Diffusion partial

If we postulate that molecules move independently, the concentration c (2, t) at some point 2 is proportional to the probability density p(z,t) of finding a molecule there. Thus, the diffusion partial differential equation (2.16) holds when probability densities are substituted for concentrations ... 

Undoubtedly, the most promising modehng of the cardiac dynamics is associated with the study of the spatial evolution of the cardiac electrical activity. The cardiac tissue is considered to be an excitable medium whose the electrical activity is described both in time and space by reaction-diffusion partial differential equations [519]. This kind of system is able to produce spiral waves, which are the precursors of chaotic behavior. This consideration explains the transition from normal heart rate to tachycardia, which corresponds to the appearance of spiral waves, and the fohowing transition to fibrillation, which corresponds to the chaotic regime after the breaking up of the spiral waves, Figure 11.17. The transition from the spiral waves to chaos is often characterized as electrical turbulence due to its resemblance to the equivalent hydrodynamic phenomenon. 

Nijsii [1964] Organic coolant in nuclear reactors Hydrodynamic boundary layer and diffusion partial differential equations Product diffiision back to the fluid bulk Solution with diffiision control fits plant data, fouling rate predicted to increase with velocity... 

Here, if ris known in advance, D can be directly obtained from Eq. (15). If it is unknown, x and D are determined fix)m the measurements with varied A. As the striation period can be as narrow as 1 pm, even slight movement and slow diffrision can be measured. The measurement area can also be small. For measurement purposes, a molecule that can be excited by a laser light as a probe or a probe that is tagged by a chromophore must be used. Studies on polystyrene diffusion partially labeled by 4-bromomethyl-azobenzene in gelatin [22] and spiropyrane in polystyrene gel [23] have been reported. 

Gas transport can occur in porous membranes by four different idealized mechanisms Knudsen diffusion, partial condensation/diffiision, selective adsorption/diffusion, and molecular sieving (Rao and Sirkar, 1993a,b). Knudsen diffusion occurs when the mean free path of the molecule is greater than the size of the pore therefore, the di sing gas molecule collides more often with the pore wall than with other molecules. Knudsen diffusion can be described using Eq. (23.1) (Hines and Maddox, 1985) ... 

If tire diffusion coefficient is independent of tire concentration, equation (C2.1.22) reduces to tire usual fonn of Pick s second law. Analytical solutions to diffusion equations for several types of boundary conditions have been derived [M]- In tlie particular situation of a steady state, tire flux is constant. Using Henry s law (c = kp) to relate tire concentration on both sides of tire membrane to tire partial pressure, tire constant flux can be written as... 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

In practice it would not be reasonable to solve the balances at the limit of Knudsen diffusion control by considering the n simultaneous boundary value problems (11.7). All the partial pressures can be expressed in terms of by integrating equations (11,25), with the result... 

Hite s treatment is based on equations (5.18) and (5.19) which describe the dusty gas model at the limit of bulk diffusion control and high permeability. Since temperature Is assumed constant, partial pressures are proportional to concentrations, and it is convenient to replace p by cRT, when the flux equations become... 

Cyclohexane, produced from the partial hydrogenation of benzene [71-43-2] also can be used as the feedstock for A manufacture. Such a process involves selective hydrogenation of benzene to cyclohexene, separation of the cyclohexene from unreacted benzene and cyclohexane (produced from over-hydrogenation of the benzene), and hydration of the cyclohexane to A. Asahi has obtained numerous patents on such a process and is in the process of commercialization (85,86). Indicated reaction conditions for the partial hydrogenation are 100—200°C and 1—10 kPa (0.1—1.5 psi) with a Ru or zinc-promoted Ru catalyst (87—90). The hydration reaction uses zeotites as catalyst in a two-phase system. Cyclohexene diffuses into an aqueous phase containing the zeotites and there is hydrated to A. The A then is extracted back into the organic phase. Reaction temperature is 90—150°C and reactor residence time is 30 min (91—94). 

A molecule colliding with the pore wall is reflected in a specular manner so that the direction of the molecule leaving the surface has no correlation with that of the incident molecule. This leads to a Fickian mechanism, known as Knudsen diffusion, in which the flux is proportional to the gradient of concentration of partial pressure. The Knudsen diffusivity is independent of pressure and varies only weaMy with temperature ... 

Absorption. Some inks (eg, oil-based newspaper inks) dry by penetration or absorption into the pores of the printed stock, which has a blotter or sponge effect. This is accompHshed by the gross penetration of the ink vehicle into the pores of the substrate, the partial separation of the vehicle from the pigment, and the diffusion of the vehicle throughout the paper. The abiHty of an ink to penetrate into paper depends on the number and size of the air spaces present in the paper, the affinity or receptivity of the stock for the ink, and the mobiHty of the ink. 

Although microporous membranes are a topic of research interest, all current commercial gas separations are based on the fourth type of mechanism shown in Figure 36, namely diffusion through dense polymer films. Gas transport through dense polymer membranes is governed by equation 8 where is the flux of component /,andare the partial pressure of the component i on either side of the membrane, /is the membrane thickness, and is a constant called the membrane permeability, which is a measure of the membrane s ability to permeate gas. The ability of a membrane to separate two gases, i and is the ratio of their permeabilities,a, called the membrane selectivity (eq. 9). 

---