Yang Fluidization, Solids Handling, and Processing

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There are cases when the dominant mechanism of attrition is not granule erosion. This occurs when the process zone is small in comparison to the granule size and leads to granule fracture or breakage becoming the dominant attrition mechanism. This mechanism of attrition is observed for Bladex 90 DF. Although, this mechanism has not been studied as much as erosion, related work by Yuregir et al. (1987) on crystal attrition, has shown that the volumetric wear rate for breakage, Vbreakage, is given by:

where ac is the crystal length, ρ is the crystal density, and Ui is impact velocity. It is interesting to note that hardness plays an opposite role for breakage than it does for wear, primarily due to the fact that it concentrates stress for fracture. In addition, breakage rate is a stronger function of toughness.

Controlling Breakage in Granulation and Coating Processes. The parameters which, in general, control the attrition occurring in granulation and coating equipment are given in Table 6. From Eqs. (36) and (37), it can be shown that expressions for breakage and wear in fluidized beds are given by:

Both fracture toughness, Kc, and hardness, H, are strongly influenced by the compatibility of the binder with the primary particles. However, these material properties also are a very strong function of granule voidage. Therefore, both hardness and toughness increase with decreasing voidage and are strongly influenced by previous consolidation of the granules. From Eqs. (38) and (39) above, it is clear that increased fluidization velocity and bed height will increase both the wear and breakage of dried granules. However,

406 Fluidization, Solids Handling, and Processing

these same process variables may act to increase consolidation, lower the granule voidage and therefore counteract, to some extent, the breakage of the granules.

Table 6. The Effect of Operating Variables on Attrition in Granulation Processes

3.6Modeling of Granulation Processes

Most process engineering problems involve mass and energy balances. However, in particulate processes, especially in cases where particle number rather than mass is of primary importance, a balance over the population of materials of a given size in the system is often necessary. This is particularly

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true of granulation systems where the size distribution, in addition to granule structure and voidage, is a key property of the final product. The population balance is a statement of continuity that describes how the particle-size distribution changes with time and position. It is widely used to model particle formation and growth in a variety of processes such as crystallization (Randolph & Larson, 1988), granulation (Adetayo et al. 1995), pelletization (Sastry & Fuerstenau, 1973) and aerosol reactors (Friedlander & Wang, 1966; Landgrebe & Pratsinis, 1989). A review of the various applications of the population balance is given by Ramkrishna (1985).

The population follows the change in the granule size distribution as granules are born, die, grow, or enter or leave a control volume, as illustrated in Fig. 34. The number of particles between volume v and v + dv is n(v)dv, where n(v) is the number frequency size distribution, or the number density. The population balance for granulation is then given by

Eq. (40)

The left-hand side of Eq. (40) is the accumulation of particles of a given size. The terms on the right-hand side are, in turn, the bulk flow into and out of the control volume, the convective flux along the size axis due to layering and attrition, the birth of new particles due to nucleation, and birth and death of granules due to coalescence.

The granule size distribution (GSD) is a strong function of the balance between different mechanisms for size change—nucleation, layering, coalescence, attrition by erosion, and attrition by breakage (see Fig. 35). For example, Fig. 36 shows the difference in GSD for a doubling of the mean granule size due to (i) layering only, or (ii) due to coalescence only.

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Attrition is the direct opposite of layering. It is a major mechanism when drying occurs simultaneously with granulation and granule velocities are high, e.g., fluidized beds and spouted beds. In a fluid bed, attrition rate is proportional to excess gas velocity and inversely proportional to granule fracture toughness, described earlier. Typical forms of attrition were given previously in Sec. 3.5.

Effect of Mixing. The degree of mixing within the granulator has an important effect on the granule size distribution. In general, the exit size distribution is broadened by good mixing in the granulator. In general, fluidbeds are modeled as well-mixed for granulation, and may either operate continuously with draw-off or batch. Some fluid-bed drying systems with simultaneous granulation may also be modeled as plug-flow. Spouted beds may be modeled as well mixed, with a two-zone model.

Solutions of the Population Balance. Solution of the population balance is not trivial. Analytical solutions are available for only a limited number of special cases. Table 8 lists analytical solutions for some special cases of practical importance. In general, analytical solutions are only available for specific initial or inlet size distributions. However, for batch coalescence, at long times the size distribution may become “self preserving.” The size distribution is self preserving if the normalized size distributions at

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis & Adetayo (1994).