U

Lift force, FL

Added mass = Pc CVMVP

Displaced volume =

CVMVP

Added mass = Pc CVMVP

Displaced volume =

FIGURE 4.4 Lift (a) and virtual mass forces, (b) on dispersed phase particles.

CVMVP

and the direction of the curl of the continuous phase velocity field (Fig. 4.4). For locally homogeneous flows, the lift force is given by

where CL is an empirical lift coefficient. For potential flow and spherical particles, Drew and Lahey (1979) report the value of lift coefficient as 0.5. Recently Oesterle (1994) investigated lift forces in the Reynolds number range 1 to 1000, which is typical for reactor engineering applications. This study shows that lift forces are not negligible and tend to increase with increasing particle diameter. Two other situations may lead to transverse force: (1) when the dispersed phase particle is rotating and (2) when the particle is moving in the vicinity of a wall. Most studies related to these situations are restricted to small Reynolds number and have recently been summarized by Peirano (1998).

When a dispersed phase particle accelerates relative to the continuous phase, some part of the surrounding continuous phase also is accelerated. This extra acceleration of the continuous phase has the effect of added inertia or 'added mass' (Fig. 4.4).

This 'added mass' effect is modeled by introducing the virtual mass term, FVM ( DI

Fvm = - I 'vucj ••• I = cvmpcvp(up- uc) (4.16)

The material derivative, D/Dt, in this equation should be the derivative pertaining to the dispersed phase particle. The virtual mass coefficient, CVM, may be a function of the volume fraction of neighboring bubbles. For a single dispersed particle, it is in the range 0.25 to 0.5. For gas-liquid flows, van Wijngaarden (1976) recommended following expression to estimate CVM:

where CVMP indicates the value of CVM for a single particle of dispersed phase.

There may be some additional forces, such as Basset force (due to development of a boundary layer around the dispersed phase particles), thermophoretic force (due to large temperature gradient) and Brownian force. The Basset force (denoted by FH in Eq. 4.9) is relevant only for unsteady flows and in most cases, its magnitude is much smaller than the interphase drag force. Basset force involves a history integral, which is time-consuming to evaluate. Moreover, Basset force decays as t-n with n > 2 (Mei, 1993) for intermediate time. Therefore, it is very often neglected while integrating the equation of motion of the particle. Picart et al. (1982) discussed specific conditions under which the Basset term may be neglected. For most reactor engineering flows, the other two forces, thermophoretic and Brownian forces, are also quite small compared to some of the terms discussed earlier.

Once the velocity field is calculated from the force balances discussed above, the trajectories of all the particles can be calculated using:

When simulating the trajectories of dispersed phase particles, appropriate boundary conditions need to be specified. Inlet or outlet boundary conditions require no special attention. At impermeable walls, however, it is necessary to represent collisions between particles and wall. Particles can reflect from the wall via elastic or inelastic collisions. Suitable coefficients of restitution representing the fraction of momentum retained by a particle after a collision need to be specified at all the wall boundaries. In some cases, particles may stick to the wall or may remain very close to the wall after they collide with the wall. Special boundary conditions need to be developed to model these situations (see, for example, the schematic shown in Fig. 4.5).

As particles move within the solution domain, solution of the force balance of each particle requires information about the flow field of the continuous phase. The continuous phase flow is described using the volume-averaged (overall) mass and momentum conservation equations:

d(acpc)

+ v ■ (a c pcUcUc) = - a cVp - V ■ (a ctc) + a cpcg + Scm (4.20)

0 0

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