Transport limited pattern formation in catalytic fluid–particle systems

Abstract

We develop and analyze a two-phase model of a system of n catalyst particles in an adiabatic reactor where the fluid phase is well mixed and a single exothermic reaction of the type A → B occurs on the solid phase. In contrast to prior models, we include the flow rate or velocity dependence of the inter-phase heat and mass transport coefficients. This eliminates many non-physical solutions and leads to a unique and globally stable uniform state in the limit of very small or large residence time. The stability of the uniform state of the particles with respect to non-uniform perturbations in the solid and fluid phases was examined to determine the possibility of non-uniform states in the solid phase (for finite residence times). Linear instability theory revealed that when transport resistances are significant, steady non-uniform patterned states emerge from the unstable middle branch of the uniform state of the particles. In contrast to the Turing patterns, these transport limited patterns exist only in the region of multiple homogeneous states and always emerge as unstable branches but become stable after a limit point bifurcation. Moreover, the stable branches emerge from the unstable branches after going through a hysteresis on the patterned branch. For n = 2 , the non-uniform states appear sub-critically near the ignition point and super-critically near the extinction point, while, for n > 2 , the bifurcation to non-uniform state is either super-critical or transcritical near the ignition point and sub-critical or transcritical near the extinction point. For a fixed set of kinetic constants, the range of Damköhler numbers over which stable patterned states exist increases with increased heat and mass-transfer resistances between the phases or with decreased heat conduction between the solid particles. For the case of n particles ( n ⩾ 2 ) , the number of patterned branches emerging from the unstable homogeneous branch can be as large as ( n - 1 ) while the number of stable patterned states can be even larger (and increases exponentially with n). Some major results of this work are: (i) development of a model that tends to the thermodynamic and flow limits for the case of large and small residence times, respectively; (ii) physical interpretation of the meaning of unstable and stable patterned states (e.g. unstable patterned states describe the non-uniform perturbations that lead to transients with propagating temperature and/or concentration fronts where the entire system attains the ignited or quenched homogeneous state while stable patterned states correspond to “propagation failure” of localized perturbations); (iii) for Le f < 1 , the temperature of the solid in the patterned state can exceed not only the adiabatic temperature but also the temperature of the solid in the uniform case; and (iv) estimation of the critical heat conduction values in the solid at which stable and unstable patterned states disappear.