The NDF transport equation

We now consider the evolution of the NDF in time, phase space, and physical space. We shall see that the underlying mathematical structure for the PBE and the GPBE is very similar. 

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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 understand the DQMOM, we consider here only the univariate equations associated with the NDF transport equation ... 

Consider now the case in which the known advection velocity u(t, x, depends on For this case, the moment-transport equation is not closed, but the NDF transport equation is still given by Eq. (B.18). For the x direction, the finite-volume formula for the NDF is... 

The simplest aggregation and breakage models can be formulated in terms of the NDF (o), which uses volume as the independent variable.6 The microscopic transport equation for the NDF has the form (Wang et al., 2005a,b)... 

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. 

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 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... 

When using moment methods for inhomogeneous systems, the moment set is transported in physical space due to advection, diffusion, and free transport. Since the moment-transport equations are derived from a transport equation for the NDE, the problem of moment transport is closely related to the problem of transporting the NDF. Denoting the NDE by n(t, X, ), the process of spatial transport involves changes in n(t, x, ) for fixed values... 

In the remainder of this section, we introduce the principal modeling issues related to spatial transport using moment methods. First, we discuss the realizability of the NDF and of moment sets (which are related to the numerical errors discussed above). Second, we introduce the phenomenon of particle trajectory crossing (PTC) that occurs with the inhomogeneous KE (and is exactly captured by the NDF), and describe how it leads to a closure problem in the moment-transport equations. Next, we look at issues related to coupling between spatial and phase-space transport in the GPBE (i.e. due to correlations between velocity and internal coordinates such as particle volume). Finally, we introduce KBFVM for solving the moment-transport equations in connection with QBMM, and briefly discuss how they can be used to ensure realizability as well as to capture PTC and to treat coupled moments. 

Before looking at specific examples for which the DQMOM is likely to fail, we should note that the direct solution of the moment-transport equations using the QMOM is very robust. In the context of conservation equations, the moment-transport equations (and, for that matter, the NDF) are written in a conservative form ... 

Denoting the moment fluxes as F(n), where n is the NDF reconstructed from the transported moment set M, the moment-transport equations can be written as... 

Moreover, employing the Stokes-Einstein formula, we can write f f r(f)ndf = Fom/i-i, which is essentially the form that will be used in the numerical examples in Section 8.3.4. In summary, we will consider two variations of the moment-transport equations in the numerical examples in Section 8.3.4. The first example will use a closed moment system wherein the diffusivity does not depend on f ... 

As a final case, we will again consider the joint velocity-scalar NDF governed by Eq. (8.122), but without specifying a functional form for the scalar-conditioned velocity. The moment-transport equation is again given by Eq. (8.122). However, we will now use a scalar-conditioned multivariate EQMOM to reconstruct the joint NDE ... 

Since the moment-transport equation is closed, one might be tempted to try to design a high-order scheme directly for M. However, for such a scheme it would be difficult to ensure realizability, and thus a better approach is to work directly with the NDF. Moreover, when the advection velocity depends on f the moment-transport equation is not closed, thus working with the NDF will result in more general formulas. 

Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. In this chapter the many possible number-density functions (NDF), formulated with different choices for the internal coordinates, are presented, followed by an introduction to the PBE in their various forms. The chapter 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. 

To overcome the difficulty of inverting the moment equations, Marchisio and Fox (2005) introduced the direct quadrature method of moments (DQMOM). With this approach, transport equations are derived for the weights and abscissas directly, thereby avoiding the need to invert the moment equations during the course of the CFD simulation. As shown in Marchisio and Fox (2005), the NDF for one variable with moment equations given by Eq. (121) yields two microscopic transport equations of the form... 

The extension of DQMOM to bivariate systems is straightforward and, for the surface, volume NDF, simply adds another microscopic transport equation as follows ... 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are... 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

As mentioned above, macroscale models are written in terms of transport equations for the lower-order moments of the NDF. The different types of moments will be discussed in Chapters 2 and 4. However, the lower-order moments that usually appear in macroscale models for monodisperse particles are the disperse-phase volume fraction, the disperse-phase mean velocity, and the disperse-phase granular temperature. When the particles are polydisperse, a description of the PSD requires (at a minimum) the mean and standard deviation of the particle size, or in other words the first three moments of the PSD. However, a more complete description of the PSD will require a larger set of particle-size moments. 

This example illustrates the classical problem faced when working with macroscale models (Struchtrup, 2005). No matter how the transport equations for the moments are derived, they will always contain unclosed terms that depend on higher-order moments (e.g. Up depends on 0p, etc.). In comparison, the solution to the kinetic equation for the NDF contains information about all possible moments. In other words, if we could compute n t, x, v) directly, it would not be necessary to work with the macroscale model equations. The obvious question then arises Why don t we simply solve the kinetic equation for the mesoscale model instead of working with the macroscale model ... 

When the NDF represents the particle velocity, the transport equation is hyperbolic and thus the solution need not be smooth and need not be differentiable at every point in space. In fact, the treatment of shocks in the NDF is a significant challenge. 

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