Coalescence rate, granulation

Fig. 19. Mean granule volume as a function of the agglomeration time, showing the time dependence of specific random-coalescence rate function. [From Kapur (K2).]...

Coalescence Coalescence is the most difficult mechanism to model. It is easiest to write the population balance [Eq. (21-158)] in terms of number distribution by volume n v) because granule volume is conserved in a coalescence event. The key parameter is the coalescence kernel or rate constant P(m,u). The kernel dictates the overall rate of coalescence, as well as the effect of granule size on coalescence rate. The order of the kernel has a major effect on the shape and evolution of the granule size distribution. [See Adetayo Ennis, AIChE J. 1997.] Several empirical kernels have been proposed and used (Table 21-38). 

The number densities are time dependent in general and so m(m, t), the number density of granules of volume u at time t and n(.v — u, t) is the number density of granules of volume (p — u) at time t. The constant in this proportionality is fi, the coalescence kernel or coalescence rate constant, which is in general assumed to be dependent on the volumes of the colliding granules. Hence, f (u, v — u,t) is the coalescence rate constant for collision between granules of volume u and (u — v) at time t. 

The exponents a and in Equation (13.19) are dependent on granule deform-ability and on the granule volumes u and v. In the case of small feed particles in the non-inertial regime, P reduces to the size-independent rate constant Po and the coalescence rate is independent of granule size. Under these conditions the mean granule size increases exponentially with time. Coalescence stops ( = 0) when the critical Stokes number is reached. 

Monte Carlo methods for the artificial realization of the system behavior can be divided into time-driven and event-driven Monte Carlo simulations. In the former approach, the time interval At is chosen, and the realization of events within this time interval is determined stochastically. Whereas in the latter, the time interval between two events is determined based on the rates of processes. In general, the coalescence rates in granulation processes can be extracted from the coalescence kernel models. The event-driven Monte Carlo can be further divided into constant volume methods... 

Fig. 23. (I) Effect of water content on the growth rate of agglomerates sand granules grown by crushing and layering mechanism [from Capes and Danckwerts (C5)]. (II) Limestone nuclei by random coalescence [from Kapur (K2)]. (Ill) Limestone balls by nonrandom coale-scene [from Kapur (K4)]. (IV) Iron ore pelletized in a disk.[From Kanetkar (K1)].

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