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and 12.16b. Smaller particle diameters lead to a flatter profile for solids velocity. As expected, the predicted slip velocity increases with particle diameter. All slip velocity profiles exhibit a sharp peak near the riser wall. The predicted solids holdup profiles for all four cases are not significantly different. The predicted granular temperature increases with particle diameter and particle density, though the shape of the profile remains the same. Results of the numerical experiments to examine the influence of gas and solids flux are shown in Fig. 12.16c and 12.16d. It can be seen that for the same superficial gas velocity, the decrease in solids flux leads to higher granular temperature and higher slip velocities. For the same solids flux, a decrease in gas velocity significantly reduces the granular temperature. Predicted profiles of slip velocity exhibit a maximum in the near-wall region, the magnitude of which increases with increasing superficial gas velocity and decreasing solids flux. The computational model, however, failed to predict any significant downflow of solids even for the highest solids flux (400 kg m-2 s-1) and lowest gas velocity (5 ms-1) case. The role of gas phase and secondary solids phase turbulence on radial segregation of solids needs to be studied systematically to evaluate the currently used KTGF-based models. The predicted values of pressure drops (Table 12.2) show the expected trends. However, unless the downflow near the wall is captured, quantitative comparison with the experimental data will be difficult.

Pita and Sundaresan (1991) reported results of numerical experiments using their computational model (without including a turbulence model). They reported the existence of multiplicity for large diameter risers. In order to examine the possibility of multiple solutions, numerical experiments were initiated with several different initial guess fields. However, multiplicity was not detected in any of the cases discussed above. The computational model always converged to the same result from any initial guess field. The computational model used in the present work predicted monotonic decrease in pressure drop with increase in riser diameter for specific values of gas and solid fluxes. This trend is in line with the observations of Yerushalmi and Avidhan (1985). Pita and Sundaresan's model, however, predicted a reversal in the trend: it predicts an increase in pressure drop with increase in riser diameter, if it increases beyond a certain value (about 0.1 m). Such a reversal in trend may occur if the model predicts the downflow of solids near wall for the large diameter risers. The model used in the present work did not predict any downflow even for the 0.5 m riser. It is necessary to generate systematic data concerning radial segregation of solids and effective slip velocity of solids (clusters and particles) by conducting experiments at different riser diameters covering the range of particle diameters and gas and solid fluxes relevant to industrial riser flows. The data will also be useful to understand cluster formation and to quantify its influence on the dynamics of gas-solid flows. Understanding gained through interpretation of experimental data and the results of numerical experiments may be translated into appropriate sub-models to represent cluster formation and their effect on gas-solid dynamics. Such sub-models may be able to capture the downflow of solids near riser walls with adequate accuracy. Instead of empirically adjusting the values of restitution coefficient and speculiarity coefficient, independent measurements of these parameters should accompany the experimental data suggested above.

In light of these comments, some recent work (Kuipers et al., 1998; Dasgupta et al., 1998; Mathiesen et al., 1999, 2000; Neri and Gidaspaw, 2000) on the application of granular models to simulating gas-solid flows in riser reactors is briefly reviewed here. Mathiesen et al. (1999, 2000) developed a multifluid model to account for the particle size distribution. Each solid phase is characterized by a diameter, form factor, density and restitution coefficient. Their model was able to predict axial segregation by particle size quite well (Fig. 12.17a). The model was, however, not able to capture radial segregation by particle size adequately. This may be because the model ignored some of the external forces acting on solid particles. The predictions of solids volume fraction were also not very good (Fig. 12.17b). The model of Neri and Gidaspaw (2000) could capture the oscillatory motion of dense clusters reasonably well. Predicted results show the well-known core-annulus flow regime in the time-averaged sense. The values of solid volume fraction near the wall, however, were underpre-dicted (Fig. 12.17c). Apart from the time-averaged results, the predicted dynamic characteristics were found to be in reasonable agreement with experimental data. Their results show that imposition of the symmetry boundary condition at the riser axis is not justified because of the strongly asymmetric instantaneous flows. Benyahia et al. (2000) also report simulations of oscillatory behavior and asymmetric flow in risers. The calculated solids volume fraction deviated from the experimental data at the wall region. These results indicate that state of the art CFD models are not yet able to capture the influence of clusters without empiricism. It is essential to establish a systematic database to develop useful empirical relationships for immediate use and to guide further development of computational models. Herbert et al. (1999) systematically stored data collected over the past several years at ETH in an Oracle relational database. Databases of this type will be valuable for further development and for fine tuning CFD-based models for reactor engineering applications. Some applications of such CFD models to reactor engineering are discussed in the next section.

Simulations of bubbling/turbulent bed reactors

Bubbling/turbulent fluidized bed reactors are characterized by excellent solids mixing and gas-solids contact. Most of these properties can be attributed to the presence of bubbles. An understanding of the formation and motion of gas bubbles and their influence on various transport rates is of crucial importance for reactor engineering of bubbling/turbulent bed reactors. In a bubbling bed, distinct gas voids or bubbles exist. In turbulent beds, voids and large clusters of solids particles are distributed all over the reactor. A Eulerian-Eulerian approach is particularly suitable to simulate industrial bubbling/turbulent bed reactors (which may contain billions of solid particles). Some of the key issues in developing computational models for such reactors are discussed below.

Bubbling/turbulent beds contain regions of steep voidage gradients, which are difficult to handle numerically. Most early work on the simulation of dense bubbling beds was restricted to simulation of a single or few bubbles in a two-dimensional fluidized bed for a short time (Kuipers et al., 1991). Although this work is useful, for industrial applications it is more relevant to simulate vigorously bubbling fluidized beds and obtain predictions of bubble frequency, bubble volume fraction, bubble size and bubble rise velocities. When reviewing these attempts, Clift (1993) invoked the so-called 'Occam's razor' and recommended that such CFD models of dense bubbling beds may only be used as 'learning models', and conventional or discrete bubble Lagrangian models may be used for the design and scale-up of bubbling fluidized beds. His recommendation was mainly based on state of the art results in 1993. Since

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