3 Pl V

where Vb is effective bubble velocity in a swarm. For the swarm of large bubbles, Krishna et al. (1999) measured rise velocity and proposed the following correlation:

where UG is superficial gas velocity (ms-1) and VblXJ is the rise velocity of a single bubble. This expression is recommended to estimate the drag coefficient of large bubbles in swarms.

It must be noted that although Eq. (11.4) describes the time-averaged form of interphase coupling terms, how turbulence (and dispersed phase volume fraction) affect several empirical coefficients appearing in this expression, is seldom known. (Note that, fortunately, influences of higher turbulence and higher volume fraction of dispersed phase on interphase drag coefficient are opposite to each other.) In the absence of quantitative information, it is not really meaningful to rigorously include various terms appearing in the interphase coupling forces due to time averaging. In most cases, therefore, only those terms containing mean values of variables are retained and all other effects are lumped by suitably modifying values of empirical coefficients appearing in this equation. Unfortunately, no systematic data or study is available to independently validate values of empirical coefficients used in practice. Such an effort is essential to make further progress in modeling interphase coupling terms. When turbulence effects are neglected and interphase drag force dominates the overall interphase coupling terms, the interphase coupling term can be written (for multidimensional flows):

It also must be kept in mind that all consistent two-phase momentum equations should reduce to single-phase equations if there is no slip between the two phases or the volume fraction of dispersed phase is zero.

Apart from interphase coupling terms, time-averaged transport equations demonstrate the effects of turbulence via various higher order and unknown terms. The viscous shear terms normally can be neglected in comparison with the turbulent shear terms. For closure of time-averaged transport equations, the concept of eddy viscosity is generally employed. Velocity correlations (Reynolds stresses) are generally modeled following the practice of single-phase flows:

where Sj is the Kronecker delta and vt is the kinematic turbulent viscosity. Johansen (1988) reported an alternative route to anisotropic modeling of these velocity correlations. The form reported in Eq. (11.11), however, is most commonly used. Several authors proposed empirical formulae to estimate effective turbulent viscosity (Sato et al., 1981; Salcudean et al., 1985; Clark et al, 1987). Most of these formulae, however, prescribe a unique value of turbulent viscosity for the entire reactor and, therefore, fail to account for its spatial variation. To account for this variation, the desired turbulence model should be able to predict turbulence length and velocity scale correctly. Two-equation turbulence models are the simplest models that promise success for flows in which length scale cannot be prescribed empirically. Among the various two-equation turbulence models, the k-e model is the most widely tested and used. With this model, turbulence viscosity is related to local values of turbulent


kinetic energy, k and the rate of turbulent energy dissipation, e in that phase:

0 0

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