Fine grid resolution is necessary to fully demonstrate the accuracy of two-fluid model(TFM) in predicting gas-solids flows in the riser section of circulating beds.Insufficient grid resolutions will likely lead to incorrect solutions.In most industrial applications involving large devices

it is impractical to use fine grids

and therefore

constitutive models that are suitable for the coarse-grid TFM simulations are required.It has been reported in some literatures that microscopic drag laws

such as those due to Gibilaro and Gidaspow

did not yield grid-independent solutions over the range of grid sizes investigated in some test problems.Filtered two-fluid models were developed using the results of highly resolved simulations.A filtering procedure was applied to the results of these simulations to deduce the constitutive models for the residual correlations appearing in the corresponding filtered TFM equations.These equations were appropriate for coarse-grid simulations of gas-particle flows.Using the same example studied by Mckeen and Pugsley(2003)

it was first demonstrated that grid-independent predictions could be afforded by a filtered TFM

in which filtered drag and particle phase stress models were included.Then grid-independent and experimentally consistent solutions could also be obtained with filtered TFM model when filtered particle phase stress model was replaced with kinetic theory model as long as filtered fluid-particle drag force model was used.