Transversal moving-front patterns: Criteria and simulations for two-bed and cylindrical shell packed-bed reactors

Abstract

We show that a moving-front solution in a cylindrical shell packed-bed catalyzing a first-order activated reaction may bifurcate into transversal patterns when Pe C / Pe T < Δ T ad / Δ T m , i.e. when the ratio of the mass to heat Pe numbers is smaller than the ratio of the adiabatic to maximal temperature rises. This coincides with the previous condition of transversal patterns to emerge in stationary fronts [ Pe C / Pe T < 1 [Viswanathan, G., Bindal, A., Khinast, J., Luss, D., 2005. Stationary transversal hot zones in adiabatic packed-bed reactors. A.I.Ch.E. Journal 51, 3028–3038]] and extends the bifurcations condition to the case of moving fronts. The novel condition cannot be satisfied in a downstream propagating front ( Δ T m / Δ T ad > 1 ) , but for an upstream propagating front (toward the cold reactor inlet) Δ T m / Δ T ad < 1 and the symmetry breaking can be obtained within a feasible domain of operating conditions ( Pe C / Pe T > 1 ). It was also assumed that the axial and the transversal Pe numbers vary consistently, i.e. κ C = Pe C ⊥ / Pe C = κ T = Pe T ⊥ / Pe T . A similar condition was also obtained using a simplified model composed of two 1-D beds with heat and mass exchange between them. Bifurcation diagram showing domains of transversal patterns is constructed using a learning two-bed model. These predictions are verified by direct numerical simulations of the continuous 2-D cylindrical shell model showing various types of moving transversal patterns within a feasible domain of the state parameters with Pe C > Pe T . In the case of varying ratio ( κ C ≠ κ T ) the pattern domain can be significantly extended toward larger Pe C / Pe T .