Tortuosity tortuous path

Tye [38] explained that separator tortuosity is a key property determining the transient response of a separator (and batteries are used in a non steady-state mode) steady-state electrical measurements do not reflect the influence of tortuosity. He recommended that the distribution of tortuosity in separators be considered some pores may have less tortuous paths than others. He showed mathematically that separators with identical average tortuosities and porosities can be distinguished by their unsteady-state behavior if they have different distributions of tortuosity. 

One must be very careful in reviewing the older, and some more recent, literature in consideration of the tortuosity and constriction factors some work has attempted to separate these two factors however, more modem developments show that they cannot be strictly decoupled. This aspect will be particularly important when reviewing the barrier and tortuous-path theories of electrophoresis, as discussed later. 

It is relevant to note that tortuosity defined by Equation (57) is by no means the same as that defined by (JLJL), where L, is the effective tortuous path length... 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

It is noted that straight capillary tubes may not portray the complex structure of the porous medium. Thus, in practice, the factor 32 in Eq. (5.321) is commonly replaced by an empirical parameter known as tortuosity. The tortuosity accounts for the tortuous paths of the porous medium. 

To apply Fick s laws to solute fluxes in sediments, adjustments have to be made to these equations to account for the negative interference effects that sediment particles have on the diffusion of solutes in pore waters (Lerman, 1979 Berner, 1980). For example, tortuosity, defined as the length of the tortuous path that a solute travels around particles across a distance across a certain depth interval can be described by the following equation (Berner, 1980 Krom and Berner, 1980a) ... 

Tortuosity the length of the tortuous path that a solute travels around particles across a distance across a certain depth interval. 

Equation 2.1 defines the flux in a bulk liquid or gas phase (with unit porosity or in a straight capillary). The effect of tortuous paths has to be considered in a porous medium. We use the effective diffusion coefficient, D, to replace Do in Eq. 2.1 to consider the effect of tortuosity. The relationship between D, and Do may be defined (Childs, 1969) using Eq. 2.4,... 

For the situation where the voids are filled completely with liquid water, the concentration of salt in water at boundary 1 is c, and at point 2 isc j- The salt in diffusing through the water in the void volume takes a tortuous path which is unknown and greater than (zj — z,) by a factor t, called tortuosity. Diffusion does not occur in the inert solid. For a dilute solution using Eq. (6.3-5) for diffusion of salt in water at steady state,... 

In addition, the presence of impermeable filler in a polymer forces the diffusant molecule to travel further around the filler particles. This physical blocking effect is known as tortuosity, because the filler forces the diffusant to take a more indirect, or tortuous, path through the material. The degree of tortuosity imposed is dependent upon the anisotropy and orientation of the filler particles with respect to the direction of diffusion. For example, platy particles oriented perpendicularly to the diffusion vector will be particularly effective in retarding diffusion. The permeability of a composite can be calculated using an equation that allows for the reduction in permeant solubility and for the tortuosity (Equation 8.3). Where P and Pp are the permeability of the composite and the unfilled polymer, respectively. The terms w and t refer to the width and thickness of the filler and (pp and (pf represent the volume fraction of polymer and filler. 

The parameter Tp is referred to as the tortuosity of the catalyst particle. Originally, Tp was intended to correct for the fact that the pores in a catalyst are not straight and are not all parallel to the direction of diffusion. As a result, the diffusing molecules must follow a tortuous path and must travel a longer distance than a straight line in the direction of the net diffusive flux. In reality, Tp corrects for many other nonidealities, such as the variation in cross section along the length of a pore. 

Usually, nanoparticles in polymeric systems, are supposed to increase the barrier properties of such materials. One of the most important effects of clays on the polymer matrix properties is the dramatic improvement of the barrier properties of polymers, since clay sheets are namrally impermeable. The extent of the improvement in the barrier properties depends on the degree of tortuosity created by the nanoparticles layers, and the manner by which diffusion of molecules through the polymer film occurs. The ratio of the actual distance to molecule diffusion to the shortest distance to diffusion (polymer film thickness) determines the tortuous factor. This is done by creating a maze or tortuous path that will ordinarily slow down or retard the progress of gas molecules through the polymeric matrix. 

Carrera et al. [51] proposed that the increase in barrier properties which were found at low contents of clay were due to defects at the interface which had less resistance to permeation. At higher content of OMt, and since the crystallinity was maintained as that of the pure polymer, the barrier properties were increased as a result of the tortuous path created by the exfoliated structure of the OMt. Monsivais-Barro et al. [52], proposed that the tortuosity effect caused by the dispersed nanolayers in HDPE improve the gas barrier properties in some cases while in other cases the interfacial regions or the free... 

The values of k, D, and a used in Equations 3, 12, and 4, lespeetively, are reduced within the porous electrode relative to their bulk values due to the tortuous path which the ions in solution must make around the solid particles or which the electrons must make around the electrolyte-fiUed pores. Generally, the effect of volume fraction e, and tortuosity of the phase of interest on property P in that phase is accounted for by the Bmggeman relation [82], P = EpPoo/T where x is the tortuosity and is usually related to the porosity by t = s . Because the flux N is defined with respect to superficial area as opposed to electrolyte area, D already gets multiplied once by e in the mass balance, so D = Dj ... 

The effective diffusivity is obtained from D, but must also take into account the two features that (1) only a portion of the catalyst particle is permeable, and (2) the diffusion path through the particle is random and tortuous. These are allowed for by the particle voidage or porosity, p, and the tortuosity, rp, respectively. The former must also be measured, and is usually provided by the manufacturer for a commercial catalyst. For a straight cylinder, rp = 1, but for most catalysts, the value lies between 3 and 7 typical values are given by Satterfield. 

The tortuosity is also included in the geometric factor to account for the tortuous nature of the pores. It is the ratio of the path length which must be traversed by molecules in diffusing between two points within a pellet to the direct linear separation between those points. Theoretical predictions of r rely on somewhat inadequate models of the porous structure, but experimental values may be obtained from measurements of De, D and e. 

Support tortuosity should be minimized to reduce the diffusional path length. However, many membrane preparation techniques, such as casting, produce support materials with tortuous pores. Operating Pressure Considerations... 

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