Internal coordinate fluid

The CFD model described above is adequate for particle clusters with a constant fractal dimension. In most systems with fluid flow, clusters exposed to shear will restructure without changing their mass (or volume). Typically restructuring will reduce the surface area of the cluster and the fractal dimension will grow toward d — 3, corresponding to a sphere. To describe restructuring, the NDF must be extended to (at least) two internal coordinates (Selomulya et al., 2003 Zucca et al., 2006). For example, the joint surface, volume NDF can be denoted by n(s, u x, t) and obeys a bivariate PBE. 

If an appropriate relation for the contact area as a function of the internal coordinates is available, the particle growth term due to interfacial mass transfer can be modeled in accordance with the well known film theory (although still of semi-empirical nature) and the ideal gas law [68]. The modeling of the source and sink terms due to fluid particle breakage and coalescence is less familiar and still on an early stage of development. Moreover, the existing theory is rather complex and not easily available. Further research is thus needed in order to derive consistent multifluid-population balance models. 

For flows where compressibility effects in a gas are important the use of the particle mass as internal coordinate may be advantageous because this quantity is conserved under pressure changes [11]. In this approach it is assumed that all the relevant internal variables can be derived from the particle mass, so the particle number distribution is described by the particle mass, position and time. Under these conditions, the dispersed phase flow fields are characterized by a single distribution function /(m, r,t) such that f m,r,t)drdm is the number of particles with mass between m and m+dm, at time t and within dr of position r. Notice that the use of particle diameter and particle mass as inner coordinates give rise to equivalent population balance formulations in the case of describing incompressible fluids. 

The growth terms need some further attention. Considering that for fluid particles the mass change rate (or an imaginary velocity in the internal coordinates) of the particle, is negative for the case of condensation or particle dissolution, and is positive in the case of evaporation or mass diffusion into the particle the distribution function was discretized using an upwinding approach. 

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... 

A special case of considerable interest occurs when the internal-coordinate vector is the particle-velocity vector, which we will denote by the phase-space variable v. In fact, particle velocity is a special internal coordinate since it is related to particle position (i.e. external coordinates) through Newton s law, and therefore a special treatment is necessary. We will come back to this aspect later, but for the time being let us imagine that otherwise identical particles are moving with velocities that may be different from particle to particle (and different from the surrounding fluid velocity). It is therefore possible to define a velocity-based NDF nv(t, x, v) that is parameterized by the velocity components V = (vi, V2, V3). In order to obtain the total number concentration (i.e. number of particles per unit volume) it is sufficient to integrate over all possible values of particle velocity Oy ... 

The fluid could also change the internal coordinates so that additional source terms for the internal coordinates are needed. For the present discussion, we will ignore this possibility. 

In order to derive a transport equation of the fluid-phase volume fraction af, we will let the internal coordinate be equal to the fluid volume seen by a particle. The fluid-phase volume fraction is then defined by... 

The transport equations for g and are used in two-fluid models for multiphase flows. A fc-fluid model can be developed by treating all particles with the same internal coordinates as a fluid. Thus, for example, if all particles are identical except that some have mass Ml and the others have mass M2 (which implies that they have different solid densities), then we can treat the particle phase as two fluids. Mathematically, this follows directly from the form of the NDF for this case. 

In the literature on turbulent two-phase flow (Minier Peirano, 2001 Peirano Minier, 2002 Simonin et al, 1993), the fluid phase is usually treated using a separate distribution function whose integral over phase space leads to the fluid-phase mass density. Here, we use a different approach starting from n(f, x, Vp, p, Vf, f). In this approach, we let the internal coordinate be equal to the fluid mass seen by a particle. The fluid-phase mass density is then given by... 

If the particles are composed of multiple chemical species, then usually the fluid phase will be also. In such cases, it is necessary to introduce a vector of internal coordinates whose components are the mass of each chemical species seen by a particle. Obviously, the sum of these internal coordinates is equal to the fluid mass seen by a particle. By definition, if is the mass of component a, then integration over phase space leads to a component fluid-phase mass density ... 

The second example is the case in which the disperse-phase velocity is equal to the conditional expected disperse-phase velocity given the other mesoscale variables Vp = (Upl p, Vf, f). In this case, fluctuations in the disperse phase are slaved to the fluid-phase fluctuations and can depend on the particle internal coordinates (e.g. particle size). The NDF in this case is n(Vp, p, Vf, f) = n( p, Vf, f)(5(Vp - [Pg.131]

Ap is the difference in material density between the liquid and gas phases. This situation is typically handled by describing the bubbles with a single internal coordinate (i.e. the equivalent-sphere diameter) and by introducing an aspect ratio, defined as the ratio between the minor and the major axes of the bubble. This aspect ratio E can be calculated by using the empirical equation proposed by Moore (1965) as a function of the Morton number E = 1/(1 + 0.043RCp Mo ). An alternative to this is the use of the correlation proposed by Wellek et al. (1966) for liquid-liquid droplets E = 1/(1 + 0.1613Eo° ), which is valid for Eo < 40 and Mo < 10 , whereas for Eo > 40 and RCp > 1.2 fluid particles are typically of spherical shape. Once the characteristic E value is known, the ratio of the real area of the bubble Ap and the area Aeq of a sphere with an equivalent volume can be calculated as follows ... 

In summary, the Eulerian two-fluid model is represented by Eqs. (5.112) and (5.113) in addition to a constitutive model for the fluid stress tensor Tf. As already mentioned, Eq. (5.112) was derived under the assumption that the particle-velocity distribution is very narrow (i.e. small particle Stokes number), and the particles must have the same internal coordinates. If these simplifications do not hold, for example under dense conditions when particle-particle collisions become important, then particle-velocity fluctuations must be taken into account, as discussed at the end of Chapter 4. 

It is important to stress here that in Eq. (7.145) there are explicit dependences on spatial coordinates in the normalized NDF F and fluid velocity V, but also implicit dependences on the rates of change of internal coordinates the rate of formation of the disperse phase J, and the kernels jS and b. In fact, as described in Chapter 5, these rates and the kernels depend on the flow properties, which change from point to point in the system. 

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