Statistical foundation of EMMS-based two-fluid models for heterogeneous gas-solid flow

•Statistical foundation of EMMS-based two-fluid model.•Effect of heterogeneous structures on the constitutive laws was considered.•The interphase mass, momentum and energy transfer between dilute and dense phases. 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.