Full PDF models

The concept of the full PDF approaches is to formulate and solve additional transport equations for the PDFs determining the evolution of turbulent flows with chemical reactions. These models thus require modeling and solution of additional balance equations for the one-point joint velocity-composition PDF. The full PDF models are thus much more CPU intensive than the moment closures and hardly tractable for process engineering calculations. These theories are comprehensive and well covered by others (e.g., [8, 2, 26]), thus these closures are not examined further in this book. 

The effect of turbulence on scalars in the flow (c, T, reaction kinetics) is strong, and is sensitive to the details of the velocity and turbulence fields. Models that have been formulated to solve the combination of velocity and scalar fields have not yet accounted for the multiplicity of interactions between the fields, especially when complex reaction kinetics exist. Steady progress continues in the application of full PDF models to these problems. 

Particular attention is to be paid to closure models exploiting various types of PDFs such as beta, presumed, or full PDFs (e.g., Baldyga, 1994 Fox, 1996, 2003 Ranade, 2002). While PDFs have successfully been exploited for describing chemical reactions in turbulent flames, tubular reactors (Baldyga and Henczka, 1997), and a Taylor-Couette reactor (Marchisio and Barresi, 2003), they have never been used successfully in stirred reactors so far. 

Phenomenological models-EB, ESCIMO, multigroup E model PDF-based models presumed/ full PDF algorithm ... 

For the PDF model of the SMB process, full discretization was used, that is, both temporal and spatial variables were discretized leading to a huge system of algebraic equations. The standard SMB optimization problem has 33 997 decision variables and 33 992 equality constraints while the superstructure SMB optimization problem has 34 102 decision variables and 34 017 equality constraints. Note that there are many more degrees of freedom in the superstructure formulation (altogether 85) than in the standard SMB formulation (5 degrees of freedom). 

However, if the correlation matrix p is rank-deficient, but the scalar dissipation matrix is full rank, the IEM model cannot predict the increase in rank of p due to molecular diffusion. In other words, the last term on the right-hand side of (6.105), p. 278, due to the diffusion term in the FP model will not be present in the IEM model. The GIEM model violates the strong independence condition proposed by Pope (1983). However, since in binary mixing the scalar fields are correlated with the mixture fraction, it does satisfy the weak independence condition. The expected value on the left-hand side is with respect to the joint PDF (c, f x, t). 

Mathematically, the PPDF method is based on the Finite Volume Method of solving full Favre averaged Navier-Stokes equations with the k-e model as a closure for the Reynolds stresses and a presumed PDF closure for the mean reaction rate. 

Animal Models, Pages 138-140, Shayne C. Gad SummaryPlus Full Text + Links PDF (64 K)... 

The evaluation of the degrees of freedom of a nonlinear model built on a data set so close to full rank can only be possible if the degrees of freedom associated with each model can be estimated reliably. Van der Voet (1999) suggested a method of defining pseudo-degrees of freedom (pdf) based on the performance of a model, as in (14)... 

Probability provides a classical model for dealing with uncertainty (Halpem 2003). The basic elements of probability theory are a) random variables and b) events, which are appropriate subsets of the sample space Q. A probabilistic model is an encoding of probabilistic information that allows the probability of events to be computed, according to the axioms of probability. In the continuous case, the usual method for specifying a probabilistic model assumes a full joint PDF over the considered random variables. 

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