  Using the kinetic-theory-based two-fluid models as a starting point, we
develop filtered two-fluid models for a gas-particle flow in the presence of an
isothermal, first-order, solid-catalyzed reaction of a gaseous species. As a
consequence of the filtering procedure, terms describing the filtered reaction
rate and filtered reactant dispersion need to be constituted in order to close
the filtered species balance equation. In this work, a constitutive relation
for filtered reaction rate is developed by performing fine-grid, two-fluid
model simulations of an isothermal, solid-catalyzed, first-order reaction in a
periodic domain. It is observed that the cluster-scale effectiveness factor,
defined as the ratio between the reaction rate observed in a fine-grid
simulation to that observed in a coarse-grid simulation, can be substantially
smaller than unity, and it manifests an inverted bell shape dependence on
filtered particle volume fraction in all simulation cases. Moreover, the
magnitude of the deviation in the cluster-scale effectiveness factor from unity
is a strong function of the meso-scale Thiele modulus and dimensionless filter
size. Thus coarse-grid simulations of a reacting gas-particle flow will
overestimate the reaction rate if the cluster-scale effectiveness factor is not
accounted for.
