Nakamura equation

Based on Equation (7.33) and assuming that nP G[t(T) i l3 = k(T), one can derive Equation (7.31) with n = 4. The Nakamura equation (Eq. 7.31) has been widely used in the modeling of crystaUization in the injection molding process, for example. References [33-36], because it allows application of the results of isothermal crystallization analysis to the description of nonisothermal crystallization. It can be noticed that in isothermal conditions, for a given temperature T, k(T) = th(Ty In 2, where h is the crystallization halftime. Therefore, for nonisothermal conditions. Equation (7.31) assumes the form ... 

FIC models for FS often apply the Nakamura equation [114] to describe the crystallization dynamics (see for example References [115-117]). However, the use of the Nakamura equation is inappropriate because, first, the isokinetic assumption is not satisfied during FIC, and, second, a change in the number density of crystals cannot be described [3]. 

With the advent of sophisticated simulation techniques, the physics of the flow-enhanced nucleation process at the molecular level are gradually being unraveled (see Chapter 6). The results of such investigations can serve to validate and/or improve continuum-level FIC models. Some of the most advanced of these are compared here in terms of the formulation of flow-enhanced nucleation kinetics. A description of flow-induced oriented structure formation and application to IM are discussed in Section 14.4.2 and Section 14.4.3, respectively. We focus on models that calculate the number density and dimensions of nuclei since this is necessary to predict morphological features beyond merely the degree of crystallization or the volume fraction of semicrystalline material. Therefore, approaches based on a (modified) Nakamura equation are left out of consideration. 

Several groups developed numerical models for the simulation of the IM process. Early works modeled the effect of temperature on the crystallization rate only and included the crystallization heat in the energy equation (see for example Hieber [171]). The effect of shear was taken into account by, for example a modified Nakamura equation where the kinetic parameters were made (shear) stress dependent [22,76,77,116,172]. 

Pantani et al. [11] gave an extensive review on available models to predict and characterize the morphology of injection-molded parts. The authors themselves proposed a model to predict the morphology of injection-molded iPP, in which flow kinematics are computed using a lubrication approximation. Polymorphism was accounted for, using the Avrami-Evans-Nakamura equation to describe the crystallization kinetics of the mesomorphic phase, while the evolution of the a phase was modeled using Kolmogorov s model [122]. 

However, Hidrer [7] asserted that there is no need to explicitly incorporate an induction time when modeling is based on the Nakamura equation. In this study, the view will be examined. 

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

The activity of the water is derived from this expression by use of the Gibbs-Duhem equation. To utilize this equation, the interaction parameters fif ) and BH must be estimated for moleculemolecule, molecule-ion and ion-ion interactions. Again the method of Bromley was used for this purpose. Fugacity coefficienls for the vapor phase were determined by the method of Nakamura et al. (JO). 

Recently, Nakamura and coworkers described a related reaction of the zinc enolates derived from /3-aminocrotonamides of type 395256. In the presence of a stoichiometric amount of Et2Zn, the latter underwent smooth addition to terminal alkynes upon heating in hexane and afforded the corresponding tetrasubstituted 2-alkylidene acetoacetamides 396 (after acidic hydrolysis of the imine) with high (Z)-stereoselectivity (equation 173). 

Besides the activation of the olefinic partner by a metal, the unfavorable thermodynamics associated with the addition of an enolate to a carbon—carbon multiple bond could be overwhelmed by using a strained alkene such as a cyclopropene derivative286. Indeed, Nakamura and workers demonstrated that the butylzinc enolate derived from A-methyl-5-valerolactam (447) smoothly reacted with the cyclopropenone ketal 78 and subsequent deuterolysis led to the -substituted cyclopropanone ketal 448, indicating that the carbometallation involved a syn addition process. Moreover, a high level of diastereoselectivity at the newly formed carbon—carbon bond was observed (de = 97%) (equation 191). The butylzinc enolates derived from other amides, lactams, esters and hydrazones also add successfully to the strained cyclopropenone ketal 78. Moreover, the cyclopropylzincs generated are stable and no rearrangements to the more stable zinc enolates occur after the addition. 

This equation was advanced originally by Horiuti and Nakamura (26) in a... 

Although cobalt catalysts have been rarely used in cyclopropanation reactions, Nakamura and coworkers2 1 have developed the camphor-based complex (35) as a useful asymmetric catalyst, as shown in a typical example in equation (16). High yields were obtained with dienes and styrenes but cyclopropanation did not occur with simple alkenes. Studies with cu-ife-styrene showed that, unlike other catalytic systems, the reaction was not stereospecific with respect to alkene geometry. 

A detailed two-dimensional numerical analysis of nonisothermal spinning of viscoelastic liquid with phase transition was carried out recently by Joo et al. (15). They used a mixed FEM developed for viscoelastic flows (16) with a nonisothermal version of the Giesekus constitutive equation (17), the Nakamura et al. (18) crystallization kinetics... 

Expressions (29)-(32) give correct results for the case of weak coupling (/ <, f dt 1) and for exact resonance (co = 0, 3co = 0). Neither is correct at the adiabatic limit [f- /dt I, - /exp (icot)dt 1] when P must be found from equation (15). For intermediate cases these expressions have been investigated mainly with respect to certain interactions/(t). Skinner [35] discussed equations (29) and (30) for an exponential interaction and compared them with computed values of P. It was found that equation (29) is more reliable than equation (30), but a collision must not be too adiabatic (tor < ). A comparison of equations (29), (30), and (31) for the dipole-dipole interaction was made by Nakamura [36], who showed that equation (31) gave the lowest transition probability under adiabatic conditions (cot 1). Equation (30) has been discussed also by Mori and Fujita [37]. 

Judging from the calculation of the coupled equation (without the rotating atom approximation) in the resonant case and the discrete-continuum case, the error from the rotating atom approximation would not change the order of the magnitude (67, 68, 70). With these approximations, the process can be described in terms of the resonance defect [o> = (Et — Ef)/h] and the transition matrix dipole moments of A and B fiA and /xB. By Nakamura s calculation (50), the transfer probability is written as... 

C. Zhu and H. Nakamura, Stokes constants for a certain class of second-order ordinary differential equations, J. Math Phys. 33 2697 (1992). 

Under the aforementioned hypotheses, as two different crystalline phases are formed (a and mesomorphic), at least two kinetic processes take place simultaneously. The simplest model is a parallel of two kinetic processes non-interacting and competing for the available molten material. The kinetic equation adopted here for both processes is the non-isothermal formulation by Nakamura et al. (Nakamura et al., 1973, Nakamura et al., 1972) of the Kolmogoroff Avrami and Evans model (Avrami, 1939,1940,1941, Evans, 1945). 

Transformation kinetics according to Nakamura and Ziabicki Nakamura (3) extended Avrami theory to non-isothermal transformations and proposed the following equation ... 

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