Two-Fluid Model with Kinetic Theory of Granular Flow Closures

1 Two-Fluid Model with Kinetic Theory of Granular Flow Closures 

The moment method can then be employed to derive a generalized equation of change for a mean particle property 4 )m in the same manner as described in Chap. 2 for molecular systems. In particular, the generalized transport equation for 0)m is derived multiplying (4.1) by a microscopic quantity V ir, c, t) and integrating the resulting relation over the whole velocity space. 

Suppose that i) 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 thoroughly some further comments are needed. If we multiply (4.1) by i (t, 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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In the literature numerous two-fluid models of different complexity have been proposed to predict the fluidized bed reactor cold flow and reactive flow behaviors. Four decades ago emphasis was placed on the modeling of the velocity fluctuation co-variance terms in the dispersed particle fluid phase momentum equations. The early one-dimensional models were normally closed by an elasticity modulus parameterization for the particle phase collisional pressure and a constant viscosity parameter for the corresponding shear stresses. Later, with the improved computer memory and speed capacities, multi-dimensional flow models and more advanced model closures were developed based on the kinetic theory of granular flow (KTGF). Moreover, the... 

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