Darcy’s equation

Pressure drop in static mixers depends very strongly on geometric arrangement of the inserts. It is simply defined in relation to the pressure drop AP in an empty tube given by Darcy s equation ... 

From Darcy s equation we can determine a formula for the counterforce produced by the porous material to the flowing or diffusing component A, If this counterforce is found, it can be added to the diffusion resistance force caused by component B to component A hence the sum of these two forces represents the total diffusion resistance. 

Darcy s equation can be used to describe flow in this region however, the value of permeability varies as a function of saturation. Also, the value of moisture potential is a function of saturation. The total potential for flow (hydraulic gradient in Darcy s equation) can be defined as the difference between the moisture potential (minus) and the elevation potential (plus). When the potential for flow is positive, flow can occur. 

Fluid dynamics in a tubular fuel cell has significant effects on different other phenomena such as the electrochemical reactions, which need species to be transported to the reaction site, and mass and heat transfer. In the porous structures, an equation that accounts for the fluid flowing in the pores - such as Brinkman s or Darcy s equations (Equation (3.4)) - must be used usually the velocities are low. Fluid-dynamic is modelled through Navier-Stokes equation in the channels (Equation (3.3)). 

Work of Richards—L. A. Richards (1931) gave an excellent theoretical presentation of the factors involved in a study of capillary constants. He succeeded in establishing a general capillary equation and gave a method for determining conductivity. The conductivity was defined as the constant contained in Darcy s equation (Eq 13-2)... 

The local hydroconductivity (analogous to the filtration coefficient in Darcy s equation) represents the liquid volume flowing through all borders in unit time per unit area... 

Gas mobility depends on the permeability of the porous medium. In the presence of foam gas mobility is the mobility of the continuous gas phase through the free channels and the mobility of the confined gas along with the liquid. Formally the relative permeability of each phase (liquid or gas) can be expressed by Darcy s equation. 

The driving force behind the liquid transport has been determined by Kramer and Bubeck to be capillary pressure [17]. The velocity of liquid flow V is expressed by Darcy s equation... 

Microfiltration and ultrafiltration are the two main filtration techniques for which ceramic membranes have been widely used to date. As described in Section 6.2.1.2, MF and UF ceramic membranes exhibit macro- and mesoporous structure, respectively, which result from packing and sintering of ceramic particles. Liquid flow in such porous media is convective in nature and the simplest description of permeation flux, J, is given by the Darcy s equation [20] ... 

The three-dimensional flow of groundwater through the subsurface can be described by a combination of Darcy s equation for groundwater flow with a continuity equation (or mass balance equation) and equations of state for the groundwater and the porous medium. A detailed theoretical overview of equations for groundwater flow is given by e.g. Barenblatt et al. (1990), De Marsily (1986), Domenico and Schwartz (1990) and Freeze and Cherry (1979). 

By introducing the appropriate expression for the hydraulic head, i.e. Equation 1.3 or 1.4, into Darcy s equation 1.8, it describes the flow for compressible and incompressible groundwater, respectively. The flow of compressible groundwater can thus be described by a more general form of Darcy s equation, in which the gradient of the groundwater potential is given by Equation 1.5... 

For steady-state flow of groundwater, a combination of the continuity Equation 1.11 with Darcy s equation 1.8 yields... 

Introducing Darcy s equation 1.8 into the simplified continuity Equation 1.27 yields the general equation for unsteady-state flow through a water-saturated anisotropic medium under the assumed conditions listed in Section 1.2.2. 

The characteristics of the groundwater and the porous medium that appear in Darcy s equation vary independently on a geological time scale. Equation 1.9 should be used when dealing with large temporal and spatial scales. 

The equations describing the conservation of groundwater mass (Equation 1.12), solid mass (Equation 1.34), chemical mass (Equation 1.36) and heat (Equation 1.35) and Darcy s equation for groundwater flow (Equations 1.8 and 1.9) are coupled and nonlinear (Bredehoeft and Norton, 1990 Garven, 1985). 

Our sample is thin, so that we take it to be uniform vertically. Then Darcy s equation is... 

Another expression derived from Darcy s equation that is often used in the literature is the column permeability B (Eq. 3.9). 

In addition to the Reynolds-number limitations discussed above, a shortcoming of Darcy s law is that it does not allow boimdary conditions to be imposed at porous media interfaces (e.g., a no-slip condition at a reactor wall). This problem is addressed by Brinkman s equation, which is a combination of the Stokes equation and Darcy s equation ... 

In Darcy s equations [Eqs. (3) and (7)], the permeability K and the specific resistance a are functions of the effective stress ps in the particulate structure. Replacing dpL by -dps in accord with Eq. (12), and involving both spatial coordinate x and material... 

As a model for this transport process, consider the axisymmetric spreading of a fluid of density p + Ap in a porous media containing a fluid of density/ ). Assume the fluid spreads out over an impermeable bottom and that the volume of the dense fluid or gravity current is given by Qta, where t is time. The viscosity of the gravity current is p. and the permeability of the porous medium is k. Also, neglect the effects of capillary forces and assume the flow is dominated by a balance between buoyancy and viscous forces. This balance of forces in a porous medium is described by Darcy s equations, which are given by... 

The third major factor in dynamics is packed bed flow behavior. The pressure drop in the adsorbent media is caused by drag due to fluid flowing through the interstices. For slow flow, the pressure drop, AP, across a bed of length Az and superflcial velocity v, can be represented by Darcy s equation ... 

Another measure of length is the equivalent hydraulic radius, rh, determined inversely from saturated or unsaturated flow experiments. By comparing Darcy s equation with Poiseuille s law, and invoking a capillary bundle model, one obtains the following definition of rh (Kutflek Nielsen, 1994),... 

Here pj- is the fluid density, and Q is the source or sink term for fluid. The flow velocity, is calculated from Darcy s equation for variable density fluid flow ... 

Absolute and Effective Permeability. The permeability of a porous medium is a measure of the fluid conductivity of the medium that determines the flow rate at a given pressure gradient. The permeability of a porous medium can be determined using Darcy s equation as follows. The porous medium, usually a core, is saturated under vacuum with a fluid of a known viscosity. The fluid is then injected into the core in a linear mode at a constant flow rate under isothermal flow conditions. If the porous medium is homogeneous, isotropic, the core is placed in horizontal direction, and Reynolds number of the injected fluid is less than unity, then Darcy s law can be used to determine permeability. Darcy s law is... 

The functional form of the static filtration curve (Figure 33) may be readily derived if it is assumed that the flow of the filtrate through the filter cake is governed by Darcy s law (126). The flow rate of the filtrate dVfe/df is given by Darcy s equation... 

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