Fick’s diffusion equation

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). 

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... 

In previous studies, it was presumed that a GI POP fabricated by this process would have a refractive index distribution with a tail at the core/cladding boundary, not as steep as that prepared by the conventional batch process [50]. Therefore, the bandwidth of the GI POF obtained by dopant diffusion coextrusion would be inferior to that obtained by the batch process. However, contrary to this expectation, a GI POF without any tail prepared by the process was reported in 2007 [57]. If low-molecular-weight molecules diffuse in a radial direction, their diffusion can be expressed by Fick s diffusion equation... 

Another illustration of time-dependent FEM analysis is the use of SECCM (double-barrel ion conductance probe) to investigate ionic crystal dissolution. A transient Fick s diffusion equation. 

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. 

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). 

Fick s laws Equations that describe the relationship between gradients in concentration and the rate of diffusion. See Eqs. (13) and (16). 

The steady-state transport of A through the stagnant gas film is by molecular diffusion, characterized by the molecular diffusivity DAg. The rate of transport, normalized to refer to unit area of interface, is given by Fick s law, equation 8.5-4, in the integrated form... 

Fick s law ( Equation 20 ) shows that it is important to construct a flow cell in such a way that the diffusion layer is as thin as possible. This will give the highest signal currents, independent of the inner geometry of the cell. 

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] ... 

Diffusion controlled processes are measured by Fick s law. (Equation 1)... 

The analytical solutions to Fick s continuity equation represent special cases for which the diflusion coefficient, D, is constant. In practice, this condition is met only when the concentration of diffusing dopants is below a certain level ( 1 x 1019 atoms/cm3). Above this doping density, D may depend on local dopant concentration levels through electric field effects, Fermi-level effects, strain, or the presence of other dopants. For these cases, equation 1 must be integrated with a computer. The form of equation 1 is essentially the same for a wide range of nonlinear diffusion effects. Thus, the research emphasis has been on understanding the complex behavior of the diffusion coefficient, D, which can be accomplished by studying diffusion at the atomic level. 

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. 

FIGURE 22.2 Schematic illustration of a diffusion-controlled matrix system for which the diffusion process is typically governed by Fick s Law (Equation 22.9). 

The important conclusion illustrated by Figures 2.5 and 2.6 is that, although the fluids on either side of a membrane may be at different pressures and concentrations, within a perfect solution-diffusion membrane, there is no pressure gradient—only a concentration gradient. Flow through this type of membrane is expressed by Fick s law, Equation (2.13). 

It is convenient to correlate transport data in terms of a diffusivity defined according to Fick s first equation,... 

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. 

A set of Fick s diffusional equations are applied to the case, linear diffusion normal to the electrode with simultaneous chemical reactions. 

The relationship between these four parameters is given by a derivation of Fick s laws (Equation (1)), where J is the overall rate (flux) of diffusion across the skin (conventionally expressed in units of g cm 2 h 1) ... 

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. 

In catalytic cracking the gas oil feed reacts to much lighter compounds, which causes a high convective flux from the catalyst surface to the bulk of the fluid. Therefore Fick s diffusion law is not applicable (assumes equimolar counter-diffusion) in the mass transfer calculations and as a result rigorous Maxwell-Stefan equations must be used. Due to the... 

The various terms in the equation are as follows i is the current, n is the number of electrons transferred, F is the Faraday, A is the electrode area, C is the concentration, D is the diffusion coefficient, and t is the time. Thus the technique may be used to estimate, among other things, the charge transport parameter. The reader interested in the solution of the Fick s law equation using the Laplace transformation should consult Ref. 6. A closely related technique is chronocoulometry, in which the excitation function is still the potential pulse, but instead of monitoring the current, the integrated charge is monitored as a function of time. This entails less error as a cumulative measurement is made. The equation for chronocoulometry is... 

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