Analytic prediction of the effective reaction rate for methane cracking in molten catalysts: Transition from kinetics-dominated to diffusion-limited regimes

Molten catalytic media are increasingly attracting interest as a suitable reaction environment where cracking of hydrocarbons and separation of the solid carbon product can be achieved simultaneously, ideally avoiding catalyst regeneration. In conditions relevant to industrial production, multiscale approaches are needed to predict the reactor performance, where the macroscopic dynamics of two-phase flow is combined with the evolution of the reaction inside individual bubbles. We here develop a thermodynamically-consistent, analytically solvable model for predicting the effective reaction rate resulting from the interplay between diffusive transport within the bubble and the surface reaction at the gas/liquid interface. The analytical solution is based on a spectral (eigenvalue/eigenfunction) expansion of the concentration profiles. We show how in practically affordable conditions, the effective characteristic time of the reaction can be estimated based only on the dominant eigenvalue of the diffusion-reaction problem. The closed-form knowledge of the dominant eigenvalue provides key insight in establishing the range of existence of diffusion-limited, mixed, and kinetics-controlled regimes. Based on the limiting solutions of the model, an approximate model for the conversion, yielding the explicit dependence of conversion on the equilibrium, kinetics and transport parameters is derived. The approach proposed is validated by comparing its predictions with the solution of the fully-coupled mass/momentum local balance equations obtained through a finite-element commercial solver enforced on a moving boundary domain to account for the bubble expansion due to the increasing number of moles in the gas phase.