Internal coordinates continuous

With pure advection processes, we refer to continuous phenomena that cause continuous changes in the external and internal coordinates. Continuous changes of the particle s position in real space are quantified by the real-space advection (or free-transport) term ... 

The full dynamical treatment of electrons and nuclei together in a laboratory system of coordinates is computationally intensive and difficult. However, the availability of multiprocessor computers and detailed attention to the development of efficient software, such as ENDyne, which can be maintained and debugged continually when new features are added, make END a viable alternative among methods for the study of molecular processes. Eurthemiore, when the application of END is compared to the total effort of accurate determination of relevant potential energy surfaces and nonadiabatic coupling terms, faithful analytical fitting and interpolation of the common pointwise representation of surfaces and coupling terms, and the solution of the coupled dynamical equations in a suitable internal coordinates, the computational effort of END is competitive. 

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... 

An n-valued surface can be constructed from the eigenvalues of an n x n matrix whose elements are continuous functions of the internal coordinates of the molecules. Thus the Jahn-Teller surfaces can be represented by the eigenvalues of a 2 x 2 matrix such as (23), which are given by expression (24). At the point of intersection the diagonal elements of this matrix are equal and the off-diagonal elements are zero. 

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 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. 

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... 

Note that left-hand side of this expression is, in fact, a continuity equation for which states that the multi-particle joint PDF is constant along trajectories in phase space. The term on the right-hand side of Fq. (4.32) has a contribution due to the Alp-particle collision operator, which generates discontinuous changes in particle velocities Up" and internal coordinates p", and to particle nucleation or evaporation. The first term on the left-hand side is accumulation of The remaining terms on the left-hand side represent... 

Likewise, in the case of heat exchange between phases, continuous changes for the internal coordinates are induced. If, for example, one of the internal coordinates is the temperature of the particle, Tp, the rate of change of particle enthalpy can be calculated as... 

Another possible scenario is that as a consequence of mass transfer only the number of primary particles changes, whereas their size remains more or less constant. This hypothesis seems to be realistic in the case of negative molar flux, J <0, or, in other words, in the case of shrinking particles. In fact, in this case it is more likely that the external particles will be consumed before the internal ones. The resulting expressions for the continuous rate of change of the two internal coordinates therefore read as... 

Another interesting example is that of gas bubbles dispersed in a continuous liquid phase with which mass is exchanged. Also for this case the rate of change of the internal coordinates due to mass transfer is written starting from a simple mass balance for a single bubble. Following the standard notation for gas-liquid systems, the single-particle mass balance becomes... 

The relationship among ij, rj, and rj can be derived by relating continuity statements written in terms of pre- and post-event values for the internal coordinates and depends on the second-order point process under investigation. The positive source term can also be written in terms of the frequency (rather than the kernel) ... 

The rate of continuous change can also be a function of time. However, in this chapter we will discuss only internal-coordinate dependences, since these are the ones which cause the most significant numerical issues. 

The drift term represents a continuous variation of the internal coordinate f due to a rate of change f ... 

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