Carman-Kozeny model

The formation of ceramic membranes for microfiltration, ultrafiltration or nanofiltration by association of various granular layers is now a common procedure [10]. Each layer is characterized by its thickness, h, its porosity, 8, and its mean pore diameter, dp. These parameters are controlled by the particle size, d, and the synthesis method. Each layer induces a resistance which may be predicted through the classical Carman-Kozeny model ... 

Permeability data seem to be more reliable when modehng gas transport phenomena. Measured data for compressed and uncompressed GDLs were successfully fitted to the Carman-Kozeny model. ... 

It was shown in Chapter 2 that the simplest models of solid-liquid separation are those based of the Carman-Kozeny equation for filtration in whieh the bed permeability (filtrability), F, may be expressed by... 

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 ... 

The Carman-Kozeny equation is also a bundle-of-tubes model,but is usually written in terms of permeability and, more importantly, retains specific surface area as an adjustable parameter. The latter difference makes it somewhat more general than the Ergun equation, and it has found widespread application in areas such as soil science and materials science. 

Although the early versions of capillary models, namely, the Blake and the Blake-Kozeny models, are based on the use of V = Vq/s and = LT, it is now generally accepted [Dullien, 1992] that the Kozeny-Carman model, using Vi = VoT/e provides a more satisfactory representation of flow in beds of particles. Using equations (5.36), (5.38) and (5.40), the average shear stress and the nominal shear rate at the wall of the flow passage (equations 5.34 and... 

The present model development is based on a semi-heuristic model of flow through solid matrices using the concept of hydraulic diameter, which is also known as the Carman-Kozeny theory [7]. The theory assumes the porous medium to be equivalent to a series of parallel tortuous tubules. The characteristic diameter of the tubules is taken to be a hydraulic diameter or... 

For a given system, Eq. (6.9) predicts that the rate of consolidation increases with an increase in the permeability of the cast Kc. Various models have been put forward to account for the permeability of porous media. One of the most popular, based on its simplicity and its ability to outline the key parameters, is the Carman-Kozeny equation ... 

How through packed beds under laminar conditions can be described by a model in which the flow is assumed to be through capillaries whose surface equals that of the solids comprising the bed. The capillary volume is set equal to the void volume of the bed. The model leads to the well-known Carman-Kozeny equation as follows ... 

As was briefly pointed out in section 18.3.1, the Carman-Kozeny equation does not work well towards the limit of e = 1 Carman himself stated that the equation should not be used for e > 0.8. Several researchers have attempted to derive a model with a more realistic and general outcome. Perhaps the most significant attempt is that by Rudnick who used a ffee-surface cell model by Happel in which each particle is assumed to be a sphere at the centre of a cell, the volume of which is such that the porosity of each cell is the same as that of the bed. If the tangential stresses at the boundaries of adjoining cells are set to zero, an exact solution of the general Navier-Stokes equations exists, assuming that the inertial terms are negligible. 

Kozeny modeled a packed bed as a series of parallel, small diameter tubes of equal length and diameter [1]. Carman applied the work of Kozeny to experimentally determine pressure drops for the flow through packed beds [2]. This work produced the Carman-Kozeny equation for gas-phase pressure drop ... 

The coefficient of proportionality K is called the permeability of the reinforcement. According to theory [5] K is only dependent on the geometry between the fibers in the reinforcement (the pore space ). Several models for the dependence of K on the fiber volume fraction Vf has been proposed. The most-cited model is the so-called Kozeny-Carman model [16,17], which predicts a quadratic dependence on the fiber radius R in addition to the dependence on Vf... 

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 ... 

The best way to use the Kozeny-Carman model and other permeability models (e.g. the anisotropic model by Gebart) [18], is to use them as interpolation formulas for intermediate volume fractions between known values. Extrapolation should be done with extreme caution because the models are developed for idealized reinforcements. Typical values for the permeability of different types of reinforcement are given in Table 12.1. 

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 correction on the tortuosity term in Kozeny-Carman equation was initiated by Foscolo et al. (89) however, they repeated the error of Kozeny s work, which was pointed out later by Epstein (65). The tortuosity model of Foscolo et al. (89) was not in agreement with experiments, although the same line of investigation was continued by Puncochar and Drahos (66). 

A porous medium is modeled as made up of uniformly distributed straight circular capillaries of the same diameter. The flow through each capillary is an inertia free Poiseuille flow. By comparing the Poiseuille pressure drop and the Darcy pressure drop formulas, deduce an expression for the permeability. Discuss the difference between the result obtained and the Kozeny-Carman permeability. 

Table 8.5.1 tabulates the Happel values of the Kozeny constant, and the results are remarkable for the fact that, for a porosity between 0 and 0.6, the constant has a range only between about 4.4 and 5.1. Above a porosity of about 0.7 the Kozeny constant increases rapidly and becomes indeterminate. This is not surprising, since a condition of isolated particles is approached rather than a packed bed. What is, however, most encouraging is the agreement with the early Carman value in the range of porosity for which the model would most likely be expected to hold. 

To analyze the flow through a porous medium, we can, as before, model the medium as a collection of parallel cylindrical microcapillaries. As noted in Section 4.7, the actual sinuous nature of the capillaries may be accounted for by the introduction of an empirical tortuosity factor. The results for electroosmotic flow through a capillary are then readily carried over to the porous medium by using Darcy s law (Eq. 4.7.7) and, for example, the Kozeny-Carman permeability (Eq. 4.7.16). 

The Kozeny-Carman model can be used for membrane considered as a compacted noncharged packed bed, the permeate flow through the filtration medium can be assumed laminar when the mean free path of fluid molecules is very small compared to the pore diameter. 

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