Internal coordinate volume

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

Polymer Particle Balances (PEEK In the case of multiconponent emulsion polymerization, a multivariate distribution of pjarticle propierties in terms of multiple internal coordinates is required in this work, the polymer volume in the piarticle, v (continuous coordinate), and the number of active chains of any type, ni,n2,. .,r n (discrete coordinates), are considered. Therefore... 

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. 

Figure 6.2 represents the behavior of quasi-monodisperse double emulsions in (C , (pf) coordinates, where (pf is the initial volume fraction of droplets in the globules. Sorbitan monooleate (SMO) was used for the stabilization of the primary W/O emulsion and sodium dodecyl sulfate (SDS, CMC = 8 10 mol/1) was used for the stabilization of the oil globules in the external aqueous phase. Three different compositions zones, referred to as A, B, and C can be defined they differ by their qualitative behavior observed via microscopy. Moderate internal droplet volume fractions are considered first (cpf < 20%). If Ck = CMC/10, the system does not exhibit any structural evolution after a few days of storage (zone A). If C is... 

Preferential sorption in the sinusoidal channels was confirmed by Nicholas et al. (67) in an MD study of methane and propane adsorption. This preference was most noticeable at infinite dilution at a loading of 12 molecules per unit cell the distribution of molecules over the channels was found to be close to that expected from the relative volumes of the channel segments. The propane molecules were predicted to spend more time in the intersections than the straight channel at infinite dilution. This result is rationalized by considering the slow motion of the molecules and the conformational changes necessary to move from one channel type to another via an intersection. The distribution of propane backbone bond angles was predicted to be similar to that of gas-phase propane, indicating the rather minor effect of the zeolite on the internal coordinates of propane. 

The integration over z, y, z is to be carried over the volume of the container and gives simply a factor V. The integrations over px, py, ps arc carried from — oo to oo, and can be found by Eqs. (2.3) of Chap. IV. The integral depending on the internal coordinates and momenta will not be further discussed at present we shall abbreviate it... 

In this section the population balance modeling approach established by Randolph [95], Randolph and Larson [96], Himmelblau and Bischoff [35], and Ramkrishna [93, 94] is outlined. The population balance model is considered a concept for describing the evolution of populations of countable entities like bubble, drops and particles. In particular, in multiphase reactive flow the dispersed phase is treated as a population of particles distributed not only in physical space (i.e., in the ambient continuous phase) but also in an abstract property space [37, 95]. In the terminology of Hulburt and Katz [37], one refers to the spatial coordinates as external coordinates and the property coordinates as internal coordinates. The joint space of internal and external coordinates is referred to as the particle phase space. In this case the quantity of basic interest is a density function like the average number of particles per unit volume of the particle state space. The population balance may thus be considered an equation for the number density and regarded as a number balance for particles of a particular state. 

When the population balance is written in terms of one internal coordinate (e.g., particle diameter or particle volume), the closure problem mentioned above for the moment equation has been successfully relaxed for solid particle systems by the use of a quadrature approximation. 

O-H bond. Such extreme cases are characterized as short strong hydrogen bonds (SSHBs) [12]. Two geometrical parameters, being in principle the internal coordinates, were introduced 1 = dl—d4 and q2 = di dl. Linear relationship between them was found. The g-parameter Q = q - - ql) may be treated as the descriptor of the ir-electron delocalization. Gilli and coworkers also introduced the A-parameter (see Chapt. 2 of this volume). 

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 ... 

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. 

In order to derive a transport equation of the disperse-phase volume fraction ap, we will let the first internal coordinate pi be equal to the particle volume (Vp). The disperse-phase volume fraction is then defined by... 

---