Fick’s equations

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 influence of temperature on coating thickness is shown in Fig. 12.21 which relates to a 4h treatment at temperature. Figure 12.22 shows the variations of thickness as a function of time at a constant temperature of 1 1(X)°C. This curve is in good agreement with the third of Fick s equations (12.15) ... 

We can progress from here provided that we can find expressions for the partial derivatives of equation (2.99). Provided that the concentration of supporting electrolyte is sufficiently high that all the potential difference across the interface is accommodated within the Helmholtz layer, then transport of O and R near the electrode will only take place via diffusion (i.e. we can neglect migration). The equation of motion for either O or R is given by the differential form of Fick s equation, as discussed in chapter I ... 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

In the case of mass transport by pure diffusion, the concentrations of electroactive species at an electrode surface can often be calculated for simple systems by solving Fick s equations with appropriate boundary conditions. A well known example is for the overvoltage at a planar electrode under an imposed constant current and conditions of semi-infinite linear diffusion. The relationships between concentration, distance from the electrode surface, x, and time, f, are determined by solution of Fick s second law, so that expressions can be written for [Ox]Q and [Red]0 as functions of time. Thus, for... 

If Fick s equation is used and the diffusivities of the ion pair DAb and DBa are assumed to be equal, as for binary gas diffusivities, predicted exchange rates are the same, regardless of direction of diffusion. Thus, the newer theory taking into account the electric potential is clearly an improvement over the Fick s law approach. However, if an empirical view of this simple theory is used, allowing each diffusivity to assume a... 

Free diffusion columns are arranged to be sufficiently long for the initial concentrations at the extreme ends of the cell to remain unaltered during the course of the experiment. For a monodispersed system under these conditions the concentration gradient curves (Figure 2.4b) can be shown, by solving Fick s equations, to take the shape of Gaussian distribution curves represented by the expression... 

So far we have concentrated on the behavior of particles in translational motion. If the particles are sufficiently small, they will experience an agitation from random molecular bombardment in the gas, which will create a thermal motion analogous to the surrounding gas molecules. The agitation and migration of small colloidal particles has been known since the work of Robert Brown in the early nineteenth century. This thermal motion is likened to the diffusion of gas molecules in a nonuniform gas. The applicability of Fick s equations for the diffusion of particles in a fluid has been accepted widely after the work of Einstein and others in the early 1900s. The rate of diffusion depends on the gradient in particle concentration and the particle diffusivity. The latter is a basic parameter directly... 

Huggins and colleagues proposed the PITT in 1979 [17]. This technique also aims to determine the diffusion coefficient of diffusing species from Fick s equations. The solid particles are supposed to have initially a homogeneous concentration [18]. 

Then, the effective diffusivity of the macro-porous granular material is evaluated. The transport is again governed by the Fick s equation (23), but diffusion also takes place in the solid phase, which formally represents the nano-porous material, cf. Fig. 15. The diffusion coefficient in the solid phase is Dsohd = i//nanoD — 0.112D, where D is the bulk diffusivity. The concentration field in the macro-porous media is the solution of Eq. (23) with a diffusivity... 

Dell Era. A., and Pasquali, M. 2009. Comparison between different ways to determine diffusion coefficient by solving Fick s equation for spherical coordinates. Journal of Solid State Electrochemistry 13, 849-859. 

Therefore, the transport process in liquid-filled macropores can often be represented simply by Fick s equation... 

In this equation obtained by integrating Fick s equation, r is defined by... 

This follows from the analogy between Fick s equation and Laplace s equation for the potential (22). 

If the volume fraction of the matrix is boundary conditions used to solve Fick s equations. However, tp is often unknown, nor readily determined. For one thing, the effective tp may be larger than the nominal value, because some of the solvent is not available to the solute (see Section 8.3). In such cases, what will be experimentally observed from the mass transport rate is a smaller D. ... 

The moisture permeation rate of a sugar coating composed of sucrose, talc, and other minor components was reported to conform to Fick s equation.729 The permeation rate appeared to be rate-controlling for moisture adsorption by sugar-coated tablets. 

The rate of coagulation is affected by interaction between particles separated by distances greater than twice their radius. According to N.A. Fuchs [56], one has to look at the diffusion of particles of radius r in their interaction field, described as a function of energy (negative ) on distance, u(R). Fick s equation in this case can be written as... 

Hint By making the substitution v(r,t) = rC(r, t) in Fick s equation and in the boundary conditions, the problem becomes essentially the same as that for linear diffusion.]... 

Fick s equation for the diffusion of salts in solutioh must be modified to allow for the decreasing diffusivity of the salt with increasing concentration and finally, van der Waals, Clausius, Rankine, Sarrau, etc., have attempted to correct the simple gas equation pv = BT, by making certain assumptions as to the internal structure of the gas. 

I. To find the concentration, V, of the dissolved substance at different levels, x, of the diffusion vessel after the elapse of any Stated interval of time, t. This is equivalent to finding a solution of Fick s equation, which will satisfy the conditions under which the experiment is conducted. These so-called limiting conditions ara (i) when... 

Examples.—(1) T. Graham s diffusion experiments Phil. Trans., 151,188, 1861). A cylindrical vessel 152 mm. high, and 87 mm. in diameter, contained 0 7 litre of water. Below this was placed 0T litre of a salt solution. The fluid column was then 127 mm. high. After the elapse of a certain time, successive portions of 100 c.c., or of the total volume of the fluid, were removed and the quantity of salt determined in each layer. Here x = 0 at the bottom of the vessel, and x = H at the top x — h at the surface separating the solution from the liquid when t = 0. The vessel has unit area. The limiting conditions are At the end of a certain time t, (i) when p = 0, dV/dx = 0 and (ii) when x — H, dVjdx = 0 (iii) when t = 0, V= F0 between x = 0 and x=h (iv) when t = 0, F = 0 between x — h and x — H. To adapt these results to Fourier s solution of Fick s equation, first show that (6) is a particular integral of Fick s equation. Differentiate (8) with respect to x and show that for the first condition we must have b zero, and condition (i) is satisfied. For condition (ii), sin fiH must be zero but sin mr is zero hence we can put... 

Figure 4.18. Diffusion of deuterated polystyrene (d-PS) in normal polystyrene (h-PS), measured by forward recoil spectrometry (FReS). A 10-20 nm film of d-PS of relative molecular mass 225 000 was floated onto a 2 pm film of h-PS analysis of the deuterium depth profile shows that the d-PS layer was localised, to within the resolution of the technique, at the surface. After atmealing at 170 °C for 3600 s FReS revealed that substantial diffusion into the film had occurred the solid line is a fit to the solution of Fick s equation assmning that > = 8X10 cm s". After Mills et al. (1984).

For this solution of Fick s equation the boundary conditions... 

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