Application of the energy-minimization multi-scale method to gas–liquid–solid fluidized beds

Abstract

A model for gas–liquid–solid three-phase fluidized beds with concurrent gas–liquid up-flow is proposed, which is formulated on the basis of the energy-minimization multi-scale (EMMS) method for gas–solid two-phase flow. The three-phase fluidization system is resolved into the suspending and transporting subsystem and the energy dissipation subsystem, and the former is further divided into three sub-subsystems: liquid–solid phase, gas phase and inter-phase. Force balance is analyzed at three different scales: micro-scale of particles, meso-scale of bubbles and macro-scale of the whole system. In addition to the analysis of multi-scale interactions, the energy consumption in the system is analyzed to establish the stability condition for the system, which is considered indispensable due to the multiplicity of three-phase fluidized beds. The total energy of the system consumed with respect to unit mass of particles is resolved into two portions: suspending and transporting energy and dissipated energy. The stability condition is reached when the suspending and transporting energy of the system, N st, is at its minimum. The model first formulated as a nonlinear programming problem consisting of six variables and seven constraints, is solved by using the general reduced gradient (GRG) algorithm. The calculated results show that the stability condition, N st=min, can be stated alternately as d b=d b max . Thus, the model is finally simplified to a set of nonlinear algebraic equations. The model has been used to calculate the hydrodynamic parameters in gas–liquid–solid fluidized beds with a wide range of physical properties of the liquid and the solid phases. The model predictions show good agreement with experimental data available in the literature.