Darcy’s velocity

To simplify further the description of foam flow, bubble velocity is removed from the set of internal variables. It is assumed that a fraction, Xr, of all bubbles is carried with velocity, v(x,t TT,o) as flowing foam (f). This assumption is justified because flowing bubbles remain in contact most of the time, and their average velocity, , can be related to Darcy s velocity of the gas phase, = u /([Pg.328]

The average bubble velocity is related to Darcy s velocity of the gas phase through relation (A-11)... 

Volume-averaged Darcy s velocity through a porous fiber preform. (Source Reprinted with permission from reference 1, copyright 2010 CRC Press,Taylor Francis Group.)... 

Substitution of the Darcy s velocities in Eq. [9.6] results in the following second order partial differential equation on the resin pressure, P ... 

Control volume (CV) is used to calculate the Darcy s velocity components and fluxes between the adjacent control volumes to advance the flow front position for one time step as follows. The mesh is divided into small control volumes (i.e., control area times the local part thickness) such that each node has its own control volume bounded by the neighbor elements ... 

Figure 4.15 shows the relationship between Si concentration and Darcy s velocity (Lasaga et al. 1994). These theoretical curves can roughly explain analytical... 

Darcy s law is considered valid for creeping flow where the Reynolds number is less than one. The Reynolds number in open conduit flow is the ratio of inertial to viscous forces and is defined in terms of a characteristic length perpendicular to flow for the system. Using four times the hydraulic radius to replace the length perpendicular to flow and conecting the velocity with porosity yields a Reynolds number in the form ... 

In a steady-state situation when gas flows through a porous material at a low velocity (laminar flow), the following empirical formula, Darcy s model, is valid ... 

Sephadex G-lO-G-50 behave as rigid spheres and the maximum linear velocity may be calculated by Darcy s law where U = K i p/L) U is the linear velocity... 

Estimation of the pressure-drop The system is designed to work within a given pressure limit thus, one needs a relation giving the pressure-drop in the column (per unit length). Darcy s law gives the relation of AP/L versus the mobile phase velocity u. However, the Kozeny-Carman equation is best adapted for laminar flows as described ... 

The filtration, or superficial face, velocities used in fabric filters are generally in the range of 1 to 10 feet per minute, depending on the type of fabric, fabric supports, and cleaning methods used. In this range, pressure drops conform to Darcy s law for streamline flow in porous media, which states that the pressure drop is directly proportional to the flow rate. The pressure drop across the... 

The physical process of melt ascent during two-phase flow models is typically based on the separation of melt and solid described by Darcy s Law modified for a buoyancy driving force. The melt velocity depends on the permeability and pressure gradients but the actual microscopic distribution of the melt (on grain boundaries or in veins) is left unspecified. The creation of disequilibria only requires movement of the fluid relative to the solid. 

Under chromatographic conditions, the flow profile Is usually laminar and therefore the mobile phase velocity can be described by Darcy s law... 

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 ... 

U.S. EPA specifies that the minimum detection time for the leachate entering the LDS of an LDCR system is <24 h. Response time calculations are based on velocity in the geonet and/or granular soil drainage layer. Darcy s law is used to calculate flow velocity in the geonet, and a true velocity must be used for granular soil. 

In other cases, researchers assume that the inertial resistance to flow in DLs adjacent to conventional flow fields is negligible and they tend to lump both viscous and inertial coefficients together. This may not be a correct assumption, especially when dealing with flow fields like the interdigitated design [129,212], in which higher velocities are experienced in the pores of the DL. The Forchheimer equation is an extension of Darcy s law and takes into account the inertial resistance at high velocities [211,213] ... 

In eq 51, the first term represents a convection term, and the second comes from a mass flux of water that can be broken down as flow due to capillary phenomena and flow due to interfacial drag between the phases. The velocity of the mixture is basically determined from Darcy s law using the properties of the mixture. The appearance of the mixture velocity is a big difference between this approach and the others, and it could be a reason the permeability is higher for simulations based on the multiphase mixture model. 

The role of the viscosity is quite significant in actual o ration due to two effects. At a fixed flow velocity of the eluent, the pressure drop across a column, AP, is proportional to the viscosity, t), of the mobile phase according to Darcy s law... 

Darcy s law asserts that the average flow velocity, v, of a fluid through a packed bed is proportional to the pressure gradient, AP, across the bed and is inversely proportional to the length, /, of the bed. Thus,... 

In this equation ut should be interpreted as the volumetric flux density (directional flow rate per unit total area). The indexes range from 1 to 3, and repetition of an index indicates summation over that index according to the conventional summation convention for Cartesian tensors. The term superficial velocity is often used, but it is in our opinion that it is misleading because n, is neither equal to the average velocity of the flow front nor to the local velocity in the pores. The permeability Kg is a positive definite tensor quantity and it can be determined both from unidirectional and radial flow experiments [20], Darcy s law has to be supplemented by a continuity equation to form a complete set of equations. In terms of the flux density this becomes ... 

Two matrix flow submodels have been proposed the sequential compaction model [15] and the squeezed sponge model [11], Both flow models are based on Darcy s Law, which describes flow through porous media. Each composite layer is idealized as a fiber sheet surrounded by thermoset resin (Fig. 13.9). By treating the fiber sheet as a porous medium, the matrix velocity iir relative to the fiber sheet is given as (Eq. 13.5) ... 

The flow velocity in the filter v is again related to the pressure and the electric potential gradients via a generalized Darcy s law of the form... 

This expression describes the fastest and most important mode of transport in groundwater. In fact, an important task of the hydrologist is to develop models to predict the effective velocity u (or the specific flow rate q). Like the Darcy-Weis-bach equation for rivers (Eq. 24-4), for this purpose there is an important equation for groundwater flow, Darcy s Law. In its original version, formulated by Darcy in 1856, the equation describes the one-dimensional flow through a vertical filter column. The characteristic properties of the column (i.e., of the aquifer) are described by the so-called hydraulic conductivity, Kq (units m s"1). Based on Darcy s Law, Dupuit derived an approximate equation for quasi-horizontal flow ... 

A considerable difference exists between flow through an open column and packed column as illustrated by Figure 2.11. Darcy s law, which governs flow through packed columns, states that flow velocity is proportional to the pressure gradient. 

Surprisingly enough, it is possible for the steady-state assumption in a moving-boundary system to be essentially exact. This occurs in the deposition of a filter cake under the influence of a constant pressure difference. An interesting example is given by Brenner (B13), in a study of the unconfined growth of a filter cake on a circular cloth-covered aperture in a plane wall. It is assumed that the velocity vector of the filtrate within the cake is everywhere and at all times proportional to the pressure gradient (Darcy s law). 

---