Experimental flow in various porous media and reconciliation of Forchheimer and Ergun relations

Flow characteristics and pressure drop in traditional porous media, e.g., packed beds of spheres, and in modern man-made fibrous media, e.g., metal foam, are critical in many naturally-occurring and engineered applications. Pressure drop parameters such as permeability and form/inertial drag coefficients reported in the literature are very divergent for both classes of porous media. The choice of an appropriate characteristic length; and the selection of a way for correlating pressure drop data have also varied among researchers. In the current study a large set of experimental data for pressure drop of water flow in three different porous media was collected. The porous media were packed spheres of 1. mm, packed spheres of 3. mm and aluminum foam having 20 pores per inch. The porosity of both sets of packed spheres was practically the same at about 35%, while the porosity of the foam was 87.6%. The internal structure of the two classes (packed spheres and foam) of porous media investigated here are markedly different. The range of flow velocity covered Darcy, Forchheimer and turbulent flow regimes. It is shown that the same porous medium exhibited different values of permeability in different flow regimes. The widely-used equations of Ergun and Forchheimer for the post-Darcy regimes were revisited. An apparent difference between the two famous equations was presented and explained. The two equations were reconciled using the hydraulic radius theory, and the fact that the same porous medium exhibits different values of its permeability in different flow regimes. The multipliers of the viscous term and the inertial/form drag term in the post-Darcy regimes were shown to be connected. The square root of the permeability determined in the Darcy regime is shown to be appropriate length scale for defining and correlating the friction factor and the Reynolds number.