Transport diffusion coefficient, calculation

Fixed framework-flexible adsorbate calculations were also reported by Dumont and Bougeard (68, 69). The diffusion coefficient calculated from a 42-ps calculation of 4 molecules per unit cell at 300 K was 1.60 x 10 8 m2/s. Once again the anisotropy of the diffusive process was calculated to be strong transport through the straight channel was found to be twice as fast as through the sinusoidal channel and an order of magnitude faster than motion parallel to the z-axis. 

Solutions for this type of kinetics can only be achieved numerically. Model calculations with constant kinetic parameters have been made [H. Wiedersich, et al. (1979)], however, the modeling of realistic transport (diffusion) coefficients which enter into the flux equations is most difficult since the jump rate vA vB. Also, the individual point defects have limited lifetimes which determine the magnitude of correlation factors (see Section 5.2.2). Explicit modeling for dilute or non-dilute alloys can be found in [A.R. Allnatt, A.B. Lidiard (1993)]. 

The seriousness of this oversight is apparent in Sefcik and Schaefer s analysis of Toi s transport data (24) in terms of their NMR results (28) The value of the so-called "apparent" diffusion coefficient calculated from Toi s time lag data increases by 25% for an upstream pressure range between 100 mm Hg and 500 mm Hg On the other hand, the value of Deff(c) calculated from Toi s data changes by 86% over the concentration range from 100 to 500 mm Hg The difference in the two above coefficients arises from the fact that Da is an average of values corresponding to a range of concentrations from the upstream value to the essentially zero concentration downstream value in a time lag measurement Deff > on t le other hand, has a well-defined point value at each specified concentration and is typically evaluated (independent of any specific model other than Fick s law) by differentation of solubility and permeability data (22) ... 

The temperature and pressure dependence of the FR responses was determined using sample Z57, which consists of nearly identical crystals. The transport diffusion coefficients (D) were calculated assuming that particles are spheres of 10-pm... 

ZSM-5 zeolite. Transport D values were Consequently, the diffusion coefficient calculated from FR results assuming that particles are spheres of lO-pm diameter (0) Data determined by the MEMBRANE (A), TAP( ), and QENS (H) methods were taken from refs. 13 and 14 and are given for comparison. 

Figure 21.39. NMR imaging pictures of the cross-section of a cylindrical silica gel (pH 7, target density =80 kg/m ) taken in situ at different times after purging the autoclave with liquid CO2 the pictures are showing the progress of the solvent exchange of methanol for liquid CO2. The methanol concentration is represented by the white pixels. Right inset Integrated methanol concentration calculated from the series of images on the left-hand side as a function of time the line is a fit to determine the transport diffusion coefficient I>r of the methanol in the pores of the aerogel [97] (Courtesy W. Behr).

The usefulness of Eq. (2) is that it embodie.s the effects of an angular leakage algorithm in a form conqiatible with the diffusion equation for the total flux. In practice, this formulation 11 provide economic benefits for parametric studies in which the transport diffusion coefficient is insensitive to parameter change. The procedure is to perform a base-case transport calculation to obtain the flux moments for use in Eq. (2). The spatially varying transport diffusion coefficients can then be used to perform parametric diffusion calculations. The results should be improved accuracy over the standard diffusion theory calculations and a considerable savings in computational time. 

To illustrate this technique, a PWR fuel storage pool calculation was performed to determine the Ke-eigenvalue as a function of assembly spacing. A transport calculation was performed for the base-case separation spacing of 12 cm to furnish data to determine the transport diffusion coefficient. This transport diffusion coefficient was utilized in subsequent dlffusioin calculations in which the... 

Experimental diffusion coefficients, as obtained from time-lag measiu ements, report a transport diffusion coefficient which carmot be obtained from equilibrium MD simulation. Comparisons made in the simulation literatme are typically between time-lag diffusion coefficients (even calculated for glassy polymers without correction for dual-mode contributions and self-diffusion coefficients. As discussed above, mutual diffusion coefficients can be obtained directly from equilibrium MD simulation but simulation of transport diffusion coefficients require the use of NEMD methods, that are less commonly available and more computationally expensive [117]. 

One alternative approach to the calculation of the diffusion and other transport coefficier is via an appropriate autocorrelation function. For example, the diffusion coefficie... 

Water Transport. Two methods of measuring water-vapor transmission rates (WVTR) ate commonly used. The newer method uses a Permatran-W (Modem Controls, Inc.). In this method a film sample is clamped over a saturated salt solution, which generates the desired humidity. Dry air sweeps past the other side of the film and past an infrared detector, which measures the water concentration in the gas. For a caUbrated flow rate of air, the rate of water addition can be calculated from the observed concentration in the sweep gas. From the steady-state rate, the WVTR can be calculated. In principle, the diffusion coefficient could be deterrnined by the method outlined in the previous section. However, only the steady-state region of the response is serviceable. Many different salt solutions can be used to make measurements at selected humidity differences however, in practice,... 

Carbon Dioxide Transport. Measuring the permeation of carbon dioxide occurs far less often than measuring the permeation of oxygen or water. A variety of methods ate used however, the simplest method uses the Permatran-C instmment (Modem Controls, Inc.). In this method, air is circulated past a test film in a loop that includes an infrared detector. Carbon dioxide is appHed to the other side of the film. AH the carbon dioxide that permeates through the film is captured in the loop. As the experiment progresses, the carbon dioxide concentration increases. First, there is a transient period before the steady-state rate is achieved. The steady-state rate is achieved when the concentration of carbon dioxide increases at a constant rate. This rate is used to calculate the permeabiUty. Figure 18 shows how the diffusion coefficient can be deterrnined in this type of experiment. The time lag is substituted into equation 21. The solubiUty coefficient can be calculated with equation 2. 

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

The mass transfer coefficient is calculated for a given diffusivity coefficient and reaction rate constant at the equilibrium concentration of oxygen. When oxygen is continuously transported and removed from the liquid phase we may write ... 

Overall, the RDE provides an efficient and reproducible mass transport and hence the analytical measurement can be made with high sensitivity and precision. Such well-defined behavior greatly simplifies the interpretation of the measurement. The convective nature of the electrode results also in very short response tunes. The detection limits can be lowered via periodic changes in the rotation speed and isolation of small mass transport-dependent currents from simultaneously flowing surface-controlled background currents. Sinusoidal or square-wave modulations of the rotation speed are particularly attractive for this task. The rotation-speed dependence of the limiting current (equation 4-5) can also be used for calculating the diffusion coefficient or the surface area. Further details on the RDE can be found in Adam s book (17). 

Specific heat of each species is assumed to be the function of temperature by using JANAF [7]. Transport coefficients for the mixture gas such as viscosity, thermal conductivity, and diffusion coefficient are calculated by using the approximation formula based on the kinetic theory of gas [8]. As for the initial condition, a mixture is quiescent and its temperature and pressure are 300 K and 0.1 MPa, respectively. 

Diffusion of ions can be observed in multicomponent systems where concentration gradients can arise. In individnal melts, self-diffnsion of ions can be studied with the aid of radiotracers. Whereas the mobilities of ions are lower in melts, the diffusion coefficients are of the same order of magnitude as in aqueous solutions (i.e., about 10 cmVs). Thus, for melts the Nemst relation (4.6) is not applicable. This can be explained in terms of an appreciable contribntion of ion pairs to diffusional transport since these pairs are nncharged, they do not carry cnrrent, so that values of ionic mobility calculated from diffusion coefficients will be high. 

In order to calculate the rates for electron impact collisions and the electron transport coefficients (mobility He and diffusion coefficient De), the EEDF has to be known. This EEDF, f(r, v, t), specifies the number of electrons at position r with velocity v at time t. The evolution in space and time of the EEDF in the presence of an electric field is given by the Boltzmann equation [231] ... 

The disadvantage of the fluid model is that no kinetic information is obtained. Also, transport (diffusion, mobility) and rate coefficients (ionization, attachment) are needed, which can only be obtained from experiments or from kinetic calculations in simpler settings (e.g. Townsend discharges). Experimental data on... 

Thus, the time that is necessary to attain a certain coverage, 6, or the time necessary to cover the surface completely (9 = 1) is inversely proportional to the square of the bulk concentration (cf. Fig. 4.10b). Assuming molecular diffusion only, 8 is of the order of 2 minutes for a concentration of 10 5 M adsorbate when the diffusion coefficient D is 10 5 cm2 s1 and rmax = 4 1010 mol cm 2 1). Considering that transport to the surface is usually by turbulent diffusion, such a calculation illustrates that the formation of an adsorption layer is relatively rapid at concentrations above 10 6 M. But it can become slow at concentrations lower than 10 6 M. 

The diffusion coefficients for Rb, Cs and Sr in obsidian can be calculated from the aqueous rate data in Table 1 as well as from the XPS depth profiles. A simple single-component diffusion model (9j characterizes onedimensional transport into a semi-infinite solid where the diffusion coefficient (cm2-s 1) is defined by ... 

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