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The above equation has been proposed for single-phase flows. The interaction of various turbulence length scales may affect the validity of this equation or values of parameters for two-phase flows. Lahey (1987) proposed a correlation of C^ in terms of gas hold-up. However, his correlation is valid only for small diameter tubes with large liquid throughput. For large diameter columns, with low liquid throughput, no systematic study is available. Therefore, following general practice, the same value of C^ may be used as that for single-phase flows (0.09).

The correlation between fluctuating velocity and hold-up is modeled using a gradient transport approximation as:

where aG is the turbulent Schmidt number for the gas bubbles. Simple gradient transport is strictly valid only when the size of energy-containing eddies is much smaller than the distance over which the volume fraction gradient varies appreciably. More general formulations are available (Lumley, 1975; Elghobashi and Abou-Arab, 1983), when this condition is not satisfied. The value of turbulent Schmidt number for bubbles will, in principle, depend on bubble size and scale of turbulence. Turbulent eddies smaller than bubble size will not contribute to the bubble dispersion. However, no systematic data or theory is available to quantitatively estimate the values of turbulent Schmidt number for gas bubbles. Recently, Ranade and Mashelkar (1993) attempted to subtract the effect of eddies from a particular range of length scales from the overall turbulent transport. A similar model may be able to predict the value of Schmidt number for a given bubble size. At this stage, however, the general practice is to set the value of turbulent Schmidt number to unity.

The pressure coupling term has the same magnitude but opposite sign in continuous and dispersed phase momentum equations. This term, therefore, implies a transfer of momentum between the two phases. Elghobashi and Abou-Arab (1983) developed a closure approximation for the correlation of fluctuating volume fraction and fluctuating pressure. However, values of the constants appearing in their model are known only approximately due to lack of relevant experimental data. In this situation, an approximation derived by Johansen (1988) may be used to represent pressure coupling term:

dp' ——d Vki T7 d ak dX = -Pkakvkj -¡£7 - PkVk dXakvki (1U4)

Modeling of correlations of fluctuating volume fraction and fluctuating velocity appearing in this equation have already been discussed.

In order to close the set of modeled transport equations, it is necessary to estimate turbulent viscosity or if the k-e model is used, the turbulent kinetic energy, k and turbulent energy dissipation rate, e. The modeled forms of the liquid phase k and e transport equations can be written in the following general format (subscript 1 denotes the continuous phase):

- («1<) + —(aiVki$) = — a — ^ + S< (11.15)

where < can be either k or e, and a< is the model parameter describing turbulence dispersion of <. The corresponding source terms for k and e can be written as:

k where C1 and C2 are model parameters of the k-e model. The general practice is to use the same values of these parameters as proposed for single-phase flows (Launder and Spalding, 1972) used to estimate turbulence in two-phase flows (these values are listed in Table 3.1). G is turbulence generation based on the single-phase mechanism and Ge is an extra turbulence generation due to the presence of dispersed phase. The turbulent generation term, G, is given by:

Correlation of fluctuating velocities appearing in this expression is modeled using Eq. (11.11). Extra turbulence generation Ge can be modeled in different ways. Johansen (1988) modeled it as

where fV is a virtual mass coefficient. Svendsen et al. (1992) related the extra turbulence generation with the interphase drag force:

where Cb is an empirical constant. Kataoka et al. (1992) carried out a detailed analysis of these extra terms in the source of turbulent kinetic energy. Their analysis suggests that the extra generation of turbulence due to large bubbles (represented by Eq. 11.20) is almost compensated by the extra dissipation due to the small-scale interfacial structures. Based on comparison of predicted turbulent kinetic energy with experimental data, published computational studies also indicate that the value of parameter Cb is almost zero (Ranade, 1997), which indirectly confirms the analysis of Kataoka et al. (1992). Before we discuss some published results, it is essential to discuss special boundary conditions required to simulate flow in bubble columns.

Application of model equations to simulate flow in bubble columns

The first step in the application of model equations to simulate flow in a bubble column is to select an appropriate solution domain and formulate corresponding boundary conditions. Consider a typical bubble column configuration, where gas is introduced in a plenum below the sparger (Fig. 11.5). Figure 11.5 shows semi-batch

Gas-liquid interface (may not be flat)

Gas-liquid dispersion (gas as dispersed phase)

Open to surroundings

Open to surroundings

Gas-liquid interface (may not be flat)

Gas-liquid dispersion (gas as dispersed phase)

Liquid drops may get entrained in overhead space

Sparger

FIGURE 11.5 Outline of two-dimensional bubble column.

Liquid drops may get entrained in overhead space

Sparger

FIGURE 11.5 Outline of two-dimensional bubble column.

operation. The gas pressure in the plenum is sufficient to support the liquid head above the sparger and to maintain the required gas flow through the pool of liquid. Higher plenum pressure prevents liquid from weeping through the sparger. As gas is introduced in the column, gas bubbles rise through the liquid pool and raise the liquid level in the column. Gas bubbles burst at the gas-liquid interface, and is shown schematically in Fig. 11.6. Bursting bubbles may throw some liquid droplets into the overhead space. Depending on the gas flow rate, droplet size (settling velocity) and overhead space, these droplets may or may not escape the column from the top opening. If there is a net liquid flow, a gas-liquid separator needs to be configured at the top region of the column. One has now to select a suitable solution domain and formulate appropriate boundary conditions to translate this physical picture into a mathematical framework.

Generally, the gas pressure in the plenum is assumed to be uniform and plenum is not considered in the solution domain. The region above the sparger is considered to be within the solution domain. In most published simulations, the sparger is assumed to distribute the gas uniformly through the sparger region and was modeled as an inlet. To model the sparger as a velocity inlet, it is necessary to specify the velocity and volume fraction at the inlet boundary. Since the gas volume fraction below the sparger is unity, it is tempting to specify gas volume fraction at the sparger boundary as unity,

FIGURE 11.6 Schematic diagram of bubble bursting at the interface (from Newitt et al., 1954).

FIGURE 11.6 Schematic diagram of bubble bursting at the interface (from Newitt et al., 1954).

and to specify gas phase velocity so as to ensure the desired net gas flow. However, it must be remembered that gas enters the solution domain in the form of bubbles, which rise with a velocity different than the superficial gas velocity. It is therefore advantageous to set the gas velocity at the sparger inlet equal to the estimated bubble rise velocity. The volume fraction at the inlet can then be specified in a way that ensures the desired net gas flow. Thus, if the sparger is assumed to distribute gas uniformly, the boundary conditions can be written as:

where {Uo) is superficial gas velocity over the sparged area. For special sparger configurations such as ring spargers, a similar approach can be used by considering appropriate sparger area in the computational grid. When sparger type and resistance is not adequate to ensure uniform distribution, it will be necessary to include a sparger model in the overall flow model in order to account for non-uniform gas sparging at the sparger. Ranade (1993b) developed a simple model to simulate non-uniform gas sparging. In this model, the gas velocity through any location in the sparged area is assumed to be a function of pressure drop across the sparger at that location. The plenum pressure (pressure below the sparger) can be assumed to be uniform. The pressure above the sparger, p0, is the sum of the overhead pressure, ptop, and hydrostatic head above the sparger. Thus, the pressure balance across the sparger can be written:

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