Porous solids tortuosity

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. 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

In real porous solids, the pores are not straight, and the pore radius can vary. Two parameters are used to describe the diffusion path through real porous solids the void fraction, e, defined as fhe ratio of pore area to fofal cross-sectional area, and the tortuosity, r, which corrects for the fact that pores are not straight. The resulting effective diffusivity is then... 

Tortuosity values range from about 1.5 to over 10. A reasonable range of values for many commercial porous solids is about 2-6. 

It is common in many practical battery designs to immobilize a liquid electrolyte phase within a porous solid insulator. The electrolyte conductivity and ohmic loss in such a system are determined by the number of pores, their size, shape and tortuosity. The tortuosity coefficient, /3, is defined as the ratio of the mean distance covered by an ion traversing a porous matrix, to the direct distance of one side of the matrix to the other. The relative reduction in the conductivity of an electrolyte solution caused by confining it in a porous solid is called the conductivity attenuation, 0. For a matrix of uniform cylindrical pores it is given by... 

Tortuosity Factors for High Pressure Extraction of Porous Solids (39)... 

In equations (5) and (6), DM and DK are the molecular and Knudsen diffusivities, respectively, and e and x are the void fraction and the tortuosity of the porous solid, respectively. For pore dimensions significantly larger than the mean free path of the diffusant in the gas phase, the diffusivity is governed by molecular diffusion, but when the pore diameter becomes smaller than the mean free path, diffusion is properly described by Knudsen diffusion. When the pore diameter approaches that of the diffusing species, around 10, one enters the configurational regime. 

When volume V is occupied by a porous solid, eq 11 is generally made to include a tortuosity coefficient [), the pore fraction e (fraction of the support grain volume occupied by the pore space) and an interaction coefficient K between the precursor and the support (K = 1 if there is no interaction) ... 

Here e(r) is the measured void fraction in pores of radii less than r. The permeability B and the tortuosity K(r) are symmetric second-order tensors according to the preceding two equations they can be predicted as integrals over r and f but are better obtained by fitting Eq. (3.4-11) to material flux measurements for the given porous solid. 

Obviously, if one of the molecular or Knudsen diffusion coefficients is vastly greater than the other it may be ignored. Several formulae have been given for the transition between the two, but perhaps the simplest of them derives from thinking of the reciprocal of the diffusion coefficient as a resistance and the two modes of diffusion as being in parallel. Then, allowing for the area and tortuosity, the effective diffusion coefficient in the porous solid can be taken to be... 

Estimate the effective diffusivity of hydrogen in ethane in a porous solid with an average pore size of 4000 A, 40% porosity, and tortuosity of 2.5. The gas mixture is at a pressure of 10 atm and a temperature of 373 K. For this system, the ordinary diffusion... 

For Knudsen diffusion in porous solids of porosity e and tortuosity t,... 

This simple equation for pure gas gives us a useful tool to study the structure of a porous solid. Making use of information such as the Knudsen diffusivity is proportional to square root of temperature and inversely proportional to the molecular weight, we can carry out experiments with different gases having different molecular weights and at different temperatures to determine the value for the tortuosity in the Knudsen relation and the viscous parameter Bq. 

Capillary theory uses the simplest model, whereby pores within a solid material are represented as parallel capillaries of equal diameters in the porous solid. The analogy is between the tortuous pore-system of the solid and the cylindrical pores of the capillaries. The equation for k is then derived from the Hagen-Poiseuille equation for streamline flow through straight circular capillaries taking account of the tortuosity of material s pores. The tortuosity is defined as the ratio of the actual length of the flow channel to the length of the porous medium. 

A sintered solid of silica 2.0 mm thick is porous with a void fraction e of 0.30 and a tortuosity t of 4.0. The pores are filled with water at 298 K. At one face the concentration of KCl is held at 0.10 g mol/liter, and fresh water flows rapidly by the other face. Neglecting any other resistances but that in the porous solid, calculate the diffusion of KCl at steady state. 

The rate of diffusion of the solute through the solid and solvent to the surface of the solid is often the controlling resistance in the overall leaching process and can depend on a number of different factors. If the solid is made up of an inert porous solid structure with the solute and solvent in the pores in the solid, the diffusion through the porous solid can be described by an effective diffusivity. The void fraction and tortuosity are needed. This is described in Section 6.5C for diffusion in porous solids. 

Figure 3.2.28 Influence of the porosity of a porous solid on the tortuosity according to correlations given in the literature (1) Hugo (1974), (2) Weisz and Schwartz (1962), (3) Wheeler (1955),...

The diffusivity of a component in a porous solid, where the pores are filled with a fluid phase, is of course lower than the diffusivity in the fluid phase itself. If the void fraction in the particles is e, and the tortuosity factor is the effective diffusivity, related to the total cross sectional area of the solid particle, would be reduced by a factor of e. The tortuosity factor should not only account for the fact that the diffusing molecules follow a zig-zag path, but also that the individual pores do not have a constant cross section. One might attempt to describe the diffusion in the porous structure using geometrical parameters, but it is in fact simpler to measure the effective diffusivity experimentally. In the following it is assumed that the effective diffusivity of 0 of reactant A in the porous solid is known. 

Walton [102] also included a porosity tortuosity factor in the diffusion coefficient to take into account the presence of porous solids and/or gas bubbles in the crevice gap. 

Little information is available, however, on experimentally measured tortuosities and Knudsen diffusion coefficients for noncatalytic gas-solid reaction systems [32, 33]. The techniques available for the characterization of porous solids and the measurements of pore diffusion phenomena will be discussed in Chapter 6. 

Stefan-Boltzmann constant Lennard-Jones 6-12 parameter tortuosity factor characteristic of porous solid condensation coefficient collision integral... 

In this work solid-gas chromatography is used to measure dynamic diffusion coefficients of argon in various porous solids. Mercury porosimetry is used to study the internal macroporosity and macro-morphology of these solids. Finally, an attempt is made to elucidate a relationship between the tortuosity measured from the transport experiment and the internal structure of the porous medium as characterized by porosimetry. 

The variation of stress along the thread due to diameter variation will confound the dynamic sensitivity above and complicate the correlation of tortuosity considerably. Unfortunately, in most real porous solids, a large PS/TS ratio is the norm rather than the exception. It has been observed in some porous solids (e.g., compressed Aerosils) that there is no scanning-rate influence on retained mercury. 

Gas transport in porous solids of intermediate porosity do not possess void network sensitivity. The tortuosity is well-conelated to porosity. 

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