Gases dense, diffusion coefficients

This relative invariance of signal may be a consequence of a compensation effect. That is, as temperature increases, the test gas becomes slightly less dense on a moles CO/cm3 basis, but the diffusion coefficient increases slightly. 

The diffusion coefficient as defined by Fick s law, Eqn. (3.4-3), is a molecular parameter and is usually reported as an infinite-dilution, binary-diffusion coefficient. In mass-transfer work, it appears in the Schmidt- and in the Sherwood numbers. These two quantities, Sc and Sh, are strongly affected by pressure and whether the conditions are near the critical state of the solvent or not. As we saw before, the Schmidt and Prandtl numbers theoretically take large values as the critical point of the solvent is approached. Mass-transfer in high-pressure operations is done by extraction or leaching with a dense gas, neat or modified with an entrainer. In dense-gas extraction, the fluid of choice is carbon dioxide, hence many diffusional data relate to carbon dioxide at conditions above its critical point (73.8 bar, 31°C) In general, the order of magnitude of the diffusivity depends on the type of solvent in which diffusion occurs. Middleman [18] reports some of the following data for diffusion. 

It should be pointed out that for a low pressure gas the radial- and axial diffusion coefficients are about the same at low Reynolds numbers (Rediffusion effects may be important at velocities where the dispersion effects are controlled by molecular diffusion. For Re = 1 to 20, however, the axial diffusivity becomes about five times larger than the radial diffusivity [31]. Therefore, the radial diffusion flux could be neglected relative to the longitudinal flux. If these phenomena were also present in a high-pressure gas, it would be true that radial diffusion could be neglected. In dense- gas extraction, packed beds are operated at Re > 10, so it will be supposed that the Peclet number for axial dispersion only is important (Peax Per). 

Dm denotes the molecular diffusion coefficient F denotes the interphase mass exchange rate between the dense and the dilute phases and Fc = — F, which can be directly calculated with EMMS/matrix model parameters if the reaction source term, Sk, is negligible compared to the bulk gas conservation. For vaporization of A, the source term reads... 

Due to vibrational anharmonicity, this transfer is resonant only for the K = 1 exchange, which has been considered in a previous section, but it remains, in the liquid, faster by several orders of magnitude than V-T relaxation for diatomics. Relaxation of highly excited I2 and Br2 close to the vibrational dissociation limit has been observed in the dense gas (at liquid densities). These indirect measurements of T, were correlated with the gas diffusion coefficient and should hence be more reasonably accounted for in the framework of an isolated binary interaction model. This interesting exjjerimental system raises the question of the influence of the change in molecular dimensions in higher excited states, due to anharmonicity, on the efficiency of collisional deexcitation. This question could jjerhaps be answered by more precise direct relaxation measurements. 

This process governs the rate of deposition of the molecules of nonvolatile compounds on the surface of gas ducts, and contributes to broadening of the chromatographic zones. Being of the order of 0.1 pm at STP, the mean free path of molecules, which is inversely proportional to pressure, reaches 1 cm only at about 0.01 mmHg. In dense enough gas, in the absence of convective flow, the macroscopic picture of migration of molecules (as well as of aerosol particulates) is described by the equations of diffusion. The mean squared diffusional displacement z2D of molecules, the time of diffusion t and the mutual diffusion coefficient >i 2 are related by ... 

This balance equation can also be derived from kinetic theory [101], In the Maxwellian average Boltzman equation for the species s type of molecules, the collision operator does not vanish because the momentum mgCs is not an invariant quantity. Rigorous determination of the collision operator in this balance equation is hardly possible, thus an appropriate model closure for the diffusive force is required. Maxwell [65] proposed a model for the diffusive force based on the principles of kinetic theory of dilute gases. The dilute gas kinetic theory result of Maxwell [65] is generally assumed to be an acceptable form for dense gases and liquids as well, although for these mixtures the binary diffusion coefficient is a concentration dependent, experimentally determined empirical parameter. 

Sousa et al [5.76, 5.77] modeled a CMR utilizing a dense catalytic polymeric membrane for an equilibrium limited elementary gas phase reaction of the type ttaA +abB acC +adD. The model considers well-stirred retentate and permeate sides, isothermal operation, Fickian transport across the membrane with constant diffusivities, and a linear sorption equilibrium between the bulk and membrane phases. The conversion enhancement over the thermodynamic equilibrium value corresponding to equimolar feed conditions is studied for three different cases An > 0, An = 0, and An < 0, where An = (ac + ad) -(aa + ab). Souza et al [5.76, 5.77] conclude that the conversion can be significantly enhanced, when the diffusion coefficients of the products are higher than those of the reactants and/or the sorption coefficients are lower, the degree of enhancement affected strongly by An and the Thiele modulus. They report that performance of a dense polymeric membrane CMR depends on both the sorption and diffusion coefficients but in a different way, so the study of such a reactor should not be based on overall component permeabilities. 

The D j ate the binary diffusion coefficients (or an / — j mixture thus, no additional information is required for computations of multicomponent diffusion in dilute gas mixtures although one might prefer a form in which the fluxes appeared explicitly. Generalization of the Stefan-Maxweli form to dense gassa and to liquids has been suggested. 1 but in these cases there is no rigorous relationship to the binety diffusivilies. Furthermore, Ihe form of Eq- (2.3-10) in which the fluxes du not appear explicitly has little to recommend it. 

DENSE GAS DIFFUSION COEFFICIENTS FOR THE METHANE-PROPANE SYSTEM. 

The estimation of low pressure diffusivity is based on the corresponding states theory. The dense gas diffusion coefficient estimation is based on the Enskog theory. The binary diffusion coefficient D jj at high pressures as modeled by the Dawson-Khoury-Kobayashi correlation, is next given as a representative model. For a binary system, the equations are ... 

The second of Tick s law (Equation 9.2) can be used as a sort of the reference point for describing a permeation process of one component gas (N2,02) through a dense polymeric membrane with no external field. Functional dependence of the diffusion coefficient on coordinate, time or concentration can result from different types of circumstances, i.e. when the membrane is heterogeneous [28], is subjected to some relaxation phenomena [29,30], or membrane transport is accompanied by some other processes, like adsorption, etc. [9,12]. 

A very large body of data on the gas permeability of many rubbery and glassy polymers has been published in the literature. These data were obtained with homopolymers as well as with copolymers and polymer blends in the form of nonporous dense (homogeneous) membranes and, to a much lesser extent, with asymmetric or composite membranes. The results of gas permeability measurements are commonly reported for dense membranes as permeability coefficients, and for asymmetric or composite membranes as permeances (permeability coefficients not normalized for the effective membrane thickness). Most permeability data have been obtained with pure gases, but information on the permeability of polymer membranes to a variety of gas mixtures has also become available in recent years. Many of the earlier gas permeability measurements were made at ambient temperature and at atmospheric pressure. In recent years, however, permeability coefficients as well as solubility and diffusion coefficients for many gas/polymer systems have been determined also at different temperatures and at elevated pressures. Values of permeability coefficients for selected gases and polymers, usually at a single temperature and pressure, have been published in a number of compilations and review articles [27—35]. 

In Table 3 we summarize the computed self-diffusion coefficients, reduced by the Enskog dense-gas value... 

Bulk Diffusion Bulk or ordinary diffusion, with the diffusion coefficient Dab or DA.mix> occurs in the void space of the pores as a result of collisions between gas molecules and is likely to dominate when (i) the pores are macropores larger than 100 A in radius, (ii) the gas is relatively dense, that is, at high pressures, and (iii) the pores are filled with a liquid. 

The kinetics of isotope exchange of O between dense Laq.7SrQ.30003.5 ceramics and the gas phase was studied at 950 to 1130K under O pressures of 0.8 to 1.6kPa. The diffusion coefficient was described by ... 

The mobile phase must be a fluid A gas, a dense gas, or a liquid, where the rale of mass transfer through the mobile phase, characterized by the diffusion coefficient, decreases in the order listed. To compensate for this marked decrease, finer and finer particles are used as the mobile pha.se becomes more dense. The practical lack of compressibility of liquids supports this trade-off. since small particles demand higher pressures. However.viscous liquids (e.g.,glycerol,oligomers, or concentrated solutions of polymers) are excluded, because the associated pressure requirements would be unrealistic. 

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