Moment equation, population

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

In applying the resulting state space model for control system design, the order of the state space model is important. This order is directly affected by the number of ordinary differential equations (moment equations) required to describe the population balance. From the structure of the moment equations, it follows that the dynamics of m.(t) is described by the moment equations for m (t) to m. t). Because the concentration balance contains c(t)=l-k m Vt), at I east the first four moments equations are required to close off the overall model. The final number of equations is determined by the moment m (t) in the equation for the nucleation rate (usually m (t)) and the highest moment to be controlled. 

Finally, all the bubble generation and destruction functionals in Equations (2) and (3) must be known in advance to calculate their moments. It is vastly more difficult to find these functionals than to construct approximate generation and destruction functions in Equations (5) and (6). If we start from the zeroth moment equations, however, we forfeit the ability to calculate the higher order moments of the generation and destruction functionals that in turn are necessary to solve (5) and (6). To break this vicious circle without solving the full-blown population balances (2) and (3), we need to make guesses about shape of the bubble size distribution, and then iteratively solve Equations (5) and (6) until some specified criteria are met. 

Equations involving ouly a finite number of may be derived from Eq. (S) under appropriate closure aj roximations. As only one variable (t) is then involved, the moment equations may be readily solved and the populations generated. This method will often fail, however, when Eq. (5) is nonlinear (e.g., if reentry is important) or if large numbers of moments are equired (e.g., for a polymodal PSD). 

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. 

The mathematical modeling of polymerization reactions can be classified into three levels microscale, mesoscale, and macroscale. In microscale modeling, polymerization kinetics and mechanisms are modeled on a molecular scale. The microscale model is represented by component population balances or rate equations and molecular weight moment equations. In mesoscale modeling, interfacial mass and heat transfer... 

When the population balance is written in terms of one internal coordinate (e.g., particle diameter or particle volume), the closure problem mentioned above for the moment equation has been successfully relaxed for solid particle systems by the use of a quadrature approximation. 

Alopaeus, V., Laakkonen, M. Aittamaa, J. 2006 Numerical solution of moment-transformed population balance equation with fixed quadrature points. Chemical Engineering Science 61, 4919-4929. 

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. 

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. 

These equations are based on the assumption that at any given moment the population of micro-organisms (bacteria) in a culture will multiply as long as either there is nutrient available or, the concentration of the inhibitory product is not limiting. The rate of multiplication within the biofilm will vary according to these criteria. Belkhadir et al [1988] described the fundamental growth phases in a biofilm. The growth of active biomass is assumed to have an order of zero in relation to the nutrient and an order of unity in relation to the active bacteria so that ... 

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... 

The initial conditions of the moment equations are derived directly from the initial population density n(L,0). Equations (10.12) through (10.16), together with the nucleation and growth kinetic expressions [Eqs. (10.5) and (10.6)], can be solved numerically to give the moments of the CSD for a batch or semibatch system as a function of time. The CSD can be reconstructed from its moments by the methods described by Hulbert and Katz (1964) and Randolph and Larson (1988). 

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... 

It seems clear that, if we were to proceed in this fashion, developing population balances and moment equations for copolymerization would be a procedure even more tedious (as hard as it may be to imagine ) than the one used for homopolymerization. But, do not despair. A few concepts will be introduced that will allow us to translate the homopolymerization equations to copolymerization equations with minimum effort. 

Alternatively, a reconstruction can be performed using the correct moment equations, that is, including explicit population-weighted rate coefficients (cf. Eqs. (10.8)-(10.11) corrected with Eqs. (10.12), (10.13) Bentein et al., 2011). Each time, step convergence is required... 

Not infrequently, practical needs can be fulfilled by calculating the (generally integral) moments of the number density function. The calculation of such moments can occasionally be accomplished by directly taking moments of the population balance equation producing a set of moment equations. 

One encounters similar constraints with aggregation processes to generate closed integral moment equations, i.e., a constant aggregation rate and at most a linear growth rate. In order to demonstrate the moment equations for this case, we recall the population balance equation (3.3.5) for the constant aggregation rate, a x, x ) = a, incorporate a linear growth term X(x, t) = kx, and take moments. The result is... 

It is of interest to examine the differential equation satisfied by Such a differential equation may be obtained in either of two ways. First, it may be recovered directly from the population balance equation by multiplying Eq. (4.5.1) by i " and integrating over the subinterval to get the sectional moment equation in... 

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. 

In the following model development, we will use the polymerization kinetics mechanism described by Eqs. (1)-(14) to derive the population balances and moment equations for homopolymerization with a catalyst containing only one site type. Catalysts containing two or more site types are handled similarly by defining a set of equations with distinct polymerization kinetic constants for each different site type. 

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. 

Next, we vill derive the higher moment equations. For N = 1 and M = Owe obtain the set of population balance equations for the pseudo-distributions of order (1,0) as listed in Table 9.13. 

Example Population Balance to Moment Equations The method of moments can be used to convert the differential difference equation (or infinite set of ordinary differential equations) represented by Equations 16.7 and 16.8 into a finite set of ordinary differential equations in the moments of the distribution. Then, using Equations, the important statistics of the NCLD can be derived. 

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