Tortuosity, separators

With increasing tortuosity factor T and lower porosity P, R increases sharply. The electrical resistance of a separator is proportional to the thickness d of the membrane and is subject to the same dependence on temperature or concentration as... 

Ideally, separators would present no resistance to ion transport. In practice, some resistance must be tolerated. Still, the resistance of the separator is usually insignificant relative to the transport limitations in the electrodes. Separator permeability is typically characterized by air permeability. The Gurley number expresses the time required for a specific amount of air to pass through a specific area of separator under a specific pressure (e.g., 10 mL through 1 in2 (6.45 cm2) at 2.3 cm Hg). This measurement depends on porosity, pore size, thickness, and tortuosity according to Eq. (1) [17] ... 

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. 

Boyack and Giddings [45] considered the tortuosity effects to be separable from the constriction effects. In their derivation they assumed that the electric field in the constricted channel decreased proportionally to the decrease in cross-sectional area and changes in path length. The field was assumed not to penetrate the barrier. The effect of constriction can be written in terms of porosity and L as... 

Influence on Electrolyte Conductivity In porous separators the ionic current passes through the liquid electrolyte present in the separator pores. Therefore, the electrolyte s resistance in the pores has to be calculated for known values of porosity of the separator and of conductivity, o, of the free liquid electrolyte. Such a calculation is highly complex in the general case. Consider the very simple model where a separator of thickness d has cylindrical pores of radius r which are parallel and completely electrolyte-filled (Fig. 18.2). Let / be the pore length and N the number of pores (all calculations refer to the unit surface area of the separator). The ratio p = Ud (where P = cos a > 1) characterizes the tilt of the pores and is called the tortuosity factor of the pores. The total pore volume is given by NnrH, the porosity by... 

Diffusion Through Separators Like current flow, the diffusion of dissolved components through separators will be delayed by decreasing porosity and increasing tortuosity. The attenuation factor of diffusion, 8d (= DID f), usually coincides with that of conduction. 

The separation efficiency (e.g. permselectivity and permeability) of inorganic membranes depends, to a large extent, on the microstructural features of the membrane/support composites such as pore size and its distribution, pore shape, porosity and tortuosity. The microstructures (as a result of the various preparation methods and the processing conditions discussed in Chapter 2) and the membrane/support geometry will be described in some detail, particularly for commercial inorganic membranes. Other material-related membrane properties will be taken into consideration for specific separation applications. For example, the issues of chemical resistance and surface interaction of the membrane material and the physical nature of the module packing materials in relation to the membranes will be addressed. 

Tortuosity. Tortuosity is the ratio of mean effective capillary length to separator thickness. The tortuosity factor, r of a separator can be expressed... 

This parameter is widely used to describe the ionic transport by providing information on the effect of pore blockage. A tortuosity factor r = 1, therefore, describes an ideal porous laody with cylindrical and parallel pores, whereas values of r > 1 refer to more hindered systems. Higher tortuosity is good for dendrite resistance but can lead to higher separator resistance. 

While a good equivalent-circuit representation of the transport processes in a fuel cell can lead to an increased understanding, it is not as good as taking a 1-D sandwich model and taking it into the frequency domain. These models typically analyze the cathode side of the fuel cell. °2.3i3 3i4 pj g j ost comprehensive is probably that of Springer et al. °2 The use of impedance models allows for the calculation of parameters, like gas-phase tortuosity, which cannot be determined easily by other means, and can also allow for the separation of diffusion and migra-... 

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. 

In the case of gel permeation or size-exclusion HPLC (HP-SEC), selectivity arises from differential migration of the biomolecules as they permeate by diffusion from the bulk mobile phase to within the pore chambers of the stationary phase. Ideally, the stationary phase in HP-SEC has been so prepared that the surface itself has no chemical interaction with the biosolutes, with the extent of retardation simply mediated by the physical nature of the pores, their connectivity, and their tortuosity. In this regard, HP-SEC contrasts with the other modes of HPLC, where the surfaces of the stationary phase have been deliberately modified by chemical procedures by (usually) low molecular weight compounds to enable selective retardation of the biosolutes by adsorptive processes. Ideally, the surface of an interactive HPLC sorbent enables separation to occur by only one retention process, i.e., the stationary phase functions as a monomodal sorbent. In practice with porous materials, this is rarely achieved with the consequence that most adsorption HPLC sorbents exhibit multimodal characteristics. The retention behavior and selectivity of the chromatographic system will thus depend on the nature and magnitude of the complex interplay of intermolecular forces... 

Thermal and hydrothermal exposures can change the ix>re size and its distribution, porosity and tortuosity of a porous membrane which in turn influence the separation properties of the membrane such as permeability and permselectivity. Several ceramic membranes have been investigated for their responses to thermal and hydrothermal environments. 

H.J.C. le Hennepe, C.A. Smolders, D. Bargeman, and M.H.V. Mulder, Exclusion and tortuosity effects for alcohol/water. separation by zeolite filled PDMS membranes, Sep. Sci. Technol. 26(4) 585 (1991). 

Fig. 16. Numerical analysis evaluating the connectivity of faults with a damage zone. The analysis is aimed at identifying the critical density of structural features needed to separate domains with (a) fault-rock controlled flow behaviour and (b) tortuosity controlled flow behaviour. In the example shown, a transition from an open array of faults with tortuosity flow to a connected array occurs when the structural frequency is equivalent to between 200 features/100 m of core and 400 features/100 m of core.

Several models use the mass balance in Eq. 2.2 (ideal and equUibrimn-disper-sive models. Sections 2.2.1 and 2.2.2) as derived here without combining it with kinetic equations. In the latter case, Di in Eq. 2.2, which accounts only for axial diffusion, bed tortuosity, and eddy diffusion, is replaced with Da, which accoimts also for the effect of the mass transfer resistances. This is legitimate imder certain conditions, as explained later in Section 2.2.6. Other simple models account for a more complex mass transfer kinetics by coupling Eq. 2.2 with a kinetic equation (lumped kinetic models. Section 2.2.3) in which case Di is used. More complex models write separate mass balance equations for the stream of mobile phase percolating through the bed and for the mobile phase stagnant inside the pores of the particles (the general rate model and the lumped pore diffusion or FOR model, see later Sections 2.1.7 and 2.2.4). 

The transport properties (i.e. permeation and separation efficiency) of inorganic membrane systems depend, to a Icirge extent, on the microstructural features of the membrane and the architecture of membranes and modules. The microstructural features, such as pore shape and morphology, pore size (distribution), interconnectivity/tortuosity, as well as the architecture of the membrane and membrane-support combinations will be briefly described. Here, architecture means the way the different parts of the membrane system or module are shaped and combined. 

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