Micro-PDF Transport Equations

In the Lagrangian framework derivation of a population balance equation or PDF transport equation, a number of particles or fluid elements is considered. The external coordinates of a particle are the spatial location z and the time t. The internal coordinates (p are volume or mass, composition, velocity, temperature,. .. — the so-called phase space. The velocity ris usually written separately from the other internal coordinates, with g = (v,. The distribution of the particles 

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 

By definition, the random variables u and Y are independent if their joint-PDF is the product of their so-called marginal PDFs  

For a function Q u, Y), the mean or expectation in point z, Q u, Y)) z), can be obtained from the first-order moment 

It reflects the macro-scale behavior of Q. In statistically stationary flow (Fig. 12.3-1), time averaging can be applied for the calculation of the mean, for example  

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