Moment-transport equation derivation

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. 

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

For a monodisperse system the moment-transport equation derived from Eq. (6.1) is... 

In contrast to moment closures, the models used to close the conditional fluxes typically involve random processes. The choice of the models will directly affect the evolution of the shape of the PDF, and thus indirectly affect the moments of the PDF. For example, once closures have been selected, all one-point statistics involving U and 0 can be computed by deriving moment transport equations starting from the transported PDF equation. Thus, in Section 6.4, we will look at the relationship between (6.19) and RANS transport equations. However, we will first consider the composition PDF transport equation. 

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 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. 

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... 

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. 

In summary, we have demonstrated in this section how quadrature-based moment methods can be used to evaluate the terms in the moment-transport equations arising from collisions. The principal observation is that it suffices to know the functional forms for the terms which are derived and tabulated in Section 6.1. We also observed that, unlike traditional moment closures, the closures developed in this section are applicable to highly non-equilibrium flows. 

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... 

This fact implies that the moments computed from their definition will be different from the moments computed by their transport equations derived using Eq. (7.7). 

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... 

The quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM) were introduced in Chapter 3 as equivalent methods for solving a homogeneous GPBE. In fact, the DQMOM was derived by Marchisio Fox (2005) primarily for the purpose of solving spatially inhomogeneous multivariate moment-transport equations. Unlike for the univariate case, where the moment-inversion algorithm is uniquely defined for a given set of moments, the QMOM in the multivariate case is much... 

In the DQMOM, the weights and abscissas are found by solving transport equations derived from Eq. (C.20) by defining the moments as... 

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... 

In developing closures for the chemical source term and the PDF transport equation, we will also come across conditional moments of the derivatives of a field conditioned on the value of the field. For example, in conditional-moment closures, we must provide a functional form for the scalar dissipation rate conditioned on the mixture fraction, i.e.,... 

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

In transported PDF methods (Pope 2000), the closure model for A, V, ip) will be a known function26 ofV. Thus, (U,Aj) will be closed and will depend on the moments of U and their spatial derivatives.27 Moreover, Reynolds-stress models derived from the PDF transport equation are guaranteed to be realizable (Pope 1994b), and the corresponding consistent scalar flux model can easily be found. We shall return to this subject after looking at typical conditional acceleration and conditional diffusion models. 

The most inaccurate assumption, especially at low band-bending, is undoubtedly the first. However, the mathematical derivation leading to eqns. (396) and (397) cannot be easily modified to take account of depletion-layer recombination owing to the way in which the depletion layer is considered. In order to develop the theory to take into account recombination in the depletion layer, it is necessary to solve explicitly the transport equation in the depletion layer as well as the bulk. If we persevere, for the moment, with the Schottky barrier model and we continue, for the moment, with the assumption that recombination does not occur in the depletion layer (x < VF) then the transport equations for x < VF, x > VF are... 

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