Tortuosity

The disposable parameters in these equations are the permeability 0, the surface diffusion factor y, the tortuosity function . (a) (which also... 

Here c denotes the void fraction and T a tortuosity factor, while G... 

Fig. 3. Microporous membranes are characterized by tortuosity, T, porosity, S, and their average pore diameter, d. (a) Cross-sections of porous membranes containing cylindrical pores, (b) Surface views of porous membranes of equal S, but differing pore size.

Gravity, or centrifugation rarely provide enough hydrostatic pressure to force Hquid into nonwetting pores. If the Hquid wets the soHd the clump density increases as gas is displaced from the interior. It is best if submersion does not occur until the Hquid has completely displaced gas from the pores (Fig. 2). For wetting Hquids and pores with average diameter, D-pQ- y and tortuosity,the length, to which Hquid is pulled into a bed of powder by... 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

Dpi is smaller than the diffusivity in a straight cylindrical pore as a result of the random orientation of the pores, which gives a longer diffusion path, and the variation in the pore diameter. Both effects are commonly accounted for by a tortuosity factor Tp such that Dp = DjlXp. In principle, predictions of the tortuosity factor can be made if the... 

For catalyst particles, Satterfield (Heterogeneous Cataly.si.s in Frae-tiee, McGraw-Hill, 1980) recommends the use of a value of tp = 4 when no other information is available, and this can be used for many adsorbents. In general, however, it is more rehable to treat the tortuosity as an empirical constant that is determined experimentally for any particular adsorbent. 

For adsorbent materials, experimental tortuosity factors generally fall in the range 2-6 and generally decrease as the particle porosity is increased. Higher apparent values may be obtained when the experimental measurements are affected by other resistances, while v ues much lower than 2 generally indicate that surface or solid diffusion occurs in parallel to pore diffusion. 

Ruthven (gen. refs.) summarizes methods for the measurement of effective pore diffusivities that can be used to obtain tortuosity factors by comparison with the estimated pore diffusion coefficient of the adsorbate. Molecular diffusivities can be estimated with the methods in Sec. 6. 

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

Since theoretical calcination of effectiveness is based on a hardly realistic model of a system of equal-sized cylindrical pores and a shalq assumption for the tortuosity factor, in some industrially important cases the effectiveness has been measured directly. For ammonia synthesis by Dyson and Simon (Ind. Eng. Chem. Fundam., 7, 605 [1968]) and for SO9 oxidation by Kadlec et aJ. Coll. Czech. Chem. Commun., 33, 2388, 2526 [1968]). 

For the effective diffusivity in pores, De = (0/t)D, the void fraction 0 can be measured by a static method to be between 0.2 and 0.7 (Satterfield 1970). The tortuosity factor is more difficult to measure and its value is usually between 3 and 8. Although a preliminary estimate for pore diffusion limitations is always worthwhile, the final check must be made experimentally. Major results of the mathematical treatment involved in pore diffusion limitations with reaction is briefly reviewed next. 

There are essentially four important physical parameters that characterize a filter media and are used as a basis for relating the characteristics of the material to the system flow dynamics. These are porosity, permeability, tortuosity and connectivity. 

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

Permeability and porosity are related to each other if the porosity is zero the permeability is zero. Although a correlation between these two parameters may exist, permeability cannot be predicted from porosity alone, since additional parameters that contain more information about the pore structure are needed. These additional parameters are tortuosity and cormectivity. 

Tortuosity is defined as the relative average length of a flow path (i.e., the average length of the flow paths to the length of the medium). It is a macroscopic measure of both the sinuosity of the flow path and the variation in pore size along the flow... 

Coimectivity is a term that describes the arrangement and number of pore coimections. For monosize pores, coimectivity is the average number of pores per junction. The term represents a macroscopic measure of the number of pores at a junction. Connectivity correlates with permeability, but caimot be used alone to predict permeability except in certain limiting cases. Difficulties in conceptual simplifications result from replacing the real porous medium with macroscopic parameters that are averages and that relate to some idealized model of the medium. Tortuosity and connectivity are different features of the pore structure and are useful to interpret macroscopic flow properties, such as permeability, capillary pressure and dispersion. 

Tortuosity t is basically a correction factor applied to the Kozeny equation to account for the fact that in a real medium the pores are not straight (i.e., the length of the most probable flow path is longer than the overall length of the porous medium) ... 

The simplest model ignores tortuosity and assumes the bed equivalent hydro-dynamieally to a matrix of straight tubes - like a bundle of drinking straws e.g. as in Figure 2.10. 

The size distribution of the particles (with particles of a uniform size, there is increased voidage between the particles and a lower tortuosity, that is deviation from the linear path for the fluid flowing between the particles, making for improved filterability), and... 

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