Distribution nucleation rate

One can introduce a distributed nucleation rate j R, t)d for nucleating clusters of radius between R and R + dR. Its integral over R is the total nucleation rate J t). Equation (A3,3,103) can be viewed as a radius-dependent droplet energy which has a maximum at i = 7 . . If one assumes j R, t) to be a Gaussian function, then... 

Correlations of nucleation rates with crystallizer variables have been developed for a variety of systems. Although the correlations are empirical, a mechanistic hypothesis regarding nucleation can be helpful in selecting operating variables for inclusion in the model. Two examples are (/) the effect of slurry circulation rate on nucleation has been used to develop a correlation for nucleation rate based on the tip speed of the impeller (16) and (2) the scaleup of nucleation kinetics for sodium chloride crystalliza tion provided an analysis of the role of mixing and mixer characteristics in contact nucleation (17). Pubhshed kinetic correlations have been reviewed through about 1979 (18). In a later section on population balances, simple power-law expressions are used to correlate nucleation rate data and describe the effect of nucleation on crystal size distribution. 

A pair of kinetic parameters, one for nucleation rate and another for growth rate, describe the crystal size distribution for a given set of crystallizer operating conditions. Variation ia one of the kinetic parameters without changing the other is not possible. Accordingly, the relationship between these parameters determines the abiUty to alter the characteristic properties (such as dominant size) of the distribution obtained from an MSMPR crystallizer (7). 

A plot of In n versus L is a straight line whose intercept is In and whose slope is —l/Gt. (For plots on base-10 log paper, the appropriate slope correc tion must be made.) Thus, from a given product sample of known shiny density and retention time it is possible to obtain the nucleation rate and growth rate for the conditions tested if the sample satisfies the assumptions of the derivation and yields a straight hne. A number of derived relations which describe the nucleation rate, size distribution, and average properties are summarized in Table 18-5. 

Crystallizers with Fines Removal In Example 3, the product was from a forced-circulation crystallizer of the MSMPR type. In many cases, the product produced by such machines is too small for commercial use therefore, a separation baffle is added within the crystallizer to permit the removal of unwanted fine crystalline material from the magma, thereby controlling the population density in the machine so as to produce a coarser ciystal product. When this is done, the product sample plots on a graph of In n versus L as shown in hne P, Fig. 18-62. The line of steepest ope, line F, represents the particle-size distribution of the fine material, and samples which show this distribution can be taken from the liquid leaving the fines-separation baffle. The product crystals have a slope of lower value, and typically there should be little or no material present smaller than Lj, the size which the baffle is designed to separate. The effective nucleation rate for the product material is the intersection of the extension of line P to zero size. 

S(l) is the nucleation rate for non-interacting nuclei and is further interpreted as the probability distribution for a crystal to have thickness l. Notice that for 2xsJAF < 1, S([) is negative, which corresponds to the statement that a lamella of this thickness is unstable. The total flux, ST, in an ensemble of crystals is obtained by summing S(l) over all possible values of l ... 

Above Ccrlt (i.e. E or F in Figure 1), there are two real roots to the equation, so there is a minimum and a maximum in the A G function. If a pit is nucleated at the core, the pit should spontaneously open until its radius fulfills the condition that A G is at a minimum ("10 A). There is then an activation barrier Ag (=AGmax m -A Gm. jmul) toward further opening of the pit into a macroscopic etch pit. Monte Carlo simulations of etch pit formation have shown that such hollow tubes should be stable for some materials, including guartz (27). Above Ccr t, the height of the activation barrier (Ag ) will determine the rate of formation of etch pits. If metastable equilibrium is assumed for the pit nuclei size distribution, the rate of formation of pits per unit area, J, for concentrations above critical should have the form ... 

The polymerization temperature, through its effects on the kinetics of polymerization, is a particularly effective means of control, allowing the preparation of macroporous polymers with different pore size distributions from a single composition of the polymerization mixture. The effect of the temperature can be readily explained in terms of the nucleation rates, and the shift in pore size distribution induced by changes in the polymerization temperature can be accounted for by the difference in the number of nuclei that result from these changes [61,62]. For example, while the sharp maximum of the pore size distribution profile for monoliths prepared at a temperature of 70 °C is close to 1000 nm, a very broad pore size distribution curve spanning from 10 to 1000 nm with no distinct maximum is typical for monolith prepared from the same mixture at 130°C [63]. 

Many high-pressure reactions consist of a diffusion-controlled growth where also the nucleation rate must be taken into account. Assuming a diffusion-controlled growth of the product phase from randomly distributed nuclei within reactant phase A, various mathematical models have been developed and the dependence of the nucleation rate / on time formulated. Usually a first-order kinetic law I =fNoe fi is assumed for the nucleation from an active site, where N t) = is the number of active sites at time t. Different shapes of the... 

If we take the steady-state distribution function nj t) to be close to for N 1 and to be zero for N oo, the right-hand side of Eq. (1.69) is unity so that the flux in the steady state, called the nucleation rate, is given by... 

Depending upon the parameters that influence the supersaturation ratio, the nucleation rate and the rate of growth, particle size, size distribution, and shape, can be varied over a wide range. 

In the above discussion it is assumed that the volume displaced from the cavity by diffusion is the sole contribution to the creep rate. However, an additional elastic displacement associated with cavities also contributes to the creep rate. This form of cavitation creep is important in fibrous composites, in which crack-like cavities propagate along the fiber interface. Termed elastic creep, this type of creep was first analyzed by Venkateswaran and Hasselman112 and later by Suresh and Brickenbrough.113 As with other forms of cavitation, both the nucleation rate and the growth rate are important. From Venkateswaran and Hasselman, the creep rate, e, for a body containing penny-shaped cavities distributed throughout the volume is... 

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