#### Filtered models for scalar transport in gas –particle ﬂows

We employ a kinetic-theory based two-ﬂuid model to develop a ﬁltered two-ﬂuid model for scalar transport in gas – particle ﬂows. The ﬁltering procedure gives rise to terms describing the ﬁltered interphase heat/mass transfer and ﬁltered scalar diffusion, which need to be constituted in order to close the ﬁltered transport equations. In this work, the closure for these terms is accomplished by performing ﬁne-grid simulations of the two-ﬂuid model in a two-dimensional periodic domain. Filtered scalar diffusion is investigated by imposing a lateral mean gradient in the scalar for each phase. Interphase energy/mass transport is investigated by prescribing a heat/species source (sink) in the solids (gas) phase, such that the energy/species content of the mixture is preserved. The variation of the ﬁltered transport coefﬁ cients with respect to ﬁltered particle volume fraction, and scaling with respect to ﬁlter size, ﬁltered scalar shear rate, and ﬁltered slip velocity is discussed. We ﬁnd the ﬁltered interphase heat transfer coefﬁcient to be as much as two orders of magnitude smaller than the microscopic interphase heat transfer coefﬁcient. The model for ﬁltered scalar diffusion is found to have a form very similar to that calculated for single phase turbulent ﬂows. We also calculate the ﬁltered Prandtl number for each phase.

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