Internal coordinate vector

The first step in the symmetry determination of the dynamic properties is the selection of the appropriate basis. Appropriate here means the correct representation of the changes in the properties examined. In the investigation of molecular vibrations (Chapter 5), either Cartesian displacement vectors or internal coordinate vectors are used. In the description of the molecular electronic structure (Chapter 6), the angular components of the atomic orbitals are frequently used... 

Let us consider a population of disperse entities such as solid particles or liquid droplets inside an infinitesimal control volume located at the physical point x = (xi,X2,X3) and of measure dx = dxi dX2 dxs. Let . m) be the internal-coordinate vector,... 

It is straightforward that the quantity ( )d represents the number density of disperse entities contained in the phase-space volume d centered at per unit of physical volume. If we integrate the NDF over all possible values of the internal-coordinate vector we obtain the total number concentration N(t, x) ... 

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 ... 

Note that, although we treat x and f in the same manner, they are in fact different types of vectors. The vectors x and v are the standard vectors for position and velocity used in continuum mechanics. The internal-coordinate vector on the other hand, is a generalized vector of length N in the sense of linear algebra. 

Equation (2.14) must be coupled with initial conditions given for the starting time and with boundaries conditions in physical space O and in phase space O. Analytical solutions to Eq. (2.14) are available for a few special cases and only under conditions specified by some very simple hypotheses. However, numerical methods can be used to solve this equation and will be presented in Chapters 7 and 8. The numerical solution of Eq. (2.14) provides knowledge of the NDE for each time instant and at every physical point in the computational domain, as well as at every point in phase space. As has already been mentioned, sometimes the population of particles is described by just one internal coordinate, for example particle length (i.e. f = L), and the PBE is said to be univariate. When two internal coordinates are needed, for example particle volume and surface area (i.e. = (v, a)), the PBE is said to be bivariate. More generally, higher-dimensional cases are referred to as multivariate PBEs. Another important case occurs when part of the internal-coordinate vector is equal to the particle-velocity vector (i.e. when the particles are characterized not by a unique velocity field but by their own velocity distribution). In that case, the PBE becomes the GPBE, as described next. 

When developing models for polydisperse multiphase flows, it is often useful to resort to conditioning on particle size. For example, in gas-solid flows the momentum-exchange terms between the gas phase and a solid particle will depend on the particle size. Thus, the conditional particle velocity given that the particle has internal-coordinate vector will... 

When performing the Reynolds averaging, the internal-coordinate vector is not affected. Thus, the Reynolds average commutes with time, space, and phase-space gradients. See Fox (2003) for a discussion of this topic. 

The particle mass can be written as Mp = PpVp, where pp is the material density of the particle and Vp is its volume. Note that, in addition to mass, either material density or volume could be included in the internal-coordinate vector. (In general, we will use mass and volume as the internal coordinates.) Thus, for example, fixing the particle masses to be equal does not imply that all particles have the same volume. 

Although the particle velocity is often included as one of the variables of the internal-coordinate vector, it clearly plays an important and different role, which is related to the... 

Depending on the number of internal coordinates (univariate versus multivariate) and depending on whether the particle velocity is part of the internal-coordinate vector, very different solution methods have been developed. As a consequence, in this chapter the methods for cases with and without particle velocity are discussed separately. First, the methods developed for the solution of a univariate PBE (i.e. one internal coordinate) are discussed. Second, the approaches for the solution of bivariate and multivariate PBE (i.e. 

The cumulative distribution Fj(t) represents the probability that the quiescence time T is smaller than t. By assuming that the effect of the continuous change of the internal-coordinate vector is negligible, for the calculation of this probability, and by including also collision and aggregation as an additional discontinuous event, the following expression is obtained ... 

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