41 Introduction

Multiphase flow processes are key elements of several important reactor technologies. These technologies cover a wide range, from very large-scale operations such as fluid catalytic cracking reactors, to specialized reactors to produce high value, low volume specialty chemicals. The presence of more than one phase raises several additional questions for the reactor engineer. Multiphase flow processes exhibit different flow regimes depending on the operating conditions and the geometry of the process equipment. It is often necessary to develop tools to evaluate the operability of the multiphase flow process under specified conditions and to identify the operating regime. The fluid dynamics and transport processes occurring in multiphase reactors are especially sensitive to reactor configurations and operating conditions. Even small-scale hardware details such as the design of a feed nozzle or distributor may have a dramatic influence on the resulting flow structure. It is, therefore, of paramount importance to develop understanding and predictive tools to simulate multiphase flow processes to develop better reactor technologies.

The subject of the modeling of multiphase flow processes is quite vast and covers a wide range of sub-topics. It is virtually impossible to treat all the relevant issues in a single book, let alone in a single chapter. Here we attempt to provide a brief review of modeling approaches and cite the key references for further details. An attempt is made to provide sufficient information to develop a baseline model. The first section discusses various flow regimes and their key features. Various approaches to modeling these multiphase flow processes are then reviewed. Because of the importance of dispersed multiphase flows in reactor engineering, modeling of dispersed flows is discussed in more detail, in Section 4.2. Three basic approaches to modeling multiphase flows, namely, volume of fluid (VOF), Eulerian-Lagrangian (EL) and Eulerian-Eulerian (EE) are discussed with reference to dispersed flows. Some comments are made to guide the selection of an appropriate modeling approach. The topic of flow through porous media is discussed with specific emphasis on flow through fixed bed reactors. Some comments on developing tractable computational models of complex, multiphase flow processes are also included.

4.1.1. Types of Multiphase Flows

When two or more phases move relative to each other, these phases may exhibit a large number of possible flow regimes. There are several ways of classifying these multiphase flows. The simplest, first layer classification is according to the presence of thermodynamic phases: gas-liquid, gas-solid, gas-liquid-solid, liquidliquid, liquid-solid and so on. Each component of these classes can then be grouped according to the flow regimes (topology of the flow). Broadly, flow regimes are classified as dispersed flows, mixed flows and separated flows (Ishii, 1975). In dispersed flows all the phases except one exist as dispersed (discontinuous) particles flowing through the continuous fluid. Examples of this flow regime include bubbles in liquid, solid particles in gas or liquid and liquid droplets in gas or other immiscible liquid. In separated flows, none of the phases exist in discontinuous particle form. All the phases flow in a semi-continuous mode with interfaces between the different phases. Examples of this flow regime include film flow, annular flow and jet flow. In mixed flow regimes, dispersed particles as well as semi-continuous interfaces exist together. Examples of this regime include droplet annular flow (where liquid flows in the form of an annular film over the pipe as well as suspended droplets in the gas core), bubbly annular flow (where some gas bubbles flow through the annular liquid film) and slug flow. Separated or mixed flow regimes may exist in trickle bed reactors. However, in most of the other reactors, dispersed flow regimes exist. Therefore, modeling of dispersed flows is discussed here in detail. The dispersed flow regime can be further divided into several sub-regimes. Some commonly encountered (in reactor engineering applications) gas-solid and gas-liquid flows are shown in Fig. 1.9. Key features of these different regimes are discussed here with reference to gas-solid and gas-liquid reactors.

In gas-solid reactors when solid particles are held stationary (so-called fixed bed reactor), gas flows through a porous medium comprising macropores existing between pellets or packed solid particles and micropores within the catalyst pellets (or other porous solids). Issues such as isotropy of the porous medium, initial distribution of gases, characteristics of solid particles, ratio of characteristic length scale of solid particles and that of the reactor and so on, influence the flow within fixed bed reactors. Support screens are often used to cover the bed of solid particles to avoid fluidization and carry-over of bed particles. These reactors are extensively used in process industries. Some examples and illustrative flow simulations are discussed in Chapter 13.

In other gas-solid reactors (fluidized reactors), gas is the continuous phase and solid particles are suspended within this continuous phase. Depending on the properties of the gas and solid particles, the geometry of the reactor and operating flow rates of gas and solid phases, several different sub-regimes of dispersed two-phase flows may exist as shown in Fig. 1.9. For relatively small gas flow rates, the reactor may contain a dense bed of fluidized solid particles. The bed may be homogeneously flu-idized or gas may pass through the bed in the form of large bubbles. Further increase in gas flow rate decreases the bed density and the gas-solid contacting pattern may change from dense bed to turbulent bed, then to fast-fluidized mode and ultimately to pneumatic conveying mode. In all these flow regimes the relative importance of gas-particle, particle-particle and particle-wall interaction is different. It is, therefore, necessary to identify these regimes to select an appropriate mathematical model. Details of flow regime identification are discussed in Part IV while discussing the application of computational flow modeling to specific reactor types. In this chapter, governing equations for general flow types will be discussed.

For many gas-liquid or gas-liquid-solid reactors, the liquid phase is a continuous phase in which gas bubbles and solid particles are dispersed (bubble column or multiphase stirred tank reactors). Bubble column reactors may also exhibit different sub-regimes, namely homogeneous bubbly flow, churn-turbulent flow and slug flow depending on the geometry, operating conditions such as flow rates, pressure, temperature and physical properties of individual phases. The characteristics of these regimes are quite different from each other and each regime may require specialized models and boundary conditions. When there is further increase in gas flow rate, in some cases, frothing may occur. Beyond frothing, further increase in gas flow rate may make gas a continuous phase with liquid drops dispersed in it. When an additional flow-modifying element, such as the rotating impeller in stirred reactors, is present in the reactor, one may have to use a different classification for the flow regimes. See, for example, Fig. 1.8 for some flow regimes observed in a gas-liquid stirred tank reactor. As long as one phase exists as a continuous phase and the others as dispersed phases (this includes liquid-liquid and gas-liquid-liquid reactors), the general model equations discussed in the next section may be used.

Other special types of reactor may have different flow regimes specific to those particular configurations. For example, as mentioned earlier, in trickle bed reactors, liquid and gas flow through a packed bed of solid particles. Gas and liquid phases maintain a free interface and flow over solid particles. Several flow regimes may occur in such trickle bed reactors. It will not be possible to discuss modeling of flow in all these different types of multiphase reactors in a single chapter. Because of the importance of the dispersed flow regime, modeling of this flow regime is discussed here in detail. Modeling of other types of multiphase flow is briefly discussed and key references are cited for further reading. Some details are also discussed in Part IV of this book.

4.1.2. Modeling Approaches

It will be useful to discuss here different modeling approaches and some of the key issues in modeling multiphase flow processes. In general, there are three main issues one needs to address:

• Definition of 'phase'/Flow regime/Required resolution

• Formulation of governing equations

• Solution of governing equations

An obvious definition of 'phase' is 'thermodynamic state' (gas, liquid or solid). However, it is possible to define different 'phases' for computational purpose, although the thermodynamic state is not different. For example, while modeling dispersed gassolid flows when there is a wide distribution of particle sizes, it is convenient to define multiple phases representing the solid phase. Each such phase may be associated with a specific narrow band of particle sizes, having more or less identical properties (such as drag coefficient). It is also sometimes useful to treat two thermodynamically distinct phases as one phase for computational purposes. For example, in a gas-liquid-solid slurry reactor, if the solid particles are fine enough to essentially follow liquid flow, it will be convenient to treat the liquid-solid mixture as a 'slurry' phase and model the three-phase system as a two-phase system (gas-slurry). Computational phases can even be defined based on local flow characteristics such as turbulent or irrotational fluid (Spalding, 1983). Judicious definition of computational phases and consideration of possible flow regimes are often the first crucial steps in selecting the modeling approach. Once the phases are defined, relevant flow regimes can be identified (see a brief discussion in the previous section, and Part IV). Depending on the required resolution, different modeling approaches may be used.

There are three main approaches for modeling multiphase flows:

(a) Volume of fluid approach (Eulerian framework for both the phases with reformulation of interface forces on volumetric basis).

(b) Eulerian framework for the continuous phase and Lagrangian framework for all the dispersed phases.

(c) Eulerian framework for all phases (without explicitly accounting for the interface between phases).

Basic concepts of these approaches are shown schematically in Fig. 4.1 and are briefly discussed below.

The first approach, the volume of fluid (VOF) approach, is conceptually the simplest. In this approach, the motion of all phases is modeled by formulating local, instantaneous conservation equations for mass, momentum and energy. Such local instantaneous conservation equations can be solved using appropriate jump boundary conditions at the interface. However, the interface between different phases may not remain stationary and imposing boundary conditions at such an interface becomes a very complicated moving boundary problem. To avoid this, instead of directly tracking the deforming and moving interface, the VOF approach tracks motion of all the phases, from which motion of the interface is inferred indirectly. All the interfacial forces, therefore, have to be replaced by smoothly varying volumetric forces. If the shape and flow processes occurring near the interface are of interest, the VOF approach should be used. Some interface-related forces, such as surface or adhesion forces, can be modeled accurately using this approach. This approach is, however, naturally limited to modeling the motion of only a few dispersed phase particles. For simulations of dispersed multiphase flows in large equipment, this approach is not suitable, as it requires huge computational resources to resolve flow

Eulerian framework is used for continuous phase. Influence of dispersed phase: via averaging over large number of trajectories

Simulations of trajectories of dispersed phase particles: from equations for individual particles

Simulations of trajectories of dispersed phase particles: from equations for individual particles

Phase volume fractions and velocities are defined over a suitable control volume

Averaging methods to derive time and/or volume averaged equations for all phases

FIGURE 4.1 Modeling approaches for multiphase flows. (a) 'Volume of fluid' approach, (b) Eulerian-Lagrangian approach, (c) Eulerian-Eulerian approach.

processes around each dispersed phase particle. VOF-based models can be very useful as learning tools and can provide valuable information to develop appropriate closure models for Eulerian-Lagrangian and Eulerian-Eulerian approaches.

In the Eulerian-Lagrangian approach, explicit motion of the interface is not modeled. This means small-scale fluid motions around individual dispersed phase particles are not considered. Their influence is modeled indirectly while considering the motion of dispersed phase particles. In this approach, motion of the continuous phase is modeled using a Eulerian framework. The motions of dispersed phase particles (trajectories) are explicitly simulated in a Lagrangian framework. Averaging over a large number of trajectories is then carried out to derive the required information for the modeling of the continuous phase. In this approach, particle-level processes such as reactions, heat and mass transfer etc. can be simulated in adequate detail. In the case of turbulent flows, it is necessary to simulate a very large number of particle trajectories to obtain meaningful averages. Therefore, even with this approach, when the number of particles to be simulated increases, computational resources become stretched. The approach is, therefore, suitable for simulating dispersed multiphase flows containing a low (<10%) volume fraction of the dispersed phases. For denser dispersed phase flows, it may be necessary to use a Eulerian-Eulerian approach.

The Eulerian-Eulerian approach models the flow of all phases in a Eulerian framework based on the interpenetrating continuum assumption. In this approach, trajectory simulations and averaging are not carried out at a computational level but are implicitly achieved at a conceptual level. The discrete character of the underlying process is, therefore, averaged out to provide a model involving a continuum associated with the dispersed phase particles. This approach is the most difficult one to understand conceptually, requiring extensive modeling efforts. Various averaging issues will have to be addressed while formulating the governing equations in this approach. If modeled successfully, this approach can be applied to multiphase flow processes containing large volume fractions of dispersed phase. It may, therefore, be extended to modeling and simulation of complex industrial multiphase reactors consisting of a large number of dispersed particles.

In a given situation, there is no simple answer to the question as to which of these approaches is preferable. Depending upon the complexity of the dispersed multiphase flows, more often than not, it may be necessary to use multiple modeling approaches to develop an adequate understanding of the flow processes under consideration. These three approaches are complementary in many respects (Berlemont et al., 1995; Delnoij et al., 1997). Application of these approaches to model dispersed multiphase flows is discussed below. Computational aspects of solving these model equations, including special treatment of interphase coupling terms, are discussed in Chapter 7.

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