Single-particle joint PDF

The simplest useful representation of the fluid-particle system is the single-particle joint PDF /j" defined by integrating out all independent variables except those associated with the nth particle ... 

The Eulerian analogy of the Lagrangian single-particle joint-PDF is the one-point joint-PDF. For the one-point joint velocity-composition PDF, fuj, used in Section 12.4.2, t)dvdy/ is the probability of having a given... 

The evolution of a general single-particle joint-PDF f q>] z) depends on interactions between particles, so that f q)] z) is not sufficient to fully describe its own evolution (similar reasoning for the Eulerian framework single-point joint-PDF, the evolution of which depends on interactions with other points). Information is needed on the probability of having specific encounters between particles. Encounters between two particles are, for example, determined by the two-particle joint-PDF or pair density function, fi. By definition,... 

The multi-particle joint-PDF transport equation (12.4.1-9) is closed. The calculation and use of the high dimensional multi-particle Joint-PDF, Fn, is computationally not tractable. In practice, the low(er)-dimensional single-particle joint-PDF, f, is used instead ... 

The transport equation for the NDF (12.4.1-1) is similar to (12.4.1-11) and obtained by summation of all the single-particle joint-PDF transport equations (12.4.1-11). In (12.4.1-11), the first term represents convection, the second the conditional acceleration, and the third the single-particle state conditional continuous changes in the internal coordinates other than the velocity. The rate of continuous changes of property in the phase space of

internal coordinates due to collisions are described by the conditional source terms B- D) ) on the right hand side of (12.4.1-11). 

The integration of the single-particle joint-PDF transport equation (12.4.1-11) is tedious. Computer requirements for standard CFD techniques rise exponentially with the dimensionality of the joint-PDF. Therefore, micro-PDF methods commonly use a Monte-Carlo approach [Spielman and Levenspiel, 1965 Kattan and Adler, 1967, 1972 Pope, 1981]. A deterministic system is constructed with stochastic particles whose joint-PDF evolves in the same way as the joint-PDF of fluid particles. The trajectories of the so-called conditional particles define a formal solution of the joint-PDF transport equation (12.4.1-11). Ramkrishna [2000] presents details on the computational methods. 

All of the age distribution formalism could be derived by starting from the single-particle joint-PDF transport equation (12.4.1-11) or the macroscopic population balance equation (12.6.1-3). Indeed, the only property of interest is the age, so that = a, the age. Now intervals of age are also intervals of clock time, so that Q = daldt= 1 and (12.6.1-3) reduces to ... 

The formal definition of the NDF given in Eq. (4.11) is mathematically consistent, but difficult to implement in practice. It is therefore useful to define methods for estimating the NDF from a single realization of the granular flow. Note that mathematically a statistical estimate is a random variable, and thus should not be confused with the NDF, which is deterministic. In order to distinguish the estimated NDF from n, we will denote the estimate by h. Thus, for example, if the estimate is unbiased then (n) = n, where the expected value is taken with respect to the multi-particle joint PDF / defined in Eq. (4.7). 

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

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