Darcy theory

Consider mechanical excitation frequencies in a liquid-saturated porous medium (Fig. 10). Two formalisms were conventionally used for high frequencies, Biot-Gassmann wave mechanics (Biot 1956) is used for low frequencies, Darcy theory is used (inertial effects assumed negligible). However, there clearly must be approximately three orders of magnitude where both diffusion and... 

Biot and Darcy theory shortcomings have been largely overcome by development of a coupled diffusion-dynamic formalism (de la Cruz et al. 1993, Spanos 2001, Spanos et al. 2(X)3). Porosity is treated as an explicit thermodynamic variable, so that dnumerical model development. Nevertheless, if they are solved subject to the assumption of the incompressibility of a liquid saturant, the existence of a slow wave is predicted. It is called the porosity dilation (PD) wave it is not a strain wav, it is a coupled liquid-solid displacement wave, and it has some interesting properties. 

Modem filtration theory tends to prefer the Ruth form of Darcy s law, ie,... 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

In the past, various resin flow models have been proposed [2,15-19], Two main approaches to predicting resin flow behavior in laminates have been suggested in the literature thus far. In the first case, Kardos et al. [2], Loos and Springer [15], Williams et al. [16], and Gutowski [17] assume that a pressure gradient develops in the laminate both in the vertical and horizontal directions. These approaches describe the resin flow in the laminate in terms of Darcy s Law for flow in porous media, which requires knowledge of the fiber network permeability and resin viscosity. Fiber network permeability is a function of fiber diameter, the porosity or void ratio of the porous medium, and the shape factor of the fibers. Viscosity of the resin is essentially a function of the extent of reaction and temperature. The second major approach is that of Lindt et al. [18] who use lubrication theory approximations to calculate the components of squeezing flow created by compaction of the plies. The first approach predicts consolidation of the plies from the top (bleeder surface) down, but the second assumes a plane of symmetry at the horizontal midplane of the laminate. Experimental evidence thus far [19] seems to support the Darcy s Law approach. 

Nakayama, A., A Unified Theory for Non-Darcy Free, Forced, and Mixed Convection Problems Associated with a Horizontal Line Heat Source in a Porous Medium , J. Heat Transfer, Vol. 116, pp. 508-513, 1994. 

Very briefly, the Dave model considers a force balance on a porous medium (the fiber bed). The total force from the autoclave pressure acting on the medium is countered by both the force due to the spring-like behavior of the fiber network and the hydrostatic force due to the liquid resin pressure within the porous fiber bed. Borrowing from consolidation theories developed for the compaction of soils 22 23), the Dave model describes one-dimensional consolidation with three-dimensional Darcy s Law flow. Numerical solutions were in excellent agreement with closed-form solutions for one- and two-dimensional resin flow cases in which the fiber bed permeabilities and compressibility, as well as the autoclave pressure, are all held constant21). 

To understand the flux decline in pressure-driven membrane operations, a number of models were developed. Two of the most widely smdied models are the resistance model and the concentration polarization model. The resistance model is the oldest and is based on the cake filtration theory, where it is assumed that a cake layer of rejected particles, which are too large to enter the membrane pores, is formed. The frictional drag due to permeation through these immobile particles leads to additional hydraulic resistance [21]. The cake layer and the membrane are considered as two resistances in series, and the permeate flux is described by Darcy s Law as... 

Mechanism of flow through porous media is fundamental in theoretical study of SLS process, such as filtration, thickening, centrifugation, expression, washing, etc. In the early study, with the development of fluid mechanics, interest was focused on flow in capillaries through incompressible sand beds. The beginning of present day theory can be traced to Hagen (1839), Poiseuille (1840), and Darcy (1856). 

Fig. 3 shows a simplified compactible filter cake. Darcy s law and a stress balance involving the accumulated friction drag on the particulate structure (Fig. 3) are used to develop basic theory of flow through compactible porous media. 

Before using Darcy s law to theoretically develop Equation (22.2), simplified procedures used by many authors (Rushton et al. 1996 Svarovsky 1990) will be employed to derive volume vs. time equations. Although widely used and valuable in interpreting data, highly significant information important to full-scale operation is missed when Equations (22.1) and (22.2) are the sole basis for developing the theory of cake filtration. 

Zhao (1994) presented a model of coupled coal deformation and methane migration based on a consolidation theory of elastic medium with Darcy fluid flow and the Terzaghi effective stress law, and its numerical solution technique and applications to practical problems. Works using similar approaches were also reported in Liang et al. (1995,1996), Sun and Xian (1999), Ding et al. 

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