**4.1 Mathematical models**

As we all know, the physical aspects of fluid flow are governed by three fundamental principles (1) mass is conserved; (2) Newton's second law; (3) energy is conserved. So N-S equations are formed.

To this paper, the suitable mathematical models were selected according to experimental conclusions. As we all know, the Reynolds Number values of gas and granule in dense phase gas solid flow are both high than that of lean phase, which lead to more turbulent motion. Therefore, the addition of solid particles greatly changed gas phase turbulent construction and meanwhile gas fluctuation affect particle motions. So the interaction between the two phases leads to the mass, momentum and energy transmission.

When the solid concentration is high enough, interaction among particles affect solid flow characteristics greatly. Consequently, the interaction among particles should be given in simulation process in addition to gas solid interaction and turbulent as the gas solid two phase flow turbulent model being set up.

Two-fluid model which based on granule dynamics theory was adapted in the study. Flow parameters such as macroscopic granule transport equations, solid pressure, viscous coefficient, diffusion coefficient, heat conductivity coefficient, granule temperature etc can be obtained through the model. This model was used comprehensively in several fields because it mentioned interation action of gas solid flow, granule turbulent viscosity and particle collision roundly.

In this study, by using k-ε two phase model, granule dynamics model and gas solid two phase coupled, the gas solid two phase turbulent model were built up. Some reasonable assumptions about the flow process must be given as following.

