Particle-fluid system energy analysis

This section will first deal with the phases in particle-fluid two-phase flow by developing a mathematical model to quantify local hydrodynamic states. This analysis will reveal the insufficiency of the conditions for the conservation of mass and momentum alone in determining the hydrodynamic states of heterogeneous particle-fluid systems, and calls for a methodology different from what is used in analyzing dilute uniform flow. For this purpose the concept of multi-scale interaction between particles and fluid and the principle of energy minimization are proposed. 

In effect, such a multi-scale analysis resolves a macro-scale heterogeneous system into three meso- to micro-scale subsystems—dense-phase, dilute-phase and inter-phase. Thus, modeling a heterogeneous particle-fluid two-phase system is reduced to calculations for the three lower-scale subsystems, making possible the application of the much simpler theory of particulate fluidization to aggregative fluidization and the formulation of energy consumptions with respect to phases (dense, dilute and inter) and processes (transport, suspension and dissipation). 

As known fluid molecules perform a thermal motion, whose kinetic energy is transfered to the suspended particles. This leads to non-directional motion of the particles themselves. For the sedimentation analysis only the influence of the diffusion mechanism in the direction of sedimentation is relevant, since in a multiparticle system with homogenous concentration distribution over the cross section of the vessel merely the concentration changes along the sedimentation height have an effect on the measuring result. Mathematically the problem is treated by the solution of the Fick s differential equation... 

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