## 1 Introduction  
With the rapid development of micro-electro-mechanical systems (MEMS), the demand for miniaturized power devices becomes increasing urgent. The present main power sources for miniaturized electronics and micro-propulsion systems are conventional electrochemical batteries [1]. However, there are many disadvantages for these batteries, such as the long recharging periods, the low energy densities and the short service life [2]. Owing to the higher energy densities of hydrogen and hydrocarbon fuels compared with batteries, micro-scale combustors were used as the heat source to provide power for miniaturized power systems, such as the micro-thermophotovoltaic (TPV) systems [3] or the thermoelectric (TE) systems [4]. Since 1991, when the concept of \"micro machinery\" was first proposed by [PERSON] and [PERSON] [5], various micro-power systems have been tested [6, 7, 8] and significant progress has been achieved in both basic and applied studies on micro-power systems and micro-combustion. Differing from macro-scale combustors, the combustor size is reduced to a millimeter and the increased surface-to-volume ratio intensifies the heat losses from the wall [9], making it difficult to sustain stabilized flames. Therefore, it is crucial to develop effective flame stabilization technologies for small-scale combustors. Many useful strategies have been suggested, such as the backward-facing step in cylindrical tubes [10; 11] to prolong the residence time and strengthen the mixing of the fuel mixture; catalytic combustion [12; 13] to suppress radical depletion; Swiss-roll combustor configuration [14]; and porous media combustion [15; 16; 17] to enhance heat recirculation. Two approaches are commonly employed in simulations to model the reaction and heat transfer in a porous combustor: the volume-averaged model (VAM) and pore-scale model (PSM). In theoretical analysis and numerical studies, the VAM is commonly applied to study porous media combustion. The VAM assumes a local thermal equilibrium between the gas and solid phases (the one-temperature model) [18; 19] or allows for a thermal nonequilibrium (the two-temperature model) [15; 20], which typically treats porous media as an equivalent fluid with similar thermophysical properties, and pore-scale geometric characteristics lumped into the equivalent fluid. The simplicity of the calculation model and lower computational expense of the VAM provide a foundation for understanding the global fluid flow and heat recirculation characteristics in porous media. Therefore, the VAM approach had made a substantial contribution to the design and operation of porous micro-combustors. Research results based on the VAM are greatly challenged by the lack of understanding of porous combustion due to the model's limitations. Although the VAM combined with the thermal nonequilibrium model can satisfactorily predict the global temperature distribution, detailed porous media structures are not used in the computational domain because detailed information at the pore-scale is filtered out [21]. A more precise numerical method, the PSM, is necessary to understand the combustion characteristics at the pore-scale and to investigate the flow [22], reaction and heat transfer [23] inside the porous media. [PERSON] et al. [23] performed a theoretical analysis of variable thermophysical features and the heat and mass transport phenomena of viscous fluid flow through a porous channel. The characteristic of the porous medium can be studied by Darcy-Forchheimer expression. [PERSON] et al. [24] pointed out that the PSM model is necessary when the pore size and channel height are comparable. [PERSON] and [PERSON] [25] studied filtration combustion in a porous medium made of discrete or connected square cylinders and found that the VAM predicted a higher burning velocity and lower superadiabatic temperature than the PSM. The limitation of the VAM was reflected in the powerless expression of detailed information in small spaces, which are comparable to the combustor characteristic length.