Estimate of the height of molten metal reactors for methane cracking

Methane Cracking represents one of the most promising routes to CO2-free hydrogen production.The methane decomposition reaction is typically carried out in fixed or fluidized catalytic beds, where the metal catalyst is supported on porous ceramic particles. By proper choice of the metal catalyst, the catalytic reaction environment allows to obtain sizeable reaction rates at operating temperatures as low as 700°C. Besides, in solid catalytic beds, the catalyst is swiftly deactivated due to the massive (i.e. stoichiometric) deposition of the solid carbon product. One way to bypass carbon deposition is to use a molten metal bath (which may or may not contain catalytic metal components) as a reaction environment, where methane bubbles are introduced at the bottom of the bath and are progressively converted as they rise through the liquid metal. The key point of this process is that, owing to a large density difference between the solid carbon phase and the molten metal, the solid product of the reaction floats on top of the liquid metal and can be thus mechanically skimmed. In this article, we develop an analytical approach to the estimate of the bath height, which constitutes one of the most critical design parameters of the process. Specifically, based on the observation that in practical applications the reacting bubble is in the kinetics-controlled regime, we obtain the conversion vs time solution for a bubble of given initial size. On the assumption of ideal gaseous mixture behaviour, the knowledge of the conversion curves allows to estimate the bubble diameter as a function of time during the rise of the bubble through the molten metal. This piece of information is then post-processed to obtain the bubble motion as a function of time. The elimination of the time parameter between the two solutions allows to construct a conversion-height map for different diameters of the bubbles.