Diffusivity values

The time constant R /D, and hence the diffusivity, may thus be found dkecdy from the uptake curve. However, it is important to confirm by experiment that the basic assumptions of the model are fulfilled, since intmsions of thermal effects or extraparticle resistance to mass transfer may easily occur, leading to erroneously low apparent diffusivity values. 

As a result of these difficulties the reported diffusivity data show many apparent anomaUes and inconsistencies, particularly for 2eohtes and other microporous adsorbents. Discrepancies of several orders of magnitude in the diffusivity values reported for a given system under apparendy similar conditions are not uncommon (18). Since most of the intmsive effects lead to erroneously low values, the higher values are probably the more rehable. 

The solvent used was 5 %v/v ethyl acetate in n-hexane at a flow rate of 0.5 ml/min. Each solute was dissolved in the mobile phase at a concentration appropriate to its extinction coefficient. Each determination was carried out in triplicate and, if any individual measurement differed by more than 3% from either or both replicates, then further replicate samples were injected. All peaks were symmetrical (i.e., the asymmetry ratio was less than 1.1). The efficiency of each solute peak was taken as four times the square of the ratio of the retention time in seconds to the peak width in seconds measured at 0.6065 of the peak height. The diffusivities obtained for 69 different solutes are included with other physical and chromatographic properties in table 1. The diffusivity values are included here as they can be useful in many theoretical studies and there is a dearth of such data available in the literature (particularly for the type of solutes and solvents commonly used in LC separations). 

Equations 4.31 and 4.32 also suggest another important fact regarding NEMCA on noble metal surfaces The rate limiting step for the backspillover of ions from the solid electrolyte over the entire gas exposed catalyst surface is not their surface diffusion, in which case the surfacediffusivity Ds would appear in Eq. 4.32, but rather their creation at the three-phase-boundaries (tpb). Since the surface diffusion length, L, in typical NEMCA catalyst-electrode film is of the order of 2 pm and the observed NEMCA time constants x are typically of the order of 1000 s, this suggests surface diffusivity values, Ds, of at least L2/t, i.e. of at least 4 10 11 cm2/s. Such values are reasonable, in view of the surface science literature for O on Pt(l 11).1314 For example this is exactly the value computed for the surface diffusivity of O on Pt(lll) and Pt(100) at 400°C from the experimental results of Lewis and Gomer14 which they described by the equation ... 

In accordance with Pick s Law, diffusive flow always occurs in the direction of decreasing concentration and at a rate, which is proportional to the magnitude of the concentration gradient. Under true conditions of molecular diffusion, the constant of proportionality is equal to the molecular diffusivity of the component i in the system, D, (m /s). For other cases, such as diffusion in porous matrices and for turbulent diffusion applications, an effective diffusivity value is used, which must be determined experimentally. 

Diffusivity values obtained from Eq. (14a) for solutions that are dilute or moderately concentrated in CuS04, differ appreciably from the molecular difTusivities hitherto used. 

Estimates of diffusive leakage using the same geometry used for pressure-driven leakage, and the diffusivity values in Table 5.1 show leak rates of 0.04 sccm/cm2 for pure hydrogen, or 0.008 sccm/cm2 for reformate or nitrogen diluted hydrogen streams. Thus, diffusive leak rates of 2 to 10 times that of hydrodynamic may be expected. 

The same analysis can be performed on the effect of the similar parameters k, rmax, 1 > k2 and rmax - The steady-state flux increases when any of them increases. As seen in Figure 5f for the particular case of rmax,i variation, there are two asymptotic limits for J s given by the fixed Ju>2 (at low rm lX)i) and by the fixed limiting diffusion value (see equation (16) and Figure 5e). 

Computed results from this model are compared to actual kiln performance in Table VI and the operating conditions taken from kiln samples are given in Table VII. There are no unit factors or adjustable parameters in this model. As with the explicit model, all kinetic data are determined from laboratory experiments. Values of the frequency factors and activation energies are given in Table VIII. Diffusivity values are also included. The amount of fast coke was determined from Eq. (49). With the exception of the T-B (5/12) survey, the agreement between observed and computed values of CO, CO2, and O2 is very good considering that there are no adjustable parameters used to fit the model to each kiln. In the kiln survey T-212/10, the CO conversion activity of the catalyst has been considerably deactivated and a different frequency factor was used in this simulation. 

Very few experimentally determined diffusivities of organic substances in water are available in the documented literature. If experimentally determined diffusivity values are not available, Hayduk and Laudie (1974) recommend the following equation for estimating this parameter ... 

Diffusivity values are reported in a modified exponential form. For example, the experimentally determined diffusivity of benzene in water is 1.09x 10 cm /sec, but this value is reported as 1.09 (x 10 cm /sec). 

Diffusivity of a species in a phase is an intrinsic property and does not depend on the experimental method. If diffusivity does not depend on the concentration of the species, then diffusivity extracted from different methods has the same meaning and hence should all agree within experimental error. ITowever, if diffusivity depends on the concentration of the species or component, the meaning of diffusivity extracted using different techniques may differ, leading to difference in diffusivity values. 

Both the normalized breakthrough time and the diffusivity values provide information which eliminates the thickness constraint and which concern the analytes permeation rate as a function of the material per se. 

Different theoretical models applied to this pore size distribution can give relatively large variations of the calculated effective diffusivity value (DelT). The most commonly used approximations are (i) random-pore model (Wakao and Smith, 1962) using two characteristic transport pores (micropores ji and macropores M)... 

Although there is considerable scatter in the diffusivity values for the treated samples, the major effect is a large increase in diffusivity, relative to the original H-mordenite, on either acid extraction or NH4NO3 exchange. On this basis we would conclude that the sodium rather than the aluminum content appears to be the factor of greatest importance. 

The bulkier 1,3,5 TIPB showed about one order of magnitude faster diffusion in Si-MCM-41 than in NaX zeolite (16). Moreover, contrary to prevalent expectation, this molecule shows higher diffusivity than the smaller xylene isomers. These observations are indicative of the diffusion occurring in larger cylindrical pores (mesopores) of the Si-MCM-41 sample. However, diffusivity values of 1,3,5 TIPB, as well as PFTBA, are of the 10 9 cm2/s order, which is more typical of diffusion in zeolites and other microporous materials. The relatively slow diffusion of these molecules in the larger mesopores could be related to some hindrance effects resulting from structural defects and/or from the presence of extra-framework materials in the cylindrical mesopores. 

An accurate determination of f can be obtained by considering all contributing vacancy trajectories to determine (cos ) by use of Eq. 8.29 [13]. For f.c.c., the accurate value of f is found to be 0.78 thus, correlations affect the diffusivity value by about 22% in Eq. 7.52.7 Correlations can have a considerably larger effect on the diffusivity for substitutional solute atoms by the vacancy mechanism. 

The ratio of the different parameters are also given. The diffusion values are given in units of y/kBTa2/m. 

Diffusivity. Diffusivity values derived from time lag measurements of permeation (D = P/6t) vs. membrane composition of PVC/EVA show the same general tendency as the permeability-composition data. The D values are, however, less reproducible, probably owing to short time lag values, and therefore are not reported here. Solubility values can also be calculated (S = P/D), but they are not reported for the same reason. No separate solubility measurements were made. 

Axial, film, and macropore Maxwell and Knudsen diffusion coefficients are estimated based on relatively standard formulations ( 6, 7, 8, 9), using estimates of physical properties shown in Table I to compute the diffusivity values for the systems studied. The effect of errors in the estimated values of these properties will be discussed later. 

No overall model applicable to the prediction of thermal conductivity/diffusivity values is available, but assuming the presence of additive contributions from the elements in coal, the following correlation has been proposed ... 

Comparisons of estimated diffusivity values on zeolites from sorption uptake measurements and those obtained from direct measurements by nuclear magnetic resonance field gradient techniques have indicated large discrepancies between the two for many systems [10]. In addition, the former method has often resulted in an adsorbate diffusivity directly proportional to the adsorbent crystal size [11]. This led some researchers to believe that the resistance to mass transfer may be confined in a skin at the surface of the adsorbent crystal or pellet (surface barrier) [10,11]. The isothermal surface barrier model, however, failed to describe experimental uptake data quantitatively [10,12]. 

Stoichiometric Coefficient, Rate Constant and Effective Diffusivity Values Obtained for Different Reactions Using the Grain Model,... 

P 21] The mixing of gaseous methanol and oxygen was simulated. The equations applied for the calculation were based on the Navier-Stokes (pressure and velocity) and the species convection-diffusion equation [57]. As the diffusivity value for the binary gas mixture 2.8 x 10 m2 s 1 was taken. The flow was laminar in all cases adiabatic conditions were applied at the domain boundaries. Compressibility and slip effects were taken into account The inlet temperature was set to 400 K. The total number of cells was —17 000 in all cases. 

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