## 1 Introduction  
Air flow through porous media is universal in a broad range of engineering activity, which includes such diverse fields as pneumatics, filtration, soil mechanics, and petroleum engineering. Pressure drops occur when such porous media are connected in an air circuit because they involve extremely complicated flow channels. Thus, accurate prediction of the correlation of pressure drop with flow rate is crucial for performance evaluation, since it determines the energy requirements of the supplying compressors. Some researchers have demonstrated the physics of flow through porous media. They modeled the flow behavior by a linear relation of pressure gradient versus flow velocity (Darcy regime) in cases where the flow is dominated by viscous effects. When the flow velocity becomes adequately large, they used a nonlinear relation to include the inertial effects [1; 2; 3; 4; 5; 6]. Some researchers attempt to use theoretical methods to correlate with experimental data. The two best-known theoretical relations are the Forchheimer equation [7; 8] and Ergun or Ergun-type equations [9; 10]. Both are widely used [8], and there is no specific reason for choosing one rather than the other. Early works by [PERSON] et al. [11; 12] reported that the flow pressure characteristics of porous media can be represented by a dimensionless expression related to the square root of the permeability. [PERSON] et al. [13] extended the applicability of an existing correlating equation to predict the pressure drop through packed beds of spheres. [PERSON] et al. [14] proposed a more precise method based on curve fitting to calculate the permeability and inertia coefficient of porous matrices in terms of a Forchheimer flow model. [PERSON] and [PERSON] (2015) found that the permeability for the Darcy regime differs from that for the Forchheimer regime. [PERSON] (2016) developed an Ergun-type empirical equation to correlate the dimensionless pressure drop with flow velocity for several types of foam matrix porous media. [PERSON] and [PERSON] (2017) correlated the permeability and the form drag coefficient using an Ergun model with the reciprocal of the surface area density, which has a unit of m\({}^{3}\)/m\({}^{2}\), indicating that it is used as an equivalent length scale. [PERSON] et al. (2018) presented experimental data of pressure drop measurements of different ceramic sponges and determined the two constants of the Ergun equation, which are verified to be independent of the material and porosity. The Forchheimer and Ergun equations look very similar in that they both have a viscous term and an inertia term. The difference is that the Forchheimer equation uses an analogy with pipe flow while the Ergun equation models the space between packed beds of spheres as parallel capillaries. An overview of the literatures is provided in Table 1.  
Air compressibility is negligible and flow velocity can be treated as a constant through porous media with a large porosity (over 0.8) ([PERSON] and [PERSON], 2011; [PERSON] and [PERSON], 2012; [PERSON] and [PERSON], 2013). Existing studies ([PERSON] and [PERSON], 2013; [PERSON] and [PERSON], 2013; [PERSON] and [PERSON], 2013; [PERSON] and [PERSON], 2013) proved that the Darcy-Forchheimer-based theory can be applied to both compressible and incompressible fluids. [PERSON] et al. (2012, 2013) measured considerable pressure differences during the operation of compressed air bearings equipped with sintered bronze porous pads. [PERSON] et al. (2014) predicted the pressure-flow relationship of the aerostatic porous bearing based on the quadratic Forchheimer assumption. [PERSON] et al. (2015), [PERSON] et al. (2016), and [PERSON] et al. (2017) have developed air conveyor systems with porous air supply pads. In these applications, the pressure drop is significant (400 kPa or more) and the air compressibility needs to be considered.