Moment equation, population balance method

Marchisio, D. L., Pikturna, J. T., Fox, R. O., Vigil, R. D. Barresi, A. A. 2003b Quadrature method of moments for population-balance equations. AIChE Journal 49, 1266-1276. 

Another objective in the study of the application of CFD in crystallization is to simulate the particle size distribution in crystallization. In order to solve this problem, the simulation should take into account the population balance. The internal coordinates of the population balance make it difficult to utilize it in the CFD environment. In addition, different-sized particles have different hydrodynamics, which causes further complications. Wei and Garside [42] used the assumption of MSMPR and the moments of population balance to avoid the above difficulties in the simulation of precipitation. In the CFX commercial application, the MUSIC model offers a method for solving the population balance equation in CFD and defines the flow velocity of different-sized particles... 

Yuan C, Laurent F, Fox RO (2012) An extended quadrature method of moments for population balance equations. J Aerosol Sci 51 1-23... 

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. 

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). 

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. 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

Hulbert and Akiyama (HI8) analyzed the population balance equation to determine coalescence and redispersion effects on extent of conversion, deriving a set of equations for the moments of the distribution. The results agreed with those of Shain (S18). The authors claim the moment method reduces computation effort although there is a loss of information on the distribution. 

The deterministic population balance equations governing the description of mass transfer with reaction in liquid-liquid dispersions present a framework for analysis. However, signiflcant difficulties exist in obtaining solutions for realistic problems. No analytical solutions are available for even the simplest cases of interest. Extension of the solution to multiple reactants for uniform drops is possible using a method of moments but the solution is limited to rate equations which are polynomials (E3). Solutions to the population balance equations for spatially nonhomogeneous dispersions were only treated for nonreacting dispersions (P4), and only a simple case was solved for a spray column (B19). Treatment of unmixed feeds presents a problem. 

This part of the chapter is devoted to a few of the popular numerical discretization schemes used to solve the population balance equation for the (fluid) particle size distribution. In this section we discuss the method of moments, the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM), the discrete method, the chzss method, the multi-group method, and the least squares method. 

Attarakih, M. M., Drumm, C. Bart, H.-J. 2009 Solution of the population balance equation using the sectional quadrature method of moments (SQMOM). Chemical Engineering Science 64, 742-752. 

Gimbun, j., Nagy, Z. K. Rielly, C. D. 2009 Simultaneous quadrature method of moments for the solution of population balance equations, using a differential algebraic equation framework. Industrial U Engineering Chemistry Research 48, 7798-7812. 

Strumendo, M. Arastoopour, H. 2009 Solution of bivariate population balance equations using the finite size domain complete set of trial functions method of moments (FCMOM). Industrial r Engineering Chemistry Research 48, 262-273. 

Su, J., Gu, Z., Li, Y., Feng, S. Xu, X. Y. 2007 Solution of population balance equation using quadrature method of moments with an adjustable factor. Chemical Engineering Science 62, 5897-5911. 

Vikas, V, Wang, Z. J. Fox, R. O. 2012 Realizable high-order finite-volume schemes for quadrature-based moment methods applied to diffusion population balance equations. Journal of Computational Physics (submitted). 

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 structure and interrelationship of the batch conservation equations (population, mass, and energy balances) and the nucleation and growth kinetic equations are illustrated in an information flow diagram shown in Figure 10.8. To determine the CSD in a batch crystallizer, all of the above equations must be solved simultaneously. The batch conservation equations are difficult to solve even numerically. The population balance, Eq. (10.3), is a nonlinear first-order partial differential equation, and the nucleation and growth kinetic expressions are included in Eq. (10.3) as well as in the boundary conditions. One solution method involves the introduction of moments of the CSD as defined by... 

Analogous equations can be used with any other instantaneous distribution. This relatively easy integration extends the use of instantaneous distributions to transient reactor operation and considerably broadens the use of this powerful technique. When compared with the method of moments, instantaneous distribution allows for the complete prediction of CLD and CCD instead of only averages when contrasted to the full solution of the population balances, the method of instantaneous distributions provides the same information at a much shorter time using a more elegant solution, allows the modeler to analyze the problem with a simple glance at the equation, and can even be implemented on simple commercial spreadsheets for easy calculation. 

Table 2.6 summarizes the equations developed for the method of moments. Notice that only nine ODEs need to be solved, instead of the very large system required for the complete solution of the population balance equations. The price paid for this simplification is that the complete CLD can no longer be modeled, only its averages DP and DPw. 

With these new definitions, the equations shown in Tables 2.5,2.6 and 2.7 can be applied to obtain the equations for the complete population balance, the method of moments, or the method of moments with the QSSA, respectively. Note that because of chain transfer to monomer. Equation 2.12 should, strictly speaking, include a new monomer consumption term, that is ... 

Thus (4.4.8) shows how any term in the population balance equation involving the number density can be expressed purely in terms of the first M integral moments. While the use of generalized Laguerre polynomials with additional parameters provided some flexibility for the method of moments, Hulburt and Akiyama have reported difficulties with this method. 

This large set of ordinary differential equations can be numerically solved in different ways. If overall concentrations, conversion and average molecular weights are needed, the system size can be drastically reduced by applying the popular method of moments, where the infinite population balances for active and dormant species are replaced by a few moment equations. Focusing on the most common average degrees of polymerization, the moments of the first three orders only need to be calculated, ie. 

Modeling is probably the tool of excellence for engineers (Chapter 9). It is used to simulate the reaction and the process system in order to shorten the time for development. It is based on models that can be physical or chemical, semi-empirical or empirical, descriptive or more fundamental. To describe the development of the molecular weight distribution upon reaction, moment methods or equations based on population balance are often used. 

The method of moments is the most well-known method for solving polymerization problems [1-3]. The equations are derived from the population balances. This is realized in a straightforward way for the radical polymerization system of Table 9.1, a ID problem. Table 9.2 presents the original population balances and Table 9.3 the resulting moment equations, up to the 4th moment. The linear part of this problem can be solved without additional assumptions, but the nonlinear part leads to a closure problem. This will be discussed next. Some results and a discussion on the validity of the method will be given in Section 9.4, in a comparison with the Galerkin-FEM method. 

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