Darcy’s law equation

The flow rate of extracted air can be determined by considering the air velocity, as determined by Darcy s Law (Equation 14.8), and the radial distribution of pressure (Equation 14.11). The solution for air velocity as a function of the radial distance is given in Equation 14.13 ... 

Relative permeability is defined as the ratio between the permeability for a phase at a given saturation level to the total (or single-phase) permeability of the studied material. This parameter is important when the two-phase flow inside a diffusion layer is investigated. Darcy s law (Equation 4.4) can be extended to two-phase flow in porous media [213] ... 

Unlike conventionally used volume-averaged equations, equation 20 does reduce to the Brinkman equation 5 when the inertial effects are negligible and to Darcy s law, equation 3, when both inertial and multidimensional effects are negligible. 

This equation is identical to Darcy s law (Equation 2.25), which is well known among chromatographers (Guiochon, Golshan-Shirazi, and Katti, 1994) ... 

Equations 40.6 and 40.7 must be consistent with Darcy s law (Equation 40.5) when one single-fluid phase occupies the porous medium. Consequently, the relative permeability functions fulflll the following conditions ... 

The mathematical descrption of the process starts with the neglect of septum resistance and the use of Darcy s law (Equation 2.3) to relate filtrate flow rate and pressure drop ... 

The filter cake concentration profile relative to actual height from the medium can be obtained from the above if the superficial filtrate flow rate through the medium q has been measured via a modified form of Darcy s law Equation (2.47) ... 

The shell and lumen velocity fields are obtained by substituting Darcy s law. Equation (16.15) for the velocity and applying appropriate boundary conditions. An appropriate equation of state also is required to calculate density from pressure. The ideal gas law is used for the relatively low pressure dehydration process considered here. 

The tortuosity t is introduced in [14.26] in order to express the Darcy law in terms of the thickness of the porous medium rather than the length of the tubes. We recover in [14.26] the first form of Darcy s law (equation [14.1]). The intrinsic permeabihty can then be expressed as a parameter depending on the geometrical characteristics of the capillary tube network ... 

A point of terminology is repeated here to remind the reader. The term apparent viscosity , pp, is used to describe the observed macroscopic rheology of the polymeric fluid in a porous medium. The quantity effective viscosity ,, refers in a rather similar way to the observed effective viscosity in a single capillary. Each quantity is defined phenomenologically—from Darcy s law (Equation 6.4) and rj ff from Poiseuille s law (Equation 3.75). This distinction should be kept clear, especially when considering porous media models based on networks of capillaries, as discussed later in this chapter. The overall viscosity of the non-Newtonian fluid in the network as a whole is whereas the viscosity in each of the capillaries may be different and is In this latter case, will be in some sense an average value of the in the individual capillaries. 

The convection velocity, u, in the porous electrolyte structure is given either by Darcy s law equation (Equation 6.17) or by Brinkman s equation (Equation 6.18) as discussed in Chapter 6. 

Consider Figure 6.3.23(a), which shows a membrane/ filter/cloth of thickness S , on the top surface of which a cake of thickness Sc has been formed at time t (starting time f = 0). We have observed in Section 3.T.2.3 that the volume flux of the liquid through a porous medium may be described by Darcy s law (equation (3.4.88), repeated here for convenience) ... 

It is worthy to note that permeability as defined in Darcy s law (equation 13.1) pertains to the steady flow of fluid in a saturated porous medium, whereas the resin transfer process involves unsteady flow of resin into an unsaturated preform. The above holds true when analyzing the macroscopic flow through a fiber preform, i.e. flow in production of a composite part. However, one must be aware that the flow behavior in a fiber preform is markedly different when analyzed at the mesoscopic length scale, which concerns die interaction between the fluid and the intricate structure of the fiber bundles. 

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