External flow porous media

This model assumes the diffusive flows combine by the additivity of momentum transfer, whereas the diffusive and viscous flows combine by the additivity of the fluxes. To the knowledge of the authors there has never been given a sound argument for the latter assumption. It has been shown that the assumption may result in errors for certain situations [22]. Nonetheless, the model is widely used with reasonably satisfactory results for most situations. Temperature gradients (thermal diffusion) and external forces (forced diffusion) are also considered in the general version of the model. The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes can be added to the diffusion fluxes in the gaseous phase. 

A fluid s motion is a function of the properties of the fluid, the medium through which it is flowing, and the external forces imposed on it. For onedimensional steady laminar flow of a single fluid through a homogeneous porous medium, the relationship between the flow rate and the applied external forces is provided by Darcy s law ... 

DGM visualises the porous medium as a collection of giant spherical molecules (dust particles) kept in space by external force. The movement of gas molecules in the space between dust particles is described by the kinetic theory of gases. Formally, the MTPM transport parameters and qr can be used also in DGM. The third DGM transport parameter characterises the viscous (Poiseuille) gas flow in pores. 

P. Cheng, Natural Convection Porous Medium External Flows, in Natural Convection Fundamentals and Applications, S. Kakac, W. Aung, and R. Viskanta eds., pp. 475-513, Hemisphere Publishing, Washington, DC, Springer-Verlag, Berlin, 1985. 

Darcy-like flows obtain except in a thin boundary layer near solid walls, where Darcy s equation is unable to satisfy the no-slip boundary condition [cf. Brinkman s treatment (B35) of flow through a porous medium bounded externally by a circular cylinder with solid walls]. 

When the porous medium saturated with the electrolyte is embodied in an external electric field E, there appears a nonzero volumetric body force within the Debye layer, which sets the ions in that region into motion. Far from the particle surfaces, this volumetric force is zero, since the solute there is neutral. However, the electrolyte is brought into motion also in the latter region as a result of the solute s viscosity. These processes lead to the appearance of an interstitial flow velocity field u(R). This velocity field, when integrated over a representative volume of the porous medium, yields a nonzero seepage velocity U in the absence of any macroscopic pressure gradient applied to the porous medium. This process is called electro-osmosis, and the velocity U is called the electro-osmotic velocity. 

On the other hand, when a macroscopic pressure gradient VP is applied to the porous medium, the fluid percolates through it with a Darcy velocity U. Additionally, the electrolyte flowing in the interstices affects the equilibrium ion distribution within the Debye layer, so that these ions are also set into motion. This results in a macroscopic electric current density I flowing through the porous medium in the absence of any external electric field. 

Most of the liquid filtration operations follow the mechanism of cake filtration. As the filtration proceeds, the particles retained on the filter medium will form a growing cake with porous structure. The small particles that are able to pass through the pores initially will be trapped at a greater depth as they traverse through this porous cake. This cake becomes the true filter medium and hence plays a very important part in the entire filtration operation. The mechanism of flow within the cake and the external conditions imposed on the cake are the basis for modeling a filtration process. 

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