Fick s law for diffusion

By Fick s law for diffusion for the component A with its diffusion coefficient Da we have... 

Comparing this equation with Fick s law for diffusion Jt = D dcjdx). the diffusion coefficient for NaCl is obtained by... 

A useful description of mixing in bubble columns is provided by the dispersion model. The global mixing effects are generally characterized by the dispersion coefficients El and Eq of the two phases which are defined in analogy to Fick s law for diffusive transport. Dispersion in bubble columns has been the subject of many investigations which have recently been reviewed by Shah et al. (45). Particularly, plenty of data are available for liquid-phase dispersion. 

Fick s law for diffusion to a spherical electrode of radius tq is written... 

The global mixing effects in tower bioreactors can conveniently be described by the dispersion coefficients of both phases which are defined in analogy to Fick s law for diffusive transport. 

Spatially reduced Ohm s law in electrodynamics, Fourier s equation of heat transfer, Fick s law for diffusion, and Newton s law in hydrodynamics are among the subjects treated and their comparison is enlightening. The various transfers tackled in this chapter are stationary diffusion,... 

When a relatively slow catalytic reaction takes place in a stirred solution, the reactants are suppHed to the catalyst from the immediately neighboring solution so readily that virtually no concentration gradients exist. The intrinsic chemical kinetics determines the rate of the reaction. However, when the intrinsic rate of the reaction is very high and/or the transport of the reactant slow, as in a viscous polymer solution, the concentration gradients become significant, and the transport of reactants to the catalyst cannot keep the catalyst suppHed sufficientiy for the rate of the reaction to be that corresponding to the intrinsic chemical kinetics. Assume that the transport of the reactant in solution is described by Fick s law of diffusion with a diffusion coefficient D, and the intrinsic chemical kinetics is of the foUowing form... 

As shown in Fig. 1.12, diffusional flow contributions in engineering situations are usually expressed by Fick s Law for molecular diffusion... 

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... 

Several important assumptions have been implicitly incorporated in Eqs. (15) and (16). First, these equations describe the release of a drug from a carrier of a thin planar geometry, equivalent equations for release from thick slabs, cylinders, and spheres have been derived (Crank and Park, 1968). It should also be emphasized that in the above written form of Fick s law, the diffusion coefficient is assumed to be independent of concentration. This assumption, while not conceptually correct, has been... 

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. 

The two flux equations of importance to subsurface transport are Darcy s law for the advective flow of water and other liquids and Fick s law for the diffusive flow of molecules and gases. These laws are independently discussed below. 

For convenience, let the flux of A within the ash layer be expressed by Fick s law for equimolar counterdiffusion, though other forms of this diffusion equation will give the same result. Then, noting that both (2a dCJdr are positive, we have... 

The potentially greater toxicity of peroxynitrite can be readily visualized by comparing the mean diffusion distances that various nitrogen and oxygen-centered species may traverse in one lifetime. The definition of lifetime (t) is the time required for 67% of the initial concentration to decompose, and is readily calculated as the reciprocal of the pseudo-first-order rate constant for the disappearance of the species in question. Distances were calculated from the following equation, which is readily derived from the Fick s laws of diffusion (Nobel, 1983 Pryor, 1992). 

At the simplest level, as Griskey [1] notes, Fick s law of diffusion for mass ttans-fer and Fourier s law of heat conduction characterize mass and heat transfer, respectively, as vectors i.e., they have magnitude and direction in the three coordinates X, y, and z. Momentum or flow, however, is a tensor, which is defined by nine components rather than three. Hence, its more complex characterization at the simplest level, in accordance with Newton s law,... 

SLIDER is a Fortran IV computer program for investigating the diffusion of a single fission-product isotope in a multilayered spherical fuel particle. This code enables one to compute, on the basis of Fick s law of diffusion, the transient and steady-state fission product concentrations and releases in multilayered spherical geometry. 

Transport Model. A general release equation is developed by considering diffusion in both the liquid and gaseous phases. Figure 1 is a sketch of a vessel containing liquid sodium, blanketed by a flowing inert gas and maintained at constant temperature. The assumed cesium concentration profiles for both phases are shown in the sketch. Fick s law of diffusion may be applied to the liquid phase ... 

Modeling diffusive transport requires appropriate constitutive relationships, such as Fourier s law for heat conduction or Fick s law for species diffusion. It is important to... 

For computing the diffusion parameter, D /ro, Fick s diffusion equation was assumed to be applicable to the system, with D independent of concentration of the diffusing species. Solving Fick s law for a spherical particle, where the external gas pressure is constant gives ... 

In this equation dW/dt is the rate of sorption, k. is the experimental rate constant for sorption, W, is the equilibrium sorption, W is the amount of sorption at time t = t, and h is the experimental rate constant for desorption. Thus, the sorption rate was found to be proportional to the square of the concentration of unoccupied sites, and the desorption rate was proportional to the square of the concentration of occupied sites. These rate equations are not general solutions to Fick s law of diffusion. The experimental rate constants for sorption were found to be non-linearly dependent on methanol pressure and seemed to correlate with the amount of surface sorbed methanol in different ways for coals of various rank. 

The resistance in each phase is made up of two parts the diffusional resistance in the laminar film and the resistance in the bulk fluid. All current theories on mass transfer, i. e. film, penetration, and surface renewal assume that the resistance in the bulk fluid is negligible and the major resistance occurs in the laminar films on either side of the interface (Figure 3-2). Fick s law of diffusion forms the basis for these theories proposed to describe mass transfer through this laminar film to the phase boundary. 

The application of Fick s law for the cases of an equimolecular counter-diffusion and a steady-state unidirectional diffusion of a component through an inert-gas film in an ideal gas system gives the two equations for the rate of mass-transfer ... 

To estimate the rate constant for a reaction that is controlled strictly by the frequency of collisions of particles, we must ask how many times per second one of a number n of particles will be hit by another of the particles as a result of Brownian movement. The problem was analyzed in 1917 by Smoluchowski,30/31 who considered the rate at which a particle B diffuses toward a second particle A and disappears when the two codide. Using Fick s law of diffusion, he concluded that the number of encounters per milliliter per second was given by Eq. 9-26. 

Table 2.1 presents corresponding well-known empirical force-flux laws that apply under certain conditions. These are Fourier s law of heat flow, a modified version of Fick s law for mass diffusion at constant temperature, and Ohm s law for the electric current density at constant temperature.5 The mobility, Mj, is defined as the velocity of component i induced by a unit force. 

This is a form of Fick s law for a chemically inhomogeneous material where the intrinsic diffusivity, designated by >1, measures the flux in the local C-frame. A similar procedure for component 2 yields an analogous Fick s-law expression, J2 = -D2dc2/dx. 

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