Tortuous pore, diffusion

Figure 18 Schematic showing a corrosion product deposit composed of interconnected tortuous pores illustrating that the diffusion path length of dissolved species or dissolved oxidant through the deposit is greater than the thickness of the deposit.

It follows that only a certain fraction of all the Rn atoms formed in the material grains can escape into the pore space. This is the emanation fraction of the material. In the pore space the Rn atom diffuses through the tortuous pore paths. This process is governed largely by diffusion, although convective processes may also assist in the transport e.g. if pressure differentials exist across the material. Water content again... 

The sources of band broadening of kinetic origin include molecular diffusion, eddy diffusion, mass transfer resistances, and the finite rate of the kinetics of ad-sorption/desorption. In turn, the mass transfer resistances can be sorted out into several different contributions. First, the film mass transfer resistance takes place at the interface separating the stream of mobile phase percolating through the column bed and the mobile phase stagnant inside the pores of the particles. Second, the internal mass transfer resistance controls the rate of mass transfer between this interface and the adsorbent surface. It is composed of two contributions, the pore diffusion, which is molecular diffusion taking place in the tortuous, constricted network of pores, and surface diffusion, which takes place in the electric field at the liquid-solid interface [60]. All these mass transfer resistances, except the kinetics of adsorption-desorption, depend on the molecular diffusivity. Thus, it is important to study diffusion in bulk liquids and in porous media. 

The concentration gradients in an asymmetric membrane are complex because the driving force for diffusion in the skin layer is the concentration gradient of gas dissolved in the dense polymer, and the driving force in the porous support layer is a concentration or pressure gradient in the gas-filled pore. When the porous layer is thick, diffusion does not contribute very much to the flux, and gas flows by laminar flow in the tortuous pores. For high-flux membranes, there may also be significant mass-transfer resistances in the fluid boundary layers on both sides. 

When treating diffusion of solutes in porous materials where diffusion is considered to occur only in the fluid inside the pores, it is common to refer to an effective diffusivity, DABeg, which is based on (1) the total cross-sectional area of the porous solid rather than the cross-sectional area of the pore and (2) on a straight path, rather than the actual pore path, which is usually quite tortuous. In a binary system, if pore diffusion occurs only by ordinary molecular diffusion, Fick s law can be used with an effective diffusivity that can be expressed in terms of the ordinary diffusion coefficient, DAB, as... 

Our discussion of the various types of diffusion has presumed that the diffusion takes place in a well-characterized pore stmcture, that is, straight cylindrical pores. However, the catalysts used in industry have extremely complex stmctures with interconnecting pores, tortuous pores, and wide variations in pore diameter as one moves along the length of the pore. Consequently, we need to convert the combined diffusivities that are appropriate for Knudsen and/or bulk diffusion into effective... 

The stratum corneum is the outer most layer of nonviable epidermis. It has a thickness of about 10 to 12 pm. The stratum corneum consists of 15 to 25 layers of flattened, stacked, hexagonal, and cornified cells known as corneocytes. Each cell is approximately 40 pm in diameter and 0.5 pm in thickness [Bouwstra, 1997]. The thickness of stratum corneum varies with the site of human body. The body extremities such as palms and soles have a thicker stratum corneum [Walters and Roberts, 2002]. The stratum corneum is characterized by an array of keratin-rich corneocytes surrounded by lipid lamella made of cholesterol, free fatty acids, and ceramides [Bouwstra, 1997]. The corneocytes are arranged in brick and mortar structure. Such structural arrangement creates a tortuous intercellular diffusion pathway for water or any other molecules that transverse the stratum corneum. The hydrophobic lipids that surround these diffusion paths or water pores are organized in tight lamellar structure. The summative effects translate to the formation of a tight permeation barrier [Menon, 2002]. 

The internal mass transfer resistances are ascribed to the slow migration of substrate inside long, tortuous pores or layers of hydrogel. Mass transfer resistances inside the particle leads to a decrease from unity in //,. This decrease is correlated with a dimensionless parameter, the Thiele modulus, relating the reaction rate and molecular diffusion inside the support particle. The Thiele modulus varies with the reaction mechanism and shape of the particle. However, it has been established that if a generalized Thiele modulus is used, ge . 

In a depth filter—Figure 9.3—the particles are smaller than the medium openings and hence they proceed through relatively long and tortuous pores where they are collected by a number of mechanisms (gravity, diffusion and inertia) and attach to the medium by molecular and electrostatic forces. 

Diffusion in a porous solid (a) straight-through pores (b) tortuous pores with branching (c) molecular diffusion (d) Knudsen diffusion. 

The internal structure of the catalyst particle is often of a complex labyrinth-like nature, with interconnected pores of a multiplicity of shapes and sizes, In some cases, the pore size may be less than the mean free path of the molecules, and both molecular and Knudsen diffusion may occur simultaneously. Furthermore, the average length of the diffusion path will be extended as a result of the tortuousity of the channels. In view of the difficulty of precisely defining the pore structure, the particle is assumed to be pseudo-homogeneous in composition, and the diffusion process is characterised by an effective diffusivity D, (equation 10.8). 

The geometry of the pore structure makes it impossible to determine accurately the effective length of the diffusion path. Interconnections within the pore structure, the tortuous character of individual pores, and variations in cross-sectional area along the pore length all contribute to the difficulty of the task. 

In the special case of an ideal single catalyst pore, we have to take into account that diffusion is quicker than in a porous particle, where the tortuous nature of the pores has to be considered. Hence, the tortuosity r has to be regarded. Furthermore, the mass-related surface area AmBEX is used to calculate the surface-related rate constant based on the experimentally determined mass-related rate constant. Finally, the gas phase concentrations of the kinetic approach (Equation 12.14) were replaced by the liquid phase concentrations via the Henry coefficient. This yields the following differential equation ... 

In this derivation, the diffusion coefficient which is used is really a parameter, since it is not certain which gas diffusion rate is controlling, that of hydrogen into a pore, or that of water vapour out of the pore. The latter seems to be the most probable, but the path of diffusion will be very tortuous through each pore and therefore the length of the diffusion path is ill-defined. 

The details concerning the determination of /)clT and the relevant equations have been presented in Section 3.9.9. Here, let us just recall that the effective diffusivity /)clT is used to represent the diffusion of a fluid inside a catalyst particle. Deff is defined on the basis that the paths inside the particle are tortuous and that the pores have various cross-sectional areas. Moreover, not all of the area normal to the direction of the flux is available for the molecules to diffuse. The diffusion can take place by one or more of three mechanisms ... 

When the denuder active surface is an inert porous membrane such that the analyte molecule must diffuse across the pores to be trapped by an absorber liquid, only a fraction of the membrane surface is porous, and the pores may also be tortuous. Consequently, collision at the membrane surface is not synonymous with uptake. Corsi et al. (45) developed a numerical solution for the collection efficiency observed for such a membrane-based diffusion denuder, hereinafter referred to as a diffusion scrubber (DS). Both groups of researchers dealing with the issue of less than unity uptake probability reached the conclusion that this value must be very much less than... 

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 ... 

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. 

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