Approximate value of the diffusivity

After the cell and test piece have been assembled and the high pressure side filled with gas at the test pressure, the increase in pressure on the low pressure side is measured as a function of time. The standard suggests a conditioning period of at least 16 h to reach steady state conditions unless an approximate value of the diffusion coefficient is known, when the minimum conditioning time can be estimated from ... 

There are several methods for obtaining an approximate value of the diffusion coefficient. These methods are preferred to direct determination because of the real and perceived difficulties of a direct measurement. The friction factor in Eqs. 

Alternatively on the basis of a diffusion type model, approximate values of the diffusion coefficient D, have been calculated for these systems. Values of D which fall in the range of 10-2 to 10-5mm/sec are of the same order of magnitude as those obtained by Hoyland (2) for the penetration of various aqueous liquids in bleached softwood paper. Although Hoyland s diffusion model takes into consideration fibre swelling during penetration it still overlooks many of the aforementioned structural complications unique to paper. 

Approximate Value of the Diffusivity for Small Times and Infinite h... 

This equation (4.30) shows that a straight line is obtained when plotting the ratio M/M versus the square root of time. But, as already stated in Chapter 1, this relationship is of value only when the coefficient of convection is infinite, with a liquid strongly stirred, thus, this fact reduces the interest of this equation in calculating the diffusivity. In fact. Equation (4.30) can be used only as a first approach to obtain an approximate value of the diffusivity. With the same approximation. Equation (4.28) can be useful for determining an approximate value of the diffusivity for long times, as it reduces to the simple Equation (4.31) ... 

Equation (4.28), or rather the more simple Equation (4.30), which is mostly appreciated by the users, perhaps because it looks simpler, can be used only from the first approach in order to get an approximative value of the diffusivity. 

Then, the approximate value of the diffusivity obtained with this simple method shown previously in case 2 with Equation (4.30) can be introduced into Equation (4.24) through the dimensionless number R expressed by the relationship (4.23). Of course, various values of the coefficient h should be tested, by following a trial and error method, and by determining the corresponding values of the in Equation (4.22). 

The approximate value of the diffusion current is obtained starting from the diffusion term in presence of magnetic field... 

A quantitative model requires knowledge of the diffusivity under reaction conditions and of the intrinsic activities for toluene disproportionation and xylene isomerization. While these are not easily obtained, the methodology has been worked out for the case of paraffin and olefin cracking (5). So far, we have obtained an approximate value for the diffusivity, D, of o-xylene at operation conditions from the rate of sorptive o-xylene uptake at lower temperature and extrapolation to 482°C (Table V). 

In conclusion, the free energy change of an ET step is already a good indicator of the feasibility of the reaction. A highly endergonic reaction, with, say, AG° > 20 kcal mol-1, corresponds to a rather slow ET reaction that is not likely to compete with other reactions of polar nature. In the region where AG° lies between 20 and -A kcal mol-1, we need to apply the Marcus approach in order to get an approximate value of the ET rate constant, whereas at AG° < - A kcal mol-1 most intermolecular ET reactions appear to be diffusion controlled. 

Another important set of observations is related to the detection limit dynamic range and sensitivity. For the expected values of the diffusion coefficient (in the gel) of approximately 10-6cm2 s and substrate molecular weights about 300, the detection limit is approximately 10 4M. This is due to the fact that the product of the enzymatic reaction is being removed from the membrane by diffusion at approximately the same rate as it is being supplied. The dynamic range of the sensor... 

Knowing an experimental value of k, it is possible to evaluate the diffusion coefficient of the atoms of a dissolving solid substance across the diffusion boundary layer at the solid-liquid interface into the bulk of the liquid phase using equations (5.6) and (5.7). Its calculation includes two steps. First, an approximate value of D is calculated from equation (5.6). Then, the Schmidt number, Sc, and the correction factor, /, is found (see Table 5.1). The final, precise value is evaluated from equation (5.7). In most cases, the results of these calculations do not differ by more than 10 %. Values of the diffusion coefficient of some transition metals in liquid aluminium are presented in Table 5.9.303... 

The midpoints xi+i/2 = (xi +xi+i)/2 and the corresponding values of the diffusion coefficient Di+i/2 = D(Xj+i/2) have been introduced to ensure appropriate centering of the implied derivatives. A convenient approximation for Di+1/2 results from considering the average of the values at the neighboring gridpoints ... 

We now discuss the second effect. Assuming a one sided diode, the dark current is inversely proportional to the diffusion length. Using the calculated values of the diffusion length under illumination, the calculated dark current is shown in Fig. 5.28. At 50 suns the dark current decreases by a factor approximately 16. Using the equation,... 

Fick s law of diffusion is also used for problems involving liquid and solid diffusion, and the main difficulty is one of determining the value of the diffusion coefficient for the particular liquid or solid. Unfortunately, only approximate theories are available for predicting diffusion coefficients in these systems. Bird, Stewart, and Lightfoot [9] discuss the calculation of diffusion in liquids, and Jost [6] gives a discussion of the various theories which have been employed to predict values of the diffusion coefficient. The reader is referred to these books for more information on diffusion in liquids and solids. 

Williams et al,92) have investigated the diffusion of the sodium ion in NMA solutions of NaCl at 40 °C. The values of the diffusion coefficient for the Na+ ion were found to be equal to approximately half the value of the self-diffusion coefficient for pure NMA. This suggests a solvation which is equivalent to three or four molecules of NMA coordinated to the sodium ion. 

These thermodynamic approaches to hydrophobic effects are complemented by spectroscopic studies. Tanabe (1993) has studied the Raman spectra manifested during the rotational diffusion of cyclohexane in water. The values of the diffusion coefficients are approximately half those expected from data for other solvents of the same viscosity, and the interpretations made are in terms of hindered rotation arising from the icebergs presumably formed (c/. Frank and Evans) around the cyclohexane. 

T iffusion in porous pellets is often the rate-limiting process in industrial adsorption or catalytic processes. Much useful work in this field has been done by Smith and coworkers (3, 5), but for molecular sieve pellets the situation is complicated by diffusion in the zeolite crystal itself, as well as through the pores formed between the crystals. Few studies have been made of zeolite crystal diffusion, but Barrer and Brook (1) reported some results on diffusion of simple gases in various cation-substituted mordenites, and Wilson (7) gives some indirect results from the study of separation of CO2 from air using a fixed bed of type 4A zeolite pellets. In the present work, results have been obtained by studying self-diffusion of CO2 in a single pellet of type 5A zeolite under controlled conditions. The experimental results were fitted satisfactorily by a very simplified model of the pellet structure, which made it possible to deduce approximate values of the self-diffusion coefficients for both pore and crystal diffusion. 

Proceeding with the standard weighted residuals approach [6] to find an approximated solution of the diffusion boundary-value problem (2)-(6) and (8), the approximation of concentration is represented in terms of a linear combination of the spatial trial functions generated over a certain sort of finite elements which discretise the solid under consideration, N,(x), where x stands for the instantaneous coordinates of material points over the volume of considered region V occupied by a testpiece, so that... 

RA Is an Extremely Fast Diffusion Process. The value of the diffusion coefficient of copper atoms in clusters is approximately nine orders of magnitude larger than that in bulk crystalline alloys. In terms of the relation x = /Zv, between the diffusion coefficient D and the time t needed to achieve diffusion of solute atoms across the distance x, YM roughly estimated the value 1.1 x 10 19 m2/s at least. Note that the value D of copper in the bulk gold is about 2.4 x 10 28m2/s at 300 K [10],... 

In electroanalysis, electrodes of millimeter dimensions are termed millielec-trodes, while the more recently developed very small area electrodes of micron dimensions are termed microelectrodes there are differences in properties beyond simply the change of dimension. Thus in millielectrode-scale experiments the enhancement of the diffusion-limited current plateau has been observed by a number of other workers—for example, in the reduction of methylviologen in aqueous acetonitrile [32], in the oxidation of bis(cyclopentadienyl) molybdenum dichloride in acetonitrile [33], as well as in several other studies on the aqueous ferrocyanide/ferricyanide couple using wire or disc millielectrodes to study diffu-sional phenomena [34—36], Typical values of the diffusion layer thickness of approximately 5 pm are found under ultrasound [35] in contrast to the normal value of approximately 500 pm in silent conditions. 

In order to obtain useful forms of the discretized equations, the GCV surface terms at the east e and west w faces are required. As in the pure diffusion problem, the diffusive terms are approximated using linear approximations to calculate the surface values of the diffusion coefficients and the gradients. The result is ... 

To further verify the above conclusion on the failure of the analogy between momentum and heat transfer in the case of viscoelastic fluids, the approximate values of the eddy diffu-sivities of momentum and heat transfer corresponding to the minimum asymptotic cases will be compared. The eddy diffusivity of momentum corresponding to the minimum asymptotic case was calculated by Kale [84] directly from Deissler s continuous eddy diffusivity model ... 

The way in which the diffusivity coefficient varies with temperature is shown in Fig. 5.17. This figure can be used to select values of D for all future calculations. Table 5.3 lists approximate values of the overall diffusivity coefficient for a number of materials typical... 

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