Momentum Equation in Porous Media

A number of different approaches are proposed and used in modeling flow through porous media. Some of the most popular approaches include (i) Darcy s law, (ii) Brinkman equation, and (iii) a modified Navier-Stokes equation. In the absence of the bulk fluid motion or advection transport, the reaction gas species can only transport through the GDL and CL by the diffusion mechanisms, which we will discuss in a later section. 

As we know, the fluid flow takes place under the influence of different body and surface forces. Darcy s law defines the fluid flow in a porous media under the influence of pressure gradient force only, and this is expressed as 

In this formulation, the presence of inertia and viscous forces is neglected and the region is assumed as homogeneous porous media characterized by the permeability. 

In order to match the solutions of Navier-Stokes equation to the soluhon of Darcy s equation at the channel-porous media interface, Darcy s equahon is modified to include a viscous force term in the momentum equation and this is given by the Brinkman s equation (Martys, 2001 Martys et al., 1994) as 

Notice that Brinkman s equation includes both the pressure force and the viscous force terms. The effective viscosity for the slower-moving fluid in the porous media is selected such that continuity in shear stress is maintained at the interface between the faster-moving gas flow in the channel and the slower-moving gas flow in porous electrode. The continuity in shear stress at the interface is given as 

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