Nucleation correction factor

The equations for estimating nucleate boiling coefficients given in Section 12.11.1 can be used for close boiling mixtures, say less than 5°C, but will overestimate the coefficient if used for mixtures with a wide boiling range. Palen and Small (1964) give an empirical correction factor for mixtures which can be used to estimate the heat-transfer coefficient in the absence of experimental data ... 

Equations 15 and 16 develop Equation (14) for gas-to-condensed-phase transitions. It would also be desirable to develop Equation (14) for liquid-to-crystalline nucleation to assign physical meaning to An. However, development of Equation (14) is difficult both because d can be a large correction factor for liquid/solid interfaces and because the molecular density differences between liquids and solids are smaller than between gases and condensed phases (Kashchiev 1982). 

Table 2. The Correction Factors TlTe, S) to the Nucleation Rates Predicted by Classical Nuclea-tion Theory as Calculated Purely Microscopically by Burton ...

Some of Burton s results (9.36) for the correction factor, T/Tc, S) are shown in Table 2. The most striking prediction of Table 2 is that for any material, it is possible to obtain nucleation rates significantly smaller than predicted by classical theory (T/Tc = 0.35, 5 = 12 = 10 in Table 2) or much larger than predicted by classical theory T/Tc=0.47, S = 72, < = 10 in Table 2). Equivalently, this means that the critical supersaturation can be either larger or smaller than predicted by classical theory. This effect seems to have been observed in a number of materials, including n-butylbenzene, as is seen in Fig. 9. 

From data for the correction factor it is possible to predict critical supersaturations as well as nucleation rates for any material. (This is discussed extensively in the original pape/ on this work as well as in a recent review. 

The first term represents the diffusion of chains to the growth front while the second is related to the secondary nucleation barrier. Go represents a preexponential factor, U is the activation energy for chain mobility, R is the gas constant, Tc is the isothermal crystallization temperature and Ar=T — TV is the supercooling (T is the equilibrium melting temperature). Too is the temperature where viscous flow ceases (AT —30A) and/is a temperature correction factor defined as 2TV / (T -)- TV), while Kg is the nucleation constant (which is proportional to the energy barrier for secondary nucleation) given by ... 

The correction factor Pend in Eq. (65) is dimensionless and accounts for noncontinuum effects. The mobility of solute molecules in the metastable gas phase is expressed by the gas-phase diffusion coefficient 2 g, and the driving force for condensation is the difference in mole fraction between the metastable phase (yj) and a gas phase that is in equilibrium with the precipitated phase (y ). Nucleation and condensation are mechanisms of precipitation that compete with each other. Condensation depends on the availability of the outer surface area of particles that have been generated by nucleation. Both processes lower the actual solute mole fraction yj and therefore diminish the driving force for either process. 

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. 

Here, Cp is the concentration of the dissolved solute in the bulk of the liquid, Cp is the concentration of the solute at the liquid-crystal interface, and Cp is the solubility. Note that the nucleation rate (Jn) and the linear growth rate (G) have been transformed into molar units by using appropriate multiplying factors. It should be emphasized that, while these equations capture the phenomena under consideration, to be correct, they should be expressed in terms of activities in stead of concentrations. 

The Zeldovich factor represents a thermodynamic correction parameter and takes into account the fact that a cluster having reached the critical size does not necessarily nucleate, but could fluctuate in size back into the sub-critical region (Schmelzer 2003). The Zeldovich factor for nncleation rate on a polymeric membrane is a function of porosity and contact angle. According to Equation 10.1 ... 

Thus, if the nucleation explanation is correct, it would only be too cold to snow when T decreases well below T /3. This would imply temperatures below 91 K, or -182 °C This is far colder than any recorded temperature on Earth, indicating that the nucleation factor is not an adequate explanation for snowfall suppression at frigid temperatures. The better explanation considers the water vapor capacity of the atmosphere, which decreases exponentially with decreasing temperature. Cold air holds exponentially less water vapor than warm air, and so the precipitation potential decreases dramatically as temperatures become more frigid. This is the main reason why warm snowstorms, where the air temperature is just slightly below the freezing point, typically produce the greatest snowfall rates. 

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