Nucleation calculation

Figure 5. Kinetics of primary particle nucleation calculated from Equation 16 for methacrylate-like monomers. Parameters for Curve 1 are kp = 350 dm3mol 1s 1 Du = 5 X 10 10 m2s 1 r = 2 X 10 9m c = 6.3 X 10 18 rrfs 1 k tw = 10 17 dm3s 1 = 0.10 mol dm 3 R = 1020 m s 1 jcr = 60. Parameters for Curves 2-6 relative to those for Curve 1 shown at upper right.

Consider that a bath of liquid Fe is equilibrated with a hydrogen atmosphere at an elevated temperature and dissolves 2 X 10 wt% H. The liquid is rapidly cooled to 1,540 °C, at which temperature the Sievert s-law constant is 2.3 x 10 wt% H/atm, and hydrogen bubbles are nucleated. Calculate the minimum bubble size which is stable at a depth of 1 m below the bath surface. 

Table 18.41. Specific frequencies of nucleation calculated starting from the model...

Because of the large surface tension of liquid mercury, extremely large supersaturation ratios are needed for nucleation to occur at a measurable rate. Calculate rc and ric at 400 K assuming that the critical supersaturation is x = 40,000. Take the surface tension of mercury to be 486.5 ergs/cm. ... 

Assuming that for water AGd is 7 kcal/mol, calculate the rate of nucleation for ice nuclei for several temperatures and locate the temperature of maximum rate. Discuss in terms of this result why glassy water might be difficult to obtain. 

As a follow-up to Problem 2, the observed nucleation rate for mercury vapor at 400 K is 1000-fold less than predicted by Eq. IX-9. The effect may be attributed to a lowered surface tension of the critical nuclei involved. Calculate this surface tension. 

Physical properties of the acid and its anhydride are summarized in Table 1. Other references for more data on specific physical properties of succinic acid are as follows solubiUty in water at 278.15—338.15 K (12) water-enhanced solubiUty in organic solvents (13) dissociation constants in water—acetone (10 vol %) at 30—60°C (14), water—methanol mixtures (10—50 vol %) at 25°C (15,16), water—dioxane mixtures (10—50 vol %) at 25°C (15), and water—dioxane—methanol mixtures at 25°C (17) nucleation and crystal growth (18—20) calculation of the enthalpy of formation using semiempitical methods (21) enthalpy of solution (22,23) and enthalpy of dilution (23). For succinic anhydride, the enthalpies of combustion and sublimation have been reported (24). 

Pressure drop due to hydrostatic head can be calculated from hquid holdup B.]. For nonfoaming dilute aqueous solutions, R] can be estimated from f i = 1/[1 + 2.5(V/E)(pi/pJ ]. Liquid holdup, which represents the ratio of liqmd-only velocity to actual hquid velocity, also appears to be the principal determinant of the convective coefficient in the boiling zone (Dengler, Sc.D. thesis, MIT, 1952). In other words, the convective coefficient is that calciilated from Eq. (5-50) by using the liquid-only velocity divided by in the Reynolds number. Nucleate boiling augments conveclive heat transfer, primarily when AT s are high and the convective coefficient is low [Chen, Ind Eng. Chem. Process Des. Dev., 5, 322 (1966)]. 

Equation (18-31) contains no information about the ciystalhzer s influence on the nucleation rate. If the ciystaUizer is of a mixed-suspension, mixed-product-removal (MSMPR) type, satisfying the criteria for Eq. (18-31), and if the model of Clontz and McCabe is vahd, the contribution to the nucleation rate by the circulating pump can be calculated [Bennett, Fiedelman, and Randolph, Chem. E/ig, Prog., 69(7), 86(1973)] ... 

Calculate the population density, growth, and nucleation rates for a crystal sample of urea for which there is the following information. These data are from Bennett and Van Biiren [Chem. Eng. Frvg. Symp. Ser., 65(95), 44 (1969)]. Slurry density = 450 g/L Crystal density = 1.335 g/cm ... 

At the conclusion of the calculation, a fragment size distribution as well as fragment number is provided. A cumulative number distribution is shown in Fig. 8.22 and compared with aluminum ring data acquired at = lO s (Grady and Benson, 1983). With the assumed fracture site nucleation law, the calculated distribution appears to agree reasonably well with the data. The calculation better predicts the tails of the distribution which have trends which deviate from strict exponential behavior as was noted in the previous section. 

On the other hand, whenever AV exceeds the value of AVq the formation of a dense monolayer film appears to be the continuous process. It has been demonstrated that the observed crossover between those two regimes is due to the changes in the mechanism of the adsorbate nucleation, as determined by the calculation of the nucleated cluster size distribution functions. For... 

Application of Eqs. (21)-(27) to the calculations of the nucleation rates J for various alloy models revealed a number of interesting results, in particular, sharp dependence of J and embryo characteristics on the supersaturation, temperature, interaction radius, etc. These results will be described elsewhere. 

Martensitic phase transformations are discussed for the last hundred years without loss of actuality. A concise definition of these structural phase transformations has been given by G.B. Olson stating that martensite is a diffusionless, lattice distortive, shear dominant transformation by nucleation and growth . In this work we present ab initio zero temperature calculations for two model systems, FeaNi and CuZn close in concentration to the martensitic region. Iron-nickel is a typical representative of the ferrous alloys with fee bet transition whereas the copper-zink alloy undergoes a transformation from the open to close packed structure. ... 

Figure 1. Calculated homogeneous nucleation rate for various values of Tyg as a function of temperature.

Calculate the correction to the nucleate boiling film coefficient for the tube bundle number of tubes in vertical row, hi,. See previous discussion. 

Growth theories of surfaces have received considerable attention over the last sixty years as summarized by Laudise et al. [53] and Jackson [54]. The well-known model of the crystal surface incorporating adatoms, ledges and kinks was first introduced by Kossel [55] and Stranski [56]. Becker and Doring [57] calculated the rates of nucleation of new layers of atoms, and Papapetrou [58] investigated dendritic crystallization. 

Although specific calculations for i and g are not made until Sect. 3.5 onwards, the mere postulate of nucleation controlled growth predicts certain qualitative features of behaviour, which we now investigate further. First the effect of the concentration of the polymer in solution is addressed - apparently the theory above fails to predict the observed concentration dependence. Several modifications of the model allow agreement to be reached. There should also be some effect of the crystal size on the observed growth rates because of the factor L in Eq. (3.17). This size dependence is not seen and we discuss the validity of the explanations to account for this defect. Next we look at twin crystals and any implications that their behaviour contain for the applicability of nucleation theories. Finally we briefly discuss the role of fluctuations in the spreading process which, as mentioned above, are neglected by the present treatment. 

The importance of twinned crystals in demonstrating that nucleation is the relevant growth mechanism has been realized since 1949 [64, 99]6. They were first investigated extensively in polymer crystals by Blundell and Keller [82] and they have recently received increased attention as a means of establishing, or otherwise, the nucleation postulate for lamellar growth [90, 91, 95,100-102]. The diversity of opinion in the literature shows that it is very difficult to draw definite conclusions from the experimental evidence, and the calculations are often founded upon implicit assumptions which may or may not be justified. We therefore restrict our discussion to an introduction to the problem, the complicating features which make any a priori assumptions difficult, and the remaining information which may be fairly confidently deduced. 

The model described in Sect. 3.5.1 is a very crude representation of a true three-dimensional lamella, and over the years modifications have been applied in order to make it more realistic. The major assumptions, however, are still inherent in all of them, that is, the deposition of complete stems is controlled by rate constants which obey Eq. (3.83). No other reaction paths are allowed and the growth rate is then given by nucleation and spreading formulae. We do not give the details of the calculations which are very similar, but more complicated, than those already given. Rather, we try to provide an overview of the work which has been done. Most of this has been mentioned already elsewhere in this review. 

From comparison with Eqn (3.44) we can see immediately that there is no nucleation barrier in place of a. Alternatively, we can calculate the free energy difference made by the deposition of v stems (cf. Eqn (3.60)) ... 

---