Continuous nucleation rate

In order to treat crystallization systems both dynamically and continuously, a mathematical model has been developed which can correlate the nucleation rate to the level of supersaturation and/or the growth rate. Because the growth rate is more easily determined and because nucleation is sharply nonlinear in the regions normally encountered in industrial crystallization, it has been common to... 

The uniformity of film thickness is dependent upon temperature and pressure. The nucleation rate rises with pressure, such that at pressures above atmospheric the high rate of nucleation can lead to comparatively uniform oxide films, while increase in temperature reduces the density of oxide nuclei, and results in non-uniformity. Subsequently, lateral growth of nuclei over the surface is faster than the rate of thickening until uniform coverage is attained, when the consolidated film grows as a continuous layer ... 

The decomposition mechanisms are difficult to understand because (i) the surface is not homogeneous with respect to its morphology and chemical composition and (ii) these features evolve continuously during the deposition process. Moreover, as has been clearly demonstrated for noble metals, autocatalytic phenomena can occur, dramatically increasing the growth rate while decreasing the nucleation rate. 

One approach which has resulted in experimental implementation is that of Randolph and co-workers f88-92 >. Using a simulation (21) Randolph and Beckman demonstrated that in a complex RTD crystallizer, the estimation of nuclei density could be used to eliminate cycling or reduce transients in the CSD. Randolph and Low (gg) experimentally attempted feedback control by manipulation of the fines dissolver flow rate and temperature in response to the estimated nuclei density. They found that manipulation of fines flow rate upset the fines measurement indicating that changes in the manipulated variable disturbed the measured variable. Partial fines dissolution resulting from manipulation of the fines dissolver temperature appeared to reduce CSD transients which were imposed upsets in the nucleation rate. In a continuation of this work Randolph et. al. < 921 used proportional control of inferred nuclei density to control an 18 liter KCl crystallizer. 

Equation 19.17 may be interpreted in a simple way. If the equilibrium concentration of critical clusters of size Afc were present, and if every critical cluster that grew beyond size Mc continued to grow without decaying back to a smaller size, the nucleation rate would be equal to J = (3CNexp[-AQc/(kT)]. However, the actual concentration of clusters of size Mc is smaller than the equilibrium concentration, and many supercritical clusters decay back to smaller sizes. The actual nucleation rate is therefore smaller and is given by Eq. 19.17, where the first term (Z) corrects for these effects. This dimensionless term is often called the Zeldovich factor and has a magnitude typically near 10-1. 

The nucleation rate is also sensitive to the magnitude of the driving energy since, according to Eq. 19.4, AQc is proportional to the inverse square of this quantity. When the temperature is changed and the system becomes metastable, the driving force increases with continued temperature change until the rate of nucleation increases explosively, as indicated in Fig. 19.11. 

The temperature dependence caused by the exp[—0/(R7 )] term is opposite to that of the (A T)2 factor. The net effect is that there is a maximum growth rate near 600 °C, as shown in Figure 11.6. Figure 11.6 also shows that the nucleation rate, N, continues to increase with lower temperatures. 

The two most important nucleation processes are continuous nucleation, that is, when the nucleation rate is temperature dependent according to an Arrhenius equation, and the site saturation process, that is, when all nuclei are present before the growth starts. The two growth processes normally considered are volume diffusion controlled and interface controlled. Finally, the process that interferes with growth is the hard impingement of homogeneously dispersed growing particles. 

These problems are avoided if a continuous process is employed for the precipitation however, this makes higher demands on the process control. In a continuous process all parameters as temperature, concentrations, pH, and residence times of the precipitate can be kept constant or altered at will. Continuous operation is, for instance, used for the precipitation of aluminum hydroxide in the Bayer process. Bayer aluminum hydroxide is the main source for the production of cata-lytically active aluminas. The precipitation step of the Bayer process is carried out continuously. An aluminum solution supersaturated with respect to Al(OH)3, but not supersaturated enough for homogeneous nu-cleation, enters the precipitation vessel which already contains precipitate so that heterogeneous precipitation is possible. The nucleation rate has to be controlled very carefully to maintain constant conditions. This is usually done by controlling the temperature of the system to within 2-3 degrees [7]. 

The first term in this partial differential equation describes the temporal change of the population tj the second term describes the atomistic growth of the particles (which assumes that G is independent of particle size r), and finally the last two terms account for the birth and death of particles of size r by an aggregation mechanism. The birth fimction describes the rate at which particles enter a particle size range r to r + Ar, and the death function describes the rate at which the particles leave this size range. In the case of continuous nucleation, an additional birth rate term is used for the production of atoms (or molecules) of product by chemical reaction. In this case, the size of the nuclei are the size of a single atom (or molecule) and the rate of their production is identical to the rate of chemical reaction, kfi, where C is the reactant concentration, giving... 

During the induction period this process continues on the surface of the nucleation site until the critical cluster has collected the next ion to be added triggers nucleation. Crystal growth then can follow. For barium sulfate, La Mer concluded that the slope of the line in Figure 8-1 is six and therefore the nucleation of barium sulfate is a seventh-order reaction overall. The critical cluster is then (Ba" " S04")3, and the addition of the seventh ion, either Ba " or 804 , constitutes the final step of the nucleation process. The question of the number of ions in the critical cluster, however, is by no means settled. Christiansen and Nielsen concluded that for barium sulfate the number is 8. Johnson and O Rourke also concluded that the nucleation rate of this salt is proportional to the fourth power of the concentration. The concept of a small critical nucleus is intuitively satisfying in that the nucleus then requires only a small number of steps for its formation. On the other hand, application of the... 

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