The Method of Moments and Weighted Residuals

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. 

In order for the moment equation to generate only terms containing moments, the breakage rate b x) must be a polynomial function of x. However, even this requirement leads to a set of unclosed moment equations because the differential equation in any moment involves higher moments. Thus the only way to get an exactly closed set of moment equations is to require that b x) = b, a constant which yields from (4.4.2) the equation 

If we allow particle growth, it is interesting to note that closed moment equations could be obtained for linear growth rate. Recalling (3.2.7) for breakage processes with X x, t) = fcx, where fc is a constant, and taking moments one obtains 

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 

Since for n = 1, the differential equation contains only the first term on the right-hand side of (4.4.4), the first moment has the solution / (t) = // (O) For higher moments, the moment equations (4.4.4) can be solved analytically by rewriting as 

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The FEM is part of a larger group of techniques that exploit the method of weighted residuals (MWR) [76]. These use a set of weighting functions to allow the approximation of a variable over a domain. The choice of weighting function leads to a number of different alternative formulations, including the collocation method, subdomain method, method of moments, and the Galerkin method. 

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 current section of the chapter on numerical methods is devoted to an outline of the most frequently used numerical methods for solving the population balance equation either for the particle number distribution function or for a few moments of the number density function. The methods considered are the standard method of moments, the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM), the sectional quadrature method of moments (SQMOM), the discrete fixed pivot method, the finite volume method, and the family of spectral weighted residual methods with emphasis on the least squares method. 

Fitting model predictions to experimental observations can be performed in the Laplace, Fourier or time domains with optimal parameter choices often being made using weighted residuals techniques. James et al. [71] review and compare least squares, stochastic and hill-climbing methods for evaluating parameters and Froment and Bischoff [16] summarise some of the more common methods and warn that ordinary moments matching-techniques appear to be less reliable than alternative procedures. References 72 and 73 are studies of the errors associated with a selection of parameter extraction routines. 

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