Models Fickian

Breakthrough Behavior for Axial Dispersion Breakthrough behavior for adsorption with axial dispersion in a deep bed is not adequately described by the constant pattern profile for this mechanism. Equation (16-128), the partial differential equation of the second order Fickian model, requires two boundary conditions for its solution. The constant pattern pertains to a bed of infinite depth—in obtaining the solution we apply the downstream boundary condition cf — 0 as NPeC, —> < >. Breakthrough behavior presumes the existence of a bed outlet, and a boundary condition must be applied there. 

Consequently, we will follow the example of Mills et al. (29) who recently presented the first measurements of local solvent concentration using the Rutherford back-scattering technique. They analyzed the case of 1,1,1-trichloroethane (TCE) diffusing into PMMA films in terms of a simpler model developed by Peterlin 130-311, in which the propagating solvent front is preceded by a Fickian precursor. The Peterlin model describes the front end of the steady state SCP as ... 

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

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... 

Lenormand R, Touboul E, Zarcone C (1988), Numerical models and experiments on immiscible displacement in porous media. J ITuid Mech 189 165-187 Levy M, Berkowitz B (2003) Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. J Contam Hydrol 64 203-226... 

THE BERENS-HQPFENBERG MODEL. The Berens and Hopfenberg model considers the sorption process in glassy polymers as a linear superposition of independent contributions of a rapid Fickian diffusion into pre-existing holes or vacancies (adsorption) and a slower relaxation of the polymeric network (swelling).(lS) The total amount of sorption per unit weight of polymer may be expressed as... 

Electrode transport is often represented by Fickian diffusion models. Such models, however, are applicable to bi-molecular transport, whereas for fuel cells it is often that more than two species exist, especially when considering the use of practical hydrocarbon fuels. For fuels such as natural gas, CH4, H2, H2O, CO and CO2 are all present, and the diffusion of one is coupled to the diffusion of all others. To properly analyze the transport across the electrode in such cases, the Stephan-... 

We determined the phase behavior of the HDPE/styrene/CC>2 using the method described by Berens et al. (1992), modeling mass uptake data as Fickian dilfusion into a planar sheet (Crank, 1975). Ethylbenzene was used as the penetrant to model styrene. 

Harland et al.12 developed a model for drug release based on mass balances of the drug and the solvent at the swelling front and the erosion front. The release profile was found to be a combination of Fickian and zero order, as shown by Eq. (5.16) ... 

The model shows that the non-isothermal uptake curve for an adsorbent mass which has low effective thermal conductivity (k ) is identical in form to that of the isothermal Fickian diffusion model for mass transport. 

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

The form of equation (21) is interesting. It shows that the uptake curve for a system controlled by heat transfer within the adsorbent mass has an equivalent mathematical form to that of the isothermal uptake by the Fickian diffusion model for mass transfer [26]. The isothermal model hag mass diffusivity (D/R ) instead of thermal diffusivity (a/R ) in the exponential terms of equation (21). According to equation (21), uptake will be proportional to at the early stages of the process which is usually accepted as evidence of intraparticle diffusion [27]. This study shows that such behavior may also be caused by heat transfer resistance inside the adsorbent mass. Equation (22) shows that the surface temperature of the adsorbent particle will remain at T at all t and the maximum temperature rise of the adsorbent is T at the center of the particle at t = 0. The magnitude of T depends on (n -n ), q, c and (3, and can be very small in a differential test. 

These equations are called the Navier-Stokes equations, and when supplemented by the state equation for fluid pressure and species transport equations, they form the basis for any computational model describing the flows in fires. For simplicity, several approximations are inherent (see Equation 20.3) (no Soret/Dufour effects, no viscous dissipation, Fickian diffusion, equal diffusion coefficients of all species, unit Lewis number). 

When Fickian diffusion in normal Euclidean space is justified, further verification can be obtained from the analysis of 60% of the release data using the power law in accord with the values of the exponent quoted in Table 4.1. Special attention is given below for the values of b in the range 0.75-1.0, which indicate a combined release mechanism. Simulated pseudodata were used to substantiate this argument assuming that the release obeys exclusively Fickian diffusion up to time t = 90 (arbitrary units), while for I, > 90 a Case II transport starts to operate too this scenario can be modeled using... 

Models that, either naturally or through approximation, can be discretized are suitable for study using Monte Carlo simulations. As an example, we give a brief outline below of the simulations of drug release from cylinders assuming Fickian diffusion of drug and excluded volume interactions. This means that each molecule occupies a volume V where no other molecule can be at the same time. 

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