In response to the general existence of mesoscale structures in gas–solid fluidized beds, two versions of EMMS-based two-Fluid Model (EFM) (Hong et al., 2012; Wang et al., 2012a) have been proposed simultaneously from the viewpoint of continuum mechanics via different options of interpenetrating continua. A statistical foundation is however not available yet. To this end, an attempt was made to lay a unified statistical foundation of EFMs: four Boltzmann equations were used to describe respectively the kinetics of particles and gas molecules in dilute and dense phases, the governing equations of EFMs and their corresponding constitutive relations were derived theoretically. It was shown that (i) all governing equations and constitutive laws are structure-dependent; (ii) the solid and gas stresses include the kinetic stress, the phase-internal and interphase collisional stresses and the pseudo-Reynolds stress; and (iii) the interphase mass, momentum and energy transfer between the dilute phase and the dense phase can be quantified by assuming that collisions between the particles or gases from the dilute phase and dense phase have a certain probability to result in the mass transfer and vice versa.

为了应对气固流化床中尺度结构的普遍存在，同时提出了两种基于EMMS的两流体模型（Hong-et al。，2012; Wang et al。，2012a）。通过互穿连续体的不同选择来观察连续体力学。但是，尚无统计依据。为此，尝试建立了EFM的统一统计基础：使用四个Boltzmann方程分别描述了稀相和致密相中的粒子和气体分子的动力学，推导了EFM的控制方程及其对应的本构关系。从理论上讲。结果表明：（i）所有控​​制方程和本构律都与结构有关；（ii）固体和气体应力包括动应力，相内和相间碰撞应力和拟雷诺应力；（iii）可以通过假设来自稀相和致密相的粒子或气体之间的碰撞具有一定的概率导致质量传递，来量化稀相和致密相之间的相间质量，动量和能量转移。反之亦然。