Conditional QMOM

The conditional quadrature method of moments (CQMOM) is based on the concept of a conditional density function (Yuan Fox, 2011). Conditional density functions represent, in turn, the probability of having one internal coordinate within an infinitesimal limit when one or more of the other internal coordinates are fixed and equal to specific values. For example, in the case of a generic NDF the expression 

Analogously we can define fhe following conditional density function  

Then it is immediately evident that the following equality holds  

These concepts can be used to construct a multivariate quadrature. 

The construction of the multivariate quadrature begins with the calculation of the univariate quadrature of order N for the first internal coordinate by using the Wheeler algorithm with the first 2N - 1 moments  

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Chapter 3 provides an introduction to Gaussian quadrature and the moment-inversion algorithms used in quadrature-based moment methods (QBMM). In this chapter, the product-difference (PD) and Wheeler algorithms employed for the classical univariate quadrature method of moments (QMOM) are discussed, together with the brute-force, tensor-product, and conditional QMOM developed for multivariate problems. The chapter concludes with a discussion of the extended quadrature method of moments (EQMOM) and the direct quadrature method of moments (DQMOM). 

Thus, it would be natural to attempt to extend the QMOM approach to handle a bivariate NDF. Unfortunately, the PD algorithm needed to solve the weights and abscissas given the moments cannot be extended to more than one variable. Other methods for inverting Eq. (125) such as nonlinear equation solvers can be used (Wright et al., 2001 Rosner and Pykkonen, 2002), but in practice are computationally expensive and can suffer from problems due to ill-conditioning. 

Dorao and Jakobsen [40, 41] did show that the QMOM is ill conditioned (see, e.g.. Press et al [149]) and not reliable when the complexity of the problem increases. In particular, it was shown that the high order moments are not well represented by QMOM, and that the higher the order of the moment, the higher the error becomes in the predictions. Besides, the nature of the kernel functions determine the number of moments that must be used by QMOM to reach a certain accuracy. The higher the polynomial order of the kernel functions, the higher the number of moments required for getting reliable predictions. This can reduce the applicability of QMOM in the simulation of fluid particle flows where the kernel functions can have quite complex functional dependences. On the other hand, QMOM can still be used in some applications where the kernel functions are given as low order polynomials like in some solid particle or crystallization problems. 

In the quadrature method of moment (QMOM) a few moments of the number distribution function/ are tracked in time directly, just as for the standard MOM, but in this approach the requirement of exact closure is replaced by an approximate closure condition that allows the method to be applied under a much broader range of conditions. This method was first proposed by McGraw [151] for modeling aerosol dynamics and has later been extended to aggregation and breakage processes in crystallization by Marchisio et al. [141, 142]. 

The SQMOM has the advantage that it is not tied to the inversion of large sized moment problems as required by QMOM. Such methods generally become ill-conditioned when a large number of moments are required to increase their accuracy. The accuracy of the SQMOM increases by increasing the number of primary particles while using a fixed number of secondary particles. Since the positions and local distributions for two secondary particles are found to have an analytical solution, no large moment inversion problems are encountered. 

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