Transport Hagen-Poiseuille equation

Figure 7 illustrates the dynamics of fluid migration through porous carbon electrodes to obey the Hagen-Poiseuille equation that is normally used to describe the transport through membranes having the pores of cylinder-like shape. Therefore, this method can probably be used for express analysis of the electrolyte dynamics in different porous carbon materials. 

The velocity profile for a Newtonian fluid in a capillary is well-described in most introductory transport texts by the Hagen-Poiseuille equation... 

The fine-pore model was developed assuming the presence of open micropores on the active surface layer of the membrane through which the mass transport occurs (10). The existence of these different pore geometries also means that different models have been developed to describe transport adequately. The simplest representation is one in which the membrane is considered as a number of parallel cylindrical pores perpendicular to the membrane surface. The length of each of the cylindrical pores is equal or almost equal to the membrane thickness. The volume flux through these pores can be described by the Hagen-Poiseuille equation. Assuming all the pores have the same radius, then we have... 

Generally, the pure solvent transporting through porous UF membranes is directly proportional to the applied transmembrane pressure (AP). The Kozeny-Carman and Hagen-Poiseuille equations describe the convection flow (J ) as follows (49) ... 

Microfluidics is a concept that describes the science and technology of design, fabrication and operation of systems of microchannels that conduct liquids and gases. T q)ically, the channels have widths of tens to hundreds of micrometers and the speed of flow of the fluids is such that the viscous forces dominate over inertial ones. The resulting - linear - equations of flow and its laminar character provide for extensive control the speed of flow obeys the simple Hagen-Poiseuille equation that relates the speed linearly to the pressure drop through the particular channel and to its inverse hydraulic resistance, which in term is a function of the dimensions of the channel and the viscosity of the fluid. This property, when combined with t5q)ically large values of the Peclet number [1] that reflect the fact that diffiisional transport is t5q)ically slow in comparison to the flow, it is possible to control the profiles of concentration [2] of chemicals and... 

For membranes with straight capillaries as pores the Hagen-Poiseuille equation (2) applies, while the Karman-Kozeny equation (3) various forms of this equation have been applied) is valid for transport through membranes with a more nodular structure. 

The Hagen-Poiseuille equation clearly shows the effect of membrane structure on transport. By comparing eq. V - 54 with the phenomenological eq. V - 43 (and writing in the latter case AP/Ax as driving force instead of AP), a physical meaning can be given to the... 

The viscous transport in membrane pores is generally calculated from the Hagen-Poiseuille equation for stationary Newtonian flow in a cylindrical capillary. This leads to the following equations for calculating permeance. 

It is indicated that these transport parameters are functions of A. The hydraulic permeability (D Arcy coefficient), /Cp (A), exhibits strong dependence on A because larger water contents result in an increased number of pores used for water transport and better connectivity in the porous network, as well as in larger mean radii of these pores. A modification of the Hagen-Poiseuille-Kozeny equation was considered by Eikerling et aU- to account for these structural effects ... 

For relatively porous nanofiltration membranes, simple pore flow models based on convective flow will be adapted to incorporate the influence of the parameters mentioned above. The Hagen-Poiseuille model and the Jonsson and Boesen model, which are commonly used for aqueous systems permeating through porous media, such as microfiltration and ultrafiltration membranes, take no interaction parameters into account, and the viscosity as the only solvent parameter. It is expected that these equations will be insufficient to describe the performance of solvent resistant nanofiltration membranes. Machado et al. [62] developed a resistance-in-series model based on convective transport of the solvent for the permeation of pure solvents and solvent mixtures ... 

The mean free path of molecules in air at atmospheric pressure is /free — 1 /(Niiyg), where Nl 2.69 10 cm is the number density of gas molecules and cTg 10 " cm is the cross section for elastic collisions of molecules. These numbers result in /free — 3.7 10 cm, or 37 nm. The mean pore radius of the GDL is in the order of 10 pm, which means that the flow in the GDL pores occurs in a continuum regime. Thus, pressure-driven oxygen transport in a dry porous GDL can be modeled as a viscous Hagen-Poiseuille flow in an equivalent duct. However, determination of the equivalent duct radius and the dependence of this radius on the GDL porosity is a nontrivial task (Tamayol et al., 2012). Much workhas recently been done to develop statistical models of porous GDLs and to calculate viscous gas flows in these systems using Navier-Stokes equations (Thiedmann et al., 2012). 

All these methods lead to a set of parameters (membrane thickness, pore volmne, hydraulic radius) which are related to the working (macroscopic) permselective membrane properties. In the case of liquid permeation in a porous membrane, macro- and mesoporous structures are more concerned with viscous flow described by the Hagen-Poiseuille and Carman-Kozeny equations whereas the extended Nernst-Plank equation must be considered for microporous membranes in which diffusion and electrical charge phenomena can occur (Mulder, 1991). For gas and vapor transport, different permeation mechanisms have been described depending on pore sizes ranging from viscous flow for macropores to different diffusion regimes as the pore size is decreased to micro and ultra-micropores (Burggraaf, 1996). 

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