The volume force described by Eq. (4.6) appears as a source in the momentum equation on the side of the interface corresponding to the secondary phase. In conjunction with this surface tension model, wall adhesion phenomena can also be modeled. The contact angle that the fluid is assumed to make with the wall is used to adjust the surface normal near the wall (Brackbill et al., 1992). For accurate implementation of surface tension and wall adhesion phenomena, the solution method has to maintain a compact and sharp interface between phases. If care is taken in the numerical implementation, the VOF approach can simulate the deformation of shape of dispersed phase particles due to the surrounding flow (Delnoij, 1999). Thus, the VOF approach should be used when small-scale processes occurring near an interface separating the fluid particle from the continuous phase play a crucial role. Knowledge of these small-scale flow processes and the deformation of gas bubbles is crucial for accurate estimation of local mass and heat transfer rates near the dispersed phase particle. The main disadvantage of VOF is that it is computationally very demanding and, therefore, difficult to apply to dispersed multiphase flows containing a large number of dispersed phase particles. It may, however, serve as a useful learning tool for understanding details of dispersed multiphase flows. Recently, vigorous efforts have been made to use the VOF approach to simulate the motion of a few bubbles or a few particles to enhance the current understanding of the local flow characteristics around these dispersed phase particles (Stover et al, 1997; Delnoij et al., 1997; Krishna and van Baton, 1999). Due to computational constraints, most of these studies have been restricted to two-dimensional simulations, so that the quantitative agreement between predicted results and experimental data is not very satisfactory. Efforts are continuing to improve the underlying modeling of physics as well as its numerical implementation.

4.2.2. Eulerian-Lagrangian Approach

Using this approach, trajectories of dispersed phase particles are simulated by solving an equation of motion for each dispersed phase particle. Motion of the continuous phase is modeled using a conventional Eulerian framework. Depending on the degree of coupling (one-way, two-way or four-way), solutions of both phases interact with each other. For two-way or four-way coupling, an iterative solution procedure needs to be adopted. For four-way coupling, additional models to simulate particle-particle interactions also need to be incorporated while simulating the trajectories of dispersed phase particles. In simple, one-way coupling, a continuous phase flow field can be obtained independent of the motion of the dispersed phase. Using such a flow field, the trajectories of dispersed phase particles can be obtained by solving the equations of motion for dispersed phase particles.

Historically, the equation of motion of a single rigid sphere in a stagnant fluid was first studied by Stokes (1851). He derived the well-known drag formula based on this study. To examine issues other than the drag force, a good starting point may be the so-called BBO (Basset-Boussinesq-Oseen) equation (Gouesbet et al., 1984). Tchen (1947) attempted to generalize the BBO equation to the case when the fluid is no longer at rest. Maxey and Riley (1983) developed equations of motion for a small rigid particle in a non-uniform flow. Development of the equation of motion for a rigid sphere which has non-stationary translational and rotational motion in a non-uniform flow is not a trivial problem. At present, a rigorous form of the equation of motion has been derived only in the case of creeping flow (Peirano, 1998). For general application, a modified form of the Maxey and Riley equation in the form of a general force balance over a single dispersed phase particle is used (Auton, 1983):

Here mP and UP represent the mass and velocity vector of the particle, respectively. The right-hand side represents the total force acting on the dispersed phase particle. The sum of forces due to continuous phase pressure gradient, Fp, and due to gravity, Fg, can be written:

wherep is pressure in the continuous phase and VP is volume of the particle. The drag force, Fd, can be written:

o where the subscript C denotes the continuous phase and P denotes the particulate phase. The drag force has been studied extensively. The drag coefficient, CD, depends on the flow regime (particle Reynolds number) and the properties of the continuous phase. Several empirical correlations have been proposed for the estimation of the drag coefficient. For a single rigid sphere, the drag coefficient is usually approximated by the correlation proposed by Schiller and Naumann (1935):

(Rep < 1000 ^ 24/Re(1 + 0.15 Re0687) Cd = (4.12)

ReP is the particle Reynolds number:

| UP - UC| represents the resultant slip velocity between the particulate and continuous phase. Some other commonly used drag coefficient correlations are listed in Appendix 4.2. For fluid particles such as gas bubbles or liquid drops, the drag coefficient may be different than that predicted by the standard drag curve, due to internal circulation and deformation. For example, Johansen and Boysen (1988) proposed the following equation to calculate CD, which is valid for ellipsoidal bubbles in the range 500 < Re < 5000:

Kuo and Wallis (1988) proposed a different equation for estimation of the drag coefficient, which is suitable for spherical and spherical cap bubbles. For the specific system under consideration, several specialized correlations may be used to estimate the drag coefficient. If the dispersed phase particles are not spherical, appropriate correction factors need to be introduced in these drag coefficient correlations (Clift et al., 1978). It may also be necessary to correct the drag coefficient to account for the influence of a wall (Brenner, 1961). Most of these correlations have been developed for the motion of a single particle. When the dispersed phase volume fraction is high, the presence of other dispersed phase particles will affect the effective value of the drag coefficient. Several corrections have been proposed to account for the influence of the surrounding particles. Some of these are discussed in the section describing the Eulerian-Eulerian approach. Some drag coefficient correlations suitable for gasliquid and gas-solid flows are discussed in Part IV while discussing the modeling of different reactor types.

Apart from the drag force, there are three other important forces acting on a dispersed phase particle, namely lift force, virtual mass force and Basset history force. When the dispersed phase particle is rising through the non-uniform flow field of the continuous phase, it will experience a lift force due to vorticity or shear in the continuous phase flow field. Auton (1983) showed that the lift force is proportional to the vector product of the slip velocity and the curl of the liquid velocity. This suggests that lift force acts in a direction perpendicular to both, the direction of slip velocity

Continuous phase velocity, Ue

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