Carman-Kozeny relation

In addition, for the description of this flow it is necessary to employ the Carman-Kozeny relation (Equation 10.19), since the Hagen-Poiseuille equation is not suitable, as the membranes were obtained by the sinterization of packed quasispherical particles [18]. 

It remains to establish, the pressure trend throughout the porous medium as the particles filter out of solution. As long as no filtration has taken place, Darcy s law can be applied. However, the pressure profile will change according to deposition of filtrate on the matrix of the porous medium. The permeability reduction can be approximated with the Carman-Kozeny relation ... 

The permeability, K, is characteristic of the medium and can be related to measurable properties by the Carman-Kozeny equation ... 

The Carman-Kozeny equation relates the drop in pressure through a bed to the specific surface of the material and can therefore be used as a means of calculating S from measurements of the drop in pressure. This method is strictly only suitable for beds of uniformly packed particles and it is not a suitable method for measuring the size distribution of particles in the subsieve range. A convenient form of apparatus developed by Lea and Nurse 22 1 is shown diagrammatically in Figure 4.4. In this apparatus, air or another suitable gas flows through the bed contained in a cell (25 mm diameter, 87 mm deep), and the pressure drop is obtained from hi and the gas flowrate from h2. 

If flow conditions within the bed are streamline, the relation between fluid velocity uc, pressure drop (—A P) and voidage e is given, for a fixed bed of spherical particles of diameter d, by the Carman-Kozeny equation (4.12a) which takes the form ... 

This relation, which is analogous to PoiseuiUe s relation, gave rise to various models taking into account the irregularity of the porous medium (tortuosity, noncircular sections, etc.). Carman-Kozeny s model is a simple and usually precise model which leads to the following expression of D ... 

All of the permeametry methods are based on the Carman-Kozeny equation given in Fig. 4 which relates the approach velocity u to the porosity of the powder e and the specific surface of the sample Sw. The specific surface calculated involves only the walls of the pores of the bed which are swept by the flow and it does not take into account the pores within the particles which do not contribute to the flow. The surface measured, therefore, is an envelope surface area and it can be very much smaller than the total surface area of the particles as measured, say, by gas adsorption. 

In the literature, there exists a relationship which relates the permeability K with the relative density p, and the mean pore size d of porous object [13]. This relation is called the Carman-Kozeni equation... 

The permeability of the composite aerogel can be estimated from the Carman-Kozeny equation applied to a gel [25], an approximation that relates the permeability to the pore size and which can be written as ... 

Carman-Kozeny equation This equation is empirical and relates the permeability to the pore size and pore volumerD = (1 — Pr)rw /4 T where is the relative density, is the hydraulic radius and is the so-called the Kozeny constant close to 5 CARS Coherent anti-Stokes Raman scattering... 

When a fluid passes vertically through a bed of particles the pressure drop APg will initially increase as the velocity u is increased, as shown in Figure 7.2. The relation between pressure drop and velocity will be that applicable to a fixed bed. For fine particles, the increasing straight line in Figure 7.2 has been properly described by a relationship known as the Carman-Kozeny equation ... 

All these methods lead to a set of parameters (membrane thickness, pore volmne, hydraulic radius) which are related to the working (macroscopic) permselective membrane properties. In the case of liquid permeation in a porous membrane, macro- and mesoporous structures are more concerned with viscous flow described by the Hagen-Poiseuille and Carman-Kozeny equations whereas the extended Nernst-Plank equation must be considered for microporous membranes in which diffusion and electrical charge phenomena can occur (Mulder, 1991). For gas and vapor transport, different permeation mechanisms have been described depending on pore sizes ranging from viscous flow for macropores to different diffusion regimes as the pore size is decreased to micro and ultra-micropores (Burggraaf, 1996). 

Since Eq. 3 indicates that the capillary pressure is inversely proportional to the pore size, the most obvious way to avoid drying stress is to prepare bodies with larger pores. In addition, the Carman-Kozeny equation (see Eq. 15 of Chapter 7) indicates that the permeability is related to the microstructure by... 

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 ... 

To calculate the reduction in the concentration of surfactant in the fluid by adsorption it is necessary to have an estimation of the inner surface area of the reservoir. This parameter is related to the porosity of the medium and to its permeability. Attempts have been made to correlate these two quantities but the results have been unsuccessful, because there are parameters characteristic of each particular porous medium involved in the description of the problem (14). For our analysis we adopted the approach of Kozeny and Carman (15). These authors defined a parameter called the "equivalent hydraulic radius of the porous medium" which represents the surface area exposed to the fluid per unit volume of rock. They obtained the following relationship between the permeability, k, and the porosity, 0 ... 

Various direct and indirect methods are generally used to determine the permeability of a sedimentary basin. The direct methods include laboratory measurements on core samples wire-line formation tests, single-well tests and interference tests. The data from the different types of well test and interference test can be analysed and interpreted by well-established procedures (Da Prat, 1990 Earlougher, 1977 Kruseman et al., 1990 Matthews and Russel, 1967). The conventional, indirect methods are theoretical, semi-empirical and empirical procedures which are based on the relation between permeability, grainsize characteristics and porosity (e.g. the Kozeny-Carman method, Domenico and Schwartz, 1990 Van Baaren method. Van Baaren, 1979). The laboratory methods and the conventional indirect methods provide permeability values which are representative of only a very small portion of the subsurface (cm-scale). The single-well test and interference test provide information representative of a larger volume of the subsurface (m - km scale). 

Phase-inversion membranes frequently show a sponge-like structure. The volume flux through these membranes is described by the Hagen-Poiseulle or the Kozeny-Carman relation, although the morphology is completely different. 

The constant K incorporates factors which determine the flow resistemce of the gel layer such as its porosity and tortuosity, pore size and shape, etc. Using the Kozeny-Carman relation, Leenaars [3] studied the relation between K, microstructure and process parameters. The value of the pressure drop across the gel layer APg is obtained from AP after correction for the pressure drop APg in the support, which is usually very small. 

Accordingly, it is not easy to predict B from the structure of the material. Numerous relations have been proposed, and the one used most is the Kozeny-Carman equation... 

In what follows we derive an empirical relation for the permeability, known as the Kozeny-Carman equation, which supposes the porous medium to be equivalent to a series of channels. The permeability is identified with the square of the characteristic diameter of the channels, which is taken to be a hydraulic diameter or equivalent diameter, d. This diameter is conventionally defined as four times the flow cross-sectional area divided by the wetted perimeter, and measures the ratio of volume to surface of the pore space. In terms of the porous medium characteristics. 

All terms of 0(1), 0(e), and O(e ) cancel identically to yield the above relation. The Kozeny-Carman formula, Eq. (4.7.16), can be similarly expanded to give... 

For easy separation of crystals from a solution, it is important that the crystals are sufficiently large. This can be shown by the Kozeny-Carman equation derived for laminar flow through an incompressible bed of particles, which describes the pressure drop in the bed. The pressure drop relates inversely to the square of the quadratic particle size and directly to the dynamic viscosity of the fluid. The influence of the particle size and solution viscosity on the pressnre required in filtration can be estimated as follows. If the viscosity increases tenfold, the pressnre drop also inCTeases tenfold. If, on the other hand, the particle diameter drops to 10% of its original valne, the pressnre drop increases by 100-fold. 

One important, but often not clearly defined variable in the characterisation of porous membranes, is the shape of the pore or its geometry. In order to relate pore radii to physical equations, several assumptions have to be made about the geometry of the pore. For example, in the Poiseuille equation (see eq. IV 4) the pores are considered to be parallel cylinders, whereas in the Kozeny-Carman equation (eq. IV - 5) the pores are die voids between the close-packed spheres of equal diameter. These models and their corresponding pore geometries are extreme examples in most cases, because such pores do not exist in practice. However, in order to interpret the characterisation results it is often essential to make assumptions about the pore geometry. In addition, it is not the pore size which is the rate-determining factor, but the smallest constriction. Indeed some characterisation techniques determine the dimensions of the pore entrance rather than the pore size. Such techniques often provide better information about permeation related characteristics. 

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