Solution diffusion model multicomponent

According to their analysis, if c is zero (practically much lower than 1), then the fluid-film diffusion controls the process rate, while if ( is infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the mechanical parameter represents the ratio of the diffusion resistances (solid and fluid-film). This equation can be used irrespective of the constant pattern assumption and only if safe data exist for the solid diffusion and the fluid mass transfer coefficients. In multicomponent solutions, the use of models is extremely difficult as numerous data are required, one of them being the equilibrium isotherms, which is a time-consuming experimental work. The mathematical complexity and/or the need to know multiparameters from separate experiments in all the diffusion models makes them rather inconvenient for practical use (Juang et al, 2003). 

Anderko and Lencka find. Eng. Chem. Res. 37, 2878 (1998)] These authors present an analysis of self-diffusion in multicomponent aqueous electrolyte systems. Their model includes contributions of long-range (Coulombic) and short-range (hard-sphere) interactions. Their mixing rule was based on equations of nonequilibrium thermodynamics. The model accurately predicts self-diffusivities of ions and gases in aqueous solutions from dilute to about 30 mol/kg water. It makes it possible to take single-solute data and extend them to multicomponent mixtures. 

A multicomponent HSDM for acid cfye/carbon adsorption has been developed based on the ideal adsorbed solution theory (lAST) and the homogeneous surface diffusion model (H SDM) to predict the concentration versus time decay curves. The lAST with the Redlich-P eterson equation is used to determine the pair of liquid phase concentrations, Q and Qj, from the corresponding pair of solid phase concentrations, q j and q jy at fha surface of the carbon particle in the binary component. 

In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measru ments improve and more interest is paid to complex... 

SOLUTION OF MULTICOMPONENT DIFFUSION PROBLEMS USING AN EFFECTIVE DIFFUSIVITY MODEL... 

Equations 9.3.15 and 9.3.16 represent an exact analytical solution of the multicomponent penetration model. For two component systems, these results reduce to Eqs. 92.1. Unfortunately, the above results are of little practical use for computing the diffusion fluxes because they require an a priori knowledge of the composition profiles (cf. Section 8.3.5). Thus, a degree of trial and error over and above that normally encountered in multicomponent mass transfer calculations enters into their use. Indeed, Olivera-Fuentes and Pasquel-Guerra did not perform any numerical computations with this method and resorted to a numerical integration technique. 

This chapter describes models of mass transfer in turbulent conditions. Beginning with a brief survey of turbulent eddy diffusivity models we develop solutions to the binary mass transport equations at length before presenting the corresponding multicomponent results. 

Olivera-Fuentes, C. G. and Pasquel-Guerra, J., The Exact Penetration Model of Diffusion in Multicomponent Ideal Gas Mixtures. Analytical and Numerical Solutions, Chem. Eng. Commun., 51, 71-88 (1987). 

In reactive flow analysis the Pick s law for binary systems (2.285) is frequently used as an extremely simple attempt to approximate the multicomponent molecular mass fluxes. This method is based on the hypothesis that the pseudo-binary mass flux approximations are fairly accurate for solute gas species in the particular cases when one of the species in the gas is in excess and acts as a solvent. However, this approach is generally not recommend-able for chemical reactor analysis because reactive mixtures are normally not sufficiently dilute. Nevertheless, many industrial reactor systems can be characterized as convection dominated reactive flows thus the Pickian diffusion model predictions might still look acceptable at first, but this interpretation is usually false because in reality the diffusive fluxes are then neglectable compared to the convective fluxes. 

Diffusion is the mass transfer caused by molecular movement, while convection is the mass transfer caused by bulk movement of mass. Large diffusion rates often cause convection. Because mass transfer can become intricate, at least five different analysis techniques have been developed to analyze it. Since they all look at the same phenomena, their ultimate predictions of the mass-transfer rates and the concentration profiles should be similar. However, each of the five has its place they are useful in different situations and for different purposes. We start in Section 15.1 with a nonmathematical molecular picture of mass transfer (the first model) that is useful to understand the basic concepts, and a more detailed model based on the kinetic theory of gases is presented in Section 15.7.1. For robust correlation of mass-transfer rates with different materials, we need a parameter, the diffusivity that is a fundamental measure of the ability of solutes to transfer in different fluids or solids. To define and measure this parameter, we need a model for mass transfer. In Section 15.2. we discuss the second model, the Fickian model, which is the most common diffusion model. This is the diffusivity model usually discussed in chemical engineering courses. Typical values and correlations for the Fickian diffusivity are discussed in Section 15.3. Fickian diffusivity is convenient for binary mass transfer but has limitations for nonideal systems and for multicomponent mass transfer. 

For a dilute solution of polymer Aina low molecular weight solvent B, the polymer molecules are modeled as bead-spring chains. Resistance in the motion of beads is characterized by a friction coefficient As the number of beads is proportional to the polymer molecular weight M, we have Dab 1 / Vm. Table 2.8 shows some values of diffusion coefficients in polymers. In a flow of dilute solution of polymers, the diffiisivity tensor is anisotropic and depends on the velocity gradient. The Maxwell-Stefan equation may predict the diffusion in multicomponent mixtures of polymers. 

Christensen and Newman [66] developed a set of ID equations for a spherical particle, which included moving boundary considerations and the consideration of multicomponent diffusion with concentrated solution. This model handles the... 

Divisek et al. presented a similar two-phase, two-dimensional model of DMFC. Two-phase flow and capillary effects in backing layers were considered using a quantitatively different but qualitatively similar function of capillary pressure vs liquid saturation. In practice, this capillary pressure function must be experimentally obtained for realistic DMFC backing materials in a methanol solution. Note that methanol in the anode solution significantly alters the interfacial tension characteristics. In addition, Divisek et al. developed detailed, multistep reaction models for both ORR and methanol oxidation as well as used the Stefan—Maxwell formulation for gas diffusion. Murgia et al. described a one-dimensional, two-phase, multicomponent steady-state model based on phenomenological transport equations for the catalyst layer, diffusion layer, and polymer membrane for a liquid-feed DMFC. 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

A mathematical model has been developed to describe the kinetics of multicomponent adsorption. The model takes into account diffusional processes in both the solid and fluid phases, and nonlinear adsorption equilibrium. Comparison of model predictions with binary rate data indicates that the model predictions are in excellent for solutes with comparable diffusion rate characteristics. For solutes with markedly different diffusion rate constants, solute-solute interactions appear to affect the diffusional flows. In all cases, the total mixture concentration profiles predicted compares well with experimental data. 

Nitric acid is a strong electrolyte. Therefore, the solubilities of nitrogen oxides in water given in Ref. 191 and based on Henry s law are utilized and further corrected by using the method of van Krevelen and Hofhjzer (77) for electrolyte solutions. The chemical equilibrium is calculated in terms of liquid-phase activities. The local composition model of Engels (192), based on the UNIQUAC model, is used for the calculation of vapor pressures and activity coefficients of water and nitric acid. Multicomponent diffusion coefficients in the liquid phase are corrected for the nonideality, as suggested in Ref. 57. 

Third, a serious need exists for a data base containing transport properties of complex fluids, analogous to thermodynamic data for nonideal molecular systems. Most measurements of viscosities, pressure drops, etc. have little value beyond the specific conditions of the experiment because of inadequate characterization at the microscopic level. In fact, for many polydisperse or multicomponent systems sufficient characterization is not presently possible. Hence, the effort probably should begin with model materials, akin to the measurement of viscometric functions [27] and diffusion coefficients [28] for polymers of precisely tailored molecular structure. Then correlations between the transport and thermodynamic properties and key microstructural parameters, e.g., size, shape, concentration, and characteristics of interactions, could be developed through enlightened dimensional analysis or asymptotic solutions. These data would facilitate systematic... 

Finally, Kvaalen et al have shown that the system of equations of the ideal model for a multicomponent system (see later, Eqs. 8.1a and 8.1b) is strictly h5q3er-bolic [13]. As a consequence, the solution includes two individual band profiles which are both eluted in a finite time, beyond the column dead time, to = L/u. The finite time that is required for complete elution of the sample in the ideal model is a consequence of the assumption that there is no axial dispersion. It contrasts with the infinitely long time required for complete elution in the linear model. This difference illustrates the disparity between the hyperbolic properties of the system of equations of the ideal model of chromatography and the parabolic properties of the diffusion equation. 

Some assumptions regarding the constancy of certain parameters are usually in order to facilitate the solution of the diffusion equations. For the binary diffusion problems discussed in Chapters 5 (as well as later in Chapters 8-10), we assume the binary Fick diffusion coefficient can be taken to be a constant. In the applications of the linearized theory presented in the same chapters, we assume the matrix of multicomponent Fick diffusion coefficients to be constant. If, on the other hand, we use Eq. 6.2.1 to model the diffusion process then we must usually assume constancy of the effective diffusion coefficient if... 

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