A few words on the spatially homogeneous PBE

Before looking at example homogeneous systems, it is useful to analyze when the hypothesis of spatially homogeneity can be used for a PBE. For a PBE, the internal coordinates do not include the particle velocity. For example, consider the following PBE  

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 (7.143) can be made dimensionless by a proper choice of characteristic length and time scales. It is easy for a specific system to identify a characteristic length L and a characteristic velocity 1/ resulting in a characteristic advection time tm = L/ Z/. Once the steady state has been reached, it is straightforward to calculate average values for the internal coordinates  

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 

It is important to stress here that in Eq. (7.145) there are explicit dependences on spatial coordinates in the normalized NDF F and fluid velocity V, but also implicit dependences on the rates of change of internal coordinates the rate of formation of the disperse phase J, and the kernels jS and b. In fact, as described in Chapter 5, these rates and the kernels depend on the flow properties, which change from point to point in the system. 

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