Beyond the effectiveness factor: Multi-step reactions with intraparticle diffusion limitations

Though common in many industrial chemical processes, multi-step reactions occurring within porous catalyst particles provide significant challenges to previously established modeling frameworks. In this work, a system of reaction-diffusion equations is solved analytically via eigenanalysis. The intraparticle concentration profiles are integrated over the particle volume to obtain effectiveness factors for each reacting species, referred to herein as a multi-step effectiveness vector (MEV). The MEV solution is tested for the case of uniform flow past a stationary catalyst particle with a state-of-the-art, multi-step reaction scheme for fluid catalytic cracking (FCC). Outputs from particle-resolved direct numerical simulation (DNS) are compared to outputs from ordinary differentials with effective reaction closures coming from the MEV solution, the classic single-step effectiveness factor (SEF) (R. Aris, “The mathematical theory of diffusion and reaction in permeable catalysts: The theory of the steady state,” Oxford University Press, 1975), and utilizing the intrinsic rates as-is. When compared to DNS, the MEV solution shows excellent agreement for the formation of all species and quantitatively captures the intraparticle concentration profiles. By contrast, the intrinsic treatment shows an over-prediction for product formation and the SEF treatment shows mixed levels of accuracy that depend upon the reaction coupling. Specifically, the SEF treatment accurately characterizes product formation if the kinetic scheme is weakly coupled but shows significant departure if stronger coupling is present. The computational cost associated with the MEV solution is associated with eigendecomposition of the Thiele matrix, an N×N matrix where N is the number of reacting components. Overall, the method is shown to be relatively easy to implement and may significantly enhance the fidelity of chemical reactor simulations that employ multi-step reaction schemes.