Nucleation rate equation

The interpretation of the nucleation rate equation in terms of its overpotential dependence is rather difficult to perceive as represented in the form of eq. (4.42). It has been shown already that the products in the denominator of this equation contain the formation energy AG(iV) of clusters of class N (cf. eq. (4.47)), and that, at a given overpotential, AG has a maximum determining the critically sized cluster, N=Ncnt-The n terms in the sum of the denominator of eq. (4.42) show a maximum for the critically sized cluster. All terms other than that for this cluster can be neglected including the unity in the denominator [4.13, 4.14). Note that/is always much smaller than the impingement rate ti att.o times the adsorption sites Z , and hence the denominator is much larger than unity. Then  

In the derivation of equation (4.42), it has been tacitly assumed that the cluster partition function Zjy remains unaffected by the flux of clusters to higher classes, i.e., Z] f = Z. A correction for the depletion of the cluster population due to this flux can be introduced by the factor r, known as the Zeldovich or non-equilibrium factor [4.16]. 

The constant k contains the ratio of the vibrational frequencies of cluster atoms in position X and in the kink position, and can be assumed to be of the order of unity [4.13]. It has to be remembered, however, that if entropy terms are neglected (cf. Section 2.1), particularly when dissociation enthalpies are used for the calculation of AGcrit, the value of the pre-exponential term in the nucleation rate-overvoltage relation becomes largely uncertain. 

The Zeldovich factor r depends on the geometrical form of the cluster [4.17, 4.18], and for liquid droplets has the value 

The dimensionless Zeldovich factor is always less then unity, and has an order of 

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Figure 5.15 Form of nucleation rate equations showing sensitivity to Ac...

In the secondary nucleation stage, the remaining amorphous portions of the molecule begin to grow in the chain direction. This is schematically shown in Fig. 16. At first, nucleation with the nucleus thickness /i takes place in the chain direction and after completion of the lateral deposition, the next nucleation with the thickness k takes place, and this process is repeated over and over. The same surface nucleation rate equation as the primary stage can be used to describe these nucleation processes. 

The first-principle method used in the present study will be outlined below. Population balance analysis on a perfectly mixed batch crystallizer with negligible crystal breakage and agglomeration yields the familiar nucleation rate equation used by Misra and White (5)... 

Correlation of Secondary Nucleation Rate. The nucleation rate equation (2) was correlated by using multiple regression analysis at 70 C. Only data corresponding to the accelerating phase of nucleation rate was used in the correlation. The rate equation obtained at 70 C is... 

The above qualitative predictions are consistent with experiments. However, in terms of absolute nucleation rate. Equation 4-9 usually predicts too low a rate by many orders of magnitude (see below). 

The most straightforward test of the nucleation rate equation is to make a plot of the number of nuclei per unit area as a function of time at different overvoltages. The double-pulse technique developed by Scheludko and Todorova [4.32] has been successfully used in a wide range of experiments [4.33-4.36]. 

A value of interest is also the pre-exponential factor A2D in the nucleation rate equation containing the number of nucleation sites. The pre-exponential factor can be estimated fi-om the intercept of the In/ - I/I7I curve. Fig. 5.8. 

Idealized representations of the variations with time of numbers of growth nuclei (N) present at time, t, for four of the nucleation rate equations given in Table 3.1. 

The type of polymorph appearing first at the induction time could be correlated by a classical nucleation rate equation attemperaturesfrom293 to311K. 

One method is to solve the population balance equation (Equation 64.6) and to take into account the empirical expression for the nucleation rate (Equation 64.10), which is modified in such a way that the expression includes the impeller tip speed raised to an experimental power. In addition, the experimental value, pertinent to each ch ical, is required for the power of the crystal growth rate in the nncleation rate. Besides, the effect of snspension density on the nucleation rate needs to be known. Fnrthermore, an indnstrial suspension crystallizer does not operate in the fully mixed state, so a simplified model, such as Equation 64.6, reqnires still another experimental coefficient that modifies the CSD and depends on the mixing conditions and the eqnipment type. If the necessary experimental data are available, the method enables the prediction of CSD and the prodnction rate as dependent on the dimensions of the tank and on the operating conditions. One such method is that developed by Toyokura [23] and discussed and modified by Palosaari et al. [24]. However, this method deals with the CTystaUization tank in average and does not distinguish what happens at various locations in the tank. The more fundamental and potentially far more accurate simulation of the process can be obtained by the application of the computational fluid dynamics (CFD). It will be discussed in the following section. 

Although several different approaches have been made to this problem in recent years, all lead basically to the same nucleation rate equation (Garside and Davey, 1980) and tend to justify the use of empirical scale-up rules for crystallizers such as keeping the specific power input constant (section 9.3.4). 

With this background, we are now ready to solve the nucleation rate equations. First we will solve them using McDonald s trick, then we will use an alternative trick. 

Consider a general set of nucleation rate equations that can be written in the form... 

The rate constants used in writing and solving the nucleation rate equations for void formation are very uncertain. Therefore, we do not think that great significance should be attached to the numerical values of the nucleation rates in Fig. 19. The crucial result of these calculations is the role played by helium. Under reactor conditions the nucleation rate without helium is expected to be much lower than that under accelerator conditions. However, a helium concentration of only 10 ° (which is produced within 5 min in a nuclear reactor) can, under reactor conditions, cause an enormous increase in the void nucleation rate. The same helium concentration is expected to alter only slightly the void nucleation rate under accelerator conditions. 

Note from Equation 17.6 that a higher gas saturation pressure lowers the activation energy for homogeneous nucleation, and accordingly results in a higher nucleation rate (Equation 17.7). 

By assuming that the number of nuclei necessary for observing the nucleation, N, is independent of the experimental conditions, the induction time becomes inversely proportional to the nucleation rate (Equation 2.12), that is, the induction time should be indirectly proportional to the supersaturation. 

In the Multipreci model, we treat the nucleation stage at each time step as follows a new class is introduced in flie histogram of size of radius / and composition according to Equations (9) and (12). The number of precipitates N of size R" is given by the classical non stationary nucleation rate equation ... 

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