The Two-Fluid Granular Flow Model

The two-fluid granular flow model is formulated applying the classical Eulerian continuum concept for the continuous phase, while the governing equations of the particle phase are developed in accordance with the principles of kinetic theory. In this theory it is postulated that the particulate system can be represented considering a collection of identical, smooth, rigid spheres, adapting a Boltzmann type of equation. This microscopic balance describes the rate of change of the distribution function with respect to position and time. 

The moment method can then be employed to derive a generalized equation of change for a mean particle property ip in the same manner as described in chap 2 for molecular systems. In particular, the generalized transport equation for ip r, t) is derived multiplying (4.1) by a microscopic quantity ip r, c, t) and integrating the resulting relation over the whole velocity space. 

Suppose that ip and / are expressed as functions of (r, c,t), then a mathematical moment can be defined by the following ensemble average  

To examine the derivation of the macroscopic equation throughly some further comments are needed. If we multiply (4.1) by r, c) and thereafter apply the chain rule to re-write the terms on the LHS, the intermediate result is  

Integration over the whole velocity space yields  

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Implicit Fractional Step Method for Solving the Two-Fluid Granular Flow Model Equations Applied to Fluidized Bed Flow... 

Step algorithm was employed for the solution of an unsteady two-fluid granular flow model [136], The latter semi-implicit procedure has also been extended in order to simulate reactive flows in fluidized beds [138],... 

Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

In the first coarse-grained approach, the discrete phase is treated as an Eulerian continuum, interpenetrating with the real continuous phase. The particle-particle interactions are then captured by an effective particle phase rheology obtained from kinetic theory of granular flows. These so-called two-fluid (Euler-Euler) models have been very successful at predicting the dynamic properties of, e.g., gas-solid fluidized beds (see Van derHoefet al, 2008 Verma et al, 2013). Despite their success, two-fluid models also have their limitations they are usually limited to idealized cases ofmonodisperse hard sphere particles, while extensions to polydisperse mixtures (e.g., in size or in contact properties) are difficult to make. Also, because no particles are explicitly tracked, it is difficult to include particle properties which may vary from particle to particle, such as particle temperature, surface moisture concentration, or chemical surface species concentrations. 

Chapter 10 contains a literature survey of the basic fluidized bed reactor designs, principles of operation and modeling. The classical two- and three phase fluidized bed models for bubbling beds are defined based on heat and species mass balances. The fluid dynamic models are based on kinetic theory of granular flow. A reactive flow simulation of a particular sorption enhanced steam reforming process is assessed. 

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