Diffusion and homogenization

The diffusion of a radioactive component is a relatively easy problem. It is discussed here to illustrate how coupled diffusion and homogeneous reaction can be treated, and to prepare for the more difficult problem of the diffusion of a radiogenic component. The diffusion of a radiogenic component, which is dealt with in Section 3.5.2, is an important geological problem because of its application in geochronology and thermochronology. 

Two-photon processes caused by absorption of photons by reaction intermediates and excited states are common under condition of high-power laser excitation. The consequence of two-photon excitation can include the formation of new reaction intermediates (electron photoejection is common) and the partial depletion of intermediates formed in monophotonic processes. To minimize this problem, do not use higher laser power then required to obtain a good signal/noise ratio, and do not focus the laser too tightly. There are in fact techniques used to obtain a more diffuse and homogenous laser beam (see below). 

The explicit, finite difference method (9,10) was used to generate all the simulated results. In this method, the concurrent processes of diffusion and homogeneous kinetics can be separated and determined independently. A wide variety of mechanisms can be considered because the kinetic flux and the diffusional flux in a discrete solution "layer" can be calculated separately and then summed to obtain the total flux. In the simulator, time and distance increments are chosen for convenience in the calculations. Dimensionless parameters are used to relate simulated data to real world data. 

Again, it is useful to handle diffusion and homogeneous reaction sequentially, so we split (B.3.11) into two parts. The diffusion effects are registered by (B.3.6) and the changes in concentration due to reaction are given by... 

Students often encounter diffusion and homogeneous reaction in a form given by... 

The sausages were considered as infinite cylinders and thermal diffusivity was calculated equal to 3.846 x 10" m s". Constant thermal diffusivity and homogeneity of the food were also the simplifying assumptions necessary to develop a model to predict internal temperature profile in chicken pieces fried under pressure [60]. For the chicken pieces a three-dimensional heat transfer was considered and the predicted results were closer to experimental ones obtained at the center of the pieces, than those obtained near the boundaries. Better simulation was also obtained for more regularly shaped pieces. 

Fig. 7.3. Patient with liver metastases and scintigraphic images of diffuse and homogeneous activity in the liver, with no hot or cold foci. This pattern makes ROI drawing of the tumoral areas in the planar images almost impossible...

F. Martinez-Ortiz, A. Molina, and E. Laborda. Electrochemical digital simulation with high expanding four point discretization Can Crank-Nicolson uncouple diffusion and homogeneous chemicai reactions , Electrochim. Acta. 56, 5707-5716 (2011). 

Diffusion and homogeneous reaction in a phase. Equation (7.5-23) was derived for the case of chemical reaction of A at the boundary on a catalyst surface. In some cases component A undergoes an irreversible chemical reaction in the homogeneous phase B while diffusing as follows, A->-C. Assume that component A is very dilute in phase B, which can be a gas or a liquid. Then at steady state the equation for diffusion of A is as follows where the bulk-flow term is dropped. 

Convolution with diffusion or with diffusion and homogeneous kinetics - kinetic... 

Martinez-Ortiz F, Molina A, Laborda E (2011) Electrochemical digital simulation with highly expanding grid four point discretization can Crank-Nicolson uncouple diffusion and homogeneous chemical reactions Electrochim Acta 56 5707-5716... 

A fundamental difference exists between the assumptions of the homogeneous and porous membrane models. For the homogeneous models, it is assumed that the membrane is nonporous, that is, transport takes place between the interstitial spaces of the polymer chains or polymer nodules, usually by diffusion. For the porous models, it is assumed that transport takes place through pores that mn the length of the membrane barrier layer. As a result, transport can occur by both diffusion and convection through the pores. Whereas both conceptual models have had some success in predicting RO separations, the question of whether an RO membrane is truly homogeneous, ie, has no pores, or is porous, is still a point of debate. No available technique can definitively answer this question. Two models, one nonporous and diffusion-based, the other pore-based, are discussed herein. 

For opaque materials, the reflectance p is the complement of the absorptance. The directional distribution of the reflected radiation depends on the material, its degree of roughness or grain size, and, if a metal, its state of oxidation. Polished surfaces of homogeneous materials reflect speciilarly. In contrast, the intensity of the radiation reflected from a perfectly diffuse, or Lambert, surface is independent of direction. The directional distribution of reflectance of many oxidized metals, refractoiy materials, and natural products approximates that of a perfectly diffuse reflector. A better model, adequate for many calculational purposes, is achieved by assuming that the total reflectance p is the sum of diffuse and specular components p i and p. ... 

For liquid/liquid separators, avoid severe piping geometry that can produce turbulence and homogenization. Provide an inlet diffuser cone and avoid shear-producing items, such as slots or holes. 

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

The attractive feature of LADM Is that once the fluid structure Is known (e.g., by solution of the YBG equations given In the previous section or by a computer simulation) then theoretical or empirical formulas for the transport coefficients of homogeneous fluids can be used to predict flow and transport In Inhomogeneous fluid. For diffusion and Couette flow In planar pores LADM turns out to be a surprisingly good approximation, as will be shown In a later section. 

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... 

Although we have covered mechanisms relating to solid state reactions, the formation and growth of nuclei and the rate of their growth in both heterogeneous and homogeneous solids, and the diffusion processes thereby associated, there exist still other processes zifter the particles have formed. These include sequences in particle growth, once the particles have formed. Such sequences include ... 

In a manner similar to that used for heterogeneous reactions, we can define diffusion and nuclei growth for a homogeneous solid. We have already stated that homogeneous nucleatlon can be contrasted to heterogeneous nucleatlon in that the former is random within a single compound while the latter involves more than one phase or compound. 

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