Influence of non-uniform activity and conductivity on stationary and moving patterns in catalytic reactors

We examine pattern formation in a two-phase system consisting of a catalytic wire or ribbon or a string of catalyst particles suspended in a well-mixed fluid phase in an adiabatic reactor. We assume that a single exothermic reaction of the type A→BA→B occurs in the solid phase and account for the variation in the interphase heat and mass transfer coefficients with the flow rate of the reactants. First, we examine the case of uniform activity, conductivity and diffusivity in the solid phase and use linear stability theory to show that steady non-uniform patterned states emerge from the unstable middle branch of the spatially uniform state. Unlike the Turing patterns, these transport limited patterns exist only in the region of multiple homogeneous (uniform) states. They always emerge as unstable branches but may gain stability through a limit point bifurcation when the fluid Lewis number (ratio of fluid thermal diffusivity to the mass diffusivity of the reacting species) Lef<1Lef<1 but the solid phase Lewis number (ratio of solid thermal diffusivity to the effective species diffusivity in the solid) can be larger than unity. For a fixed set of kinetic constants, the range of Damköhler numbers or residence times over which stable patterned states exist increases with increased interphase heat and mass transfer resistances and with decreased intraphase conduction in the solid. Local bifurcation analysis and transient simulations show that when the activity, conductivity and diffusivity in the solid phase are constant, only moving temperature and concentration fronts exist when Lef≥1Lef≥1, while both stationary and moving fronts exist when Lef<1Lef<1 for the practical range of other parameters. Next, we examine pattern formation with non-uniform activity but uniform conductivity and diffusivity within the solid and show that both stationary and moving patterns can exist for any value of LefLef. We also determine the influence of the length scale and magnitude of the heterogeneity in activity on pattern formation. The last case examined is that of non-uniform activity, conductivity and diffusivity within the solid phase with profiles of these corresponding to the case of discrete particles in a string with point contacts. In this case, we show that stationary patterns can exist for any LefLef and realistic values of all other parameters. This last result explains the differences between various literature results on pattern formation using discrete and continuum models with uniform properties. It also shows that one of main reasons for stationary pattern formation in catalytic packed-bed reactors is the spatial non-uniformity in activity and conductivity caused due to discrete nature of the solid phase (particles) with point contacts.