0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless radial coordinate, r/R

0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless radial coordinate, r/R

FIGURE 11.4 Simulations of time-averaged flow in bubble columns (from Ranade, 1997). Comparison with experimental data of (a) Grienberger and Hofmann (1992): Vg = 0-02 ms-1, and (b) Menzel et al. (1990): VG = 0.024 ms-1.

0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless radial coordinate, r/R

0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless radial coordinate, r/R

FIGURE 11.4 Simulations of time-averaged flow in bubble columns (from Ranade, 1997). Comparison with experimental data of (a) Grienberger and Hofmann (1992): Vg = 0-02 ms-1, and (b) Menzel et al. (1990): VG = 0.024 ms-1.

column wall and the other to account for the effect of bubble wakes. If the influence of local circulation cells is appropriately included in the model equations, no matter how small the computational cells used to solve the resulting equations, local circulation cells will not appear in the simulation results and the solution will directly predict the long-time-averaged flow pattern. Some results reported by Ranade (1997) are shown in Fig. 11.4, and show good agreement between experimentally measured time-averaged flow characteristics and simulated results. Whether it is necessary to capture inherently unsteady local circulation cells, or it is sufficient to predict longtime-averaged results, depends on the objective at hand. Care must be taken to use appropriate assumptions and derive appropriate model equations to suit the objectives under consideration. Once the model equations are derived, the user must ensure that simulated results are not grid dependent. With these comments, we describe here governing equations for turbulent gas-liquid flows, derived using a control volume larger than the bubbles, which are able to capture inherently unsteady local circulation cells.

The time-averaged mass conservation equation can be written:

where pk is the density of phase k, ak is the volume fraction of phase k and Vki is the mean velocity of phase k in direction i. The second term and third term represent the convective mass transport and the turbulent diffusive mass transport, respectively. The time-averaged momentum equation can be written:

d d dp dp' — — (Pk ak Vki ) + — (Pk akVkiVkj ) = -ak ---ak ---+ Pk akgi + Fki d t d Xj d Xi d Xi d

+ (viscous shear terms)

where Fki represents all the interphase coupling terms except pressure. vki, p' and a' indicate fluctuating components of velocity, pressure and volume fraction, respectively. It should be noted that pressure, p, is regarded as being 'shared' by both phases and therefore it appears in the transport equations of both phases. Formulation of the pressure term has occasioned some uncertainty (Spalding, 1978) and it is sometimes thought that different pressures 'ought' to be provided for each phase. Pressure inside an individual bubble is related to the pressure of the continuous phase via surface tension and bubble radius. However, this pressure inside the bubble has no relation to the flow of dispersed phase particles and is, therefore, irrelevant for the description of flow equations (Rietema and van den Akker, 1983). Pressure at the gas-liquid interface can be assumed to be equal to the liquid phase pressure since equations are spatially averaged over a control volume larger than individual bubbles (Johansen, 1988). For most reactor engineering situations, where speed of sound in each phase is large compared to velocities of interest, an assumption of microscopic pressure equilibration is adequate (Spalding, 1978; Drew, 1983).

Interphase coupling terms make two-phase flows fundamentally different from single-phase flows. Formulation of these terms, Fki, must proceed carefully, with attention being paid to force balance for a single bubble and to any possible inconsistencies. Interphase coupling terms must satisfy the following relation:

where subscript 1 and 2 denote liquid and gas (dispersed) phase, respectively. For dispersed two-phase flows, there are at least two transient forces acting at the interface in addition to the standard drag force, namely virtual mass force arising from the inertia effect (Ishii and Zuber, 1979; Auton, 1983; Cook and Harlow, 1986) and Basset force due to the development of a boundary layer around a bubble (Basset, 1888). In addition to this, transversal lift force, created by gradients in relative velocity across the bubble diameter, may also act on the bubble (Thomas et al., 1983). Time averaging of interphase coupling terms is tedious and involves several unknown correlations, which need to be modeled. Neglecting all third-order correlations and all correlations involving gradients of a fluctuating quantity, Johansen (1988) derived an expression for the time-averaged interphase coupling term (with an assumption of low volume fraction for the dispersed phase), which is given below:

9xj 9 Xj

4 dB

Where ersl is the Levi-Cevita tensor and

9V2i 9 Vu

The terms on the right-hand side of Eq. (ii.4) correspond to interphase drag force, virtual mass force, Basset force and lift force, respectively. /L is a transversal lift coefficient and is about 0.5 for potential flows and spherical particles (Drew et al., 1979). fB is a Basset force coefficient. For low volume fractions of dispersed phase, its value is about 1.5. fV is a virtual mass coefficient. The numerical value offV is generally shape dependent: for rigid, spherical solid particles, it is reported as 0.5 (Maxey and Riley, 1983). For bubbles in water, Cook and Harlow (1986) used a value of fV = 0.25. CDB is a drag coefficient, which will be a function of bubble Reynolds number, which should be based on the resultant slip velocity between two phases.

It must be noted that the Basset history term will be significant only for the simulation of unsteady flows. An order of magnitude analysis presented by Hunt et al. (1987) suggests that, for large bubble columns (D > 0.15 m), where square of the terminal rise velocity of the bubble would be smaller than the product of the gravitational constant and the characteristics length scale, the interphase coupling term will be dominated by the drag force term. Various correlations are available to estimate the value of drag coefficient for the dispersed phase particles (Clift et al., 1978; Ranade and Joshi, 1987). For the commonly encountered range of bubble Reynolds numbers, that is, 500 < ReB < 5000, the following correlation may be used to estimate the drag coefficient (Clift et al., 1978):

0.622 gApdj

For an air-water system, terminal rise velocity of bubbles is not very sensitive to bubble diameter. Therefore, for bubbles with diameters in the range 3 to 8 mm, the ratio of drag coefficient to the bubble diameter (CDB/dB) can be considered as approximately constant. Ranade (1997) carried out simulations by setting the ratio of drag coefficient to bubble diameter (CDB /dB) equal to 290 m-1.

For the swarm of bubbles, it is necessary to modify this equation to account for the interaction between bubbles and bubble wakes. Measurements by Tsuji et al. (1984) for two spheres in the Reynolds number range 100 to 200 can be expressed as

where LB is the distance between centers' of two moving spheres (which can be related to the volume fraction of dispersed phase) and CDB0 is the drag coefficient of an isolated bubble. In many cases, it is difficult to estimate the influence of the presence of other bubbles on the interphase drag coefficient under operating conditions. In such cases, it is often beneficial to use empirical information about the velocity of bubble swarms to back-calculate the interphase drag coefficient:

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