## 2

for a computational cell is equivalent to the sum of local momentum transport rates of all the dispersed phase particles within that computational cell. In principle, local momentum transport rate for each dispersed phase particle (at its center of mass) must be distributed to all the surrounding nodes of the continuous phase (Fig. 7.14). Since the local values of continuous phase properties are obtained by the reverse procedure, this method will ensure momentum conservation rigorously (as Newton's third law is strictly obeyed). Suitable area or volume averaging (similar to Eqs. (7.5) and (7.6)) can be employed to implement this method.

It should be noted here that although Eulerian-Lagrangian simulations do not require substantially new algorithms, the overall solution procedure is complex and interdependent. Computational load increases substantially as the number of dispersed phase particles increases. The usual considerations of number of computational cells, discretization schemes etc. will be important in multiphase flows as well. Advanced post-processing tools are generally required to interpret and to use the simulated results. Validation becomes a complex task since the simulated results are functions of several underlying sub-models and numerical approximations (constitutive equations of interphase transport equations, various interpolations from continuous to dispersed and vice versa). In most cases, available experimental data is of time-averaged quantities (such as velocity or volume fraction profiles). Calculation of time-averaged quantities from full transient simulations is memory intensive. Special procedures need to be employed during transient simulations to allow calculation of time-averaged quantities. For most dispersed phases, true time-averaged quantities (independent of further increase in averaging time) require long computational times (a few days to weeks of dedicated CPU time on reasonably powerful processors are common). Obviously, before one initiates such lengthy calculations, several numerical experiments should be carried out to finalize the appropriate selection of parameters.

Since Eulerian-Lagrangian simulations involve many different numerical parameters (choices of integration of trajectory calculations, method of accounting for the influence of turbulence on particle trajectories and so on) in addition to the usual numerical issues, it is difficult to discuss these issues here in detail. To illustrate the results of a Eulerian-Lagrangian simulation, the case of the reactor with a downward impeller is considered. Gas is introduced through a sparger located below the impeller. Net flow of liquid through the reactor was set to zero for these simulations. The location of the sparger and boundary conditions used for the simulations are shown in Fig. 7.16. Trajectories of gas bubbles introduced from the sparger were modeled using two methods: (1) a discrete random walk with no velocity interpolation within computational cells; and (2) a continuous random walk with velocity interpolation within computational cells. Simulations were carried out until the predicted results are almost independent of additional time steps. Predicted results (in the form of iso-lines of gas volume fraction) are shown in Fig. 7.17. Both the methods predict accumulation of gas near the 'eyes' of circulation loops. Although the predicted overall gas volume fraction from these two methods is not significantly different, trajectories and local details are quite different. It is indeed essential to validate (either directly or indirectly) predictions of such complex multiphase flow simulations before they are used for engineering applications. Examples of such validation are discussed when describing applications of the Eulerian-Lagrangian approach to simulating gas-liquid flows in bubble column reactors in Chapter 11.

FIGURE 7.16 Geometry and boundary conditions used for Eulerian-Lagrangian simulations.

FIGURE 7.16 Geometry and boundary conditions used for Eulerian-Lagrangian simulations.

With a Eulerian-Lagrangian approach, processes occurring at the particle surface can be modeled when simulating particle trajectories (for example, the process of dissolution or evaporation can be simulated). However, as the volume fraction of dispersed phase increases, the Eulerian-Lagrangian approach becomes increasingly computation intensive. A Eulerian-Eulerian approach more efficiently simulate such dispersed multiphase flows.

### 7.2.3. Eulerian-Eulerian Approach

In this approach, the governing equations are formulated based on the concept of interpenetrating continua. The governing equations are discussed in Chapter 4. It can be seen that the form of governing equations is similar to that for single-phase flows. Simulation of multiphase flows, however, requires solution of a larger number of equations (governing as well as auxiliary). The increase in the number of equations may not increase the difficulty of obtaining solutions. The main difficulties in simulating multiphase flows lie in handling the pressure-velocity coupling and non-linearity and strong coupling between various equations, which cause extremely slow convergence.

For single-phase flows, pressure is shared by three momentum equations and requires special algorithms to compute the pressure field. Most of these algorithms (discussed in the previous chapter) use one continuity equation and three momentum equations to derive pressure and/or pressure correction equations. However, for multiphase flows, there is more than one continuity equation. Answers to questions such as which continuity equation should be used to derive pressure equations are not obvious. As discussed in the previous chapter, it is customary to employ iterative techniques to solve single-phase flow equations. Such iterative techniques can, in principle, be extended to simulate multiphase flows. In practice, however, the process

is often found to converge with intolerable slowness. A suitable remedy to accelerate convergence needs to be devised, and some possible ways of overcoming these difficulties are discussed here.

Several alternatives may be used to derive suitable pressure or pressure correction equations. In this section, we will discuss a specific option based on the work of Spalding (1980) and Carver (1984). This option has been used to simulate gasliquid flows in stirred vessels (Ranade and van den Akker, 1994) and bubble columns (Ranade, 1992; 1997) and was found to be quite robust. The method is illustrated here for two-fluid models. It can be extended to more than two phases following the same general principles. The overall method is an extended version of the SIMPLER

FIGURE 9.5 Simulated flow field for two alternative reactor configurations (red: high values; blue: low values; legend not shown due to confidentiality constraints). (a) Four pitched blade turbine, (b) two pitched blade turbine with cage (left: vector plots; right: contours of turbulent kinetic energy.)

FIGURE 9.5 Simulated flow field for two alternative reactor configurations (red: high values; blue: low values; legend not shown due to confidentiality constraints). (a) Four pitched blade turbine, (b) two pitched blade turbine with cage (left: vector plots; right: contours of turbulent kinetic energy.)

High

FIGURE 9.10 Flow and mixing in the mixing cup (from Ranade, 1999). (a) Contours of stream function. (b) Left side: contours of oxygen mass fraction; right side: contours of ethylene mass fraction. (Legend not shown due to confidentiality constraints).

FIGURE 10.10 Presence of trailing vortices (Rushton turbine). (a) Turbulent kinetic energy (impeller center plane; impeller rotation: counter-clockwise), (b) Z-vorticity (r/T = 0.165; impeller rotation: from left to right).

coD/Utip 202 162 12 82 42 02 —42 -82 -12 -162 -202

FIGURE 10.10 Presence of trailing vortices (Rushton turbine). (a) Turbulent kinetic energy (impeller center plane; impeller rotation: counter-clockwise), (b) Z-vorticity (r/T = 0.165; impeller rotation: from left to right).

k/UtiP2

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