Fick model

A comparison of DGM and the extended Fick model for the transition region has been made by Veldsink et al. [46] and is illustrated by many transport data and applied to describe transport in a macro-porous membrane reactor. Their main conclusion is that for ternary mixtures the use of the DGM model is necessary and predicts the transport of a gas mixture within a few percent (5%). For binary gases usually the extended Fick model is sufficient, but with an overall pressure over the membrane the accuracy is less than that obtained by use of the DGM. A further discussion will be given in Section 9.7. 

In 1855 and 1856, physician Adolph Fick built on the previous work of Thomas Graham to develop a theory for the transfer of dilute solutes in physiological fluids. Since Fick was familiar with Fourier s analysis of thermal conduction and since his experimental apparatus was analogous to Fourier s apparatus, Fick modeled his theory on the analogous thermal conduction theory fCussler. 20091. With constant density and heat capacity, Fourier had shown that for one-dimensional heat conduction with no convection and no thermal radiation,... 

Modified Fick model, also known as mixed diffusion approach is the simplest of all approaches. It is easy to program and is less computationally expensive. In this method an equivalent Fickian diffusion coefficient is derived by considering the mixture diffusion coefficient Dktn acting in series with Knudsen diffusion coefficient Dkn as follows... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

Dispersion model is based on Fick s diffusion law with an empirical dispersion coefficient substituted for the diffusion coefficient. The material balance is... 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

The diffusional transport model for systems in which sorbed molecules can be divided in two populations, one formed by completely immobilized molecules and the other by molecules free to diffuse, has been developed by Vieth and Sladek 33) in a modified form of the Fick s second law. However, if linear isotherms are experimentally found, as in the case of the DGEBA-TETA system in Fig. 4, the diffusion of the penetrant may be described by the classical diffusion law with constant value of the effective diffusion coefficient,... 

Perhaps the simplest Fick s law permeation model consists of two aqueous compartments, separated by a very thin, pore-free, oily membrane, where the unstirred water layer may be disregarded and the solute is assumed to be negligibly retained in the membrane. At the start (t = 0 s), the sample of concentration CD 0), in mol/cm3 units, is placed into the donor compartment, containing a volume (Vo, in cm3 units) of a buffer solution. The membrane (area A, in cm2 units) separates the donor compartment from the acceptor compartment. The acceptor compartment also contains a volume of buffer (VA, in cm3 units). After a permeation time, t (in seconds), the experiment is stopped. The concentrations in the acceptor and donor compartments, CA(t) and C (t), respectively, are determined. 

The book is organized into eight chapters. Chapter 1 describes the physicochemical needs of pharmaceutical research and development. Chapter 2 defines the flux model, based on Fick s laws of diffusion, in terms of solubility, permeability, and charge state (pH), and lays the foundation for the rest of the book. Chapter 3 covers the topic of ionization constants—how to measure pKa values accurately and quickly, and which methods to use. Bjerrum analysis is revealed as the secret weapon behind the most effective approaches. Chapter 4 discusses experimental... 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

Payer80 states that the UNSAT-H model was developed to assess the water dynamics of arid sites and, in particular, estimate recharge fluxes for scenarios pertinent to waste disposal facilities. It addresses soil-water infiltration, redistribution, evaporation, plant transpiration, deep drainage, and soil heat flow as one-dimensional processes. The UNSAT-H model simulates water flow using the Richards equation, water vapor diffusion using Fick s law, and sensible heat flow using the Fourier equation. 

UNSAT-H uses the Richards equation, Fick s law, and the Fourier equation to estimate the flow of soil-water, vapor, and heat. This may be the strongest part of the model because these are the most rigorous, currently known, theoretical methods for estimating these parameters. 

This is Fick s second law of diffusion, the equation that forms the basis for most mathematical models of diffusion processes. The simple form of the equation shown above is applicable only to diffusion in one dimension (x) in systems of rectangular geometry. The mathematical form of the equation becomes more complex when diffusion is allowed to occur in more than one dimension or when the relationship is expressed in cylindrical or spherical coordinate geometries. Since the simple form shown above is itself a second-order partial differential equation, the threat of added complexity is an unpleasant proposition at best. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

The diffusion model assumes that (1) Fick s law is valid as modified by reactions (2) reactant concentrations are interpreted as probability densities (3) specific rates correspond to otherwise homogeneous reactions—Monchick et al. (1957) show that this implies neglect of interparticle correlation, which is... 

It has been demonstrated that transport rate and selectivity may be modelled using the basic concepts of Fick s law of diffusion (Behr, Kirch Lehn, 1985). Analyses of this type allow a greater appreciation of the interplay of factors influencing such membrane transport phenomena, and enable a clear theoretical differentiation between diffusion-limited and complexation rate-limited cases. 

A formal derivation of diffusion in a restricted, high diffusivity path which uses no atomic model of the grain boundary is that due to Fisher, who made a flux balance in unit width of a grain boundary having a thickness of S. There is flux accumulation in the element according to Fick s second law given by... 

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

We have used Fick s law of diffusion with separate molecular diffusivities for each species. However, most PDF models for molecular mixing do not include differential-diffusion effects. 

Dissolution time, tdi (for tablet) Tablet mass, m Diffusivity, D Grain particle size, dp Tablet size, 77 Porosity, e Order-of-magnitude model derived from Fick s and Darcy s laws [6] 2m2 td x2d2pH4Ds(l-s)2... 

We will now describe the application of the two principal methods for considering mass transport, namely mass-transfer models and diffusion models, to PET polycondensation. Mass-transfer models group the mass-transfer resistances into one mass-transfer coefficient ktj, with a linear concentration term being added to the material balance of the volatile species. Diffusion models employ Fick s concept for molecular diffusion, i.e. J = — D,v ()c,/rdx, with J being the molar flux and D, j being the mutual diffusion coefficient. In this case, the second derivative of the concentration to x, DiFETd2Ci/dx2, is added to the material balance of the volatile species. 

The second approach employs a detailed reaction model as well as the diffusion of EG in solid PET [98, 121-123], Commonly, a Fick diffusion concept is used, equivalent to the description of diffusion in the melt-phase polycondensation. Constant diffusion coefficients lying in the order of Deg, pet (220 °C) = 2-4 x 10 10 m2/s are used, as well as temperature-dependent diffusion coefficients, with an activation energy for the diffusion of approximately 124kJ/mol. 

When the pollutant concentration difference between the source and collection reservoir becomes smaller (i. e., when the concentration of pollutants in the collection reservoir approaches that of the source reservoir), the flux rate of pollutants decreases, and a near steady state flux (Js) is obtained (Fig. 3c). At this time, the diffusion parameter (D) can be calculated using Fick s model as follows ... 

Sklarew, R. C., A. J. Fabrick, and J. E. Prager. Mathematical modeling of photochemical smog using the FiCK method. J. Air Pollut. Control Assoc. 22 865-869, 1972. 

The approach is based on the universal transformation of solutions of rate equations for constant concentration conditions to those of variable concentration conditions as published earlier [93,94]. The isothermic case of Fick s diffusion in a fluid mixture consisting of N components is considered for any geometry of the sorbing medium, e.g. NS crystals, at variable surface concentration. The model is described by the following equations and initial conditions [94] ... 

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