## 112 Cfdbased Modeling Of Bubble Column Reactors

Most flow models published before 1990 to predict the flow characteristics of the heterogeneous regime in a bubble column reactor were restricted to one-dimensional approximations (reviewed by Ranade, 1992). These models require experimental information about the radial gas hold-up profile and turbulent viscosity and, therefore, lack generality (Kumar et al., 1994). Application of a computational fluid dynamics (CFD) based approach is being increasingly adopted to predict the detailed fluid mechanics of bubble columns. In a bubble column reactor, gas is sparged at the bottom of the liquid pool, through which gas bubbles rise upwards. While rising, these gas bubbles may interact with each other and may generate complex, re-circulating turbulent flow. The resulting flow is characterized by many distinct flow structures of various length scales (from tiny vortices shed by the bubble to macroscopic circulation covering the whole reactor).

Depending on the required resolution, various approaches to modeling dispersed multiphase flows have been developed. For example, when it is essential to resolve small-scale fluid dynamics around individual bubbles, it is necessary to use a volume of fluid (VOF) approach. With VOF, it is possible to resolve small-scale vortices behind bubbles, bubble-bubble interactions (coalescence/breakup) and mass and heat transfer between bubbles and surrounding liquid. These simulations can, therefore, be used to predict mass transfer coefficients and other interphase exchange terms. However, application of VOF is usually restricted to simulations of a few bubbles due to the huge computational requirements. If it is reasonable to model the small-scale flow around individual bubbles using lumped parameters such as drag coefficient or mass transfer coefficient, but it is necessary to simulate trajectories of individual bubbles, a Eulerian-Lagrangian approach needs to be used. This approach allows one to simulate bubble-scale phenomena accurately. However, it becomes computationally too demanding if millions of bubbles (which may exist in any typical industrial bubble column reactor) need to be simulated over a long period of time. For such cases, it is necessary to use a Eulerian-Eulerian approach, which invokes extensive modeling to simulate the behavior of gas-liquid dispersions with high gas volume fractions. As in most industrial applications of bubble column reactors, dispersed phase hold-up is not small (and often the dispersed phase is introduced through a distributed sparger rather than a single nozzle), a Eulerian-Eulerian approach will be most suitable and, therefore, it is discussed in more detail here.

### 11.2.1. Eulerian-Eulerian Approach

Most earlier attempts at understanding and modeling the fluid dynamics of bubble columns were aimed at characterizing the flow with the help of one or two characteristic parameters (circulation velocity and/or average turbulent intensity) (see, for example, the widely cited paper of Joshi and Sharma, 1979). Recent advances in computational fluid dynamics encouraged vigorous application of CFD to modeling flow in bubble columns. Professor Svendsen and coworkers (Torvik and Svendsen, 1990; Svendsen et al., 1992) and Professor Hofmann and coworkers (Grienberger and Hofmann, 1992; Hillmer et al., 1994) published initial results of such CFD approaches, apart from Ranade (1992; 1993a). Jakobsen et al. (1997) and Delnoij (1999), among others, have reviewed some of the recent modeling attempts. Model equations, their application to the simulation of flow in bubble columns and a brief review of recent simulations of bubble columns using a Eulerian-Eulerian approach are discussed in this section.

### Model equations

Generally a two-fluid approach is used to derive governing continuity and momentum transport equations (discussed in Chapter 4) for dispersed multiphase flows. Invariably, some kind of averaging method needs to be employed to derive these governing equations. Several different averaging methods are used (Drew, 1983; Ahmadi, 1987; Besnard and Harlow, 1988; Lahey and Drew, 1989). Because a variety of flow structures co-exist in bubble columns, it will be useful to make some comments on the relationship between the averaging method, governing equations, fluid dynamics of bubble columns and possible simulated results. The starting point for the derivation of governing equations is definition of a control volume. To simulate dispersed multiphase flows, careful definition of the control volume will provide guidance for the interpretation of simulated results.

If the considered control volume size is smaller than the dispersed phase particle, it will be necessary to track the gas-liquid interface as is done in the VOF approach. In the Eulerian-Eulerian approach, control volume is assumed to be large enough to define local phase volume fractions. For a meaningful definition of phase volume fractions in a control volume, control volume should be large enough to contain a sufficiently large number of dispersed phase particles. In bubble columns, when there is a wide bubble size distribution, control volume should be large enough to contain a sufficiently large number of the biggest size bubbles. The mean values of different variables of interest are then defined on the basis of such a control volume.

Obviously, approximations employed for the terms comprising deviations from mean values and the resulting equations will depend on the assumed size of the control volume. When developing model equations by considering such a large (with respect to bubble size) control volume, all the small-scale (of the order of bubble size) flow structures need to be modeled. The small-scale flow around individual bubbles and its effects are generally modeled by introducing interphase coupling terms in the governing equations, defined in terms of flow properties averaged over control volume (and its faces). Small-scale turbulence is also modeled using an appropriate turbulence model. It is important to note here, that there is no relationship between size of control volume assumed when deriving the governing equations and size of computational cells used to solve these governing equations numerically. Size of the assumed control volume affects the terms appearing in the governing equations. Once the governing equations are derived (modeled), their numerical solution can be carried out using a computational grid of any size to ensure grid independence.

In bubble columns, the flow is inherently unsteady. Significant flow structures, which are larger than the typical bubble size but smaller than column diameter, exist in the column. Such structures are clearly evident in visualizations of flow in bubble columns (Chen et al., 1994). If it is intended to resolve these unsteady structures, model equations should be based on a control volume larger than the bubbles but smaller than the characteristic scale of such internal flow structures. If it is sufficient to simulate only the long-time averaged flow, even larger control volumes (larger than the characteristic scales of internal flow structures) may be used. In such a case, additional terms, representing the influence of transient internal structures, will appear in the model equations. Two recent modeling studies based on these two options are discussed briefly below.

Recently, Pfleger et al. (1999) simulated gas-liquid flow in an apparent two-dimensional bubble column. The expected bubble size in the system investigated by these authors is 2 to 5 mm. The governing equations were derived by assuming control volumes larger than gas bubbles but smaller than the expected size of the internal circulation cells. The local flow around individual bubbles was modeled using appropriate interphase coupling terms. The resulting model equations were solved using different sizes of computational cells. Small-scale turbulence was modeled using a standard k-e turbulence model (with and without dispersion). Their results indicate that results obtained with a computational cell volume of about 0.1 cm3 (which is of the same order as bubble volume) are almost grid independent and agree quite well with experimental data. Samples of their results are shown in Fig. 11.3a (time averaged) and 11.3b (transients). Long-term averaging of these simulated results shows the well-known flow pattern with upward motion at the column center. In addition to correct prediction of time-averaged results, their model was also able to adequately capture inherently transient oscillations of the bubble plume. The standard k-e turbulence model was found to capture the inherent dynamics of gas-liquid flows in bubble columns.

While deriving the time-averaged governing equations, if the characteristic time scale is defined to be larger than the characteristic time scales of local circulation cells, additional terms representing the influence of inherently unsteady circulation cells on the long-time averaged flow pattern will appear in the governing equations. For example, Ranade (1997) introduced two additional terms in the momentum transport equations to simulate long-time-averaged effects: one to account for the effect of n

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