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V/Vtip turbulent energy dissipation rates, e are listed below:

— (Pl «l k) + — (pl aL Vuk) = — aL — — + aL(G - pLe) (10.11)

(PlaLe) + —(PLaLVLie) = — aL — — + aL-(CiG - C2PLe)

where G is turbulence generation rate and m is turbulent viscosity, which are given by:

G (d v} ^ d v \ d v pl ck G = M + TT7 Mt =--(10.13)

In the absence of better knowledge, standard values of k-e model parameters are generally used in these multiphase simulations. Wall functions were used to specify wall boundary conditions. Gas was introduced at the sparger by defining an appropriate source of gas at the sparger cells. The top surface of the dispersion was assumed to be flat and was modeled as a free slip wall. Bubbles escaping from the vessel were simulated by specifying an appropriate sink at the top row of computational cells (see Chapter 11 for detailed discussion on boundary conditions used for simulating dispersed gas-liquid flows).

The snapshot approach for gas-liquid flows was implemented using a commercial CFD code, FLUENT (Fluent Inc., USA). User-defined subroutines were used for this purpose. Half of the vessel was considered as a solution domain. The solution domain and details of the finite volume grid used was similar to those used for singlephase flows discussed earlier (however, the number of cells in the 0 direction were half of that used in single-phase simulations). A QUICK discretization scheme with SUPERBEE limiter function was used to integrate all the equations (Fluent User Guide, 1997). Simulations were carried out for three values of dimensionless gas flow rates (Qg/ND3), 0.01, 0.02 and 0.03.

Predicted gas-liquid flow fields for a dimensionless gas flow number 0.01 at the typical r-z planes are shown in Figs 10.27 and 10.28. The simulations indicate significant upward inclination of the radial jet issuing from the impeller in the presence of gas, which is in agreement with the published experimental evidence. It can be seen that even at such a low gas flow rate, simulations indicate that gas bubbles are not dispersed in the lower circulation loop (left side of Fig. 10.27). Significant upward inclination in the presence of gas is also obvious from the contours of turbulent kinetic energy shown in Fig. 10.28. Contours of gas hold-up confirm that the impeller is unable to re-circulate gas bubbles in the lower loop. Contours of predicted gas holdup on horizontal plane passing through the impeller are shown in Fig. 10.29 (impeller rotation direction is counterclockwise). It can be seen that snapshot simulations of gas-liquid flows clearly show the presence of gas accumulation at the locations of trailing vortices behind the impeller blades. The gas hold-up just behind the blade is orders of magnitude larger than the average gas hold-up. Such gas accumulation significantly modifies the flow around impeller blades. Predicted contours of gas hold-up at different r-z planes near the impeller region are shown in Fig. 10.30. It can be seen that just behind the leading blade, gas accumulates in the core of two trailing vortices. In the present computational model, coalescence was not modeled

FIGURE 10.27 Typical predicted flow field. Left: vectors of gas phase; Right: vectors of liquid phase (from Ranade et al., 2001c).

and hence the model is not able to simulate the formation of gas cavities behind impeller blades. However, even in the absence of a coalescence model, computations could capture significant gas accumulation in the region of the trailing vortices. As one moves away from the leading blade, the lower region of gas accumulation shifts upwards and eventually merges with the upper region as observed in the experimental data. Thus, it can be said that the computational snapshot approach can capture the essential features of gas accumulation behind the impeller blades. If a suitable coalescence model is combined with the present computational model, formation of gas cavities may be simulated. Coalescence and break-up models, which may be used to simulate the evolution of bubble size distribution in dispersed gas-liquid flows, are discussed in Chapter 11. Similar models can be used to estimate bubble size distributions and interfacial area in gas-liquid stirred reactors. It is, however, necessary to obtain detailed experimental data for gas-liquid flows in stirred vessels to quantitatively validate these computational models. Even in the absence of such quantitative validation, these models may be used to qualitatively evaluate different configurations of gas-liquid stirred reactors. Simulated results may be used to identify regions of high mass transfer coefficient and high interfacial area within the reactor to guide locations of feed pipes etc.

Two-fluid or multifluid models can be extended to simulate not only gas-liquid flows but also any combinations of different phases present in stirred reactors. To simulate gas-liquid-solid, slurry reactors, liquid and solid phases are often lumped together and treated as a slurry phase with effective properties. This approximation is reasonable as long as the solid volume fraction is low 1% ). For higher solid loading,

FIGURE 10.28 Typical predicted flow field. (Left: contours of dimensionless turbulent kinetic energy; Right: contours of gas hold-up). Ten uniform contours, maximum value = 0.1 (white); minimum value = 0 (black) (from Ranade et al., 2001c). Reproduced in colour plate section between pages 210 and 211.

FIGURE 10.28 Typical predicted flow field. (Left: contours of dimensionless turbulent kinetic energy; Right: contours of gas hold-up). Ten uniform contours, maximum value = 0.1 (white); minimum value = 0 (black) (from Ranade et al., 2001c). Reproduced in colour plate section between pages 210 and 211.

FIGURE 10.29 Contours of gas hold-up on horizontal plane located at a distance of BW/3 from impeller center plane (impeller rotation is counter-clockwise). Ten uniform contours between 0 and 0.1; white: 0.1; black: 0 (from Ranade et al., 2001c). Reproduced in colour plate section between pages 210 and 211.

FIGURE 10.29 Contours of gas hold-up on horizontal plane located at a distance of BW/3 from impeller center plane (impeller rotation is counter-clockwise). Ten uniform contours between 0 and 0.1; white: 0.1; black: 0 (from Ranade et al., 2001c). Reproduced in colour plate section between pages 210 and 211.

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