Fick’s diffusivity

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... 

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

A basis is Fick s diffusion law adapted to dispersion which states that the rate of mass transfer by dispersion is proportional to the concentration gradient. A material balance is made on a hollow cylindrical element of radii r and r+dr and length dx. This element is sketched. 

Fick s diffusion laws, depending on the shape of the liquid as determined by the particle surfaces (1,31-33). However, in such systems the rate of depletion of a supersaturated solution by diffusion controlled growth would be very fast, and if the rate is actually slow (measurable) the rate control is likely to be a surface process. 

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] ... 

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 ... 

Dispersion. The movement of aggregates of molecules under the influence of a gradient such as concentration, temperature, density, etc. The effect is represented by Fick s diffusion equation with a dispersion coefficient substituted for molecular diffusivity. Thus, Rate of Transfer = -DeOC/3z). 

Here Dk is Fick s diffusion coefficient, kB Boltzmann s constant, T the temperature, p the total mass density, vk the partial specific volume, g the acceleration of gravity. By working out the imperfection theory for... 

As the electrolysis proceeds, there is a progressive depletion of the Ox species at the interface of the test electrode (cathode). The depletion extends farther and farther away into the solution as the electrolysis proceeds. Thus, during this nonsteady-state electrolysis, the concentration of the reactant Ox is a function of the distance x from the electrode (cathode) and the time /, [Ox] =f(x,t). Concurrently, concentration of the reaction product Red increases with time. For simplicity, the concentrations will be used instead of activities. Weber (1) and Sand (2) solved the differential equation expressing Fick s diffusion law (see Chapter 18) and obtained a function expressing the variation of the concentration of reactant Ox and product Red on switching on a constant current. Figure 6.10 shows this variation for the reactant. 

First we treat diffusion processes within the homogeneous phase. The presence of a temperature gradient in binary fluid mixtures and polymer blends requires an extension of Fick s diffusion laws, since the mass is not only driven by a concentration but also by a temperature gradient [76] ... 

Cross-flow FFF (F1FFF) utilizes a second fluid flow to transport sample components across the channel thickness to the accumulation wall, and the position of individual species in the laminar carrier profile corresponds to their ordinary (Fick s) diffusion coefficient. As the particle size increases, the diffusion coefficient (decreases until it becomes a relatively insignificant transport process. For micron-size particles, the extent of protrusion into the channel becomes the decisive factor in determining the order of elution. 

Case B If Fick s diffusion theory can be applied to the entry of radicals into micelles and particles, we have ... 

Fick s diffusion law is used to describe dispersion. In a tubular reactor, either empty or packed, the depletion of the reactant and non-uniform flow velocity profiles result in concentration gradients, and thus dispersion in both axial and radial directions. Fick s law for molecular diffusion in the x-direction is defined by... 

Using much smaller semiconductor particles (d dsc), the photogenerated electrons and holes can be easily transferred to the surface and react with the electron and hole acceptors, provided that the energetic requirements are fulfilled. The average transit time % within a particle of diameter d can be obtained by solving Fick s diffusion law as... 

The DGM by Mason and Malinauskas (1983) is the frequently used alternative to effective Fick s diffusion for the calculation of the multicomponent diffusion and convection in the porous media. A number of special features of multicomponent diffusion and convection in the pore space have been outlined by Krishna (1993). 

Fick s diffusion cannot be applied to the description of the evolution of multiphase systems because it acts to smooth concentration differences and drives the system toward the uniform concentration profiles of species. Instead of Fick s diffusion, where the concentration gradient acts as a driving force, the intensity of the diffusion flux Na of the species A can be considered to be... 

Fio. 16. Effective diffusivity of the examined porous sample G1 calculated for (i) Fick s diffusion both in macro- and nano-pores, and (ii) Fick s diffusion in macro-pores only. The experimental value of ij/ = Dcii/D = 0.199 was determined in Grahams diffusion cell. Only grains larger than 10 pm were used in the reconstruction of macro-porous media (from Salejova et al., 2004). 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

The internal mass transfer is modeled with Fick s diffusion inside the (macro) pores (Eq. 6.82) as well as surface or micropore diffusion in the solid phase (Eq. 6.83). Note that Eqs. 6.82 and 6.83 no longer include the number of particles (Eq. 6.20) and therefore represent the balance in one particle. 

The governing equation for ceU-impedance-controlled lithium transport is Fick s diffusion equation. The initial condition (I.C.) and the boundary conditions (B.C.) are given as... 

For translational long-range jump diffusion of a lattice gas the stochastic theory (random walk, Markov process and master equation) [30] eventually yields the result that Gg(r,t) can be identified with the solution (for a point-like source) of the macroscopic diffusion equation, which is identical to Pick s second law of diffusion but with the tracer (self diffusion) coefficient D instead of the chemical or Fick s diffusion coefficient. 

However, these diffusion coefficients are applicable only for diluted binary solutions. In real natural water the processes of molecular diffusion are affected by the temperature, pressure, contents and charge of the other components. This effect is defined by phenomenological reciprocity coefficients in Onsager s linear law (equation 3.9). B.P. Boudreau (2004) believes that in hydrochemistry exist two approaches to the evaluation of such effect from top, i.e., from the position of Onsager s linear law, and bottom, i.e., from the position of Fick s diffusion law. We will limit ourselves to a simpler solution of the problem based on the laws of diffusion and thermodynamics. 

The geometric factor a is introduced as a modification to Fick s diffusion law in combination with Faraday s law, if the hemispherical electrode does not have a convex (a = ) but a concave geometry (a = 3) (Eq. 5). This factor has been estimated [6, 29] and later confirmed by computer simulation [30, 31]. 

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