Mass nucleation rate

Hence at the metastable limit, the mass nucleation rate (or rate of change of concentration) is equal to the super saturation rate due to cooling. Equation (2.20) can be expressed in terms of the cooling rate, b (dT/dt) ... 

Material Balances. The material (mass) balances for the ingredients of an emulsion recipe are of the general form (Accumulation) = (Input) - (Output) + (Production) -(Loss), and their development is quite straightforward. Appendix I contains these equations together with the oligomeric radical concentration balance, which is required in deriving an expression for the net polymer particle generation (nucleation) rate, f(t). 

As with nucleation, classical theories of crystal growth 3 20 2135 40-421 have not led to working relationships, and rates of crystallisation are usually expressed in terms of the supersaturation by empirical relationships. In essence, overall mass deposition rates, which can be measured in laboratory fluidised beds or agitated vessels, are needed for crystalliser design, and growth rates of individual crystal faces under different conditions are required for the specification of operating conditions. 

An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desupersaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and kg in Equation 3 explicitly. Estimates of the first and second derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not estimated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. 

Figure 7. Effect of the mass accumulation on the response of the L50 (A) and on that of the growth rate G and the nucleation rate B (B) on a step in the heat input of the ciystallizer. The cutsize of the fines was 150 /im.

The atmospheric situation is complicated by varying conditions of temperature, relative humidity, and concentrations of other gases such as NH3 which can enhance nucleation rates over those expected for a well-mixed air mass at a fixed temperature and RH (e.g., see Nilsson and Kulmala, 1998). However, there is a general consensus that the observed rates of nucleation of H2S04 often, indeed usually, exceed those expected from classical binary homogeneous nucleation theory. (Note that this is not always the case. For example, Pirjola et al. (1998) reported that the measured formation of nuclei in the Arctic boundary layer... 

The nucleation rate must generate one nucleus for every crystal present in the product. In terms of M , the total mass rate of production of crystals,... 

No study has been made to discover which of the several resistances is important, but a simple rate equation can be written which states that the rate of the over-all process is some function of the extent of departure from equilibrium. The function is likely to be approximately linear in the departure, unless the intrinsic crystal growth rate or the nucleation rate is controlling, because the mass and heat transfer rates are usually linear over small ranges of temperature or pressure. The departure from equilibrium is the driving force and can be measured by either a temperature or a pressure difference. The temperature difference between that of the bulk slurry and the equilibrium vapor temperature is measured experimentally to 0.2° F. and lies in the range of 0.5° to 2° F. under normal operating conditions. 

I J l K k k Lo M m N Electric current (A) Nucleation rate (s 1 m 3) Length of one chain link (m) in a polymer chain Spring constant (Nm-1), equilibrium constant (e.g. mol/L) Bending rigidity (J) Saddle-splay modulus (J) Thickness of a polymer brush (m) Molar mass (kg mol-1), torque in Ch. 11 (Nm) Mass (kg), molecular mass (kg per molecule) Number of molecules (dimensionless or mol), number of segments in a linear polymer chain... 

As shown by Eq. (54), growth rate G can be obtained from the slope of a plot of the log of population density against crystal size nucleation rate B° can be obtained from the same data by using the relationship given by Eq. (57), with n° being the intercept of the population density plot. Nucleation rates obtained by these procedures should be checked by comparison with values obtained from a mass balance (see the later discussion of Eq. (66)). 

Figure 8 Dimensionless relaxation rate l/pf, as a function of the dimensionless mass action variable X according the kinetic Landau model discussed in the main text, with I, the nucleation rate. Shown is the prediction in the limit where the nucleation reaction is rate limiting. Inset experimental results from measurements on actin (Attri et al., 1991). Notice the zero growth atX = l, the critical polymerization point.)...

For a mass production rate of 15kg/hr of crystals, C = 15, the nucleation rate is... 

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