Fickian transport

The species-B balance equation includes advective transport, Fickian diffusion, and depletion by chemical reaction. The binary diffusion coefficient D represents downstream diffusion of reactant species-B relative to upstream diffusion of product species-C. The expression for Ys, the surface mass fraction of B (gas side), is obtained from a species balance at the surface on B which includes advective transport of pure B to the interface on the condensed phase side and both advective and diffusive transport of B away from the surface on the gas side. The downstream condition K(oo)=0 represents the assumption of complete conversion... 

Sorption Rates in Batch Systems. Direct measurement of the uptake rate by gravimetric, volumetric, or pie2ometric methods is widely used as a means of measuring intraparticle diffusivities. Diffusive transport within a particle may be represented by the Fickian diffusion equation, which, in spherical coordinates, takes the form... 

Transport in the polymeric system is assumed to be by Fickian diffusion, although the diffusivity of the various species depends on the extent of hydrolysis of the polymeric linkages. 

The rate and type of release can be analyzed by the expression Mt/Moo=ktn (76). In the case of pure Fickian diffusion n = 0.5, whereas n > 0.5 indicates anomalous transport, i.e., in addition to diffusion another process (or processes) also occurs. If n = 1 (zero order release), transport is controlled by polymer relaxation ("Case II transport") (76). The ln(Mt/Mco) versus In t plots, shown in Figure 4, give n = 0.47 and 0.67 for samples A-9.5-49 and A-4-56, respectively. Evidently theophylline release is controlled by Fickian diffusion in the former network whereas the release is... 

For a classical diffusion process, Fickian is often the term used to describe the kinetics of transport. In polymer-penetrant systems where the diffusion is concentration-dependent, the term Fickian warrants clarification. The result of a sorption experiment is usually presented on a normalized time scale, i.e., by plotting M,/M versus tll2/L. This is called the reduced sorption curve. The features of the Fickian sorption process, based on Crank s extensive mathematical analysis of Eq. (3) with various functional dependencies of D(c0, are discussed in detail by Crank [5], The major characteristics are... 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

This solution is valid for the initially linear portion of the sorption (or desorption) curve when MtIM is plotted against the square root of time. These equations also demonstrate that for Fickian processes the sorption time scales with the square of the dimension. Thus, to confirm Fickian diffusion rigorously, a plot of MJM vs. Vt/T should be made for samples of different thicknesses a single master curve should be obtained. If the data for samples of different thicknesses do not overlap despite transport exponents of 0.5, the transport is designated pseudo-Fickian. ... 

SH Gehrke, D Biren, JJ Hopkins. Evidence for Fickian water transport in initially glassy poly(2-hydroxyethyl methacrylate). J Biomater Sci Polym Ed 6 375-390,... 

NM Franson, NA Peppas. Influence of copolymer composition on non-Fickian water transport through glassy copolymers. J Appl Polym Sci 28 1299-1310, 1983. 

Hydrodynamic dispersion is in many cases taken to be a Fickian process, one whose transport law takes the form of Fick s law of molecular diffusion. If flow is along x only, so that vx = v and vy = 0, the dispersive fluxes (mol cm-2 s-1) along x and y for a component i are given by,... 

The major pathway of drug transport across buccal mucosa seems to follow simple Fickian diffusion [17]. Passive diffusion occurs in accordance with the pH-partition theory. Considerable evidence also exists in the literature regarding the presence of carrier-mediated transport in the buccal mucosa [18,19]. Examination of Eq. (1) for drug flux,... 

Therefore, Eq. 10.5 is limited in its applicability, as are variations of this equation such as the mobile-immobile one (see Sect. 10.2). We discuss non-Fickian transport in detail in Sect. 10.3. 

Fig. 10.3 Photographs of a homogenous, saturated sand pack with seven dye tracer point injections being transported, under a constant flow of 53 mL/min, from left to right times at (a) t=20, (b) t= 105, (c) t= 172, (d) t=255 min after injection. Internal dimensions of the flow cell are 86 cm (length), 45cm (height), and 10cm (width). Reprinted from Levy M, Berkowitz B (2003) Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. J Contam Hydrol 64 203-226. Copyright 2003 with permission of Elsevier...

Regardless of the transport equation considered, the major effect of sorption on contaminant breakthrough curves is to delay the entire curve on the time axis, relative to a passive (nonsorbing) contaminant. Because of the longer residence time in the porous medium, advective-diffusive-dispersive interactions also are affected, so that longer (non-Fickian) tailing in the breakthrough curves is often observed. 

The analysis was limited in part by the scarcity of measurements, and clear discrepancies between measured and calculated values may be observed. As discussed in Chapter 10, tailing effects often are due to non-Fickian transport behavior, which was not accounted for in this model. Interestingly, the field-scale retardation coefficient values of the reactive contaminants were smaller by an order of magnitude than their laboratory values, obtained in an accompanying experiment. 

Berkowitz B, Emmanuel S, Scher H (2008) Non-Fickian transport and multiple rate mass transfer in porous media Water Resour Res 44, D01 10.1029/2007WR005906 Bijeljic B, Blunt MJ (2006) Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 42, W01202, D01 10.1029/2005WR004578 Blunt MJ (2000) An empirical model for three-phase relative permeability. SPE Journal 5 435-445... 

Fitting the swelling curves of Fig. 7a to the form Q(t) — kt yields values of a greater than or equal to 0.8. Thus the swelling must be considered anomalous, or non-Fickian. In the absence of ionic interactions, this would not be expected since BMA/DMA 70/30 is initially not far below its Tg at 25 °C. Indeed, swelling measurements of this copolymer in hexane show kinetics that are nearly Fickian (a 0.55), as shown in Fig. 7b. Therefore, the anomalous swelling observed in Fig. 7a must be attributed to ion transport and binding rates in the gel. We will return to this point later. 

The concentration of a compound at a given location depends on (1) the rate of transformation of the compound (positive for production and negative for consumption), and (2) the rate of transport to or from the location. In Part III we discussed different kinds of transformation processes. Internal transport rates were introduced in Chapter 18. Remember that we have divided them into just two categories, the directed transport called advection and the random transport called diffusion or dispersion. The second Fickian law (Eq. 18-14) describes the local rate of change due to diffusion. The corresponding law for advective processes will be introduced in Chapter 22. In Chapter 19 we discussed transport processes across boundaries. 

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