In this work, the doctoral thesis of G.S.J. Sturm on microwave reactors was continued. Microwave reactors are developed, because they have potential to increase safety and economically reduce waste. This work focused on microwaves and chemical processes, instead of just heating water, by designing a travelling microwave reactor for non-oxidative methane dehydroaromatization. The travelling microwave reactor was designed using a structured approach. First, the objective was set to an economical conversion to aromatics of otherwise flared methane. Therefore, a successfully designed reactor is able to reduce the CO2 emissions and make many other reactions economical. Secondly, twenty-one challenges for microwave reactors were found using check-lists and a self-developed phenomena exploration method. This method was used to find unknown challenges on the intersection of established engineering fields. Then sub-solutions solving these challenges were extracted from existing reactors and reactor concepts. Finally, a few sub-solutions were selected and forced to work together to obtain the designed reactor. The designed reactor consists of a high performance coated asymmetric annular monolith in an inert container with narrowing conductors in axial direction and an anisotropic porous media. The achieved production volume is three orders of magnitude larger than of a mono-mode microwave reactor at same operation frequency. The designed reactor is capable of obtaining another five orders of magnitude by increasing the temperature up to 1500K, pressure up to 50 atm, catalyst activity with at least two orders of magnitude and lowering the operating frequency. However, only four orders of magnitude were reached, because the designed reactor hit a flow limit. The designed reactor with the three plus four order of magnitude improvement is not yet economical feasible. More advanced designs such as the spiralling narrowing rectangular microwave reactor might be. Furthermore, the designed reactor could become economically feasible in case of a more valuable product. Eventually, this work revealed some unsolved problems and opportunities of microwave reactors as well as information gaps. Moreover, it brought microwave reactors closer to industrial application.