Mass transfer binary diffusion coefficient

The estimation of the film thickness is obtained from the average values for the binary mass transfer and diffusion coefficient, estimated by traditional correlation (Onda (1968) correlation for the mass transfer coefficient. Fuller and Wilke Chang correlation for the vapour and liquid diffusion coefficient). 

According to Maxwell s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. 

The variation of efficiencies is due to interaction phenomena caused by the simultaneous diffusional transport of several components. From a fundamental point of view one should therefore take these interaction phenomena explicitly into account in the description of the elementary processes (i.e. mass and heat transfer with chemical reaction). In literature this approach has been used within the non-equilibrium stage model (Sivasubramanian and Boston, 1990). Sawistowski (1983) and Sawistowski and Pilavakis (1979) have developed a model describing reactive distillation in a packed column. Their model incorporates a simple representation of the prevailing mass and heat transfer processes supplemented with a rate equation for chemical reaction, allowing chemical enhancement of mass transfer. They assumed elementary reaction kinetics, equal binary diffusion coefficients and equal molar latent heat of evaporation for each component. 

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. 

TABLE 3 Binary Diffusion Coefficients and Mass Transfer Coefficients... 

Phase Binary diffusion coefficients Mass transfer correlations... 

Single Sphere Model II (Equations 4, 5, 8, 9 and 10 in reference 6) In this model allowance is made for the resistance to mass transfer offered by the surface film surrounding the herb particles. The mass transfer coefficient kf was obtained from correlations proposed by Catchpole et al (8, 9) for mass transfer and diffusion into near-critical fluids. An average of the binary diffusivities of the major essential oil components present was used in calculating kf (these diffusivities were all rather similar because of their similar structures). 

Fig-1 Concentration profiles in gas and liquid for a chemical reaction influenced by mass transfer in the liquid phase at various ratios of the reaction rate to the rate of mass transfer. D = binary diffusions coefficient. 

Neglecting enthalpy change due to viscosity, assuming the same forces fa for all components, and taking the binary diffusion coefficients as Dap = D + ap), ap < 1, the energy and mass transfer equations can be rewritten as... 

Experimental studies were carried out to derive correlations for mass-transfer coefficients, reaction kinetics, liquid holdup and pressure drop for the new catalytic packing MULTIPAK (see [9,10]). Suitable correlations for ROMBOPAK 6M were taken from [70] and [92], The vapor-liquid equilibrium is calculated using the modification of the Wilson method [9]. For the vapor phase, the dimerization of acetic acid is taken into account using the chemical theory to correct vapor-phase fugacity coefficients [93]. Binary diffusion coefficients for the vapor phase and for the liquid phase are estimated via the method purposed by Fuller et al. and Tyn and Calus, respectively (see [94]). Physical properties like densities, viscosities and thermal conductivities are calculated from the methods given in [94]. 

The ternary diffusion coefficient strongly depends on the solution concentration. In order to calculate accurate mass transfer coefficients, experimental data of diffusion coefficients at the interest concentrations and temperatures are necessary. However, data are not available at concentrations and temperature used at the present study, it was assumed that the ternary diffusion coefficients were equal to the binary diffusion coefficients. The binary diffusion coefficients of the KDP-water pairs and the urea-water pairs were taken from literature (Mullin and Amatavivadhana, 1967 Cussler, 1997). The values were transformed into the Maxwell-Stefan diffiisivities using the thermodynamic correction factor. 

The film transfer coefficient kf is a measure of the resistance to mass transfer in the fluid phase. The coefficient is a function of the solvent velocity t/, density p, viscosity p and binary diffusion coefficient Dn. The coefficient is usually predicted from correlations using dimensionless numbers that have been developed from mass transfer studies using gases and liquids at nearambient conditions. The correlations are of the form... 

Another analogous relationship is that of mass transfer, represented by Pick s law of diffusion for mass flux, J, of a dilute component, I, into a second fluid, 2, which is proportional to the gradient of its mass concentration, mi. Thus we have, J = p Du Vmt, where the constant Z)/2 is the binary diffusion coefficient and p is density. By using similar solutions we can find generalized descriptions of diffusion of electrons, homogeneous illumination, laminar flow of a liquid along a spherical body (assuming a low-viscosity, non-compressible and turbulent-lree fluid) or even viscous flow applied to the surface tension of a plane membrane. 

Grashof number for mass transfer L is a characteristic dimension, i.e., the diameter of a spherical particle, or the equivalent diameter of a nonspherical particle, etc. v is the kinematic viscosity D is the binary diffusion coefficient U is the linear velocity of the gas stream flowing past the particle (measured outside the boundary layer surrounding the particle) g is the acceleration due to gravity is a characteristic concentration difference, and... 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

Finally, intraparticle diffusion appears to be an important factor in adsorption kinetics for many types of systems. In the past it has been customary to define such mass transfer quantitatively in terms of an effective diffusivity. However, even in gas-solid systems more than one process can be involved for porous particles. Thus, two-dimensional migration on the pore surface, surface diffusion, is a potential contribution. Liquid systems appear to be more complex, and, with electrolytes, it has been shown that the electric potential induced by counter-diffusing ions should be taken into account. A realistic description of intraparticle mass transfer in such cases requires more than a single rate coefficient for a binary system. 

In order to calculate the multicomponent diffusion matrices [D], the binary diffusivities in both phases should be known. The him thickness representing an important model parameter is estimated via the mass transfer coefficients (57,83). The binary diffusivities and mass transfer coefficients were calculated from the correlations summarized in Table 3. 

The multicomponent diffusion matrices [D] in both phases are determined via the binary diffusivities. The latter can be estimated using different correlations summarized in Tab. 9.3. The film thicknesses which represent important model parameters (cf. Section 9.4.4) are estimated via the mass transfer coefficients [16, 23]. 

Table 9.3. Binary diffusion and mass transfer coefficients.

In a binary mixture, diffusion coefficients are equal to each other for dissimilar molecules, and Fick s law can determine the molecular mass flows in an isotropic medium at isothermal and isobaric conditions. In a multicomponent diffusion, however, various interactions among the molecules may arise. Some of these interactions are (i) diffusion flows may vanish despite the nonvanishing driving force, which is known as the mass transfer barrier, (ii) diffusion of a component in a direction opposite to that indicated by its driving force leads to a phenomenon called the reverse mass flow, and (iii) diffusion of a component in the absence of its driving force, which is called the osmotic mass flow. 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

The diffusivities used in this example were for the light key—heavy key pair of components. For systems similar to this one, the diffusion coefficients of all binary pairs in the mixture would be expected to have similar values. This will not be the case for mixtures of components that differ sharply in their fundamental properties (e.g., size, polarity). For these more highly nonideal mixtures, it is necessary to estimate the mass-transfer coefficients for each of the binary pairs. 

Individual component efficiencies can vary as much as they do in this example only when the diffusion coefficients of the three binary pairs that exist in this system differ significantly For ideal or nearly ideal systems, all models lead to essentially the same results. This example demonstrates the importance of mass-transfer models for nonideal systems, especially when trace components are a concern. For further discussion of this example, see Doherty and Malone (op. cit.) and Baur et al. [AIChE J. 51,854 (2005)]. It is worth noting that there exists extensive experimental evidence for mass-transfer effects for this system, and it is known that nonequilibrium models accurately describe the behavior of this system, whereas equilibrium models (and equal-efficiency models) sometime... 

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