Random coalescence

Modeling and Simulation subsection.) It is necessary to determine both the mechanism and kernels which describe growth. For fine powders within the noninertial regime of growth, all collisions result in successful coalescence provided binder is present. Coalescence occurs via a random, size-independent kernel which is only a func tion of liquid loading, or... 

It is important to note that we assume the random fracture approximation (RPA) is applicable. This assumption has certain implications, the most important of which is that it bypasses the real evolutionary details of the highly complex process of the lattice bond stress distribution a) creating bond rupture events, which influence other bond rupture events, redistribution of 0(microvoid formation, propagation, coalescence, etc., and finally, macroscopic failure. We have made real lattice fracture calculations by computer simulations but typically, the lattice size is not large enough to be within percolation criteria before the calculations become excessive. However, the fractal nature of the distributed damage clusters is always evident and the RPA, while providing an easy solution to an extremely complex process, remains physically realistic. 

The coalescence-redispersion (CRD) model was originally proposed by Curl (1963). It is based on imagining a chemical reactor as a number population of droplets that behave as individual batch reactors. These droplets coalesce (mix) in pairs at random, homogenize their concentration and redisperse. The mixing parameter in this model is the average number of collisions that a droplet undergoes. 

Equation (17) indicates that the entire distribution may be determined if one parameter, av, is known as a function of the physical properties of the system and the operating variables. It is constant for a particular system under constant operating conditions. This equation has been checked in a batch system of hydrosols coagulating in Brownian motion, where a changes with time due to coalescence and breakup of particles, and in a liquid-liquid dispersion, in which av is not a function of time (B4, G5). The agreement in both cases is good. The deviation in Fig. 2 probably results from the distortion of the bubbles from spherical shape and a departure from random collisions, coalescence, and breakup of bubbles. 

Almost all flows in chemical reactors are turbulent and traditionally turbulence is seen as random fluctuations in velocity. A better view is to recognize the structure of turbulence. The large turbulent eddies are about the size of the width of the impeller blades in a stirred tank reactor and about 1/10 of the pipe diameter in pipe flows. These large turbulent eddies have a lifetime of some tens of milliseconds. Use of averaged turbulent properties is only valid for linear processes while all nonlinear phenomena are sensitive to the details in the process. Mixing coupled with fast chemical reactions, coalescence and breakup of bubbles and drops, and nucleation in crystallization is a phenomenon that is affected by the turbulent structure. Either a resolution of the turbulent fluctuations or some measure of the distribution of the turbulent properties is required in order to obtain accurate predictions. 

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

Kapur and Fuerstenau (K6) have presented a discrete size model for the growth of the agglomerates by the random coalescence mechanism, which invariably predominates in the nuclei and transition growth regions. The basic postulates of their model are that the granules are well mixed and the collision frequency and the probability of coalescence are independent of size. The concentration of the pellets is more or less fixed by the packing... 

Fig. 17. Size distributions of pellets generated by the random-coalescence mechanism in the nuclei and transition regions. [From Kapur and Fuerstenau (K6).]...

Fig. 18. Mean granule volume as a function of the agglomeration time in the random-coalescence mechanism. [From Kapur (K2).]...

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)].

Spectroscopic ellipsometry is a non-destructive, interface sensitive, in situ technique for interface characterization. Time resolved ellipsometric spectroscopy was used to determine the mechanism of electrochemical deposition of photoresists on copper electrodes under potentiostatic, anodic conditions. Nucleation of photoresist deposition occurs randomly. During the early stages of nucleation the semi-spherical particles are separated by about 100 A. The deposits tend to grow like "pillars" up to 50 A. Further growth of the "pillars" lead to coalescence of the photopolymer deposits. 

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