Advection phase space

This PBE is written in a general form and contains the terms representing accumulation, real-space advection, phase-space advection, phase-space diffusion, and second-, first-, and zeroth-order point processes. (See Chapter 5 for more details on these processes.) Let us... 

Equation (2.89) is a generalized advection equation stating that / remains unchanged if the observer moves along with the system points in phase space. 

The PBE is a simple continuity statement written in terms of the NDE. It can be derived as a balance for particles in some fixed subregion of phase and physical space (Ramkrishna, 2000). Let us consider a finite control volume in physical space O and in phase space with boundaries defined as dO. and dO., respectively. In the PBE, the advection velocity V is assumed to be known (e.g. equal to the local fluid velocity in the continuous phase or directly derivable from this variable). The particle-number-balance equation can be written as... 

In general, the phase-space advection terms are modeled as the sum of contributions due to pure advection and to phase-space diffusion (Gardiner, 2004) ... 

In Section 4.3, example macroscale transport equations are derived for selected moments of the NDF. Having introduced the precise forms of the mesoscale advection models in Eq. (5.2), it is of interest to derive explicitly some example moment source terms resulting from these models. In order to do so, we will use the rules presented in Section 4.3.1 for phase-space integration. For simplicity, we consider only the advection term involving (Afp)i and assume that the only phase-space variables of interest are v and Vf, and that the model in Eq. (5.2) reduces to... 

Phase-space advection mass and heat transfer... 

The rate of change of particle size is also indicated as Gp and can be positive, in the case of growing particles, or negative, in the case of shrinking particles. This is probably the most popular way to indicate the rate of phase-space advection due to mass exchange, perhaps because it is quite easy to measure the change in particle size at different instants, for example by simple imaging techniques. If the internal coordinate is instead particle volume (i.e. = Vp), the definition becomes... 

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... 

Phase-space advection mass and heat transfer where the Peclet number is defined by... 

By using these values, it is possible to estimate the characteristic rate of change for the internal coordinates, which can in turn be used to define characteristic time scales for phase-space advection for each internal coordinate = [ i/ i( i),..., m/ m( m)]-Analogously for diffusion a characteristic time scale is easily defined tdjj = jlDij. Also for the source term for point processes some characteristic time scales can be defined. For... 

The mixed advection term in Eq. (B.4) involves both a spatial and a phase-space gradient of the NDF, and thus it is more complicated than the pure advection term described in Section B.3 of this appendix. Nevertheless, by working with the transport equation for the moments, we can show that it can be treated like an advection term with special properties. 

By removing the phase-space integration, Eq. (B.46) can be rewritten as a ki-dependent advection term for the NDF ... 

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