The moment-closure problem

In high-Reynolds-number turbulent flows, the molecular diffusion term Fp( ) will be negligible compared with Ft (Fox, 2003). Except for the last term, Eq. (2.49) is closed. 

To conclude this chapter, we look briefly at the moment-closure problem using the NDF transport equation in Eq. (2.49). For clarity, we will drop the angle brackets and assume that the source terms in the PBE have been closed. The PBE then reads 

In order to reduce the number of independent variables, we use the definition of the moments in Eq. (2.2) to find the moment-transport equation corresponding to Eq. (2.50)  

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With [D ] [P ] there is little difference in magnitude between vt and ft hut, as described below, the introduction of ft eliminates the moment closure problem created by the LCB mechanism. These moment definitions collapse the infinite set of equations for polymeric species into a manageable subset, and many of the moments have precise physical meanings. The zeroth live moment, [/cq], is the concentration of polymer radicals in the system (denoted by [Ptot] in Section 3.2.1), and the first live moment, [/ri],is the concentration of monomer units contained in all growing radicals. Similarly, [fo] is the concentration of all polymer chains in the system, and [f i ] is the concentration of monomer units bound in all polymer chains. These values are used to calculate MW averages, where Wn, is the molecular weight of the monomeric repeat unit ... 

With [D ] [P ] there is little difference in magnitude between fij and Cj. Its introduction, however, eliminates the moment closure problem created by the LCB mechanism [40, 81, 85]. Many of the moments have precise physical meanings. 

Equations for the evolution of the updated PDF (i.e., after assimilating the measurements) similar to the ones given in section Markov Vector Methods can be derived. These are called the Kushner-Stratonovich equations. For most practical problems of interest, with nonlinear process and measurement equations, non-Gaussian noises, this equation remains theoretical in nature and suffers from the moment closure problem (Maybeck 1982). Thus updating reliability models is mostly carried out through the simulation-based methods. 

Note that we need to know Pj, P2, and in order to solve Xq, X, and X2. However, we also note that the relation for P2 involves P3, the one for /I3 involves X, and so on. This is known as the moment closure problem it is present in all reversible polymerization analyses. For nylon 6 polymerization, we assume the following relations ... 

Conditional moments of this type cannot be evaluated using the one-point PDF of the mixture fraction alone (O Brien and Jiang 1991). In order to understand better the underlying closure problem, it is sometimes helpful to introduce a new random field, i.e.,15... 

Inserting the same quadrature approximation to the source term integrals provides an approximate numerical type of closure avoiding the higher order moments closure problem on the cost of model accuracy [131]. This numerical approximation actually neglects the physical effects of the higher order moments. No reports applying this procedure to bubbly flows have been found so far. 

In Chapter 4 the GPBE is derived, highlighting the closures that must be introduced for the passage from the microscale to the mesoscale model. This chapter also contains an overview of the mathematical steps needed to derive the transport equations for the moments of the NDF from the GPBE. The resulting moment-closure problem is also throughly discussed. 

The set of moment expressions to be substituted into reactor balances consists of either the live and dead moments (Equations 3.64-3.69) or the live and bulkmoments (Equations 3.64, 3.65 and 3.70-3.72, substituting [( i] and [ 2 for [i ] and [V2] in Equation 3.65). Choosing the former, while common practice in the literature, suffers from a moment closure problem, as ( 2] and [02] depend on [V3]. The Hulburt and Katz [59] method assumes that the molecular weight distribution can be represented by a truncated series of Laguerre polynomials, approximating [V3] as... 

Use of the bulk moments eliminates this moment closure problem. It also reduces the number of equations, as Equation 3.66 is not required for solution of Equation 3.72. An additional balance is needed to track the concentration of LCB formed by the transfer to polymer mechanism ... 

These equations require the fourth-order moment , which is unknown an equation can be derived for the fourth-order moment, but it would involve the sixth-order moment, and so forth. We therefore need to express in terms of A = . This is the standard closure problem that always arises in statistical mechanics. There have been a number of studies to derive the best closure, but for our purposes it suffices to take the simplest form and to write... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

For instantaneous reactions the problem is thus reduced to the calculation of the presumed PDF of a passive scalar or tracer. A large number of alternative presumed PDFs have been listed and discussed by [2, 60, 67]. Each presumed PDF has its advantages and disadvantages, but none of them are generally applicable. The concept of the full PDF approaches is to formulate and solve additional transport equations for the PDFs determining the evolution of turbulent flows with chemical reactions. These models thus require modeling and solution of additional balance equations for the one-point joint velocity-composition PDF. The full PDF models are thus much more CPU intensive than the moment closures and hardly tractable for process engineering calculations. These theories are comprehensive and well covered by others (e.g., [2, 8, 26]), thus these closures are not examined further in this book. For Unite rate chemical reaction processes neither of the asymptotic simplifications explained above are applicable and appropriate elosures for 5c (w) are very difficult to achieve. 

The closure problem thus reduces to finding general methods for modeling higher-order moments of the composition PDF that are valid over a wide range of chemical time scales. 

Thus, the turbulent-reacting-flow problem can be completely closed by assuming independence between Y and 2, and assuming simple forms for their marginal PDFs. In contrast to the conditional-moment closures discussed in Section 5.8, the presumed PDF method does account for the effect of fluctuations in the reaction-progress variable. However, the independence assumption results in conditional fluctuations that depend on f only through Tmax(f ) The conditional fluctuations thus contain no information about local events in mixture-fraction space (such as ignition or extinction) that are caused by the mixture-fraction dependence of the chemical source term. 

When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the probability of arbitrary sequences t//j in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively. 

This implies that higher order moments are introduced, thus the system of PDEs cannot be closed analytically. It is possible to show that similar effects will occur for the other source terms as well. This problem limits the application of the exact method of moments to the particular case where we have constant kernels only. In other cases one has to introduce approximate closures in order to eliminate the higher order moments ensuring that the transport equations for the moments of the particle size distribution can be expressed in terms of the lower order moments only (i.e., a modeling process very similar to turbulence modeling). 

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. 

As described in earlier chapters, for univariate QMOM the closure problem appears in different parts of the transport equations for the moments of the NDF. However, it can often be reduced to the following integral ... 

While EQMOM allows us to capture an additional moment, the use of kernel density functions can lead to a closure problem when evaluating integrals such as Eq. (3.9) ... 

The MOM was introduced for particulate systems by Hulburt Katz (1964). In their pioneering work these authors showed how it is possible to solve the PBF in terms of the moments of the NDF and to derive the corresponding transport equations. A similar approach can be used for the solution of the KF, and a detailed discussion on the derivation of the moment-transport equations can be found in the works of Struchtrup (2005) and Truesdell Muncaster (1980). The main issue with this technique is in the closure problem, namely the impossibility of writing transport equations for the lower-order moments of the NDF involving only the lower-order moments. Since the work of Hulburt Katz (1964) much progress has been made (Frenklach, 2002 Frenklach Harris, 1987 Kazakov Frenklach, 1998), and different numerical closures have been proposed (Alexiadis et al, 2004 Kostoglou Karabelas, 2004 Strumendo Arastoopour, 2008). The basis... 

A second class of methods for overcoming the closure problem is to make a functional assumption regarding the NDF. The simplest is to assume that the NDF is composed of a delta function centered on the mean value of ffie internal coordinate (e.g. mi/mo), or, in other words, assume that the population of particles is monodisperse. On resorting to this approach the missing moments can be readily calculated (e.g. mk = as illustrated... 

Quadrature-based moment methods (QBMM) constitute a particular class of very successful moment methods that overcome the closure problem by using quadrature approximations. A general (multivariate) quadrature of N nodes for M internal coordinates requires knowledge of N weights, w , and N nodes (or abscissas), y M,a),... 

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