Nucleation-growth equation

First, we often forget to pay adequate attention to the more detailed temperature-dependence of the integration of basic nucleation-growth equations under non-isothermal conditions. It has been already proved [529] that the... 

J. Sestak, key lecture Integration of nucleation-growth equation when considering non-isothermal regime and shared phase separation at the. 2" ESTAC (Europ. Symp. on Thermal Analysis) in Proc. Thermal Analysis (D. Dolimore, ed) Heyden. London 1981, p. 115... 

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

We now turn to the question of developing a CFD model for fine-particle production that includes nucleation, growth, aggregation, and breakage. Applying QMOM to Eq. (114) leads to a closed set of moment equations as follows ... 

Solid PET feedstock for the SSP process is semicrystalline, and the crystalline fraction increases during the course of the SSP reaction. The crystallinity of the polymer influences the reaction rates, as well as the diffusivity of the low-molecular-weight compounds. The crystallization rate is often described by the Avrami equation for auto-accelerating reactions (1 — Xc) = cxp(—kc/"), with xc being the mass fraction crystallinity, kc the crystallization rate constant and n a function of nucleation growth and type. 

Nucleation rates are sensitive to the presence of foreign solid particles, because these objects may act as catalysts. If a nucleus is created on a solid particle, it will remain attached during part of the subsequent growth process. The growth equations for bubbles attached to solids have not been worked out mathematically, but it is rather obvious that interfacial tensions will be important as long as the bubbles are small. 

The different combinations of nucleation, growth, and impingement processes give rise to the Johnson-Mehl-Avrami kinetic model [4], which results in the following equation... 

The method is based on classical nucleation and growth equations for amorphous materials and a derived expression for AG based on the expressions of Turnbull,Hoffman, and Thompson and Spaeten. Using the calculated AG value, published material property data such as modulus and surface energy, and the measured crystallization or glass transition temperature (T ) obtained from differential scanning calorimetry (DSC), analytical expressions for nucleation rate and growth rate can be written. These expressions are then used as the basis for a pixel-by-pixel modeling approach for visualization of the microstructural evolution of the cross-section of a thin... 

Idealized representations of the variations with time of numbers of growth nuclei (N) present at time, t, for four of the nucleation rate equations given in Table 3.1. 

The population balance concept was first presented by Hulburt and Katz [37]. Rather than adopting the standard continuum mechanical framework, the model derivation was based on the alternative Boltzmann-t q)e equation familiar from classical statistical mechanics. The main problems investigated stem from solid particle nucleation, growth, and agglomeration. 

The general evolution equation for the homogeneous system undergoing nucleation, growth, diffusion, aggregation, and breakage is... 

Kumar, S. Ramkrishna, D. 1997 On the solution of population balance equations by discretization - III. Nucleation, growth and aggregation of particles. Chemical Engineering Science 52, 4659 679. 

The instantaneous nucleation-growth-precipitation model [39] assumes that the film is formed directly on the substrate, without previous dissolution however, it was observed that active dissolution of the metal occurs. Therefore, Equations 8.11 through 8.13 were examined and rewritten considering metal dissolution, that is, terms corresponding to dissolution were added to the mathematical expressions ... 

One method is to solve the population balance equation (Equation 64.6) and to take into account the empirical expression for the nucleation rate (Equation 64.10), which is modified in such a way that the expression includes the impeller tip speed raised to an experimental power. In addition, the experimental value, pertinent to each ch ical, is required for the power of the crystal growth rate in the nncleation rate. Besides, the effect of snspension density on the nucleation rate needs to be known. Fnrthermore, an indnstrial suspension crystallizer does not operate in the fully mixed state, so a simplified model, such as Equation 64.6, reqnires still another experimental coefficient that modifies the CSD and depends on the mixing conditions and the eqnipment type. If the necessary experimental data are available, the method enables the prediction of CSD and the prodnction rate as dependent on the dimensions of the tank and on the operating conditions. One such method is that developed by Toyokura [23] and discussed and modified by Palosaari et al. [24]. However, this method deals with the CTystaUization tank in average and does not distinguish what happens at various locations in the tank. The more fundamental and potentially far more accurate simulation of the process can be obtained by the application of the computational fluid dynamics (CFD). It will be discussed in the following section. 

A continuous cooling crystallizer is required to produce potassium sulphate crystals (density pc = 2660kgm, volume shape factor a = 0.7) of 750pm median size Lm at the rate Pc = 1000 kg h. On the basis of pilot-plant trials, it is expected that the crystallizer will operate with steady-state nucleation/ growth kinetics expressed (equation 9.39 with j = 1 and i = 2) as B = 4 X IO MtG m s. Assuming MSMPR conditions and a magma density Mt = 250 kg m, estimate the crystallizer volume and other relevant operating conditions. 

Discuss the kinetic and thermodynamic factors governing liquid-solid and solid-solid phase transformations. Explain and predict nucleation, growth, and time-temperature-transformation (1 IT) processes in solid-state systems both qualitatively (through diagrams) and quantitatively (through equations). 

Kumar, S. and D. Ramkrishna, On the Solution of Population Balance Equations by Discretization-III. Nucleation, Growth and Aggregation of Particles, Chem. Eng. ScL, 24, 4659-4679 (1997). 

The kinetics of nucleation/growth and separation processes in an ensemble of nanoparticles under time-dependent temperature is presented. Grounding upon our numerical studies, we conclude that under temperature cycling one should observe a hysteresis behavior. Such hysteresis is conditioned by the finite size and depletion effects and is described on the basis of the kinetic equation approach. The model shows that the width of hysteresis loop depends on (i) thermodynamic... 

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