Crystallization crystal number density function

Consider a continuous crystallizer of volume V, as shown in Figure 6.4.2(a). A feed stream having a particle (crystal) number density function ra/(rp) (which is also the population density function), volumetric flow rate Qf and species i mass concentration enters the crystallizer continuously. Product stream 1, having a particle (crystal) number density function n (tp), volumetric flow rate Qi and species i mass concentration Pf, leaves the crystallizer continuously. The particle (crystal) number density function n rp) in the well-mixed crystallizer is the same throughout the crystallizer. The macroscopic population balance equation for a stirred tank separator may be written using equations (6.2.60) and (6.2.61) as follows ... 

Define rP as the crystal number density function value for crystals of near-zero size, the embryos (see equation (3.3.99)). Integration of this equation for particular values of G and yields... 

Example 6.4.1 For the MSMPR crystal number density function (6.4.4), determine the expressions for Nt,F rp),fp, Ar and Mp. Develop expressions for the cumulative crystal size distribution fraction and the cumulative surface area distribution fraction. 

Equation (6.4.4) provides an expression for the crystal number density function (zp) in an MSMPR crystallizer. We now need information about the distribution functions of other properties of MSMPR crystallizers, e.g. the total number of crystals per unit system volume, iV, (equation... 

When b = 0 and y = 1, this relation is simplified to the standard equation (6.4.4) for G(tp) = constant. The dashed lines in Figure 6.4.2(b) illustrate the nature of these crystal number density functions. Figures 6.4.5(a) and (b) illustrate schematically size-independent and size-dependent crystal growth rates. 

It can be demonstrated that the total number of crystals, the total length, the total area, and the total volume of crystals, all in a unit of system volume, can be evaluated from the zero, first, second, and third moments of the population density function. 

Here r is the distance between the centers of two atoms in dimensionless units r = R/a, where R is the actual distance and a defines the effective range of the potential. Uq sets the energy scale of the pair-interaction. A number of crystal growth processes have been investigated by this type of potential, for example [28-31]. An alternative way of calculating solid-liquid interface structures on an atomic level is via classical density-functional methods [32,33]. 

The DECP model successfully explained the observed initial phase of the fully symmetric phonons in a number of opaque crystals [24]. The absence of the Eg mode was attributed to an exclusive coupling between the electrons photoexcited near the r point and the fully symmetric phonons. A recent density functional theory (DFT) calculation [23] demonstrated this exclusive coupling as the potential energy surface (Fig. 2.4). The minimum of the potential surface of the excited state shifted significantly along the trigonal (z) axis,... 

Both the number and weight basis probability density functions of final product crystals were found to be expressed by a %2-function, under the assumption that the CSD obtained by continuous crystallizer is controlled predominantly by RTD of crystals in crystallizer, and that the CSD thus expressed exhibits the linear relationships on Rosin-Rammler chart in the range of about 10-90 % of the cumulative residue distribution. 

Considering the tracer entering the vessel at a given instant of time to be the nuclei formed at that time, C, (Do and C(0) can be converted to the number density of nuclei in the whole vessel, n, in the 1st tank, no and that of crystals in the exit stream from the vessel, n(0), respectively. The crystals having the residence time of 0 grow up to the size L, which is given by Equation 1. Therefore, by using Equations a-1 or a-2 and 1, the number basis probability density function of final product crystals, fn(L) is obtained, as follows. 

Marsh (1988), Cashman and Marsh (1988), and Cashman and Ferry (1988) investigated the application of crystal size distribution (CSD) theory (Randolph and Larson, 1971) to extract crystal growth rate and nucleation density. The following summary is based on the work of Marsh (1988). In the CSD method, the crystal population density, n(L), is defined as the number of crystals of a given size L per unit volume of rock. The cumulative distribution function N(L) is defined as... 

Fourier representation of electron density suggests the possibility of direct structure analysis. If all structure factors, F(hkl), are known, p(xyz) can be computed at a large number of points in the unit cell and local maxima in the electron-density function are interpreted to occur at the atomic sites. A typical single-crystal diffraction pattern of the type used for measuring structure factor amplitudes is shown in Figure 6.12. 

Before any computational study on molecular properties can be carried out, a molecular model needs to be established. It can be based on an appropriate crystal structure or derived using any technique that can produce a valid model for a given compound, whether or not it has been prepared. Molecular mechanics is one such technique and, primarily for reasons of computational simplicity and efficiency, it is one of the most widely used technique. Quantum-mechanical modeling is far more computationally intensive and until recently has been used only rarely for metal complexes. However, the development of effective-core potentials (ECP) and density-functional-theory methods (DFT) has made the use of quantum mechanics a practical alternative. This is particularly so when the electronic structures of a small number of compounds or isomers are required or when transition states or excited states, which are not usually available in molecular mechanics, are to be investigated. However, molecular mechanics is still orders of magnitude faster than ab-initio quantum mechanics and therefore, when large numbers of... 

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