Internal coordinate surface area

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. 

Fig. 6 Porous coordination polymer (PCP) developed by Kitagawa and coworkers. The pores, which extend throughout the array, can be filled by C02 molecules (grey and red), allowing these materials to employ their high internal surface area as gas adsorbents [13]. Reprinted with permission...

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. 

But mi is usually not zero when the internal coordinate represents particle mass, surface area, size, etc. In these cases the PD algorithm can be safely used. The case of null mi occurs more often when the internal coordinate is a particle velocity that, ranging from negative to positive real values, can result in distributions with zero mean velocity. Another frequent case in which the mean is null is when central moments (moments translated with respect to the mean of the distribution) are used to build the quadrature approximation. These cases will be discussed later on, when describing the algorithms for building multivariate quadratures. 

When the elements of the disperse phase can be classified as equidimensional, namely they have nearly the same size or spread in multiple directions, and have constant material density, typically a single internal coordinate is used to identify the size of the elements. This could be particle mass (or volume), particle surface area or particle length. In fact, in the case of equidimensional particles these quantities are all related to each other. For example, in the trivial cases of spherical or cubic particles, particle volume and particle surface area can be easily written as Vp = k d and Ap = k d, or, in other words, as functions of a characteristic length, d (i.e. the diameter for the sphere and the edge for the cube), a volume shape factor, k, and a surface-area shape factor, k. For equidimensional objects the choice of the characteristic length is straightforward and the ratio between kp, and k is always equal to six. The approach can, however, be extended also to non-equidimensional objects. In this context, the extension turns out to be very useful only if... 

In the case of non-equidimensional particles, typically more than one internal coordinate is used to describe the shape, morphology, and size of the particles. To illustrate the issues related to phase-space advection due to mass transfer, let us analyze a simple example. We consider needle-like particles described as rectangular parallelepipeds with length and equal width and depth 2- Clearly the particle volume can easily be calculated from these two quantities (i.e. Vp = particle surface area... 

If the particle density is constant, this description is equivalent to one based on the particle volume. On the other hand, consider a system with two internal coordinates (such as, for example, particle mass pi and particle surface area p2) that are additive during an aggregation event p = + p, where p = ( pi, p2), = and p = ( pi, p2). This... 

If the internal coordinate is the particle size, the second-order moment can be related to the mean particle surface area. 

Metal-organic frameworks (MOFs) are three-dimensional extended structures in which metal ions or clusters are linked through organic molecules that have two or more sites through which links can be formed. Unlike coordination polymers, MOFs are exclusively crystalline the trademarks of MOFs are their extremely high surface areas, tunable pore size, and adjustable internal surface properties. Functional groups that link metals or metal clusters within MOFs are molecules or ions that have two or more Lewis basic sites (for example, carboxylates, triazolates, tetrazolates, ° and pyrazolates (Figure 9.36)). 

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