Diffusion cell models

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

Flow in stirred vessels was also investigated by Holmes et al. (H5), who simulated mass transfer in a diaphragm diffusion cell stirred by magnetic stirrer bars. This is a good example of a simple model study with a direct practical purpose. A minimum stirring speed in such cells is necessary to avoid appreciable errors in the cell constant. The experiment permits this stirring speed to be related to the solution properties. 

CA in which many filled cells execute a random walk but never interact with one another, cannot give rise to stable pattern formation since the cells will move at random forever. However, if cells can interact when they meet, so that one diffusing cell is allowed to stick to another, stable structures can be created. These structures illustrate the modeling of diffusion-limited aggregation (DLA), which is of interest in studies of crystal formation, precipitation, and the electrochemical formation of solids. 

Here G %) is the modulus that results from the particle diffusing a distance / (t). This can be represented by a cell model with a time dependent separation ... 

The Sartorius Absorption Model (26), which served as the forerunner to the BCS, simulates concomitant release from the dosage form in the GI tract and absorption of the drug through the lipid barrier. The most important features of Sartorius Absorption Model are the two reservoirs for holding different media at 37°C, a diffusion cell with an artificial lipid barrier of known surface area, and a connecting peristaltic pump which aids the transport of the solution or the media from the reservoir to the compartment of the diffusion cell. The set-up is shown in Figures 7a and b. 

The two media typically used include Simulated Gastric Fluid (pH 1-pH 3) and Simulated Intestinal Fluid (pH 6-pH 7). The drug substance under investigation is introduced, and its uptake in the diffusion cell ( absorption ) is governed by its hydrophilic-lipophilic balance (HLB). The absorption model proposed by Strieker (26) in the early 1970s therefore effectively took into consideration (in an experimental sense) all aspects considered by the theory of the BCS, which was introduced more than 20 years later. 

The beginning of modeling of polymer-electrolyte fuel cells can actually be traced back to phosphoric-acid fuel cells. These systems are very similar in terms of their porous-electrode nature, with only the electrolyte being different, namely, a liquid. Giner and Hunter and Cutlip and co-workers proposed the first such models. These models account for diffusion and reaction in the gas-diffusion electrodes. These processes were also examined later with porous-electrode theory. While the phosphoric-acid fuel-cell models became more refined, polymer-electrolyte-membrane fuel cells began getting much more attention, especially experimentally. 

Electrochemical reactions take place at the catalyst layers of the fuel cell. At the anode and cathode, hydrogen is oxidized (eq 1) and oxygen is reduced (eq 2), respectively. These layers are often the thinnest in the fuel-cell sandwich but are perhaps the most complex because this is where electrochemical reactions take place and where all of the different types of phases exist. Thus, the membrane and diffusion media models must be used in the catalyst layer along with additional expressions related to the electrochemical kinetics on the supported electrocatalyst particles. 

The simplest way to treat the catalyst layers is to assume that they exist only at the interface of the diffusion media with the membrane. Thus, they are infinitely thin, and their structure can be ignored. This approach is used in complete fuel-cell models where the emphasis of the model is not on the catalyst-layer effects but on perhaps the membrane, the water balance, or multidimensional effects. There are different ways to treat the catalyst layer as an interface. 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

The importance of materials characterization in fuel cell modeling cannot be overemphasized, as model predictions can be only as accurate as their material property input. In general, the material and transport properties for a fuel cell model can be organized in five groups (1) transport properties of electrolytes, (2) electrokinetic data for catalyst layers or electrodes, (3) properties of diffusion layers or substrates, (4) properties of bipolar plates, and (5) thermodynamic and transport properties of chemical reactants and products. 

A very brief description of biological membrane models, and model membranes, is given. Studies of lateral diffusion in model membranes (phospholipid bilayers) and biological membranes are described, emphasizing magnetic resonance methods. The relationship of the rates of lateral diffusion to lipid phase equilibria is discussed. Experiments are reported in which a membrane-dependent immunochemical reaction, complement fixation, is shown to depend on the rates of diffusion of membrane-bound molecules. It is pointed out that the lateral mobilities and distributions of membrane-bound molecules may be important for cell surface recognition. 

Hastings et al. [55] used this same in vitro technology to assess the enhancement delivery of dexamethasone using the Visulex iontophoretic system. The Visulex applicator and a freshly excised rabbit sclera were positioned between two halves of a side-by-side diffusion cell with the conjunctival side of the sclera facing the applicator (Figure 26.10). The donor drug solution (1 mg of dexamethasone phosphate) was present in the applicator, and diluted vitreous humor was modeled in the receptor cell. One milliampere direct current was applied for 60 min, and samples were collected during different treatment periods. It was demonstrated that the Visulex system produced a twofold increase in the amount of dexamethasone phosphate delivered after 60 min, compared with a standard iontophoretic administration (without the Visulex applicator). 

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