Diffiisivity correlation

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

For conditions in industrial production reactors and in corresponding recycle reactors, the mass transfer coefficients of Gamson et al (1943) will be used. These are approximately correct and simple to use. There may be better correlations for specific cases and especially for larger molecules, where diffiisivity is low and Schmidt number is high. In such cases literature referring to given conditions should be consulted. 

S.-W. Lo, G. J. Hirasaki, W. V. House, R. Kobayashi 2002, Mixing Rules and Correlations of NMR Relaxation with Viscosity, Diffiisivity, and Gas/OU Ratio of Methane/Hydrocarhon Mixtures, presented at SPEJ, March 24-34, 2002. 

The treatment can be modified to include effects of the temperature development and tilting of the susceptor by using the temperature dependence of the diffiision coefficient and adjusting d and (191). In this manner, the experimental data can be correlated, but the model has limited capability for predicting behavior beyond the particular set of experiments used to fit the model. In fact, because of the low values of the Reynolds number (<50) in typical horizontal CVD reactors, film theory and simple... 

The effective diffiisivities of DCB in SC C02 for these runs are optimized to De = 4 10 9 m /s. The corresponding tortuosity factors amount to Dm/Dc = 2.3 and 2.5 when estimating the molecular diffusivity by means of the correlation of [10]. These values are in reasonable agreement with respect to those found in the literature. For example in a related study of... 

To get a sense of the physics behind diffusion, diffusion fluxes and diffusion potentials need to be defined in terms of driving forces. Pick s first law is used to correlate the diffusion flux density with the concentration gradient of the diffiisants, which yields the diffusion coefficient as the factor of proportionality. For a system with two components and one-dimensional diffusion in the z-direction, the particle flux ji is related to the gradient of concentration, 5c,/5z, of these particles as... 

The temperature dependencies of the viscosity (Figure 5.6) and the summation of the self-diffusion coefficient (Dcation + Oanion) (Figure 5.4) interestingly show the contrasted profiles with the indication of inverse relationship between viscosity and self-diffiision coefficient. This can be explained in terms of the Stokes-Einstein equation, which correlates the self-diffusion coefficient (Dcation Danion) with viscosity (q) by the following relationship ... 

It has been demonstrated that the combined application of various NMR techniques for observing molecular rotations and migrations on different time scales can contribute to a deeper understanding of the elementary steps of molecular diffusion in zeolite catalysts. The NMR results (self-diffusion coefficients, anisotropic diffiisivities, jump lengths, and residence times) can be correlated with corresponding neutron scattering data and sorption kinetics as well as molecular dynamics calculations, thus giving a comprehensive picture of molecular motions in porous solids. 

Typical results are shown in Fig. 6 for U-methane in graphite pores of H =7.5 at T=114 K. At p/ps=l the system is solid-like at this temperature, but a discrete change in density occurs around p ps ca.0.5. The self diffiisivity along axial direction also shows drastic change at this point. Further examination of various characteristics of molecular state such as snapshots, in-plane pair correlations and static structure factors confirmed that this change in density is the result of a phase transition from solid-like state to liquid-like one, or melting. Since the critical condensation condition for this pore is far lower than this transition point to stay around p ps= ca.0.2, the liquid-like state is not on metastable branch but thermodynamically stable. Thus a solid-liquid coexistence point is found for this temperature. 

The thermal conductivity k is a transport property whose value for a variety of gases, liquids, and solids is tabulated in Sec. 2. Section 2 also provides methods Tor predicting and correlating vapor and liquid thermal conductivities. The thermi conductivity is a function of temperature, but the use of constant or averaged values is frequently sufficient. Room temperature values for air, water, concrete, and copper are 0.026, 0.61, 1.4, and 400 W/(m K). Methods for estimating contact resistances and the thermal conductivities of composites and insulation are summarized by Gebhart, Heat Conduction and Mass Diffiision, McGraw-Hill, 1993, p. 399. 

The control of reaction rates by a bulk difiusion process is not usually demonstrable by microscopic observations, but support may be obtained from measurements of diffusion coefficients of appropriate species within the structure concerned. This approach has been invaluable in formulating the mechanisms of oxidation of metals, where rates of reaction have been correlated with rates of transportation of ions across barrier layers of product. Sometimes the paths by which such movements occur correspond to regions of high difi isivity, involving imperfect zones within the barrier layer, compared with normal rates of intracrystalline diffusion across more perfect regions of material [63]. Difiusion measurements have been made for ions in nickel sulfide and it was concluded that the decomposition of NiS is diffiision controlled [50]. 

Flynn GL, Yalkowsky SH. Correlation and prediction of mass transport across membranes. I. Influence of alkyl chain length on flux-determining properties of barrier and diffiisant. J Hiarm Sci 1972 61(6) 838-852. 

Fig. 10.3.7 Pulse sequence for spin-diffusion imaging with ID spatial resolution [Wei8] and effect of mobility filters, (a) The magnetization source is selected by the dipolar filter which suppresses the magnetization in the sink. During the spin-diffiision time the magnetization dif ses from the source to the sink, (b) The dipolar filter selects magnetization from chain segments which are highly mobile and intermediately mobile. By use of a lineshape filter the signal loss is analysed only for the mobile components. IP(Tc) is the probability for a particular correlation time to arise in the sample. It is essentially the spectral density of motion.

A sesgle experimental measurement of the average diffiisivity over the whole composition range gave a vnlne of 7.4 X 0 M m2/s. These solutions do form nearly ideal liquid mixtures so the diffiisivity of a 50/50 mixture could he estimated from Vjgnes correlation ... 

The flux of each component is proportional to the concentration gradient and the diffiisivity in the dense layer. However, the concentration gradient is often nonlinear because the membrane swells appreciably as it absorbs liquid, and the diffusion coefficient in the fully swollen polymer may be 10 to 100 times the value in the dense unswollen polymer. Furthermore, when the polymer is swollen mainly by absorption of one component, the diffusivity of other components is increased also. This interaction makes it difficult to develop correlations for membrane permeability and selectivity. 

Gudmundsson [1981] has quoted a correlation for the dimensionless mass transfer coefficient K in terms of the particle Schmidt number for particles in the diffiision regime in turbulent pipe flow. 

A rigorous solution to this problem is not possible, because there are no correlations for the diffiisivity and mass transfer coefficient in... 

Correlation between diffiision and entropy Adam-Gibbs relation 157... 

Suppose that a fraction a of the total anisotropy is lost due to the segmental motion, with a fast correlation time 6/r, and the remainder decays by overall rotational diffusion of the protein (6 ). For independent segmental motions and rotational diffiision, the anisotropy is given as the product of the depolarization frctOR (Section 10.10), and... 

The theory for rotational diffusion of ellipsoids, and measurements by fluorescence polarization, can be traced to the classic reports by F. Perrin. Since these seminal reports, the theory has been modified to include a description of expected anisotropy decays. Hiis theory has been summarized in several reviews.For a rigid ellipsoid with three unequal axes, it is now agreed that the anisotropy decays with five correlation times. The correlation times depend on the three rotational diffiision coefficients, and the amplitudes depend on the orientation of the absorption and emission transition moments widiin the fluoroi iore and/or ellipsoid. While the the( predicts five correlation times, it is known diat two pairs of correlation times will be very close in magniOide, so that in practice only three correlation times are expect for a nonsf oical molecule. ... 

In these equations, Tj is the temperature at coordinates (r, t) in the macrograin, pcp is the average value of the heat capacity per unit volume of the macroparticle, is the effective thermal diffiisivity in the macrograin and (—Affr) the enthalpy of polymerization. In Equation 2.141, the parameter hp is the average convective heat transfer coefficient, usually calculated from a Nusselt number correlation. Early works tended to use the well-known Ranz-Marshall correlation for evaporation from a droplet however, it has been... 

The correlations between De and 1/p for PNP and PCP obtained in DR and CMBR are presented in Figure 2, indicating that the apparent diffiisivity is a strong function of the adsorbate molecule and temperature. 

This section provides some Qqiical experimental values of interdiffiision coefficients in gases, in solutions of nonelectrolytes, electrolytes, and macromolecules, in solids, and for gases in porous solids. Theoretical and empirical correlations for predicting diffiisivities will also be discussed for use in those cases where estimates must be made because eiqierimental data are unavailable. A critical discussion of predicting diffiisivities in a fluid phase can be found in the book by Reid et al. 

Vignes correlated the composition dependence of binaiy diffiision coefficients in terms of their inflnite-dilution values and this thermodynamic correction factor. 

Some representative curves from his paper are also shown in Fig. 2.3. The coirdation appears to be quite successful unless the mixtures are strongly associating (e.g., CHjCI-aoelone). Leffler arid CuUinan have provided a derivation of Eq. (2.3-20) based on Absolute Rale Theory while Gainer has used Absolute Rate Theoiy to provide a different correlation for the composition dependence of binaiy diffiision coefficients. 

MuilieompoiuHt mfects. Only limited experimental data are available for multicompoaeni diffiision in liquids. The binary correlations are sometimes employed for the case of a solute diffiising through a mixed solvent of uniform composition. It is clear that thermodynamic nonidealities in mullicomponent systems can cause significant effects. The reader is referred to Cussler s book for a discussion of available experimental information on diffusion in multicomponent systems. 

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