The Continuous General Dynamic Equation

Although (13.82) is a rigorous representation of the system, it is impractical to deal with discrete equations because of the enormous range of k. Thus it is customary to replace the discrete number concentration Nk(t) (cm-3) by the continuous size distribution function n(v,t) (pm-3 cm-3), where v = kv, with w, being the volume associated with a monomer. Thus n(v, t)dv is defined as the number of particles per cubic centimeter having volumes in the range from to t) + dv. If we let wo = g v be the volume of the smallest stable particle, then (13.82) becomes in the limit of a continuous distribution of sizes 

In (13.83), a stable particle has been assumed to have a lower limit of volume of wo-From the standpoint of the solution of (13.83), it is advantageous to replace the lower limits vo of the coagulation integrals by zero. Ordinarily this does not cause any difficulty, since the initial distribution n(v, 0) can be specified as zero for v vo, and no particles of volume v vo can be produced for t 0. Homogeneous nucleation provides a steady source of particles of size vq according to the rate defined by Jo(t). Then the full equation governing n(v, t) is as follows  

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An aerosol distribution can be described by the number concentrations of particles of various sizes as a function of time. Let us define Nk(t) as the number concentration (cm-3) of particles containing k monomers, where a monomer can be considered as a single molecule of the species representing the particle. Physically, the discrete distribution is appealing since it is based on the fundamental nature of the particles. However, a particle of size 1 pm contains on the order of 1010 monomers, and description of the submicrometer aerosol distribution requires a vector (N2, N-j,..., N10io) containing 1010 numbers. This makes the use of the discrete distribution impractical for most atmospheric aerosol applications. We will use it in the subsequent sections for instructional purposes and as an intermediate step toward development of the continuous general dynamic equation. 

This equation is called the continuous general dynamic equation for aerosols (Gelbard and Seinfeld 1979). Its initial and boundary conditions are... 

GENERAL DYNAMIC EQUATION FOR THE CONTINUOUS DISTRIBUTION FUNCTION... 

General Dynamic Equation for the Continuous Distribution Function 309... 

Unsolved fundamental problems of great practical importance remain in aerosol dynamics. In addition to the need for rapid chemical measurement methods mentioned above, much more research is required on the effects of turbulence on coagulation and nucleation the general dynamic equation must be extended to include factors that determine the crystal state of primary particles. We also need to continue efforts to Jink aerogel formation and aerosol dynamics as initiated by A. A. Lushnikov (Karpov Institute), Experimental and... 

In the absence of nucleation (Jq(v) = 0), sources (S(v) = 0), sinks (R(v) = 0), and growth [/( ) = 0], we have the continuous coagulation equation (13.61). If particle concentrations are sufficiently small, coagulation can be neglected. If there are no sources or sinks of particles then the general dynamic equation is simplified to the condensation equation (13.7). 

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

The structure of the system in Eq. (2.11) is formally very simple, although apart from the kinematic reversibility of the individual particle motions, which is a consequence of the time invariance of the quasistatic Stokes and continuity equations (Slattery, 1964), very little else can be said explicitly. Equation (2.11) would appear to pose a fruitful future study within the more general framework of dynamical systems (Collet and Eckmann, 1980) whose temporal evolution is governed by a system of equations identical in structure... 

A general solution of Stokes equations can be obtained by analytic continuation of the interstitial velocity and pressure fields into the interior of the regions occupied by the spheres, replacing the particle interiors by singular multipole force distributions located at the sphere centers R (Zuzovsky et al, 1983). Explicitly, (v, p) satisfies the dynamical equation... 

The general strategy of attack to the molecular dynamics outlined by this and earlier chapters appears especially promising to shed further light into this stimulating field of research In the special case of H-bonded liquids, a natural development of the ideias outlined here implies the replacement of the discrete variable t) with a continuous variable, which in turn involves the replacement of the master equation method with a suitable Fokker-PIanck equation. Moreover, this improvement of the theory is fundamental to exploring the short-time dynamics when the details of the correlation functions on the time scale of structure V of water must be accounted for. 

The prediction horizon is discretized in cycles, where a cycle is a switching time tshift multiplied by the total number of columns. Equation 9.1 constitutes a dynamic optimization problem with the transient behavior of the process as a constraint f describes the continuous dynamics of the columns based on the general rate model (GRM) as well as the discrete switching from period to period. To solve the PDE models of columns, a Galerkin method on finite elements is used for the liquid... 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

This relation enable us to simplify the formulation of the general equation of change considerably. Fortunately, the fundamental fluid dynamic conservation equations of continuity, momentum, and energy are thus derived from the Boltzmann equation without actually determining the form of either the collision term or the distribution function /. 

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