Diffusivity Fickian model

The most rigorous formulation to describe adsorbate transport inside the adsorbent particle is the chemical potential driving force model. A special case of this model for an isothermal adsorption system is the Fickian diffusion (FD), model which is frequently used to estimate an effective diffusivity for adsorption of component i (D,) from experimental uptake data for pure gases.The FD model, however, is not generally used for process design because of mathematical complexity. A simpler analytical model called linear driving force (LDF) model is often used. ° According to this model, the rate of adsorption of component i of a gas mixture... 

The Fickian diffusion models with constant effective diffusivities presented earlier and the rigorous dusty gas model presented in this section are not the only alternatives for modelling diffusion and reaction in porous catalyst pellets. Fickian models with effective diffusion coefficients which are varying with the change of concentration of the gas mixture can also be used. This is certainty more accurate compared with the Fickian model with constant diffusivities although of course less accurate than the dusty gas model. The main problem with these models is the development of relations for the change of diffusivities with the concentration of the gas mixture without solving the dusty gas model equations. Two such techniques are presented in this section and their results are compared with dusty gas model results. 

B.l SIMPLE APPLICATION OF THE ORTHOGONAL COLLOCATION TECHNIQUES FOR A FICKIAN-TYPE DIFFUSION-REACTION MODEL FOR POROUS CATALYST PELLETS... 

Techniques for a Fickian-type Diffusion-Reaction Model for Porous Catalyst Pellets 436... 

Within the FFenry s law region dlnp/dlnq= 1.0 and D = Dq. Thus one can expect that at sufficiently low loadings transport diffusion can be accurately represented by the simple Fickian model with a constant diffusivity, but at higher loadings the diffusivity may be expected to be concentration dependent. For a system obeying the Langmuir isotherm (Eq. 5), Eq. 36 implies ... 

A completely different situation arises in one-dimensional channels when the molecules are too large to pass each other. The transport can no longer be characterized by a Fickian model. Such behavior is discussed in this volume in the chapter Single-File Diffusion in Zeolites by Jorg Karger. 

The terms of interest in Eq. (5) are the time-averaged cross products, such as u c. These terms represent the volume flux per unit area due to fluid turbulence and the resulting transfer of material. It is in the treatment of this term that the two methodologies (Fickian model and diffusions by continuous movements) already mentioned can be brought to bear. In the Fickian approach, an exact analogy is made to molecular diffusion, as in Eq. (1). Specifically, this becomes here... 

The assumptions made in Eq. (10), as shown in Eqs. (7)-(9), are rather sweeping. Sayre and Chang (82) noted that experimental evidence from a number of sources implies that lateral diffusion is much better represented by the Fickian model than is longitudinal dispersion [as represented by the onedimensional form of Eq. (10), with transport only in the x direction]. Most workers have found that despite its theoretical shortcomings, the Fickian model provides a reasonable starting point and an approximate kinematic description of diffusion in open channels. 

The model representing diffusion/reaction involved solution of the transport equations for each single pore simultaneously to give concentration profile in the pore network. The calculations related to capillary condensation were performed in the same way as for the Fickian model, described in Section lll.C. 

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. 

Section 15.6 describes the deficiencies in the Fickian model and points out why an alternative model (the fourth) is needed for some situations. The alternative Maxwell-Stefan model of mass transfer and diffusivity is explored in Section 15.7. The Maxwell-Stefan model has advantages for nonideal systems and multicomponent mass transfer but is more difficult to couple to the mass balances when designing separators. The fifth model of mass transfer, the irreversible thermodynamics model fde Groot and Mazur. 1984 Ghorayeb and Firoozabadi. 2QQQ Haase. 1990T is useful in regions where phases are unstable and can split into two phases, but it is beyond the scope of this introductory treatment. The... 

Since both convection and diffusion occur, we need to separate the terms in some fashion. The usual assumption in the Fickian model for binary systems is that the effects are additive,... 

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. 

The conclusion is that use of the Fickian model for binary diffusivities is reasonable, although it maybe awkward for nonideal liquid systems. For systems with more than two conponents, the Fickian model is not the best choice. 

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... 

The proponents of Maxwell-Stefan theory claim it is a better model than Fickian theory. What makes one model better than another model First, it is nice to have a model tied to the basic physics or chemistry. A sinple explanation based on the physics or chemistry should explain the basic behavior. This is illustrated in Section 15.7.1. Second, the model should not conflict with well-accepted laws such as the first or second law of thermodynamics. The Maxovell-Stefan model incorporates thermodynamics for the analysis of nonideal systems. Third, we want a model that can explain and/or predict data and that can be exctrapolated. Except for ideal gas behavior, diffusion models invariably have to use measured constants (diffusivity values). A good model will minimize the number of constants required and minimize the variation of these constants. If the constants vary (say with concentration), the variation should be monotonic and preferably be close to linear. The constants for multiconponent systems should be predictable from binary pairs, and no inpossible or inprobable values (e.g., negative values) of the constants should be required to predict the data. Based on these criteria, the Maxovell-Stefan theory is a better theory than the Fickian model. 

Fickian model. The Fickian model is a widely accepted model for diffusion, which means no one will laugh if you use it. It works well for ideal and close to ideal binary systems and can be used for nonideal binary systems if data are available. Most diffusivity data and correlations for mass-transfer coefficients are based on the Fickian model. This model is very difficult to use for nonideal ternary systems and can require negative diffusion coefficients to predict data. This model works well for dilute binary systems. 

Maxwell-Stefan model. The Maxwell-Stefan model is generally agreed to be a better model than the Fickian model for nonideal binary and all ternary systems. However, it is not as widely understood by chemical engineers, data collected in terms of Fickian diffusivities need to be converted to Maxwell-Stefan values, and the model can be more difficult to use. Use this model, coupled with a mass-transfer model, when the Fickian model fails or requires an excessive amount of data. 

Describe how the Maxwell-Stefan model differs from the Fickian model and use the Maxwell-Stefan model for ideal and nonideal binary and ideal ternary diffusion problems... 

Whereas the value of n equals 0.5, the mechanism of diffusion described by Case I of Fickian model, when n values range between 0.5 and 1 that exhibits anomalous transport model and when n value equals 1, Case II of non-Fickian model is used to determine. Fickian model describing the rate of diffusion of penetrant molecule is much less than the relaxation rate of polymer chains while Case II of non-Fickian diffusion representing rate of diffusion is rapid than relaxation process. For the anomalous transport model, both solvent diffusion rate and polymer relaxation rate are comparable. Figure 27.2 shows various types of non-Fickian model anomalous transport and Case II of non-Fickian are included in this group. ... 

These methods are interested in studying the distinction between the pure dualmode sorption/transport model curve and the actual sorption and permeation experiment curve that seems to contain the various unsolved appearances for diffusion and sorption of gases in glassy polymer. It becomes obvious that the deviations from the Fickian model in an experimental transport or sorption/desorp-tion curve for a gas in a glassy polymer are not necessarily consistent with the onset of concomitant diffusion and relaxation [11], but are just owing to the dual-mode model. That is to say, only the combinations of parameters of the dual mode model make the curve either fit with or deviation from the Fickian model curve. 

Numerical solutions were applied to the dual-mode sorption and transport model for gas permeation, sorption, and desorption rate curves allowing for mobility of the Langmuir component. These rate curves were almost consistent with the curves predicted by Fickian-type diffusion equation, except the desorption curve in which a slight sigmoidal deviation from the Fickian model line was apparent. The sorption... 

First, Fig. 15.2 shows the diffusion profile that does match Fickian diffusion kinetic model. What expected is that the slope should become smaller and smaller with after initial linearity. Fig. 15.2 is just opposite. Fig. 15.3 demonstrates the diffusion index n is far higher than the value n = 0.5. All these facts indicate that polymer degradation is companying with the in vitro water diffusion progressing, which... 

Note that the weight gain data follow curve A in Fig. 4.2 rather than the Fickian model. At least for the case of cr = 0 this suggests the presence of a two phase diffusion process. 

Yet another complication arises if the diffusion process is triggered by a mass source. A puff of smoke emanating from a chimney, for example, constitutes such a source. If it lasts for only an instant, we speak of an instantaneous source. If the emanation persists, we speak of a continuous source. The existence of such sources must be incorporated into the Fickian model and leads to what we term solutions of source problems. Because of their importance, particularly in an environmental context, we address this topic separately in some detail. 

The assumptions in his model do not allow for the complexity of the moisture sorption isotherm and the sorption kinetics of fibers. Scientists presented two mathematical models to simulate the interaction between moisture sorption by fiber and moisture flux through the air spaces of a fabric. In the first model, they considered diffusion within the fiber to be so rapid that the fiber moisture content is always in equilibrium with the adjacent air. In the second model, they assumed that the sorption kinetics of the fiber follows Fickian diffusion. Their model neglected the effect of heat of sorption behavior of the fiber. Scientists developed a new sorption equation that takes into account the two-stage sorption kinetics of wool fibers, and incorporated this with more realistic boundary conditions to simulate the sorption behavior of wool fabrics. They assumed that water vapor uptake rate of fiber consists of a two components associated with the two stages of sorption identified by researchers. 

For dilute gases, the generalized multicomponent Fickian diffusion coefficients are strongly composition dependent. It follows that these diffusion coefficients do not correspond to the approximately concentration independent binary difffisivities, Dsr, which are available from binary diffusion experiments or kinetic theory since Dgr Dsr. In response to this Fickian model limitation it has been proposed to transform the Fickian diffusion flux model, in which the mass-flux vector, jj, is expressed in terms of the driving force, dj, into the corresponding Maxwell-Stefan form [95, 97, 142, 143] where d is given as a linear function of jj. The key idea is to rewrite the Fickian diffusion flux in terms of an alternative set of difffisivities (i.e., preferably the known binary difffisivities) which are less concentration dependent than the Fickian difffisivities. 

In some cases the Fickian model does not accurately represent moisture uptake in adhesives. This is illustrated in O Fig. 31.12a, which shows the uptake plot for an epoxide immersed in water at SO C. The experimental data appears to indicate Fickian diffusion however, the best fit of the Fickian diffusion equation to the data indicates equilibrium is reached more slowly than predicted by Fickian diffusion. This type of behavior is sometimes termed pseudo-Fickian behavior. In general, anomalous behavior is seen at high temperatures and humidity. 

Many models have been suggested to describe anomalous (non-Fickian) uptake and a number of the more relevant to structural adhesives will be discussed. Diffusion-relaxation models are concerned with moisture transport when both Case I and Case II mechanisms are present. Berens and Hopfenberg (1978) assumed that the net penetrant uptake could be empirically separated into two parts, a Fickian diffusion-controlled uptake and a polymer relaxation-controlled uptake. The equation for mass uptake using Berens and Hopfenbergs model is shown below. 

Loh et al. (2005) proposed a dual-uptake model based on the summation of two Fickian diffusion models. Physically this can be interpreted as two different uptakes processes operating in parallel, both of which are adequately described by Fickian diffusion. This model was... 

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