Carman-Kozeny equation for

The Ergun density may be determined from a well-mixed bed of porous, non-cohesive powder of known weight by fluidisation at a known pressure drop, Ap. After the gas flow has been stopped the height of the settled bed is measured. An Ergun density may then be calculated from the Kozeny-Carman equation. Use of the Kozeny-Carman equation for the determination of the Ergun particle density becomes, however, invalid with coarse powders... 

Giroud (1996) started firom the classical Kozeny—Carman equation for the hydraulic conductivity of porous media and obtained the following equation (Eq. [8.12]) to theoretically evaluate the permeability of nonwoven geotextiles ... 

The specific resistance coefficient for the dust layer Ko was originally denned by Williams et al. [Heat. Piping Air Cond., 12, 259 (1940)], who proposed estimating values of the coefficient by use of the Kozeny-Carman equation [Carman, Trans. Inst. Chem. Fng. (London), 15, 150 (1937)]. In practice, K and Ko are measured directly in filtration experiments. The K and Ko values can be corrected for temperature by multiplying by the ratio of the gas viscosity at the desired condition to the gas viscosity at the original experimental conditions. Values of Ko determined for certain dfists by Williams et al. (op. cit.) are presented in Table 17-5. 

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 determination of the specific surface area of a powder by air permeability methods essentially involves the measurement of the pressure drop across a bed of the powder under carefully controlled flow conditions. The data obtained are substituted in the Kozeny-Carman equation to estimate the specific surface area. Permeability methods have certain advantages, one of them being that the equipment used for carrying out the measurements is cheap and robust. Another advantage is that sample problems are minimized because a large sample of powder is required to be used for analysis. 

Continuous strand mats are approximately isotropic and have almost the same permeability in all directions (in the plane of the fabric). Many other fabrics, however, are strongly anisotropic and have different permeability in different directions. Gebart [18] proposed a model for this class of fabrics derived theoretically from a simplified fiber architecture. The model, which is valid for medium to high fiber volume fractions, was developed for unidirectional fabrics, but it can also be used for other strongly anisotropic fabrics. In this model the permeability in the high permeability direction (which is usually, but not always, in the direction of the majority of fibers) follows the Kozeny-Carman equation (Eq. 12.2). In the perpendicular direction, however, it is ... 

Liquid infiltration into dry porous materials occurs due to capillary action. The mechanism of infiltrating liquids into porous bodies has been studied by many researches in the fields of soil physics, chemistry, powder technology and powder metallurgy [Carman, 1956 Semlak Rhines, 1958]. However, the processes and kinetics of liquid infiltration into a powdered preform are rather complex and have not been completely understood. Based on Darcy s fundamental principle and the Kozeny-Carman equation, Semlak Rhines (1958) and Yokota et al. (1980) have developed infiltration rate equations for porous glass and metal bodies. These rate equations can be used to describe the kinetics of liquid infiltration in porous ceramics preforms, but... 

Here p is the density of the infiltrant,, > is the acceleration due to gravity (9.8 m/s2) and Patm is the pressure of the surrounding space above the liquid infiltrant. This model is useful for predicting the influence of pressure on the rate of infiltration. Another formula is the Kozeny-Carman equation [Carman, 1956] ... 

The Kozeny-Carman equation is suitable for the laminar flows met in chromatography ... 

For particles under viscous conditions and Re < 1 a more simplified form, known as the Kozeny-Carman equation, can be used to calculate the pressure drop within the chromatographic column. 

In conventional filtration systems used for cell separation, plate filters (e.g., a filter press) and/or rotary drum filters are normally used (cf. Chapter 9). The filtrate fluxes in these filters decrease with time due to an increase in the resistance of the cake Rc (m 1), as shown by Equation 9.1. If the cake on the filtering medium is incompressible, then Rc can be calculated using Equation 9.2, with the value of the specific cake resistance a (m kg-1) given by the Kozeny-Carman equation (Equation 9.3). For many microorganisms, however, the values of a obtained by dead-end filtration (cf. Section 9.3) are larger than those calculated by Equation 9.3, as shown... 

Equation (7,17) is called the Kozeny-Carman equation and is applicable for flow through beds at particle Reynolds numbers up to about 1.0. There is no sharp transition to turbulent flow at this Reynolds number, but the frequent changes in shape and direction of the channels in the bed lead to significant kinetic energy losses at higher Reynolds numbers. The constant 150 corresponds to = 2.1, which is a reasonable value for the tortuosity factor. For a given system, Eq. (7,17) indicates that the flow is proportional to the pressure drop and inversely proportional to the fluid viscosity. This statement is also known as Darcy s law, which is often used to describe flow of liquids through porous media. 

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. 

Equation 12.13 is known as the Blake-Kozeny equation or the Kozeny-Carman equations it describes the experimental data for steady flow of newtonian fluids through beds of uniform-size spheres satisfactorily for less than about 10. 

We will now look at graphs of resolution versus analysis time for various hoioes of particle size and column length. As a measure for resolution, we aply use the square root of the plate count, as shown in the resolution luation. Equation 3.1. As analysis time, we use 10 times the breakthrough of an unretained peak, as shown in Equation (32). We use the van ater equation to calculate the HETP, from which we determine the plate Dt From the Kozeny-Carman equation [Eq. (3.3)] we calculate the lire drop across the column. We set an upper pressure limit of 20 MPa 3000psi). The curves will stop when this pressure limit is reached. 

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. 

For a Newtonian fluid, m = 1, both equations (5.46) and (5.47) reduce to the well known Kozeny-Carman equation. Equation (5.47) correlates most of the literature data on the flow of power-law fluids through beds of spherical particles up to about Re 1, though most work to date has been carried out... 

It should be noted that T = -Jl and R o = 2.5 have been used in deriving equation (5.49). Again for the special case of Newtonian fluids, Tq = 0 or 0 = 0, F(0) = 1 and equation (5.49) reduces to the Kozeny-Carman equation. The scant experimental data available on pressure drop for the streamline flow for Bingham plastic fluids (Re 1) is consistent with equation (5.49). 

For n = 1, equation (5.67) reduces to the well-known Kozeny-Carman equation ... 

The steam flow can be calculated using the Kozeny-Carman equation, which is valid for laminar flow in the neighborhood of a particle ... 

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