Porosity and Tortuosity

The openness (e.g., volume fraction) and the nature of the pores affect the permeability and permselectivity of porous inorganic membranes. Porosity data can be derived from mercury porosimetry information. Membranes with higher porosities possess more open porous structure, thus generally leading to higher permeation rates for the same pore size. Porous inorganic membranes, particularly ceramic membranes, have a porosity 

As discussed in the previous chapter, the network of the inter-connected membrane pores formed during preparation and fabrication may be tortuous or nearly straight depending on the synthesis and subsequent heat treatment methods and conditions. The microstructure of a membrane, particularly the type with tortuous pores, is too complicated to be described by a single parameter or a simple model. Due to the relatively poor knowledge of flow through porous media, an empirical term called tortuosity has been introduced and used by many researchers to reflect the relative random orientation of a pore network and is based on the Kozeny-Carmen equation for the membrane flux, J  

However, the complexity of the manner in which tortuosity is determined and the general belief that it is not an intrinsic parameter directly related to material or uansport properties have limited its use only in the scientific literature discussions but not by the membrane industry. 

---

From the magnitudes of the diffusion coefficients, it is evident that under the conditions cited the majority of the mass transport will occur by Knudsen diffusion. Equation 12.2.9 and the tabulated values of the porosity and tortuosity may be used to determine the effective diffusivity. 

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. 

Because the firn is ventilated by atmospheric air while the bubbles are forming over a period of time and ice depths, the air eventually trapped in the bubbles is a time-integrated sample that is younger than the snow deposit itself. For example, in one recent study (Smith et al., 1997), the air bubbles were, on average, 220-700 years younger than the ice in which they were embedded, but the difference can be as much as several thousand years (e.g., see Rommelaere et al., 1997). These exchange processes with the atmosphere, gas diffusion, and the porosity and tortuosity of the ice pores have to be taken into account in relating the depth of the core to the age of the trapped air. 

When the effective diffusivity of solutes D g can be approximated by the diffusivity in water, D, multiplied by a constant that includes the effects of particle porosity and tortuosity of pores in particles. Equation 14.3 can be written as follows ... 

In the porous medium, diffusion is affected by the porosity and tortuosity of the medium itself therefore Knudsen diffusion is computed as well as the ordinary diffusion. Eventually, an effective diffusion coefficient is calculated that depends on the ordinary and Knudsen diffusion coefficients and on the ratio between porosity and tortuosity of the medium (Equation (3.58)). 

Equations 8.1, 8.9 and 8.13 describe the dependence of the reaction rate on the pore size, particle porosity and tortuosity. 

Since the rate constant is known, estimation of the effective diffusivity allows the calculation of particle radius. In the absence of experimental data, the porosity and tortuosity are assumed to be 0.5 and 4, respectively. Thus,... 

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. 

Higher intrapellet residence times increase the contribution of chain initiation by a-olefins to chain growth pathways. This intrapellet delay, caused by the slow diffusion of large hydrocarbons, leads to non-Flory carbon number distributions and to increasingly paraffinic long hydrocarbon chains during FT synthesis. But intrapellet residence time also depends on the effective diameter and on the physical structure (porosity and tortuosity) of the support pellets. The severity of transport restrictions and the probability that a-olefins initiate a surface chain as they diffuse out of a pellet also de-... 

Pmi, Pm2 6 the partial pressures of vapor (water) at the membrane surfaces on the feed and permeate sides, respectively Kiji is the membrane coefficient that is a function of membrane properties (pore size, thickness, porosity, and tortuosity), properties of the vapor transported across the membrane (molecular weight and diffusivity) and temperature gradient... 

For membranes with the same L and ro/ri, the membrane s porosity and tortuosity will contribute to Dm m> and consequently will determine the value of EF [48]. 

The constant K incorporates factors which determine the flow resistemce of the gel layer such as its porosity and tortuosity, pore size and shape, etc. Using the Kozeny-Carman relation, Leenaars [3] studied the relation between K, microstructure and process parameters. The value of the pressure drop across the gel layer APg is obtained from AP after correction for the pressure drop APg in the support, which is usually very small. 

Here Cq = c(z = 0,t) and Dg is the effective diffusion coefficient (porosity and tortuosity effects are incorporated in Dg). If the upstream (high) pressure is constant and much larger than the downstream (low) pressure, the slope of the asymptote will correspond to the steady state and so it is possible to determine the diffusivity under both steady state and transient conditions from a single permeation experiment. With a narrow and unimodal pore size distribution both methods yield reasonable consistent values. Large discrepancies point to strong microstructural effects (bimodal broad distribution, many dead ends, many defects). 

The aqueous or organic stagnant boundary films diffusion resistances may be combined with the diffusion resistances of the same hquid films (aqueous or organic) inside the membrane pores (taking into account the membrane porosity and tortuosity) in one-dimensional series of diffusion resistances. This assumption is related to the BLMs with membrane supports only. 

The diffusional process through a SLM is affected by the porosity and tortuosity of the polymeric support. Direct comparison of fluxes J and the corresponding diffusion coefficients when using different supports is not possible and has to be corrected for the membrane characteristics to obtain the bulk diffusion coefficient Db [51] ... 

For non-porous catalyst pellets the reactants are chemisorbed on their external surface. However, for porous pellets the main surface area is distributed inside the pores of the catalyst pellets and the reactant molecules diffuse through these pores in order to reach the internal surface of these pellets. This process is usually called intraparticle diffusion of reactant molecules. The molecules are then chemisorbed on the internal surface of the catalyst pellets. The diffusion through the pores is usually described by Fickian diffusion models together with effective diffusivities that include porosity and tortuosity. Tortuosity accounts for the complex porous structure of the pellet. A more rigorous formulation for multicomponent systems is through the use of Stefan-Maxwell equations for multicomponent diffusion. Chemisorption is described through the net rate of adsorption (reaction with active sites) and desorption. Equilibrium adsorption isotherms are usually used to relate the gas phase concentrations to the solid surface concentrations. 

---