The moment-transport equation

The results shown in Figs. 6 and 7 can be combined to compute the mean mixture fraction ( ) and its variance (c2) from Eqs. (34) and (35), respectively. Example plots are shown in Liu and Fox (2006) and, as expected, they agree with the solution found by solving the moment transport equations directly (Eqs. 28 and 29). 

The quadrature method of moments (QMOM) is a presumed PDF approach that determines the unknown parameters by forcing the lower-order moments of the presumed PDF to agree with the moment transport equations (McGraw 1997 Barrett and Webb 1998 Marchisio et al. 2003a Marchisio et al. 2003b). As with the multi-environment presumed PDF method discussed in Section 5.10, the form of the presumed PDF is... 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

Moment methods come in many different variations, but the general idea is to increase the number of transported moments (beyond the hydrodynamic variables) in order to improve the description of non-equilibrium behavior. As noted earlier, the moment-transport equations are usually not closed in terms of any finite set of moments. Thus, the first step in any moment method is to apply a closure procedure to the truncated set of moment equations. Broadly speaking, this can be done in one of two ways. 

Work directly with the unclosed terms in the moment-transport equations to find a functional form to close them (Struchtrup, 2005). For example, a spatial flux involving moment might be closed using a gradient-diffusion model involving... 

Use the transported moments to reconstruct the unknown density function and then compute the unclosed terms in the moment transport equations. For example, in Grad s 13-moment method (Grad, 1949b) the reconstructed density function uses moments up to third order, and the fourth-order moments are found by integration. 

The remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

Many disperse-phase systems involve collisions between particles, and the archetypical example is hard-sphere collisions. Thus, Chapter 6 is devoted to the topic of hard-sphere collision models in the context of QBMM. In particular, because the moment-transport equations for a GBPE with hard-sphere collisions contain a source term for the rate of change of the NDF during a collision, it is necessary to derive analytical expressions for these source terms (Fox Vedula, 2010). In Chapter 6, the exact source terms are derived... 

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of averaged quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDE transport equation by integration over phase space. Einally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations. 

The process of finding moment-transport equations starting from the PBE can be continued to arbitrary order. We should note that in most applications the resulting moment-transport equations will not be closed. In other words, the moment-transport equation of order k will involve moments of order higher than k. We will discuss moment-closure methods in Chapters 7 and 8 in the context of quadrature-based moment methods. 

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) ... 

In general, simulations carried out starting from realizable moment sets should result in realizable moment sets. Nevertheless, the moment-transport equations are integrated numerically with some finite discretization errors. As Wright (2007) clearly reports, most of the problems are caused by the approximation of the convective term, in particular with higher-order discretization schemes, which can turn a realizable set of moments into... 

The two terms on the right-hand side are the flux at infinity, which we can safely set to zero, and the flux at the origin. Depending on the forms of flux term to cancel out. Unfortunately, there are important applications in which the flux term is nonzero, so one must pay attention to how the flux term is handled in the derivation of the moment-transport equations. For example, if p represents the surface of evaporating droplets and (Gp)i is constant (i.e. the evaporation rate is proportional to the surface area), then n will be nonzero at p = 0. Physically, the nonzero flux is due to the disappearance of droplets due to evaporation, and thus it cannot be neglected. 

In summary, computing the moment-transport equations starting from Eq. (4.39) involves integration over phase space using the mles described above for particular choices of g. In the following, we will assume that the flux term at the boundary of phase space can be neglected. However, the reader should keep in mind that this assumption must be verified for particular cases. 

The moment-transport equations that we have derived up to this point are of first order in the velocity variables. In order to describe fluctuations about the first-order moments, it is necessary to derive transport equations for second- and sometimes higher-order moments. Just as before, this is accomplished by starting from Eq. (4.39) with a particular choice for g. In order to illustrate how this is done, we will consider the function g = fp2V which results in the particle-mass-average moment... 

A popular method for closing a system of moment-transport equations is to assume a functional form for the NDF in terms of the mesoscale variables. Preferably, the parameters of the functional form can be written in closed form in terms of a few lower-order moments. It is then possible to solve only the transport equations for the lower-order moments which are needed in order to determine the parameters in the presumed NDF. The functional form of the NDF is then known, and can be used to evaluate the integrals appearing in the moment-transport equations. As an example, consider a case in which the velocity NDF is assumed to be Gaussian ... 

The integral terms in parentheses are known for the family of orthogonal polynomials. With finite N and known moments, this system of linear equations has the form M = AC and can be solved to find fhe expansion coefficients Ca(t, x). Thus, the presumed NDF n (t, X, Vp) is a unique function of a finite set of moments, and the latter are found by solving the moment-transport equations using n to close the unclosed terms. The fact that... 

A increasingly popular method for closing the moment-transport equations is to assume a discrete form for the phase-space variables. Taking the velocity NDF as an example, the velocity phase space can be discretized on a uniform, symmetric lattice centered at Vp = 0. For illustration purposes, let us assume that A = 16 lattice points are used and denote the corresponding velocities as Ua. The formal definition of the discrete NDF is... 

The simplest manner to specify the fluid mass seen by a particle is fn = Pffpi/Pp, where p is the particle mass. If all particles have the same mass, then p is constant. Nevertheless, for deriving the moment-transport equations, it is easiest to treat and p as independent variables during the derivations and then to substitute n = gffpi /gp in the final result, which is equivalent to taking the Umit Tpf -> 0. 

For all other cases, it will be necessary to solve the moment-transport equations derived from the GPBE as described in Chapter 4. In Chapter 8 the numerical algorithms used to find approximation solutions to the GPBE using quadrature-based moment methods are presented in detail. 

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