Crystallizers moment equations

Properties of the distribution such as total number of crystals per unit volume, total length of crystals per unit volume, total area of crystals per unit volume, and total volume of soUds (crystals) per unit volume may be expUcitiy evaluated from the moment equations. 

MOMENT EQUATIONS. Equation (27.29) is the fundamental relation of the MSMPR crystallizer. From it diflerential and cumulative equations can be derived for crystal population, crystal length, crystal area, and crystal mass. Also, the kinetic coefficients G and are embedded in these equations. 

Use the moment equations for constant growth rate in a MSMPR crystallizer to calculate (a) the surface-volume mean size and (6) the mass average size, (c) Compare these values with the sizes where a maximum occurs in the corresponding distribution curves. 

Nevertheless, as discussed previously, the physical model for a crystallizer is an integro-partial differential equation. A common method for converting the population balance model to a state-space representation is the method of moments however, since the moment equations close only for a MSMPR crystallizer with growth rate no more than linearly dependent on size, the usefulness of this method is limited. The method of lines has also been used to cast the population balance in state-space form (Tsuruoka and Randolph 1987), and as mentioned in Section 9.4.1, the blackbox model used by de Wolf et al. (1989) has a state-space structure. 

Jones (1974) used the moment transformation of the population balance model to obtain a lumped parameter system representation of a batch crystallizer. This transformation facilitates the application of the continuous maximum principle to determine the cooling profile that maximizes the terminal size of the seed crystals. It was experimentally demonstrated that this strategy results in terminal seed size larger than that obtained using natural cooling or controlled cooling at constant nucleation rate. This method is limited in the sense that the objective function is restricted to some combination of the CSD moments. In addition, the moment equations do not close for cases in which the growth rate is more than linearly dependent on the crystal size or when fines destruction is... 

The moment model approach provides a set of ordinary differential equations (ODEs). Prom the definition of i-th moment in Equation 10.12, we can convert the population balance in Equation 10.10 to moment equations by multiplying both sides by P, and integrating over aU particle sizes. The moments of order four and higher do not affect those of order three and lower, implying that only the first four moments and concentration can adequately represent the crystallization dynamics[100j. Separate moment equations are used for the seed and nuclei classes of crystals, and are defined as follows... 

Moment equations (combining the nucleated and seeded crystals together) ... 

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

In the present case the state variables are most conveniently chosen as crystal size, moments of the distribution and solution concentration (which, as shown above, give rise to ordinary rather than partial differential equations, equations 7.14-7.18) while the control is solution temperature. The performance measure adopted is to maximize the terminal size of the S-crystals (i.e. those originating as added seeds those from nuclei being A -crystals )... 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

When the anisotropy energy is large enough it prevents any precession of the magnetic moment of super-paramagnetic crystals. The magnetic fluctuations then arise from the jumps of the moment between different easy directions. The precession prohibition is introduced into the Freed equations in order to meet that requirement every time the electron Larmor precession frequency appears in the equations, it is set to zero 12). 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

This results In a set of first-order ordinary differential equations for the dynamics of the moments. However, the population balance Is still required In the model to determine the three Integrals and no state space representation can be formed. Only for simple MSMPR (Mixed Suspension Mixed Product Removal) crystallizers with simple crystal growth behaviour, the population balance Is redundant In the model. For MSMPR crystallizers, Q =0 and hp L)=l, thus ... 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

The most familiar estimation procedure is to assume that the population mean and variance are equal to the sample mean and variance. More generally, the method of moments (MOM) approach is to equate sample moments (mean, variance, skewness, and kurtosis) to the corresponding population. Software such as Crystal Ball (Oracle Corporation, Redwood Shores, CA) uses MOM to fit the gamma and beta distributions (see also Johnson et al. 1994). Use of higher moments is exemplified by fitting of the... 

Equation (1) expresses the crystal polarization (P, C/iiF). as a function of the dipole moment (p, Cm) and the unit cell volume (V, iif). In PVDF, it suffices to express Eq. (1) in scalar form, where it is miderstood that P and p represent the components of the polarization and dipole moment vectors parallel to the ( -crystal axis. This arrangement of dipoles produces a significant local electric field in the... 

Equation (2) expresses the model of the crystal polarization used in the molecular modeling of PVDF reported in this chapter, where is the dipole of each repeat unit of the single chain in vacuum, Ap is the change in dipole moment of the repeat unit of the chain in going from the vacuum environment to the environment of the packed crystal and (cos tp) is the attenuation of the dipole moment of the repeat unit along the fe-axis due to thermally stimulated oscillations about thec-axis. Ap is directly related to the local electric field (Eioc, V/m) through the repeat unit polarizability (ot, m ) ... 

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