The Transport Diffusivities

The analogy among the three modes of transport is further reinforced by noting that all three transport coefficients v, a, and D have identical units of square meter per second (rcF/s), and all three are referred to as diffusivities. We have 

Because all three quantities are conveyed by the same molecules that, one assumes, move at the same speed, it is tempting to conclude that the diffu-sivities of momentum, heat, and mass will be identical, or at least similar, in magnitude. This is in fact the case for transport in low-density gases, but the assumption breaks down in liquids and even more so in solids. Mass diffusivity in particular begins to diverge sharply from its partners and marches on to its own drummer (very slowly). The following illustration examines this important aspect in more detail. 

Listed in Table 1.3 are experimental diffusivities for momentum, heat, and mass (taken to be oxygen) in three representation media — air, water, and glycerin. 

The first feature of nofe is fhe near-identity of values for transport in air, and this can be shown to apply to low-density gases in general. The harmony found in gases changes dramatically when we turn to liquids. In water, mass diffusivity has distanced itself from its partners by two to three orders of magnitude, and when we turn to glycerin, with a viscosity one thousand times that of water, mass diffusion has slowed to a crawl, some five to nine orders of magnitude behind its partners. 

In low-density gases, the molecules spend their time almost exclusively in transit between collisions. That transit time ( - 10 s) and the speed at 

---

The Ru surface is one of the simplest known, but, like virtually all surfaces, it includes defects, evident as a step in figure C2.7.6. The observations show that the sites where the NO dissociates (active sites) are such steps. The evidence for this conclusion is the locations of the N and O atoms there are gradients in the surface concentrations of these elements, indicating that the transport (diffusion) of the O atoms is more rapid than that of the N atoms thus, the slow-moving N atoms are markers for the sites where the dissociation reaction must have occurred, where their surface concentrations are highest. 

The concentrations of the reactants and reaction prodncts are determined in general by the solution of the transport diffusion-migration equations. If the ionic distribution is not disturbed by the electrochemical reaction, the problem simplifies and the concentrations can be found through equilibrium statistical mechanics. The main task of the microscopic theory of electrochemical reactions is the description of the mechanism of the elementary reaction act and calculation of the corresponding transition probabilities. 

Knox, J. B. Numerical modeling of the transport diffusion and deposition of pollutants for r ions and extended scales. J. Air Pollut. Control Assoc. 24 660-664, 1974. 

Based on the use of the NARCM regional model of climate and formation of the field of concentration and size distribution of aerosol, Munoz-Alpizar et al. (2003) calculated the transport, diffusion, and deposition of sulfate aerosol using an approximate model of the processes of sulfur oxidation that does not take the chemical processes in urban air into account. However, the 3-D evolution of microphysical and optical characteristics of aerosol was discussed in detail. The results of numerical modeling were compared with observational data near the surface and in the free troposphere carried out on March 2, 4, and 14, 1997. Analysis of the time series of observations at the airport in Mexico City revealed low values of visibility in the morning due to the small thickness of the ABL, and the subsequent improvement of visibility as ABL thickness increased. Estimates of visibility revealed its strong dependence on wind direction and aerosol size distribution. Calculations have shown that increased detail in size distribution presentation promotes a more reliable simulation of the coagulation processes and a more realistic size distribution characterized by the presence of the accumulation mode of aerosol with the size of particles 0.3 pm. In this case, the results of visibility calculations become more reliable, too. 

The relationship between the transport diffusivity (D), as measured under non-equilibrium conditions in an uptake experiment and the tracer self diffusivity (Ds), measured under equilibrium conditions in an NMR experiment, has been discussed by Ash and Barrer(30) and Karger(31,32)t who show that... 

In order to determine the activation energy of the difiuaon, the uptake experiments were conducted at temperatures in the range 398 to 473 K. In Table 2, results are compiled. The errors of the transport diffusion coefficient are estimated to be 0.75 10 cmVs. A thermodynamic correction [13] of the transport diffiiavity has not been applied. However, since the... 

The results described in this report compare well with data of Van-Den-Begin et al. [15] obtained on silicalite samples with an equivalent radius of 31pm by means of Single-Step Frequency-Response. The authors report a self diffusion coefficient for n-hexane of about 2 10 cmVs at a temperature of444 K. However, it has to be considered that, due to the shape of the sorption isotherm, the self-diffusion coefficient will be somewhat smaller than the transport diffusion coefficient. Caro et al [16] report a transport diffusion coefficient of 1.8 10 cmVs for the system n-hexane/HZSM-5 at 298 K, determined gravimetrically. The crystals used in that study were of prismatic shape, the dimensions being 330 pm (z-axis), 110 pm... 

The transport diffusivities determined are of the order of 4 1 O cmVs, which corresponds well to literature data obtained by various other techniques. The activation energy for the diffusion process has been found to be 6-IS kJ/mol. 

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... 

The dissolution of a mineral is a sum of chemical and physical reaction steps. If the chemical reactions at the surface are slow in comparison with the transport (diffusion) processes, the dissolution kinetics is controlled by one step in the chemical surface processes thus, rates of transport of the reactants from the bulk solution to the surface and of products from the surface into the solution can be neglected in the overall rate. It has been shown by Petrovic et al. (1976) and Berner and Holdren (1979) that the dissolution of many minerals, especially under conditions encountered in nature, are surface-controlled. 

NMR PFG measurements determine the tracer or self-diffusivity (D ) under equilibrium conditions with no concentration gradient. n any sorption rate measurement it is the transport diffusivity under the influence of a concentration gradient which is measured. In general these two quantities are not the same but the relationship between them can be established from irreversible thermodynamics. (17,18) In the low concentration limit the thermodynamic correction factor vanishes and the transport and self diffusivities should approach the same limit. Since ZLC measurements are made at low concentrations within the Henry s Law region the diffusivity values should be directly comparable with the NMR self-dif fusivities. ... 

The diffusivity (D) defined in this way is not necessarily independent of concentration. It should be noted that for diffusion in a binary fluid phase the flux (/) is defined relative to the plane of no net volumetric flow and the coefficient D is called the mutual diffusivity. The same expression can be used to characterize migration within a porous (or microporous) sohd, but in that case the flux is defined relative to the fixed frame of reference provided by the pore walls. The diffusivity is then more correctly termed the transport diffusivity. Note that the existence of a gradient of concentration (or chemical potential) is implicit in this definition. 

All sorbates exhibited transport diffusivities of the order 10 cm s increasing with increasing temperature. The errors of the transport diffusion coefficients are estimated to be 0.75 x 10 cm s in the case of n-hexane and n-heptane. Due to the lower absolute absorbance in the case of n-octane and n-nonane, the estimated error is higher, viz. 1.25 x 10 cm s The re-... 

Fig. 43 Concentration dependence of the transport diffusivity as determined from the center (a) and from the entire profile (b)...

These data are displayed in Figs. 51 and 52. The transport diffusivity in the z direction is approximately twice as large as that in the x or y direction. This correlates nicely with the difference in the critical sizes of the windows on the diffusion paths in the different directions. It is interesting to note that also the surface permeability in the z direction slightly (by about 20%) exceeds that in the X andy directions. 

In separation or catalytic applications, it is the transport diffusivity, Dt, which matters (this quantity is also named Fickian or chemical diffusivity). Transport diffusivities are traditionally obtained under non-equilibrium conditions [2], but they can be measured at equilibrium by coherent QENS [5]. Coherent neutron scattering is in principle more comphcated than incoherent scattering, but under certain conditions transport diffusivities can be extracted from the neutron data. 

In short, incoherent scattering allows one to determine the self-diffusivity, Ds, whereas coherent scattering gives access to the transport diffusivity, Du from experiments performed at equihbrium. When the scattering is both incoherent and coherent, then both diffusivities can in principle be determined simultaneously. 

Despite extensive work in the last decade, large discrepancies still persist between the various experimental techniques which measure diffusion in zeohtes. One of the difficulties is that one has to compare self-diffusivities, obtained by PFG NMR or QENS methods, with transport diffusivities derived from macroscopic experiments. The transport diffusivity is defined as the proportionahty factor between the flux and a concentration gradient (Fick s first law)... 

Coherent QENS measurements and MD simulations have been performed for N2 and CO2 in silicalite [30,31]. It has been found that the self-diffusivities of the two gases decrease with increasing occupancy, while the transport diffusivities increase. For a comparison with other systems, it is appropriate to remove the influence of the thermodynamic correction factor and to discuss the collective mobility in terms of the corrected diffusivity (also called Maxwell-Stephan diffusivity). Dq(c) is directly obtained from the Simula-... 

---