## 1 Introduction  
Over the past decades, a trend has been set in the miniaturization of mechanical devices [1]. In line with this trend, combustion of high-energy density fuels, such as hydrocarbons and hydrogen, in micro-reactors has attracted growing interest for applications in portable power generation and in process intensification for improving energy efficiency [2; 3]. The heat evolved from combustion can be converted into electricity by coupling with thermoelectrics [4], thermophotovoltaics [5], or thermoionics [6] for use in portable devices. Furthermore, it can be utilized to conduct endothermic reactions, such as in the case of the steam reforming of methane, by close thermal coupling with combustion [7]. Despite its benefits, miniaturization leads to several issues [8]. For micro-combustors, one of the most challenging issues is stability [2; 3]. Micro-combustors lose stability via two primary mechanisms, extinction and blowout [9]. As the scale of the device decreases, the surface-to-volume ratio increases, whereas the residence time decreases and, thus, heat losses increase, adversely impacting stability. In the case of extinction, most of the heat is lost through the walls to the surroundings. In the case of blowout, most of the heat is lost in the form of hot exhaust gases. Extinction occurs when heat losses lower the temperature of the system to values at which combustion does not self-sustain, i.e., when the input power is insufficient compared to the power lost from the system. Blowout, on the other hand, occurs when there is insufficient pre-heating of the reactants to ensure ignition and stabilization of combustion, i.e., when there is insufficient residence time compared to the chemical reaction time. Under the conditions of the sub-millimeter confinement of micro-reactors, homogeneous combustion is especially susceptible to quenching [10]. Hence, catalytic combustion,with its lower operating and light-off temperatures, is the focus area for research in the field of micro-power generation systems [11]. Indeed, the catalytic coating of the walls of the micro-combustors, even if only partial [12; 13; 14], favors their ignited state. Although to a lesser extent than homogeneous micro-combustors, catalytic micro-combustors still suffer from stability issues and, thus, even for them, the non-trivial task of minimizing heat losses is crucial to the development of more robust configurations [15; 16; 17; 18]. A key approach to minimize the detrimental effects of heat losses in combustion systems is based on recovering heat from the reaction zone, i.e., on recirculating part of the sensible heat of the hot exhaust gases to pre-heat the cold incoming reactants. This concept was first proposed by [PERSON] and [PERSON] [19] for conventional-sized combustors as a means to burn fuels with very low heat content in a double-spiral counter-current heat-recirculating geometry called \"Swiss roll\". Due to the large ratios of internal heat exchange area to external heat loss area, the Swiss-roll configuration causes \"excess-enthalpy\" combustion, i.e., large super-adiabaticity, in both macro-scale [20] and micro-scale [21] systems. Excess-enthalpy reactors other than the Swiss-roll configuration have been proposed and also investigated in the context of homogeneous micro-combustion. They are the porous-media [22], reverse-flow [23], and heat-recirculating [24; 25] reactors, the latter often being simplified Swiss-roll configurations. In contrast to excess-enthalpy combustors, standard straight-channel combustors exhibit worse stability, the only mechanism of heat recirculation on which they can rely being the mechanism of axial heat conduction in the upstream direction along the solid walls [24]. The concept of excess-enthalpy combustion has also been transferred to catalytic micro-reactors through studies carried out by different research groups and mainly focused on heat-recirculating configurations [26; 27; 28; 29].