Diffusion half-depth

In analogy to the reaction half-life that was discussed in Chapter 3, we can specify a diffusion half-depth in transient diffusion problems (S1/2) which is the spatial position at which the concentration of the diffusing species reaches half of its surface value. As an example, the diffusion half-depth for the semi-infinite diffusion process in Equation 4.26 can be obtained as... 

For the simplest one-dimensional or flat-plate geometry, a simple statement of the material balance for diffusion and catalytic reactions in the pore at steady-state can be made that which diffuses in and does not come out has been converted. The depth of the pore for a flat plate is the half width L, for long, cylindrical pellets is L = dp/2 and for spherical particles L = dp/3. The varying coordinate along the pore length is x ... 

At a depth l below the liquid surface, the. concentration of A has fallen to one-half of the value at the. surface. What is the. ratio of the. mass transfer rate at this depth t to the. rate, at the surface Calculate the numerical value of the ratio when l /k/D = 0.693, where. D is the molecular diffusivity and k the first-order rate constant. 

A solute diffuses from a liquid surface at which its molar concentration is C, into a liquid with which it reads. The mass transfer rate is given by Fick s law and the reaction is first order with respect to the solute, fn a steady-state process the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration C of solute at a depth z from the surface in terms of the molecular diffusivity D and the reaction rate constant k. What is the molar flux at the surface ... 

The Henry s law constant value of 2.Ox 10 atm-m /mol at 20°C suggests that trichloroethylene partitions rapidly to the atmosphere from surface water. The major route of removal of trichloroethylene from water is volatilization (EPA 1985c). Laboratory studies have demonstrated that trichloroethylene volatilizes rapidly from water (Chodola et al. 1989 Dilling 1977 Okouchi 1986 Roberts and Dandliker 1983). Dilling et al. (1975) reported the experimental half-life with respect to volatilization of 1 mg/L trichloroethylene from water to be an average of 21 minutes at approximately 25 °C in an open container. Although volatilization is rapid, actual volatilization rates are dependent upon temperature, water movement and depth, associated air movement, and other factors. A mathematical model based on Pick s diffusion law has been developed to describe trichloroethylene volatilization from quiescent water, and the rate constant was found to be inversely proportional to the square of the water depth (Peng et al. 1994). 

DBCP. The predictions suggest that DBCP is volatile and diffuses rapidly into the atmosphere and that it is also readily leached into the soil profile. In the model soil, its volatilization half-life was only 1.2 days when it was assumed to be evenly distributed into the top 10 cm of soil. However, DBCP could be leached as much as 50 cm deep by only 25 cm of water, and at this depth diffusion to the surface would be slow. From the literature study of transformation processes, we found no clear evidence for rapid oxidation or hydrolysis. Photolysis would not occur below the soil surface. No useable data for estimating biodegradation rates were found although Castro and Belser (28) showed that biodegradation did occur. The rate was assumed to be slow because all halogenated hydrocarbons degrade slowly. DBCP was therefore assumed to be persistent. 

For situations where the reaction is very slow relative to diffusion, the effectiveness factor for the poisoned catalyst will be unity, and the apparent activation energy of the reaction will be the true activation energy for the intrinsic chemical reaction. As the temperature increases, however, the reaction rate increases much faster than the diffusion rate and one may enter a regime where hT( 1 — a) is larger than 2, so the apparent activation energy will drop to that given by equation 12.3.85 (approximately half the value for the intrinsic reaction). As the temperature increases further, the Thiele modulus [hT( 1 — a)] continues to increase with a concomitant decrease in the effectiveness with which the catalyst surface area is used and in the depth to which the reactants are capable of... 

Sediments in the Mississippi River were accidentally contaminated with a low-level radioactive waste material that leaked from a nuclear power plant on the river. Pore water concentrations of radioactive compounds were measured following the spill and found to be 10 g/m over a 2-mm depth. The water contamination was 30% radioactive cesium ( Cs), with a half-life of 30 years, and 70% radioactive cobalt ( °Co), with a half-life of 6 years. Objections by the local residents are preventing clean-up efforts because some professor at the local state university convinced them that dredging the sediments and placing them in a disposal facility downstream would expose the residents to still more radioactivity. The state has decided that the sediments should be capped with 10 cm of clay and needs a quick estimate of the diffusion of radioactive material through the clay cap (Figure E2.8.1). If the drinking water limit (10 g/m ) is reached at mid-depth in the cap, the state will increase its thickness. Will this occur ... 

Figure 19.8 Diffusivity D and concentration C at wall boundary, (a) Schematic view of a wall boundary. Diffusivity drops abruptly from a very large value DB, which guarantees complete mixing in system B, to the much smaller value Da. The concentration penetrates into system A when time t grows. X(/2(

The mathematics of diffusion at flat wall boundaries has been derived in Section 18.2 (see Fig. 18.5a-c). Here, the well-mixed system with large diffusivity corresponds to system B of Fig. 18.5 in which the concentration is kept at the constant value Cg. The initial concentration in system A, CA, is assumed to be smaller than Cg. Then the temporal evolution of the concentration profile in system A is given by Eq. 18-22. According to Eq. 18-23 the half-concentration penetration depth , x1/2, is approximatively equal to (DAt)m. The cumulative mass flux from system B into A at time t is equal to (Eq. 18-25) ... 

In single step voltammetry, the existence of chemical reactions coupled to the charge transfer can affect the half-wave potential Ey2 and the limiting current l. For an in-depth characterization of these processes, we will study them more extensively under planar diffusion and, then, under spherical diffusion and so their characteristic steady state current potential curves. These are applicable to any electrochemical technique as previously discussed (see Sect. 2.7). In order to distinguish the different behavior of catalytic, CE, and EC mechanisms (the ECE process will be analyzed later), the boundary conditions of the three processes will be given first in a comparative way to facilitate the understanding of their similarities and differences, and then they will be analyzed and solved one by one. The first-order catalytic mechanism will be described first, because its particular reaction scheme makes it easier to study. 

The modeling considered 1.25 mm of the channel before the confluence and 3 mm of the channel after the confluence with a plane of symmetry at half the channel depth. The computational domain was discretized with structured hexahedral meshes, the size of the cells being 10 pm long. Diffusion was modeled. 

The F uptake of flint takes a much longer time than that for bone. Fluorine diffusion into the depth of flint material is controlled by defect clusters. The diffusion coefficient determined by implanting a model compound (amorphous silica bombarded with heavy ions and hydrated at 100°C) is 9.10—21 cm2/s at room temperature. The corresponding penetration depth of F under ambient conditions in a 1000-year-old artefact can be estimated via x — (Dt)1/2 = 0.17. im [50], Thus, F accumulates only in the first micrometre of the surface. The surface of ancient flint artefacts can be altered by dissolution. The occurrence of this phenomenon is especially important in basic media. However, in some cases, the thickness of the dissolved layer can be neglected compared to the F penetration depth at low temperatures. Therefore, Walter et al. [35] proposed relative dating of chipped flint by measuring the full width at half maximum (FWHM) of F diffusion profiles in theses cases. 

The relatively slow gas diffusion rates in rocks of low porosity at depth have brought the contribution of diffusion to long-distance gas migration into question. The half-life of Rn is so short that its persistence and detection after transport by diffusion over tens or hundreds of metres is extremely unlikely. 

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