Discrete sectional method

A method of solving many coagulation anti agglomeration problems (Chapter 8) has been developed based on the use of a similarity transformation for the size distribution function (Swift and Fricdlander, 1964 Friedlander and Wang. 1966). Solutions found in this way are asymptotic forms approached after long times, and they are independent of the initial size distribution. Closed-form solutions for the upper and lower ends of the distribution can sometime.s be obtained in this way, and numerical methods can be used to match the solutions for intermediate-size particles. Alternatively, Monte Carlo and discrete sectional methods have been used to find solutions. 

The results of the numerical calculation are shown in Fig, 7.8, where they are compared with numerical calculations carried out for the discrete spectrum starting with an initially monodisperse system. There is good agreement between the two methods of calculation. Other calculations indicate that the similarity form is an asymptotic solution independent of the initial distributions so far studied. The values of a and b were found to be 0.9046 and 1.248, respectively. By (7,75) this corresponds to a 6.5% increase in the coagulation constant compared with the value for a monodisperse aerosol (7.21). The results of more recent calculations using a discrete sectional method are shown in Table 7.2. 

Values of the Self-Preserving Size Distribution for the Continuum ((trr) and Free Molecule ( /) Regimes Calculated by a Discrete Sectional Method (Vemury and Pratsinis, 1995)... 

Three different approaches are briefly discussed here. They refer to the steady state assumption of propagating radicals, the discrete section method and the method of moments. 

During polymer decomposition, whenever a species is formed with a molecular weight not equal to that of a pseudocomponent, its amount is linearly distributed between the adjacent bins (Bi,Bi+. This lumping procedure or discrete section method was successfully applied to the thermal degradation of PS (Faravelli et al., 2001). 

As already mentioned, different modelling approaches were applied (a) the complete system of balance equations of all the species with QSSA for the single propagating radical (b) discrete section method (c) the method of moment. These comparisons demonstrate the equivalence between these techniques, not only with respect to total mass loss, but also in terms of intermediate liquid components (alkanes, alkenes and a—oj dialkenes). 

Moment methods Discrete-sectional methods Monte Carlo methods... 

A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic stroboscopic sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method (see figure 4.1). In general, an N — l)-dimensional surface-of-section 5 C F is chosen, and we consider the sequence of successive in-... 

The effect of US variabies is discussed in detaii in Chapters 3 and 4, so this section focuses on the soie difference from the effects described in them viz. not only the extraction efficiency, but aiso the undesirabie emuisification effect of US must be considered). However, most existing discrete USALLE methods have been optimized without provision for emuisification ( .e. for the time needed for the phases to separate). This is not the case with continuous methods, where emuisification is avoided or restricted to the interface, which is not monitored. 

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. 

Sectional and class methods for the solution of the collisional KE are generally called discrete-velocity methods (DVM). These methods are based on the simple idea of discretizing the velocity space into a grid constituted by a finite number of points. The existing methods are characterized by different grid structures (Aristov, 2001). For example, lattice Boltzmann methods discretize the velocity space into a regular cubic lattice with a constant lattice size (Li-Shi, 2000), whereas other methods employ different discretization schemes (Monaco Preziosi, 1990). By using a similar approach to that used with PBE, it is possible to define A,- as the number density of the particles with velocity and the discretized KE becomes... 

As with class and sectional methods for univariate PBE, the particle-velocity magnitude space is uniformly discretized over M intervals of size A < 2 < 3 < < m-i <... 

The discrete mapping method presented in Section II.B also can be used to solve Eq. (5.112). Thus, with the generalized time expressed as a power of the scale x (i.e., t = x ) we find that... 

Khlebtsov, N. G. (2001) Orientational averaging of integrated cross sections in the discrete dipole method. Opt. Spectrosc., 90,408-415. 

The group of methods collectively called sectional methods are sometimes referred to as zero order methods, group methods, discrete methods or methods of classes. 

After discretizing in the property space coordinates, all the sectional methods take the following form ... 

The discrete fixed pivot method proposed by Kumar and Ramkrishna [112] is one of the most widely used sectional method due to its generality and robustness [52, 114]. Kumar and Warnecke [114] did analyzed the performance of several sectional methods and concluded that the fixed pivot technique predicts the first two moments of the distribution very accurately. However, the method consistently over-predicts the number density as well as its higher moments. 

The discrete pivot method provides information on the particle size distribution that is needed in the multi-fluid framework and can also be used for proper validation of the source term closures. However, in order to cover a broad range in the particle size distribution, a large number of sections are often needed making these algorithms rather time consuming. To ensure mass conservation, the smallest particles are generally not allowed to break and the largest particles are normally not involved in the coalescence process. 

Filbet and Laurengot [58] developed a particular finite volume method (FVM) scheme for dicretizing the Smoluchowski equation for purely coalescing systems. For the application of the FVM they established a continuous flux form of the PBE coalescence source terms. The FVM thus ensures that the poly-disperse particle fluxes between the individual sections are conserved in the system. This approach deviates from the conventional sectional methods which are applied to the standard discrete form of the PBE source terms. Kumar et al. [113] adapted the FVM scheme for solving the transformed coalescence source terms to pure breakage and simultaneous breakage and coalescence systems. 

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