Cussler model

All of the assumptions from the Nielsen model remain in the Cussler model except for the regularity of the array. The Cussler model for relative permeability of a composite with monodisperse particles in a regular array is given by Eq. (8.7) 

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A three-fold improvement in gas barrier has been demonstrated with a mica platelet filler in EVOH. This suggests that a value of about 0.3 would be appropriate as the geometric factor in the Cussler model using irregular flakes of nonuniform size typical of commercially available fillers. 

For diffusion of liquid through rubbery polymer composites, Fickian and non-Fickian diffusion theories are frequently used to describe the mechanism of transport, but for gas or vapour, other models have been developed to fit experimental data of diffusion profiles. The models of gas transport include Maxwell s model," free volume increase mechanism," solubility increase mechanism," nanogap hypothesis," Nielsen model, " " Bharadwaj model, ° Cussler model " " and Gusev and Lusti model, " etc. 

The Cussler model focuses on the diffusion of small gas molecules through a polymer matrix which is partly filled of impermeable flakes which are oriented perpendicular to the direction of diffusion. Therefore, the diffusion process is mainly related to three factors the tortuous wiggles to get around the flake, the tight slits between the flakes, and the resistance of going from the wiggle to the slit. This model proposes that the diffusion depends on the volume fraction of the impermeable filler and the aspect ratio. Then, a permeability model of Cussler can be obtained by multiplying the diffusion by the appropriate solubility as following equation ... 

The aspect ratio calculated using the Cussler model is much higher than that calculated using the Neilsen model. This model will agree with a very low clay loading and a better homogeneous dispersion in polymer matrix as a model fit. However, it is not suitable for the latex nanocomposite system. ... 

There are two mathematical methods for formulating transport by random motion. The first, often called a mass transfer model (Cussler, 1984), relates the net flux to the difference in occupation numbers between two adjacent subsystems, A and B ... 

According to the model of random walk in three dimensions, the diffusion coefficient of a molecule i, can be expressed as one-third of the product of its mean free path A, and its mean three-dimensional velocity u, (Eq. 18-7a). In the framework of the molecular theory of gases, u, is (e.g., Cussler, 1984) ... 

Good quality RO membranes can reject >95-99% of the NaCl from aqueous feed streams (Baker, Cussler, Eykamp et al., 1991 Scott, 1981). The morphologies of these membranes are typically asymmetric with a thin highly selective polymer layer on top of an open support structure. Two rather different approaches have been used to describe the transport processes in such membranes the solution-diffusion (Merten, 1966) and surface force capillary flow model (Matsuura and Sourirajan, 1981). In the solution-diffusion model, the solute moves within the essentially homogeneously solvent swollen polymer matrix. The solute has a mobility that is dependent upon the free volume of the solvent, solute, and polymer. In the capillary pore diffusion model, it is assumed that separation occurs due to surface and fluid transport phenomena within an actual nanopore. The pore surface is seen as promoting preferential sorption of the solvent and repulsion of the solutes. The model envisions a more or less pure solvent layer on the pore walls that is forced through the membrane capillary pores under pressure. 

An interesting engineering approach was proposed by Kokini and coworkers to model viscosity-taste interactions. Kokini et al. (1982) have studied the perception of sweetness of sucrose and fructose in solutions with various tomato solids contents, and with basis on the observed decreasing of sweetness intensity as the percentage of tomato solids increased, they have proposed a more complex but rather comprehensive physical model relating viscosity and taste intensity, based on the physics and chemistry in the mouth. This model was further successfully applied (Cussler et al., 1979) to explain the effect of the presence of hydrocolloids at different levels on the subjective... 

There is a large body of literature that deals with the proper definition of the diffusivity used in the intraparticle diffusion-reaction model, especially in multicomponent mixtures found in many practical reaction systems. The reader should consult references, e.g.. Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley Sons, New York, 2002 Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993 and Cussler, Diffusion Mass Transfer in Fluid Systems, Cambridge University Press, 1997. 

The mass-transfer efficiencies of various MHF contactors have been studied by many researchers. Dahuron and Cussler [AlChE 34(1), pp. 130-136 (1988)] developed a membrane mass-transfer coefficient model (k ) Yang and Cussler [AIChE /., 32(11), pp. 1910-1916 (1986)] developed a shell-side mass-transfer coefficient model (ks) for flow directed radially into the fibers and Prasad and Sirkar [AIChE /., 34(2), pp. 177-188 (1988)] developed a tube-side mass-transfer coefficient model (k,). Additional studies have been published by Prasad and Sirkar [ Membrane-Based Solvent Extraction, in Membrane Handbook, Ho and Sirkar, eds. (Chapman Hall, 1992)] by Reed, Semmens, and Cussler [ Membrane Contactors, Membrane Separations Technology Principle. and Applications, Noble and Stern, eds. (Elsevier, 1995)] by Qin and Cabral [MChE 43(8), pp. 1975-1988 (1997)] by Baudot, Floury, and Smorenburg [AIChE ]., 47(8), pp. 1780-1793 (2001)] by GonzSlez-Munoz et al. [/. Memhane Sci., 213(1-2), pp. 181-193 (2003) and J. Membrane Sci., 255(1-2), pp. 133-140 (2005)] by Saikia, Dutta, and Dass [/. Membrane Sci., 225(1-2), pp. 1-13 (2003)] by Bocquet et al. [AIChE... 

E. L. Cussler, Diffusion Mass Transfer in Fluid Systems, second edition Liang-Shih Fan and Chao Zhu, Principles of Gas-Solid Flows Hasan Orbey and Stanley I. Sandler, Modeling Vapor-Liquid Equilibria Cubic Equations of State and Their Mixing Rules... 

A recent paper by Cussler, et al., (9) using a regular array model also predicted quite high barrier improvements. Their equation (Equation 2) predicted that aspect ratio and volume fraction of filler would be the major variables. They incorporated a universal correction designated, /j, as a geometric factor to correct for the reality that available platelet fillers were not shaped like uniform rectangular parallelepipeds of uniform size and shape. 

The present model assumes that ingredients diffuse and evaporate independently, whereas thermodynamic and mass transport considerations dictate that interactions must occur in concentrated mixtures (Cussler, 1997). Careful analysis of the evaporation rates in Vuilleumier etal. (1995) shows this to be the case The musk ingredient, compound XII in Table 10.2, depressed the initial evaporation rates of... 

As noted earlier, determination of the diffusivity requires that a model be defined so that the concentration data collected in the experiment can be analyzed. Almost all of the diffusivity data tabulated in the literature (e.g., Cussler. 2009 Demirel. 2007 Marrero and Mason. 1972 Poling et al.. 2008 Reid et al.. 1987 Sherwood et al.. 19751 were analyzed with the Fickian model. 

In 1868, 12 years after Tick s definitive publication of his theory, James Clerk Maxwell published a paper on a different approach to studying the diffusivity of gases, hi 1871 Josef Stefan extended Maxwell s theory and anticipated multiconponent effects (Cussler. 2009). Although the Maxwell-Stefan theory has had many strong adherents in the more than 140 years since its development, it always seems to be playing catch-up to the earlier Fickian theory. Three perceived difficulties have prevented wider acceptance of the Maxwell-Stefan theory. First, the Fickian model is well-entrenched in textbooks and diffusivity data collections, and it works well for many binary systems. Second, the Maxwell-Stefan theory gives one fewer flux N than is needed to conpletely solve the problem. However, this is really no different than choosing a reference velocity for Tick s law, and, as will be shown later, for most... 

How does mass transfer affect the efficiency of a tray column This is a question of considerable interest in the design of staged columns. We will develop a very sinple model following the presentations of Cussler (1997). King (1980). Lewis (1936). and Lockett (1986). 

Table 7.3. Models used to estimate kp for commonly encountered liquidisolid interface geometries adapted from Table 8.3-3 in Cussler (2009). D (mdsec) is the diffusion coefficient q (ml sec) is the Darcy velocity of the fluid and v (rtf I sec) is the kinematic viscosity.

Values of kj) are influenced by the interface geometry and the fluid flow velocity. Table 7.3 lists some k, models for geometries that might be useful to geochemists. The estimated accuracy of most of these models is on the order of 10% but much larger uncertainties are possible (Cussler, 2009). 

In this section a number of methods ate described for the experinrental determination of molecular diffitsioa coefficients. The purpose is not ottty to acquaint the reader with some of these tedmiques but also to illustrate, by means of the associated analyses, the proper fomwlation and solution of the appropriate mathematical model for the particular experirtrental dif km situation. In all cases, the governing equations follow ftom simplifying the conservation equations for total mass and particular qiecies, using the flux expression and necessary thermodynamic relationships, artd qiplying appropriate boundary conditions established from the physical situation. Further descriptions of experirtrental methods tttay be found in the books by Jost and Cussler. Many mathematical solutions of the diffusion equation ate found in Ctank. ... 

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