Diffusion coefficient pressure scaling

Pressure Dependencies Equation 3.95 predicts the binary diffusion coefficient to scale as p l, which is generally true except as the pressure approaches or exceeds the critical pressure. The Takahashi formula [392], which can be used to describe the high-pressure behavior, is discussed below. The Chapman-Enskog theory also predicts that Vji, increases with temperature as T3/2. However, it is often observed experimentally the temperature exponent is somewhat larger, say closer to 1.75 [332], An empirical expression for estimating T>jk is due to Wilke and Lee [433]. The Wilke-Lee formula is [332]... 

Supercritical fluid chromatography (SFC) refers to the use of mobile phases at temperatures and pressures above the critical point (supercritical) or just below (sub-critical). SFC shows several features that can be advantageous for its application to large-scale separations [132-135]. One of the most interesting properties of this technique is the low viscosity of the solvents used that, combined with high diffusion coefficients for solutes, leads to a higher efficiency and a shorter analysis time than in HPLC. 

Fig. 8. Graph illustrating the effect of particle size (left scale) and column length (right scale) on the pressure drop (vertical scale) required for obtaining SOOQ plates in S min with soliiies h. iving different molecular diffusion coefficients. (1) f> 3 x I0 cm /sec (2)...

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

Fig. 3.5 Scaling of the diffusion coefficient with pressure suggested by Takahashi [392] see Eq. 3.102 for definition of /.

As mentioned earlier, the diffusion coefficient scales inversely with pressure for moderate pressures. At pressures near and above the critical pressure, the exponent decreases below this value. Takahashi [392] suggested a scaling relationship to obtain Vjk(p) from the value Vjkipo) measured at some lower pressure po- His correlation is... 

The supercritical fluid extraction of analytes from solid sorbents is controlled by a variety of factors including the affinity of the analytes for the sorbent, the tortuosity of the sorbent bed, the vapor pressure of the analytes, and the solubility and the diffusion coefficient of the analytes in the supercritical fluid. Additionally, SFE efficiencies are affected by a complex relationship between many experimental variables, several of which are listed in Table I. Although it is well established that, to a first approximation, the solvent power of a supercritical fluid is related to its density, little is known about the relative effects of many of the other controllable variables for analytical-scale SFE. A better understanding of the relative effects of controllable SFE variables will more readily allow SFE extractions to be optimized for maximum selectivity as well as maximum overall recoveries. 

Many computational studies of the permeation of small gas molecules through polymers have appeared, which were designed to analyze, on an atomic scale, diffusion mechanisms or to calculate the diffusion coefficient and the solubility parameters. Most of these studies have dealt with flexible polymer chains of relatively simple structure such as polyethylene, polypropylene, and poly-(isobutylene) [49,50,51,52,53], There are, however, a few reports on polymers consisting of stiff chains. For example, Mooney and MacElroy [54] studied the diffusion of small molecules in semicrystalline aromatic polymers and Cuthbert et al. [55] have calculated the Henry s law constant for a number of small molecules in polystyrene and studied the effect of box size on the calculated Henry s law constants. Most of these reports are limited to the calculation of solubility coefficients at a single temperature and in the zero-pressure limit. However, there are few reports on the calculation of solubilities at higher pressures, for example the reports by de Pablo et al. [56] on the calculation of solubilities of alkanes in polyethylene, by Abu-Shargh [53] on the calculation of solubility of propene in polypropylene, and by Lim et al. [47] on the sorption of methane and carbon dioxide in amorphous polyetherimide. In the former two cases, the authors have used Gibbs ensemble Monte Carlo method [41,57] to do the calculations, and in the latter case, the authors have used an equation-of-state method to describe the gas phase. 

Proper understanding of solute behavior in sub-H20 systems requires evaluation of both solute-sub H2O phase behavior (particularly in engineering scale systems where solute concentrations are finite), solute solubility trends in hot pressurized water, the diffusion coefficients of solutes in water as a function of temperature and their role in facilitating mass transport, and the potential effect of pressure - often trivialized as a major factor in SWE 16) - in affecting analyte extractions fi om sample matrices via sub-H20. All of these factors ultimately impact the resultant SWE, SWF, and SWR process, reinforcing one another fortuitously as temperature is increased, leading to an increase in solute flux into the sub-H20 medium. 

With the advent of Fourier-transform NMR and powerful superconducting magnets, spin-echo NMR is now the method of choice for high-pressure intradiffusion measurements. Individual measurements take of the order of few (2-30) minutes to collect rather than the 24-50 h needed for diaphragmcell measurements. Because of the higher sensitivity and the faster collection of data it is now possible to study by NMR systems with diffusion coefficients which are as low as 10 s whereas the time scale for diaphragm-cell mea-... 

Two unusual features can be observed in these plots (and, at least for the self-diffusion coefficient, this behaviour is common to all hydrogen-bonded liquids). This ratio is a function of temperature. At constant temperature and pressure, rotation and translation reveal the same isotope effect. From simple sphere dynamics one would expect the rotation to scale as the square root of the ratio of the moments of inertia (=1.38) while for translational mobility the square root of the ratio of the molecular masses ( = 1.05) should be found. This is clearly not the case, indicating that the dynamics of liquid water are really the dynamics of the hydrogen-bond network. The hydrogen bonds in D2O are stronger than those in H2O and thus the mobility in the D2O network decreases more rapidly as the temperature decreases. 

The coefficient for molecular diffusion D is determined by the mean free path between collisions of molecules, which is inversely proportional to pressure. In the ground-level atmosphere the value for the diffusion coefficient typically is D 2x 10 s m2/s, which makes molecular diffusion unsuitable for large-scale transport. Turbulent mixing is faster. The pressure decreases with height, however, causing the diffusion coefficient to increase until it reaches a value of 100 m2/s at an altitude near 100 km. Above this level molecular diffusion becomes the dominant mode of large scale transport. 

Diffusive flow for neutrals The importance of convective vs. diffusive flow of neutrals is determined by the Peclet number Pe = uL/D, where L is a characteristic dimension of the system. Away from inlet and exit ports, the characteristic length will be on the order of the reactor dimension. The system will be primarily diffusive when Pe 1. For CI2 gas in a reactor with L 0.1 m and a neutral species diffusivity of D 5m s at 20mtorr, the Peclet number will be Pe 1 when M = 50ms. Convective gas velocities are not likely to be that high, except for a small region near the gas inlet ports. It follows that gas flow can be approximated as diffusive this obviates the need for solving the full Navier-Stokes equations which adds to the computational burden. It should be noted that both the diffusivity and the convective velocity scale inversely with gas pressure, so the Pe number is independent of pressure. However, as the pressure is lowered to the point of free molecular flow, the gas diffusion coefficient has no meaning any more. Direct Simulation Monte Carlo (DSMC) [41, 143] can then be applied to solve for the fluid velocity profiles. 

As in the case of atom recombination, a convenient pressure scale to use across the entire range is the inverse of the binary diffusion coefficient, am, of reactant A in solvent M, as compared to density p in the low-pressure gas and the inverse of solvent viscosity t in liquid solution [46]. According to kinetic theory the diffusion coefficient in a dilute Lennard-Jones gas is given by... 

Fig. 11. The inadequacy of diffusive mixing, even after geological time periods, (a) Very poor mixing in a thin reservoir because the diffusive length scale L is small relative to the lateral extent of the reservoir. (Deff> effective diffusion coefficient, m /s t, time period, s). (b) Moderate diffusive mixing in a thick gross reservoir. The small inset figures show generalized API gravity and/or GOR, and bubble point and reservoir pressure depth trends...

Two conceptual models of groundwater flow beneath A u have been simulated using PARADIGM. In the first case, the matrix model, pressure diffusion is within a 3D homogeneous matrix and the diffusion coefficient, D, is calibrated such that the peak pressure reaches 2 km depth after 4.5 months. This time delay matches the observed delay in peak seismicity for cluster a, which has the most events. The matrix mesh for the local scale model is that shown in Figure 4, without the fault plane. 

Where X is the distance that hydrogen must diffuse to contact the membrane surface, D is the gas-phase diffusion coefficient for hydrogen at the temperature and pressure of operation, and t is the residence time for the feed stream in the feed channel of the membrane. If B is less than 1, then gas phase hydrogen diffusion is relatively fast on the time scale of interest, whereas if B is greater than 1, then the gas phase hydrogen diffusion is too slow. 

Small Cr contents increase the rate of reaction, but at 20% Cr, the reaction rate starts to decrease and exhibits a minimum value at 25%-30% Cr (Figure 20.62). The minimum value depends on the pressure. More chromium is needed to stabilize a protective film since the diffusion coefficient of chromium in cobalt is lower than for chromium in nickel. However, since the adhesion strength of the film on Co-Cr alloys is poorer than on Ni-Cr alloys despite the identical oxidation rate of the Cr-containing Co alloys with Cr203 protective film, the practical oxidation resistance is lower. Other alloying elements, as Figure 20.62 shows, have little influence on scale resistance. 

The water photolysis under low O2 pressure always led to a loss of hydrogen into space. The diffusion rate of the H2 (or H after it has been broken down by photolysis) through the homopause and exobase is limited. The definition of the homopause (80-90 km altitude) is the point at which the molecular and eddy diffusion coefficients are equal or, in other words, the critical level below which an atmosphere is well-mixed. The exobase ( 550 km) is the height at which the atmosphere becomes collisionless above that height the mean free path of the molecules exceeds the local scale height (RTIg). 

The mobility and diffusion coefficient are proportional to l/P (1.3.1), and the best gap width gopt by Equation 4.29 depends on the pressure even when the dispersion field is adjusted such that E /N is fixed. In that case, the value of Ad by Equation 3.43 is independent of P, but Dn scales as l/P because D scales as l/P while Dadd is a function of Ejy/N by Equation 3.22 and thus stays constant. In the limit of high waveform frequency where Ad can be ignored, g pt would scale as P . In reality. Ad is not negligible, leading to a smaller increase of gopt at lower pressure. With the exemplary of 2 and 100 ms (4.2.4), the gop, values for medimn-size ions rise from respectively 0.36 and 2.0 mm at P = 1 atm to 1.3 and 8.7 mm at 38 Torr. Then the Dd needed for constant Ej)/N at reduced pressure would scale slower than P but somewhat faster than P. In the above examples, the Dd values for equal Ejy/N = 140 Td at P= 300 K would decrease fivefold from SSO V for 2 ms and 4900 V for 100 ms at 1 atm to lOO V and llOO V, respectively, at 38 Torr. Even with Ej)/N lifted (at low P) to 360 Td to deliver far greater specificity, we would stiU have t/o of 410 and 2800 V, respectively, or half of the values for 1 atm. 

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