Protonic Ceramic Fuel Cells (PCFCs) are an emerging mid-temperature fuel cell technology that specializes in electrochemically converting chemical energy into electrical energy. These PCFCs have been suggested as a future technology for flare mitigation through the cogeneration of power and useful chemicals. PCFCs are here proposed to accomplish this purpose by utilizing the wasted methane and C2 components at these flaring sites to produce electricity, hydrogen, and aromatics. In this study, a numerical methane-fed protonic ceramic fuel cell model is developed utilizing recent advancements in PCFC fabrication, innovation, and experimentation. The model used a tubular PCFC geometry and implements mass and energy balances, as well as electrochemical, and kinetic equations solved using Engineering Equation Solver (EES) to predict the viability of the PCFC system. The system exhibits a very small power density on the order of ~0.01 W/cm^2, which is much lower than other fuel cells due to the chosen catalyst prioritizing the slower kinetics of methane dehydroaromatization. However, the results also indicate that the production of value-added aromatics allow the system to potentially be very economically friendly if the manufacturing costs can be brought down to 65% of their current costs. The realistic PCFC model is also compared and measured against competing technologies and is found to be competitive with current power production practices. The results of this study highlight the potential of PCFC technology to transform wasted energy into economic and environmental gains, offering a significant step forward in hydrogen-based sustainable energy practices.