constrained by the necessity for it to be axially periodic, so that a periodic ﬂow ﬁeld could be used. Different cases of the same arrangement were studied, in which the particles had different void structures formed by using various numbers and sizes of holes running parallel to the axis of the cylinder and

368

ANTHONY G. DIXON ET AL.

placed symmetrically about that axis. A standard hole had diameter slightly larger than a quarter of the particle diameter, a ‘‘big’’ hole had diameter twice the standard diameter, while a ‘‘small’’ hole had diameter reduced by a factor O2 of the standard diameter. The particles studied were designated as full, 1-hole, 1-bighole, 3-hole, 4-hole, and 4-small-holes, with the obvious interpretation. Symmetry conditions were retained on the sidewalls of the segment geometry, in default of a better alternative. The focus of the simulations was intended to be the cylinders in the center (horizontally) of the segment, which were only marginally affected by the conditions on the side walls (Leising, 2005). The ﬂow and energy simulations were decoupled, so that the ﬂow ﬁeld was established under isothermal conditions and the temperature ﬁeld was then established with a ﬁxed ﬂow ﬁeld. The dependence of the ﬂow on temperature was judged to be small at industrial conditions. The simulation conditions (T, P, xi) were for the reaction gas at the inlet conditions of a typical industrial methanol plant steam reformer. Details and parameter values are available in the original reference (Nijemeisland et al., 2004). The ﬂow simulations were run under isothermal conditions at the inlet temperature of the reactor tube Tin, with periodic conditions on top and bottom surfaces. The RNG k– e model was used for turbulence, with nonequilibrium wall functions. Fig. 25a shows typical ﬂow path lines for the 1-hole particles, obtained by simulating the release of marker particles from the planes r ¼ 0.045 m and r ¼ 0.05 m. Releasing the particles from these planes emphasizes the ﬂow patterns near the tube wall. From the path lines it may be seen that the ﬂow is mostly axial, as expected; however, there are regions of ﬂow with a strong radial component, and also regions of backﬂow (i.e., ﬂow with a negative axial component). The strongly axial ﬂow is found between particles, such as to the right of the central particle. Strong radial components are found when the ﬂow is displaced by a particle, such as above the central particle. Backﬂow often occurs in the wake of an obstruction, or where two particles approach closely.

FIG. 25. Wall-segment geometry for 1-hole particles: orthographic projections showing (a) ﬂow path lines for particles released from vertical planes close to the tube wall; (b) ﬂow path lines for particles released from the bottom horizontal plane.

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN

369

In Fig. 25b, the simulated marker particles were released from the bottom surface, which generates path lines that show more detail of the ﬂow inside the WS, at lower radial coordinate values. The path lines reinforce the trends seen in Fig. 25a, and it is also possible to see some evidence of ﬂow through the center voids of the particles. Most evident is the mix of spiraling and axial ﬂow between the center front and center right particles. It is of interest to determine the extent to which there is ﬂow through the interior holes of the particles, as the reaction activity is proportional to geometric surface area under these conditions. So, it is important to know whether the extra surface area provided by the holes is accessible to the ﬂow. It is not easy to see this internal ﬂow from the path lines in Fig. 25, although there appears to be ﬂow through the center particle. To determine this more clearly we constructed a surface that passed through the midpoint of the center particle, perpendicular to its axis, for each of the particle geometries. This is shown as the dark square in Fig. 26, which illustrates the results for the 4-hole particle. The point of view for Fig. 26 is aligned with the axis of the particle, and the perspective causes the sides of the particle to appear as the outer darker ring of cells and the sides of the holes to appear as skewed rings of darker cells around each hole. Contours of velocity magnitude are shown for the midway point of each hole and demonstrate that there is clearly substantial ﬂow through the voids, but it is not symmetric inside the holes. By summing mass ﬂow rates for

FIG. 26. View of velocity contours down internal voids or holes of the center 4-hole particle.

370

ANTHONY G. DIXON ET AL.

the four internal cross-sections, we found that slightly over 10% of the total mass ﬂow for the segment actually passed through the interior voids of the center particle. Within each void, the ﬂow rate was higher in that part of the hole that was toward the top of the particle, i.e., at the higher Z-coordinate. It should be noted that the particles simulated here were quite large, and so were the internal voids, as is typical for steam reforming. This result should not be extrapolated to other applications with smaller particles and voids, which may present much higher resistance to internal ﬂow. The simulations for temperature did not use periodic axial boundary conditions, as the long-term objective was to incorporate heat sinks into the particles (see next subsection). Instead, the technique of virtual stacking was used, as has been described above. All results were from the third segment in the stack and used a tube wall boundary condition of constant wall heat ﬂux, with the inlet ﬂuid temperature for the ﬁrst stage set to the uniform Tin used for the ﬂow simulation, and the top and bottom solid surface heat ﬂuxes set to zero. The outlet ﬂuid required no thermal boundary condition. Each subsequent stage used the outlet temperature ﬁeld from the previous stage as the inlet ﬂuid temperature ﬁeld, and the ﬁrst two stages were regarded as being present to overcome thermal entry effects and to give a more developed radial temperature proﬁle for the third stage comparisons of the particles. The ﬂuids thermal properties were those of the reforming mixture, the solid thermal properties were those of alumina, and radiation was neglected, having been previously determined to be small. Fig. 27a shows the temperature ﬁeld in the ﬂuid adjacent to the tube wall, by means of a temperature contour map. The axes of the map are the axial coordinate Z and the arc length along the curved tube wall, S. The contour map shows one hot region and several colder regions in an overall temperature distribution that was quite moderate. The hotter region in the center of the map is associated with the strong axial ﬂow component found there. The cold region to the left of the center of the map (S-coordinate 0.035–0.045, Z-coordinate 0.02–0.04) corresponds to the position of the curved section of the center particle in Fig. 25. In this area the ﬂow is of average velocity, but has a uniform direction and a reasonable radial component, creating the cooler spot. The weaker hot region at the left side of the segment (S-coordinate 0.01–0.015, Z-coordinate 0.01–0.04) corresponds again to a part of the WS where a strong uniform axial ﬂow was found, but the radial component of the ﬂow was minimal. The energy being put into the system could not be easily transported into the bed, as the ﬂow was more parallel to the tube wall, which resulted in a higher near-wall temperature and lower energy uptake. Similar features could be found near the right-hand side of the temperature plot (S-coordinate 0.08). The cool spot near the right-hand side (S-coordinate 0.09–0.1, Z-coordinate 0–0.03) is a result of the radial ﬂow into the bed around the particles there. Fig. 27b gives a more quantitative comparison between the radial temperature proﬁles for the full, 1-hole, and 1-bighole particles. There is a temperature

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN 990 980 970 960 950 940 930 920 910 900 890 880 870 860 850 840 830

0.05

Z [m]

0.04 0.03 0.02 0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 S [m]

(a)

371

0.1

910 Solid cylinder 1-hole cylinder 1-big-hole cylinder

890

T (K)

870

850

830

810

(b)

0

0.2

0.4

0.6

0.8

1

r/R

FIG. 27. (a) Near-wall temperature map for the 1-hole particles; (b) radial temperature proﬁles for solid cylinders and cylinders with two different sizes of internal void.

jump over the viscous boundary layer at the tube wall that is not shown here; the proﬁles are all for ﬂuid temperatures in the fully turbulent region. The temperature proﬁles are very similar, with the full cylinders being slightly higher, which implied more effective thermal transport for the simulations shown here. There was little difference between the 1-hole and 1-bighole particles, except toward the tube wall, which was ascribed to the difference in porosity there. It is interesting that there was quite a large difference in ﬂow through the two different 1-hole geometries, which did not translate into a strong difference in heat transfer effectiveness. There is a complex interaction between changes in the amount of solid conduction and radial displacement of ﬂow, which facilitates convective heat transfer. Also, comparisons, such as the above, do not take into consideration the effect of particle geometry on reaction, which will further affect the heat transfer picture by providing heat sinks.

372

ANTHONY G. DIXON ET AL.

The wall temperature maps shown in Fig. 28 are intended to show the qualitative trends and patterns of wall temperature when conduction is or is not included in the tube wall. The temperatures on the tube wall could be calculated using the wall functions, since the wall heat ﬂux was speciﬁed as a boundary condition and the accuracy of the values obtained will depend on their validity, which is related to the y+ values for the various solid surfaces. For the range of conditions in these simulations, we get y+ E 13–14. This is somewhat low for the k– e model. The values of Tw are in line with industrially observed temperatures, but should not be taken as precise. Fig. 28a shows the wall temperature map for the 1-hole particles that results if the conduction in the tube wall is omitted from the model. It is quite similar to the temperature map for the near-wall ﬂuid shown in Fig. 27a, and is relatively rich in features such as regions of hot and cold temperatures, which correspond to the local ﬂow ﬁeld and positions of the particles. The temperatures are considerably higher than in the ﬂuid, reﬂecting the temperature difference across the viscous boundary layer, which is substantial at the industrial ﬂow rate of this study. There is, correspondingly, a fairly wide range of wall temperatures. Wall conduction was then included in the model, using thermal properties and tube thickness of a typical high-temperature reforming tube made out of a high-alloy steel. The results are presented in Fig. 28b, as a wall temperature map on the same temperature scale as in Fig. 28a. It appears virtually featureless, and all the temperature variations seem to have been smoothed out, due to the thermal conduction in the wall that is now operating. This is misleading, however, as may be seen in Fig. 28c, in which the same temperature ﬁeld is presented on a scale chosen to bring out the temperature variations. We can see that the features in this map are similar to those of Fig. 28a. The average wall temperatures are the same with or without wall conduction, but the extremes in temperature of Fig. 28a have been strongly mitigated by the wall conduction, as shown in Fig. 28c. The distribution of temperatures still reﬂects the distribution of temperatures in the neighboring ﬂuid, which in turn reﬂects the local variation in heat transfer resistance near the wall. Temperature ﬁelds within the bed have been shown to be unaffected by the inclusion of conduction in the tube wall (Dixon et al., 2005).

C. REACTION THERMAL EFFECTS

IN

SPHERES USING CFD

Steam reforming is a heterogeneously catalyzed process, with nickel catalyst deposited throughout a preformed porous support. It is empirically observed in the industry, that conversion is proportional to the geometric surface area of the catalyst particles, rather than the internal pore area. This suggests that the particle behaves as an ‘‘egg-shell’’ type, as if all the catalytic activity were conﬁned to a thin layer at the external surface. It has been demonstrated by conventional reaction-diffusion particle modelling that this behaviour is due to

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN

1-hole

Tube wall

No wall conduction

0.05

Z (m)

0.04 0.03 0.02 0.01

0.01

0.02

0.03

0.04

(a)

0.05

0.06

0.07

0.08

0.09

0.1

S (m)

1-hole

Tube wall

Wall conduction

0.05

Z (m)

0.04 0.03 0.02 0.01

0.01

0.02

0.03

0.04

(b)

0.05

0.06

0.07

0.08

0.09

0.1

S (m)

1-hole

Tube wall

373

12 00 11 90 11 80 11 70 11 60 11 50 11 40 11 30 11 20 11 10 11 00 10 90 10 80 10 70 10 60 10 50 10 40 10 30 10 20 10 10 10 00 99 0 98 0

1200 1190 1180 1170 1160 1150 1140 1130 1120 1110 1100 1090 1080 1070 1060 1050 1040 1030 1020 1010 1000 99 0 98 0

Wall conduction 1110

0.05

1105 1100

0.04

Z (m)

1095 1090

0.03

1085 1080

0.02

1075 1070

0.01

1065 1060

0.01

(c)

0.02

0.03

0.04

0.05

0.06

S (m)

0.07

0.08

0.09

0.1

1055 1050

FIG. 28. Wall temperature maps for 1-hole particles (a) without wall conduction; (b) with wall conduction, on the same temperature scale as (a); (c) with wall conduction, on a temperature scale chosen to show nonuniform temperature features.

374

ANTHONY G. DIXON ET AL.

the extremely strong diffusion limitations under steam reforming conditions (Pedernera et al., 2003). This study was carried out to simulate the 3D temperature ﬁeld in and around the large steam reforming catalyst particles at the wall of a reformer tube, under various conditions (Dixon et al., 2003). We wanted to use this study with spherical catalyst particles to ﬁnd an approach to incorporate thermal effects into the pellets, within reasonable constraints of computational effort and realism. This was our ﬁrst look at the problem of bringing together CFD and heterogeneously catalyzed reactions. To have included species transport in the particles would have required a 3D diffusion-reaction model for each particle to be included in the ﬂow simulation. The computational burden of this approach would have been very large. For the purposes of this ﬁrst study, therefore, species transport was not incorporated in the model, and diffusion and mass transfer limitations were not directly represented. The approach that was used was to represent the energetic effects of reaction through user-deﬁned volumetric heat generation terms. For this approach, the partial pressures of the species in the gas mixture were held constant at the conditions corresponding to the position of interest in the reactor tube, and reaction energy effects were allowed in the catalyst particles in an outer shell only. Thus, the WS was regarded as representing a differential slice of the reactor, with uniform species partial pressures. For each solid phase cell, if the local position of the volume centroid was within an outer shell of the catalyst particle, the reaction rates were calculated at the local solid temperature, and an energy source term was included. The volumetric heat generation rate, Qp, in units of W/m3(cat.) was given by the sum of the products of the reaction rates and the heats of reaction. A cut-off value was used so that the reaction heat effects were conﬁned to the outer shell, and a parametric study was carried out on the effects of the choice of the value. This was equivalent to setting the effectiveness factor of the particle. The simulation was run by ﬁrst determining the ﬂow solution in the periodic segment, and subsequently determining the energy solution. The solution of ﬂow and energy were decoupled, as the temperature-dependence of the gas properties was not expected to inﬂuence ﬂow at the extremely high industrial ﬂow rates simulated here. This assumption allowed the ﬂow to be treated as periodic, independently of the temperature ﬁeld. The gas heated up slightly as it passed through the segment, and as the reaction rates gave temperaturedependent sinks/sources, the temperature ﬁeld could not be treated as periodic. The ﬂux through the tube wall of the segment was kept constant. Symmetry conditions were applied to the sides of the WS. The CFD calculations of the present work used conditions and compositions from a Johnson Matthey detailed reformer model of a methanol plant steam reformer with upwards ﬂow, at typical operating conditions. Conditions were chosen corresponding to three different axial positions along the tube, to reﬂect reaction rates typical of those close to the inlet, midway down the tube and close

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN

375

to the outlet. Details of parameter values used in these simulations can be found in the original reference (Dixon et al., 2003). For our simulations with heat sinks in spheres, we chose not to stack runs, in order to minimize the change in temperature across the WS, and instead carried out a study of the effects of simpliﬁed boundary conditions. For most runs, simple boundary conditions were used in which the ﬂuid entering at the base of the segment was set uniformly to Tin, and ﬂuxes through the solid areas on the top and bottom planes were set to zero. No thermal boundary condition was required for the ﬂow outlet boundary. From knowledge of the coordinates of the center points of the spherical particles, we developed a simple criterion to select those control volumes whose centroid lay within the cut-off from the particle surface. We then veriﬁed this using a user-deﬁned marker technique as shown in Fig. 29a. In the ﬁgure, ﬂuid cells were tagged 0 and are intermediate in shade, solid particle inactive cells were tagged 1 and are the lightest, while the selected solid active cells were tagged 2 and are the darkest. As can be seen, the algorithm correctly selected the cells at the particle edges. The algorithm did not select any interior cells. The appearance of somewhat ragged edges is due to a 2D representation of a 3D situation. A closer look is shown in Fig. 29b. Some tetrahedra appear to be selected that are not at the surface of a particle. In these cases, the centroid of the tetrahedron lies close enough to the surface for the cell to be selected, even though the part of the cell that is intersected by the plane of Fig. 29b appears to lie further from the surface. A similar explanation holds for cells that appear to be next to a surface, but which are not selected, due to the position of the centroid. Examination of the geometry in 3D conﬁrms this interpretation of the picture. The simulation of the thermal effects of the steam reforming reaction was based on a published reaction model (Hou and Hughes, 2001) for methane

FIG. 29. (a) Midplane cross-section of the WS packed with spheres, showing control volumes found by selection algorithm, marked as darkest cells.; (b) close-up of gap between two spheres, showing cell selection in detail.

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reforming over a Ni/Al2O catalyst. The model is based on the performance of the steam reforming catalyst produced by ICI-Katalco (now Johnson Matthey Catalysts), and consisted of the following three main reactions: 1: CH4+H2O ¼ CO+3H2 2: CO+H2O ¼ CO2+H2 3: CH4+2H2O ¼ CO2+4H2 The ﬁnal rate expressions, which were used in the present work, were given by Hou and Hughes (2001). In these rate expressions all reaction rate and equilibrium constants were deﬁned to be temperature-dependent through the Arrhenius and van’t Hoff equations. The particular values for the activation energies, heats of adsorption, and pre-exponential constants are available in the original reference and were used in our work without alteration. The active shell thickness was determined by the cut-off value rcut, which represented how much of the spherical catalyst particle was active and provided an energy sink/source, depending on the local reaction rates. A position in the particle was regarded as inert if its distance from the particle center was less than rcut. The actual catalyst particles themselves were uniformly impregnated with active metal catalyst. The use of rcut was a device to represent the ‘‘eggshell’’ nature of the reaction and diffusion in the catalyst particles, in the present study of the energetic effects of steam reforming. It is, therefore, of interest to investigate the effect of different values of rcut, or the equivalent values of the effectiveness factor, Z. The amount of heat actually taken up by the particles was an important quantity, as tubes operate under heat transfer limited conditions near the tube inlet. Fig. 30 shows a plot of Q against Z, where Q was the total energy ﬂow into the solid particles, for the entire segment. For inlet conditions, Q varied strongly at lower Z, but was almost constant at higher values. As rcut/rp decreased from 0.95 to 0.0 and the effectiveness factor increased from nearly zero to one, the active solid volume increased by a factor of 7. If the solid temperature had remained the same, the heat sink would also have had to increase sevenfold. This could not be sustained by the heat transfer rate to the particles, so the particle temperature had to decrease. This reduced the heat sink and increased the driving force for heat transfer until a balance was found, which is represented by the curve for the inlet in Fig. 30. Subsequent to the simulations reported in Dixon et al. (2003), some runs were carried out in which the virtual stacking was used, to provide a similar length of heated tube as in the runs without heat sinks, for comparison purposes. Some results for the third segment are given in Fig. 31, which shows a horizontal plane at the vertical midpoint of the WS and a vertical plane at the horizontal midpoint of the WS, under inlet tube conditions. The level of particle activity was rcut/rp ¼ 0.95, which was closest to the egg-shell picture.

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377

500

400

Inlet

Q (W)

300

200 Mid-tube 100

0 Outlet -100 0

0.2

0.4

0.6

0.8

1

FIG. 30. Heat ﬂow into particles, as a function of effectiveness factor, for the three tube positions studied in Dixon et al. (2003).

FIG. 31. Horizontal and vertical planes through the third-stacked WS, showing the temperature ﬁelds in the ﬂuid and through the spherical catalyst particles, with 5% activity.

We can see that for these conditions, the temperature ﬁelds inside the wall particles are far from symmetric. Signiﬁcant temperature incursions appear inside the spheres, and the inﬂuence of the wall is strong. The spheres are hotter close to the tube wall than on the side facing the center of the segment. The interior particles appear to be more symmetrical in temperature. It is noticeable that the particles are considerably lower in temperature than the surrounding

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gas, except possibly near a contact point with the wall. This is an expected result of the endothermic reactions. At the inlet of the reactor tube, the gas mixture is rich in methane and steam with some carbon dioxide. It is also at a relatively low temperature. The water–gas shift reaction (reaction 2) is energetically favored because of its low activation energy, but the reactant carbon monoxide is not present to any signiﬁcant extent. Of the other two reactions, methane conversion to carbon dioxide (reaction 3) is favored over methane conversion to carbon monoxide (reaction 1) at lower temperatures, as its activation energy E3 is approximately half the value of that of reaction 1, E1. Since this is not the desired product mix for synthesis gas, the strategy would be to increase the tube temperatures as quickly as possible to a range where reaction 1 is favored. Reaction 3 is strongly endothermic, and reduces the heat-up rate of the catalyst by providing a heat sink. Thus, efﬁcient heat transfer in the early stages of the tube is essential.

D. REACTION THERMAL EFFECTS

IN

CYLINDERS USING CFD

Our initial work on reaction thermal effects involved CFD simulations of ﬂuid ﬂow and heat transfer with temperature-dependent heat sinks inside spherical particles. These mimicked the heat effects caused by the endothermic steam reforming reaction. The steep activity proﬁles in the catalyst particles were approximated by a step change from full to zero activity at a point 5% of the sphere radius into the pellet. To extend these calculations to cylinders is more complicated, as both the position and orientation of a cylinder must be obtained. To do that we followed the sequence of operations used to position each cylinder, as shown in Fig. 32a, for particle 1, the lower front particle in the wall segment (note that wire frames of the previous positions are retained in each sketch for comparison). Similar sequences were available for each of the other particles in the wall segment model. It was possible to follow the centre point and top centre point under the transformations, and we developed criteria to select those control volumes whose centroid lay within the cut-off from either the curved surface or the ﬂat ends. We then veriﬁed this using the same user-deﬁned marker technique as for the spheres, and the results are shown in Fig. 32b. Again, ﬂuid cells were intermediate in shade; solid particle inactive cells were the lightest, while the selected solid active cells were the darkest. As can be seen, the algorithm correctly selected the cells at the particle edges, whether ﬂat or curved. The algorithm did not select any interior cells. The appearance of somewhat ragged edges is again due to a 2D representation of a 3D situation. Following the veriﬁcation step, we applied the heat sink methodology to a WS with full cylinders, comprising a ﬂow ﬁeld solution followed by three virtual

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379

FIG. 32. (a) Sequence of transformations 1-2-3-4 to place bottom front cylindrical particle; (b) Midplane cross-section of the WS packed with cylinders, showing control volumes found by selection algorithm, marked as darkest cells.

stacking thermal simulations. As mentioned previously, the main reason for stacking is to mitigate the effects of a ﬂat inlet temperature proﬁle, and to develop the thermal penetration of the bed. For our simulations with heat sinks in spheres, where we studied the effects of using simpliﬁed boundary conditions at the inlet and outlet instead of virtual stacking, we found that the main qualitative features of the solution were not changed, but there was deﬁnitely some inﬂuence at the detailed level, and on quantitative results. For the work described here, we wished to compare simulations with and without heat sinks, so we chose to use virtual stacking as in our earlier work without heat sinks. This meant that the gas and solids would gradually heat up and depart from the set reactor tube conditions, while the partial pressures stayed constant. We regarded this as one of the simpliﬁcations necessary for the heat sinks methodology, which will be eliminated by the development of an improved approach with proper modeling of species diffusion and reaction in the solid particles. We are currently working on such an approach. Comparisons for the full solid cylinders, with and without heat sinks, were made in Nijemeisland et al. (2004) for conditions near the inlet of the reactor tube. These showed that temperature proﬁles changed drastically when heat sinks were included. In Figs. 33 and 34, we show a similar comparison for conditions typical of the middle of the reactor tube. We compare the planes at the midpoint of the third stage, for the cases where the outer 5% of the particle was active and where the entire particle was inactive. These are shown in Fig. 33. Both temperature ﬁelds shown are on the same temperature scale. As expected, the simulation that included heat sinks resulted in much colder temperatures. With no reaction heat sinks, the temperature ﬁeld is barely affected by the presence of the particles, despite their higher thermal conductivity, as

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ANTHONY G. DIXON ET AL.

FIG. 33. Comparison of temperature ﬁelds in WS midplane for particles with active outer shell (95% inactive, left diagram) and particles with no reaction heat effects (100% inactive, right diagram).

1100 1080

No sinks Sinks No sinks - fluid Sinks - fluid

1060 1040

T [K]

1020 1000 980 960 940 920 900 0

0.2

0.4

0.6

0.8

1

r/r

FIG. 34. Comparison of radial temperature proﬁles for WS packed with full cylinders, with and without heat sinks; solid symbols are for temperatures averaged over ﬂuid and solid, open symbols for temperatures averaged over ﬂuid alone.

shown in the picture on the right. When active pellets are simulated, there is a clear effect resulting in the particles being seen owing to their lower temperatures. It is, however, remarkable that at the level of reaction included in these models, there is still considerable lack of symmetry in the particle temperatures for the cylinders nearest the tube wall. This observation conﬁrms that the assumption of a symmetric temperature ﬁeld surrounding the catalyst particles could lead to serious errors in estimating reaction rates and modeling the reactor tube.

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381

These qualitative observations can be expressed more quantitatively by examining average radial temperature proﬁles for the WS, as shown in Fig. 34. The introduction of heat sinks into the catalyst particles decreases the radial temperatures, whether they are averaged over both solid and ﬂuid cells or over the ﬂuid cells alone. If we look at the average ﬂuid temperatures, the heat sinks cause a decrease of about 40–50 K, and the ﬂuid proﬁles are fairly ﬂat in both cases. For the situation in which there are no heat sinks, the radial temperature proﬁle is almost the same for the ﬂuid and for the ﬂuid and solid combined. This corresponds to the observation in Fig. 32b for the 100% inactive case, in which no differences between ﬂuid and solid could be seen. When heat sinks are included, the solid is much colder, and maxima and minima in the overall averaged proﬁle appear, together with much larger differences in temperature compared to the ﬂuid only proﬁle. It is expected that the shape of the overall temperature proﬁle may be related to the distribution of the solid and its active region; future work will investigate this further. The extension of this work to include catalyst particles with internal voids is more complex, as there are regions of catalytic activity adjacent to the internal holes, complicating the testing procedure. A comparison of several different catalyst conﬁgurations of internal voids has recently been completed, and a description of the method, its veriﬁcation, and the results obtained will be the subject of a future publication.

V. Future Prospects This article has attempted to review the issues in applying CFD to simulate interstitial ﬂow in packed tubes, with an emphasis on low-N tubes. The rapid changes in computational capability of today’s computers mean that the problems and limitations discussed here will also change rapidly. All we can do is to try to extrapolate where the needs and areas of interest are likely to be in the near future. There continues to be a need for improved automatic generation of 3D packings, for arbitrary shapes of particles, in cylindrical tubes, and with wall effects present. Since a CFD simulation is based on a single instance of a packing, it is necessary to have the details of that packing readily available. To extend the simulations to consider ensembles of packings, for statistical analysis, would require an easy, inexpensive method of generating the packings. The LBM has been used to obtain some very interesting results and good insight into packed tube ﬂow, dispersion, and simple reactions. This method however appears to us to have limitations for reactor applications due to its complications in addressing high-Re, nonisothermal situations. Future developments in simulations of catalysis in reactor tubes are likely to be based on the FV-CFD methods. The geometry creation, meshing, and post-processing tools

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of the commercial versions of these methods are powerful, and a growing array of turbulence model options promises adequate capability for simulating highRe ﬂows. Abilities to simulate porous regions and species ﬂows are available or are rapidly being added in several packages, in addition to the traditional base of ﬂow and heat transfer. One area that will be critical for ﬁxed bed modeling is that of wall functions. Owing to the packing, a packed tube ﬂow is dominated by interactions with solid surfaces. As discussed above, resolving turbulent boundary layers will be too expensive for a long time to come, while the demands of the geometry and accurate calculation often make it difﬁcult or impossible to obtain a mesh to satisfy the conditions for wall functions to be strictly applicable. The development of improved wall functions that can work with a range of mesh sizes would be very helpful. Alternatively, several groups have indicated reasonable results and agreement with experimental data from simulations that used meshes and wall functions that were not strictly appropriate. More research is needed on this question. The research on the ﬂow regimes in packed tubes suggests that laminar ﬂow CFD simulations should be reasonable for Re o100 approximately, and turbulent simulations for Re 4600, also approximately. Just as RANS models provide steady solutions that are regarded as time averages of the real timedependent turbulent ﬂow, it may be suggested that CFD simulations in the unsteady laminar inertial range 100 oRe o600 may provide a time-averaged picture of the ﬂow ﬁeld. As with wall functions, comparisons with experimental data and an improved assessment of what information is really needed from the simulations will inform us as to how to proceed in these areas. Simulations with representative segments and unit cells employing periodic or symmetry boundary conditions are likely to be necessary for the foreseeable future. Although simulations of complete tube cross-sections would be preferred, these are anticipated to remain too costly for some time to come. This will be especially true for turbulent ﬂows and geometries that require ﬁne meshes or boundary-layer resolution. The validation of CFD codes by comparison to reliable experiments is of the highest importance. Especially promising is the use of MRI methods to noninvasively provide ﬂow ﬁelds and dispersion data. Major challenges will be to extend MRI and similar methods such as LDV and particle tracking to a wider range of conditions, and to develop noninvasive measurements of temperature to improve veriﬁcation of heat transfer simulations. Applications of packed tube CFD to improve transport and reaction understanding and models are still in early stages. A number of studies have focused on addressing the technical issues of using CFD for packed tubes, and have presented qualitative results that have yielded insight into the phenomena. Other studies have tried to use CFD simulations to extend experimental data, and provide estimates of familiar transport parameters.

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Some of the recent work on structured packings may provide a pointer to future directions for developing models of transport in packed tubes of particles. The identiﬁcation of REUs in structured packings has provided a fruitful approach to correlating pressure drop. Since low-N packings of particles are also strongly structured due to the wall ordering, can we similarly identify REUs for these tubes, and develop contributions to dispersion and heat transfer? Some of our work on near-wall heat transfer suggests that this may be a way forward, to move on from the current approaches based on neglect of bed structure and empirical correlation of bed effective transport parameters. The simulation of reacting ﬂows in packed tubes by CFD is still in its earliest stages. So far, only isothermal surface reactions for simpliﬁed geometries and elementary reactions have been attempted. Heterogeneous catalysis with diffusion, reaction, and heat transfer in solid particles coupled to the ﬂow, species, and temperature ﬁelds external to the particles remains a challenge for the future. A promising start has been made in packed tube CFD simulation, especially at lower Re and for reduced geometries such as unit cells and bed segments. Applications to transport and catalyst particle assessment are active areas of research. We look forward to the insights that these simulations promise, to more streamlined and easier application of the CFD methods, and to wider applications such as two-phase ﬂow in trickle beds.

Nomenclature cp Cm C1e, C2e, C3e dt dp Di E Ei Fi gi Gk Gb hi H ho

ﬂuid heat capacity (J/kg?K) constant in Eqn. (6) (-) constants in Eqn. (8) (-) tube diameter (m) particle diameter (m) molecular diffusivity of species I (m2/s) empirical constant in law of the wall (-) activation energy of species i (kJ/mol) external body forces component in direction i (N/m3) acceleration due to gravity in coordinate direction i (m/s2) generation of turbulent kinetic energy due to stress (J/m3?s) generation of turbulent kinetic energy due to buoyancy (J/m3?s) ﬂuid enthalpy of species i (kJ/kg) ﬂuid enthalpy (kJ/kg) particle to ﬂuid heat transfer coefﬁcient (W/m2?K)

384 hp hw Jj k kP kf kr kt K ‘ ‘d N P P q Q Qp r rp R S Sh Sm t T Tin T0 Tw u’ u% u+ u* uP vin vi v0 X y yP y+ y* Yi Z

ANTHONY G. DIXON ET AL.

particle to ﬂuid heat transfer coefﬁcient (W/m2?K) wall-heat transfer coefﬁcient (W/m2?K) diffusion ﬂux of species i (kg/m2?s) turbulent kinetic energy (J/kg) turbulent kinetic energy at point P (J/kg) ﬂuid thermal conductivity (W/m?K) effective radial thermal conductivity (W/m?K) turbulent thermal conductivity (W/m?K) material-dependent constant (Eqn. 26) (-) eddy length scale (m) Kolmogorov dissipation scale (m) tube-to-particle diameter ratio (dt/dp) (-) static pressure (Pa) Larson-Miller parameter (K) heat ﬂux (W/m2) heat uptake into catalyst particles (W/m3) heat generation in catalyst particle (W/m3 (cat.)) radial coordinate (m) particle radius (m) tube radius (m) arc length (m) volumetric heat source (J/m3?s) volumetric mass source (kg/m3?s) time (s) temperature (K) tube inlet temperature (K) global reference temperature (K) tube wall temperature (K) ﬂuctuating velocity component (m/s) mean velocity component (m/s) dimensionless mean velocity (-) dimensionless mean velocity (Eqn. 19) (-) mean velocity of ﬂuid at point P (m/s) inlet velocity (m/s) interstitial gas velocity (m/s) superﬁcial gas velocity (m/s) coordinate (m) coordinate normal to the wall (m) distance from wall to point P (m) dimensionless distance from wall (-) dimensionless distance from wall (Eqn. 20) (-) mass fraction of species i (-) axial coordinate (m)

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN

GREEK LETTERS af b dij e e y k l lt m meff mmol mt r sk se t tw u O

relaxation factor for variable f (-) thermal expansion coefﬁcient (K–1) Kronecker delta function ( ¼ 1 if i ¼ j, 0 otherwise) (-) turbulence dissipation rate (J/kg?s) bed porosity (-) dimensionless temperature (T–Tin)/(Twall–Tin) (-) von Ka´rma´n constant (-) ﬂuid thermal conductivity (W/m?K) turbulent thermal conductivity (W/m?K) ﬂuid dynamic viscosity (kg/m?s) effective viscosity (kg/m?s) molecular viscosity (kg/m?s) turbulent viscosity (kg/m?s) ﬂuid density (kg/m3) turbulent Prandtl number for k (-) turbulent Prandtl number for e (-) deviatoric stress tensor (N/m2) wall shear stress (N/m2) ﬂuid kinematic viscosity (m2/s) speciﬁc dissipation rate (s–1)

DIMENSIONLESS NUMBERS Particle Nusselt number Nu ¼

hp d p kf

rv0 cp d p kf cp m ¼ kf

Pe´clet number Pe ¼ Prandtl number Pr

Turbulent Prandtl number Prt ¼

c p mt kf

rv0 d p m rvi d p (interstitial) Rei ¼ m rv d (effective) Reeff ¼ m0 p eff

Reynolds number (superﬁcial) Re ¼ Reynolds number Reynolds number

ABBREVIATIONS BFD CFD CR

Brinkman-Forcheimer-Extended Darcy Computational Fluid Dynamics Collective Rearrangement

385

386 CSP cv DEM DNS FD FE FV LBM LDA LES MRI RANS REU RNG RSM RTD SD TWT WS

ANTHONY G. DIXON ET AL.

Composite Structured Packings control volume Discrete Element Method Direct Numerical Simulation Finite Difference Finite Element Finite Volume Lattice Boltzmann Method Laser Doppler Anemometry Large Eddy Simulation Magnetic Resonance Imaging Reynolds-Averaged Navier–Stokes Representative Elementary Unit Renormalized Group Reynolds Stress Model Residence Time Distribution Sequential Deposition Tube Wall Temperature Wall Segment

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INDEX

A Agglomeration, 198 Aggregation, 274, 275, 277–282, 287–289 Algebraic stress model, 163 Anisotropy, 184–185

B Batchelor length scale, 167, 213 Batchelor scale, 241, 242 Beeman–Verlet algorithm, 98 Blending, 151, 153, 190, 219 Blending, suspending, 183 Boundary methods, 180, 181, 191, 219 Boussinesq, 160, 163, 196 Breakage, 252, 275, 278–281, 287–289 Break-up, 170, 199, 203–206, 209 Bubble column, 2, 8, 11, 16, 18, 21, 24, 58 Bubble dynamics, 2 Bubble number density, 204

C Carnahan–Starling Equation, 108 Catalyst particles, 308, 310–312, 315, 330, 352, 356, 363–364, 367, 372, 374, 376–378, 380–381, 384 Chemical reaction engineering (CRE), 231–233, 236, 244, 245, 253, 298, 310 Chemical reactions, 156–157, 166, 183, 190, 209–210, 217–219, 310, 315, 319, 355 Coalescence, 157, 170, 203–206, 209, 218–219 Coarse-grid simulation, 136 Coefﬁcient of normal restitution, 95 Cohesive force, 96, 110 Collision, 157, 168, 174–175, 177, 193–194, 198, 202–203, 218 391

Collision operator, 77 Compartmentalization, 199 Composition vector, 267–272, 274, 276, 277, 285, 287 Computational ﬂuid dynamics, 307, 310, 385 Computational ﬂuid dynamics (CFD) code, 232, 233, 235, 237, 238, 244, 247–253, 259, 266, 267, 269, 274, 288, 291, 298, 300 model, 232–238, 244–246, 248–250, 252, 253, 257, 261–263, 267–269, 271, 273–275, 277, 279, 281, 282, 286–288, 292–301 simulation, 238, 239, 243, 24, 252, 24, 255, 272, 279, 282, 283, 300, 301 Computer-generated packing, 325–326 Contact force, 90 Continuum, 169–170, 175, 189 Cross-correlation terms, 166, 167, 210 Crystallization, 151, 193, 197, 219 Crystallizer, 173, 193, 198, 199, 202–203

D Damko¨hler number, 209, 213–214 Direct numerical simulation (DNS), 3, 151–152, 160, 193, 217, 235, 237, 238, 239, 244, 245, 253, 287, 295 Direct quadrature method of moments (DQMOM), 249, 268–272, 277, 282–284, 286–289, 301 Discrete bubble model, 141 Discrete particle model, 72, 86 Dispersed, 167–170, 189, 204, 219 Dispersed two-phase, 167, 169 Dispersion, 310, 314, 334, 343, 352–355, 381–383 Dissipated, 180

392

INDEX

Dissipation, 154, 162–164, 170, 183, 190, 193, 203–204, 206, 214 Dissolution, 196–197, 200, 219 DNS, 3–4, 24, 156–157, 159–160, 193–194, 202, 209, 216–219 Drag, 157, 168, 195–196 Drag, virtual mass, 169 Droplet deformation, 59 Droplet evaporation, 28, 40 Droplet–particle collision, 3, 27–28, 49, 55 D3Q19 model, 79

Gas–liquid, 151, 203 Gas–liquid ﬂow, 11 Gas–liquid–solid ﬂow, 1, 3, 13 Gas–liquid–solid ﬂuidization, 2, 14, 24 Gas–solid drag force, 83 Gas-ﬂuidized beds, 66 Geldart A particles, 127 Granular temperature, 115 Growth, 252, 274–277, 279, 281, 282, 284, 287–289

E

Hamaker constant, 96, 110 Hard-sphere model, 86 Heat transfer, 28, 33–34, 39–40, 58–59, 308–310, 312–315, 318–319, 321, 323, 330, 332–334, 337–340, 342, 344–345, 356–365, 367, 371–372, 376, 378, 382–384 History, 168 Hold-up, 204–208 Hydrodynamics, 2, 4, 28, 34, 58

H Eddy viscosity, 162–165, 184 Energy spectrum scalar, 241, 242 turbulent, 240, 247 Enskog theory, 117 Ergun equation, 84 Euler–Euler, 167, 169, 170, 195, 204 Euler–Lagrangian, 167, 196 Eulerian, 152, 165–167, 169–170, 189–190, 205, 213 Excess compressibility, 107, 109

F Favre average, 294, 295, 298, FHP model, 75 Film boiling, 28–29, 31, 38–39, 58 Film-boiling evaporation, 2, 27 Filtered drag coefﬁcient, 138 Filtered particle-phase pressure, 138, 139 Finite element, 308, 312, 315, 355–356, 386 Finite volume, 151, 159, 171, 174, 176, 308, 315–316, 324, 341, 355, 386 Fixed bed reactor, 308–310 Flow transitions, 312 Fluidized bed, 2, 11, 13, 24, 58 Front-capturing method, 4 Front-tracking method, 4 FV, 159, 171, 174–178, 181, 191, 216, 219

G Gas liquid solid ﬂow, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63

I Immersed boundary method, 1, 3, 9 Impeller swept, 179–180, 182–183, 191, 204 Interaction force, 169–170, 194, 195–196, 219 Interstitial ﬂow, 307, 311–312, 315, 348, 351, 381 Isotropic, 155, 161, 184 Isotropy, 160, 170, 183–184, 218

J Jacobi matrix, 125

K k–e model, 164, 171, 183–184, 212 k–e turbulence model, 195, 207, 216 k–o model, 164 Kinetic theory of granular ﬂow, 69, 119 Knudsen number, 40, 42, 59, 67 Kolmogorov, 154, 159, 167, 195, 202, 205, 210–213

393

INDEX

Kolmogorov length scale, 238, 239, 241, 273, 274, 281 time scale, 238, 253, 281

L Lagrangian, 152, 165–170, 189, 194, 203, 211–212, 214 Large eddy simulations (LES), 151–152, 157, 160–161, 186, 194, 217, 235, 238, 240, 244, 295 Lattice, 151, 175–176, 197 Lattice Boltzmann (LB), 151, 159, 174–175, 314, 328, 343, 351, 354–355, 386 Lattice Boltzmann Model, 72, 74 Lattice-gas models, 74 LB, 159, 174–175, 181–182, 184, 186, 190–191, 197, 199–200, 209, 214, 216–217, 219 LB LES, 194 LES, 157, 159–162, 194–197, 214, 216–219 Level-set method, 2–3, 5–6, 8, 12–13, 29–30, 39, 50–51, 58 Liquid–liquid, 151, 203, 209 Local grid reﬁnement, 176, 182, 219

Moments, 199, 208 Momentum exchange coefﬁcient, 84, 102 Multi-environment model, 248, 250, 261, 278, 284, 285 Multiﬂuid model, 234, 288, 292, 293, 295, 296, 299 Multiphase ﬂow model, 234, 237, 244, 287, 291, 294–296, 299 reacting, 233, 234, 244, 288, 297, 301 Multiple frames of reference (MFR), 179

N Navier–Stokes, 159 Navier–Stokes equations, 100 Neighbor listsm, 98 Nucleation, 193, 198–199, 202–203, 274–278, 280–282, 284, 287 Number density function (NDF) bivariate, 282, 286, 301 univariate, 274, 279, 283 Numerical diffusion, 172 Nusselt number, 298

O M Macroinstabilities, 154, 158, 165, 188, 189 Magnus, 168, 194 Magnus, history, 169 Mass transfer, 193, 197, 204–206 Mesh generation, 324, 332, 336 MFR, 179, 191 Micromixing, 152, 166–167, 210–211, 214, 219, 245, 246, 248–250, 262, 265, 268, 269, 273, 275–277, 282, 284–288, 299–301 Microscopic balance equation, 233–235, 240, 301 model, 233, 234, 240, 275 transport equation, 250, 254, 259, 267, 268, 275–277, 279, 283, 284 Mixing time, 190–192, 209 Mixture fraction mean, 250, 257, 266, 299 transport equation, 257 variance, 245, 248, 250, 257, 266, 299

Obukhov–Corrsin constant, 241, 243 One-way coupling, 165, 167, 203

P Packed bed, 309–310, 313, 325, 328, 334, 342, 344, 351, 354, 357 Packed tube, 307–308, 315–317, 321, 324–328, 330, 332, 335–337, 340, 342, 344–345, 348, 352, 354, 356, 364–365, 381–383 Parallel, 159, 171, 174, 197, 200, 210, 212, 214, 219 Parallellization, 216 Particle-image velocimetry (PIV), 246 Particle size distribution (PSD), 252, 274, 277, 279, 281, 282, 287, 288 PDF, 167, 199, 210, 213, 214, 219 Periodic box, 157, 161, 193–194, 202, 209, 211, 216, 218 Planar laser-induced ﬂuorescence (PLIF), 241, 246

394

INDEX

Poincare´, 190 Point particles, 167–168, 194 Population balance, 206, 208–209, 219 Population balance equation (PBE), 274, 275, 279, 282, 289, 301 Prandtl number, 250 Precipitation, 151, 197, 219 Pressure drop, 309, 313–315, 321, 335, 337–338, 340, 342–344, 348–352, 357, 364, 367, 383 Probability density function (PDF) method, 248, 257, 268, 284 model, 281, 299 presumed, 248, 257, 261, 299 transported, 261, 268, 272, 273, 280 Product-difference (PD) algorithm, 276–278, 282

Q Quadrature method of moments (QMOM), 276, 281, 282

R Random, 168–169, 202, 211 RANS, 160, 163, 195–197, 215–217 Reaction engineering, 336 Reaction-progress variable, 257–259, 262, 266 Reaction rate, 213 Reaction thermal effects, 372, 378 Reactor, 199, 209 Relaxation, 169, 172, 204–206 Residence time distribution, 209 Reynolds average, 245, 246, 278, 294, 297–299 Reynolds averaged Navier–Stokes (RANS), 151, 159, 163, 195 Reynolds-averaged Navier–Stokes (RANS) models, 235, 238, 240, 241, 244–246, 253 Reynolds number particle, 237, 291, 296, 298 turbulent, 238–240, 244, 266, 281 Reynolds stress model (RSM), 163, 206 RSM, 171, 212 RSM turbulence model, 164

S Saffman force, 194 Saffman lift, 169 Saffman lift force, 168 Saturated droplet impact, 29 Scalar dissipation, 241, 242, 250 Schmidt number, 240–244, 247, 250, 266, 300 Selectivity, 153, 209, 214, 217 SGS, 160–162, 168–169, 176, 184–186, 188–190, 200, 202, 213–214, 216, 218 SGS eddy viscosity, 162, 165 SGS stresses, 161–162 Sherwood number, 298 SIMPLE algorithm, 120 Single-phase ﬂows model, 244, 21, 287, 291, 295 nonreacting, 233, 244, 253 reacting, 233, 234, 253, 299 Sliding, 179 Slip, 176, 180, 204 SM, 179, 191 Smagorinsky, 162, 184–185, 188 Smagorinsky SGS, 162, 183 Snapshot, 180, 207 Soft-sphere model, 87 Solids suspension, 155, 170, 193, 209 Species transport, 166, 172, 176, 191, 213 Spring–dashpot soft-sphere model, 90 Steam reforming, 308–309, 333, 358, 363–364, 367, 370, 372, 374–376, 378 Stick-boundary rules, 81 Stokes number, 273–275, 288, 295 Stresses, 162–164, 167–168, 170, 184, 191 Subgrid, 160, 171–172, 213 Subgrid-scale model, 233, 234, 243–245, 250, 251, 23, 272, 280, 298–301 phenomena, 234, 236, 238 Subgrid stresses, 162 Suspending, 151, 153, 192, 219

T The Lattice Bhatnagar–Gross–Krook model, 78 Tracking, 166–168, 170, 175, 189–190, 194–195, 211–212, 217

395

INDEX

Turbulence, 203, 314–315, 317, 319–324, 334–336, 341–342, 350–351, 368, 382, 385 Turbulence model, 165, 171, 183, 218 Turbulence spectrum, 169, 209, 217 Turbulent kinetic energy, 154, 162–164, 183–184, 186–190 Turbulent kinetic energy dissipation, 199 Two-ﬂuid, 169–170, 196, 207 Two-ﬂuid model, 71, 111 Two-Phase, 151, 161, 167, 170–171, 173, 196, 202, 208, 217 Two-phase DNS, 203 Two-phase turbulence model, 170 Two-way coupling, 168–169, 194, 202–203

V Variance mixture-fraction, 245, 248, 250, 299 scalar, 241 Virtual mass force, 142 Voke, 162, 184–185 Volume-of-ﬂuid, 212 Vorticity, 164–165, 211

W Wall effects, 309–310, 327, 343–344, 350, 354–355, 381 Wall function, 321–324, 337–338, 342, 348, 368, 372, 382 Wall segment, 330, 378, 386 Wen and Yu eq., 84

Y U Unresolved, 166, 168, 191, 193, 218–219 Unresolved SGS, 166

Yield, 153, 209–215, 217

Z Zwietering, 153, 193