Internal coordinate discretization

It is always possible to convert internal to Cartesian coordinates and vice versa. However, one coordinate system is usually preferred for a given application. Internal coordinates can usefully describe the relationship between the atoms in a single molecule, but Cartesian coordinates may be more appropriate when describing a collection of discrete molecules. Internal coordinates are commonly used as input to quantum mechanics programs, whereas calculations using molecular mechanics are usually done in Cartesian coordinates. The total number of coordinates that must be specified in the internal coordinate system is six fewer... 

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 NDF is very similar to the PDFs introduced in the previous section to describe turbulent reacting flows. However, the reader should not confuse them and must keep in mind that they are introduced for very different reasons. The NDF is in fact an extension of the finite-dimensional composition vector laminar flow where the PDFs are not needed, the NDF still introduces an extra dimension (1) to the problem description. The choice of the state variables in the CFD model used to solve the PBE will depend on how the internal coordinate is discretized. Roughly speaking (see Ramkrishna (2000) for a more complete discussion), there are two approaches that can be employed ... 

A simplified homogeneous dispersed-phase mixing model was proposed by Curl (C16). Uniform drops are assumed, coalescence occurs at random and redispersion occurs immediately to yield equal-size drops of the same concentration, and the dispersion is assumed to be homogeneous. Irreversible reaction of general order s was assumed to occur in the drops. The population balance equations of total number over species concentration in the drop were derived for the discrete and continuous cases for a continuous-fiow well-mixed vessel. The population balance equation could be obtained from Eq. (102) by taking the internal coordinate to be drop concentration and writing the population balance equation in terms of number to yield... 

Eq. (24) is applicable only to lattice models where the beads have discrete positions. To extend this methodology to the models considered in this work, continuous space must be discretized. Fortunately, this can be done using the internal coordinates of the potential function as a reference. First, bit vectors c(a) and d a) are introduced with components of 0 or 1. To each component of c[a) we assign a pair of non-adjacent phobic beads i and j. We then define the value of Cy as... 

In general, the numerical solution of PlDEs consists of a three step procedure. First the basic set of PlDEs is discretized in the internal coordinates and expressed as a set of partially differential equations (PDFs). The PDFs are then discretized in time and the physical space coordinates using standard PDF discretization techniques in a second step. Finally the resulting set of algebraic equations are solved by use of a suitable solver. 

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 disperse phase is constituted by discrete elements. One of the main assumptions of our analysis is that the characteristic length scales of the elements are smaller than the characteristic length scale of the variation of properties of interest (i.e. chemical species concentration, temperature, continuous phase velocities). If this hypothesis holds, the particulate system can be described by a continuum or mean-field theory. Each element of the disperse phase is generally identified by a number of properties known as coordinates. Two elements are identical if they have identical values for their coordinates, otherwise elements are indistinguishable. Usually coordinates are classified as internal and external. External coordinates are spatial coordinates in fact, the position of the elements in physical space is not an internal property of the elements. Internal coordinates refer to more intimate properties of the elements such as their momenta (or velocities), their enthalpy... 

As introduced in the previous section, class and sectional methods are based on a discretization of the internal coordinate so that the GPBE becomes a set of macroscopic balances in state space. Indeed, the fineness of the discretization will be dictated by the accuracy needed in the approximation of the integrals and derivative terms appearing in the GPBE. As has already been anticipated, the methods differ according to the number of internal coordinates used in the description and depend on the nature of the internal coordinates. Therefore, in what follows, we will discuss separately the univariate, bivariate, and multivariate PBE, and the use of these methods for the solution of the KE. 

Readers interested in a detailed discussion concerning the application of the different CM to this equation are referred to the specialized literature (Immanuel Doyle, 2005 Lau-renzi et al 2002 Qamar Warnecke, 2007 Vale McKenna, 2005 Xiong Pratsinis, 1991, 1993). In what follows, only a brief summary is reported. Let us now discuss a simple geometric extension to bivariate problems of the discretization presented for univariate PBE. Given the interval of the internal coordinate delimited by and... 

It is worth noticing that with this formulation the size of the particle-velocity interval, Af, appears in the discretized equation, in contrast to what happens with population balances. This is simply because in the case of population balances, Nt was defined as the integral of n( ) over the ith interval, representing therefore the finite number of particles with internal coordinate in between f and f -i- df W, = n(f)d - In the case of the KE,... 

The particles of interest to us have both internal and external coordinates. The internal coordinates of the particle provide quantitative characterization of its distinguishing traits other than its location while the external coordinates merely denote the location of the particles in physical space. Thus, a particle is distinguished by its internal and external coordinates. We shall refer to the joint space of internal and external coordinates as the particle state space. One or more of either the internal and/or external coordinates may be discrete while the others may be continuous. Thus, the external coordinates may be discrete if particles can occupy only discrete sites in a lattice. There are several ways in which the internal coordinates may be discrete. A simple example is that of particle size in a population of particles, initially all of uniform size, undergoing pure aggregation, for in this case the particle size can only vary as integral multiples of the initial size. For a more exotic example, let the particle be an emulsion droplet (a liquid) in which a precipitation process is carried out producing a discrete number of precipitate particles. Then the number of precipitate particles may serve to describe the discrete internal coordinate of the droplet, which is the main entity of population balance. 

Hounslow Discretization. Hounslow et al. (16) developed a relatively simple discretization method by employing an M-I approach (the mean value theorem on frequency). The population balance equations, such as Eq. 3.20, are normally developed using particle volume as the internal coordinate. Because of the identified advantages of length-based models, Hounslow et al. (16) performed the coordinate transformation to convert the volume-based model described by Eq. 3.20 to a length-based model as follows ... 

Attention should be paid to the selection of the internal coordinates. The original Kumar-Ramkrishna discretization should be applied to volume-based models rather than length-based models. Although both are interconvertible, it is important to check consistency in numerical computations. 

One manner of using the presented results is to incorporate them in the traditional way of tackling fluid bed granulation theoretically, namely population balance modeling. This can be achieved by expanding the population balance to more internal coordinates than just particle size (see Volume 1 of this series. Chapter 6, Section 6.9.1). The additional property in the case of the present example would be wet a lomerate composition, defined either by the mass fraction of solids within one particle or the binder/solid ratio. The latter can be further spht up to account for the spatial distribution - and, thus, accessibUity - of the hquid binder and for the thickness of the binder layer on the outer surface of the agglomerate. Alternatively, discrete models of agglomeration (see Section 7.7) could be expanded to account for non-spherical primary particles. 

The transport equation for the NDF (12.4.1-1) is similar to (12.4.1-11) and obtained by summation of all the single-particle joint-PDF transport equations (12.4.1-11). In (12.4.1-11), the first term represents convection, the second the conditional acceleration, and the third the single-particle state conditional continuous changes in the internal coordinates other than the velocity. The rate of continuous changes of property in the phase space of

internal coordinates due to collisions are described by the conditional source terms B- D) ) on the right hand side of (12.4.1-11). 

Standard methods are used to propagate each Om in time. For the z and Z coordinates we make use of the fast fourier transform [99], and for the p coordinate we use the discrete Bessel transform [100]. The molecular component of asymptotic region at each time step, and projected onto the ro-vibrational eigenstates of the product molecule, for a wide range of incident energies included in the incident wave packet [82]. The results for all ra-components are summed to produce the total ER reaction cross section, a, and the internal state distributions. 

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