The presence of porous media in the domain offers resistance to the flow because of drag effects. Porous structures are characterized by voids or pores through which the fluid flows, labyrinth or tortousity factor which defines interconnectivity of the porous structure and porosity which is the ratio of the amount of fluid present in the porous structure to the total volume including fluid and solid volumes. It can be directly deduced that the drag force increases with increase in the surface area of the porous matrix. The surface area of the matrix can further be divided into two parts. Area normal to the fluid flow and internal surface area of the matrix structure which induces viscous drag on the fluid flowing through it.
In the experiment conducted by Darcy, it was realized that the bulk shear stresses were dominant as the length of the porous media and hence the internal surface used was many times greater than its normal surface area. Darcy then found the velocity of the fluid in the porous matrix ~υdarqx,por by measuring flux of an incompressible fluid flowing through a cylindrical tube at the outlet ρA~υdarqx,por and dividing it by ρA. Pressure difference was the driving force which was directly proportional to the flux and hence the velocity in the porous region ~υdarqx,por. He realized that the proportionality constant could be expressed in terms of permeability K of the porous media and viscosity µ of the fluid. The darcy relation is [20]:
∂p
“Permeability accounts for the interstitial surface area, the fluid particle path as it flows through the matrix and other related hydrodynamic characteristics of the matrix ”[20]. In micro porous matrices, the flux at the outlet is generally more than those measured by Darcy’s experiment since an additional Knudsen diffusion flux appears because of slip flow of the molecules with the porous structure walls. Hence the additional knudsen flux has to be modelled to appropriately represent the flow condition.
Figure 3.5 shows different forms of transport of molecules in a porous matrix. As seen, momentum is lost by the molecules due to their interaction with the surface of the porous matrix. This is termed as the Darcy drag. A few molecules which move in micro pores experience knudsen slip as mentioned earlier contributing to the additional Knudsen flux. The remaining molecules retrace their path because of the inter connected pores in the porous matrix. This is accounted for by the tortuosity term.
Consider a fluid q flowing through a porous matrix. From the kinetic theory of gases, the mean molecular velocity for the component can be written as [20]:
Px
Px+dx
Molecules that undergo Darcy drag from internal surface area of porous matrix
Molecules that are transported due to Knudsen slip Molecules that are retraced due to
labyrinth/tortuosity factor of the porous matrix
Figure 3.5: Movement of molecules of a single phase in porous region
~υqm,por =
s8RT πMq
. (3.16)
R is the universal gas constant, T is the temperature in Kelvin and Mq is the molecular weight in Kg mol−1. Considering a flow through a tube which is the idealised form of a pore with diameter dp and applying momentum conservation, Shear stress at the radius of the pore is[20]:
τ(dp/2)= 3
From the above equation, an expression for ~υqkx,por is obtained,
~υqkx,por = −dp
Ideal gas behavior is assumed:
pq = nqkBT . (3.19)
Substituting ~υqm from equation 3.16:
~υqkx,por = − dp
which can be rewritten as,
is the Knudsen diffusion coefficient of a gas component.
Thus for flows in 1-dimension and from 3.21 , 3.15 we have:
~υqx,por = ~υdarqx,por + ~υqkx,por , (3.23)
and extending it to 3-dimensions,
~υq,por = −[Dqk] pq
∇p − [K]
µ ∇p . (3.25)
In this work, porosity is modelled for non deformable and isotropic media.
The permeability expression for permeability K is modelled according to the velocity profile of a Hagen-Poiseuille flow[27]. It is dependant on the diameter of the pore and is:
K = dp2
8 . (3.26)
The dimensionless tortuosity or the labyrinth factor τ is introduced into the equa-tions to compensate the extended path a fluid travels when it encounters closed paths due to interconnectivity of the porous media.
Thus the expression for pressure gradient which is the driving force for the flows in porous media is:
tions, local volume averaging is used. It is a technique introduced by Whitaker[28]
and Slattery[13]. The approach is to define a quantity φ which is in a fluid volume Vf by averaging over a volume V. The condition for averaging is that the volume over which φ is averaged must neither be large compared to the flow domain nor very small to represent the geometry of the porous structure. “The local representative elementary volume is chosen such that the smallest differential volume that results in statistically meaningful local average properties (such as local porosity) and when this volume is appropriately chosen, adding extra pores(extra volume) around it will not result in changes to the values of these local properties”[20].
A penalty approach is used in order to define the void distribution fuction or the solid distribution function. Let us name this distribution function as gp defined as,
gp(x) = 1 if x is in void region,Vf
0 if x is in solid region,Vs (3.28) Local porosity is defined as:
ǫ(x) = 1
Volume average of a quantity φ is,
< φ >= 1 V
Z
Vf
φdV = ǫφ . (3.30)
A comprehensive explanation of volume averaging including the general form of Reynolds transport theorem can be found in [20] and will not be explicitly mentioned here.
Thus by applying the stated averaging technique to the modelled source porosity terms in equation 3.30 the final term that is added to the fluid momentum equation is:
The first term on the right hand side can replace −∇p in the Eulerian momentum equation since ǫ is 1 in the absence of porous media and accounts for both porous regions and non porous regions.