Diffusion laws

Internal diffusion occurs during drying when Hquid or vapor flow obeys the fundamental diffusion laws. 

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. 

This section describes equipment for heat transfer to or from solids by the indirect mode. Such equipment is so constructed that the solids load (burden) is separated from the heat-carrier medium by a wall the two phases are never in direct contact. Heat transfer is by conduction based on diffusion laws. Equipment in which the phases are in direct contact is covered in other sections of this Handbook, principally in Sec. 20. 

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

The factor B = D/RT is the mobility and contains the diffusion coefficient D, the gas constant R, and the absolute temperature T. The equation includes a diffusion and a migration term. Correspondingly Eq. (2-23) gives the first diffusion law for Zj = 0 and Ohm s Law for grad /i, = 0. For transfer across a phase boundary ... 

Fiek s diffusion law is used to deseribe dispersion. In a tubular reaetor, either empty or paeked, the depletion of the reaetant and non-uniform flow veloeity profiles result in eoneentration gradients, and thus dispersion in both axial and radial direetions. Fiek s law for moleeular diffusion in the x-direetion is defined by... 

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). 

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,... 

The Henry s law constant value of 2.Ox 10 atm-m /mol at 20°C suggests that trichloroethylene partitions rapidly to the atmosphere from surface water. The major route of removal of trichloroethylene from water is volatilization (EPA 1985c). Laboratory studies have demonstrated that trichloroethylene volatilizes rapidly from water (Chodola et al. 1989 Dilling 1977 Okouchi 1986 Roberts and Dandliker 1983). Dilling et al. (1975) reported the experimental half-life with respect to volatilization of 1 mg/L trichloroethylene from water to be an average of 21 minutes at approximately 25 °C in an open container. Although volatilization is rapid, actual volatilization rates are dependent upon temperature, water movement and depth, associated air movement, and other factors. A mathematical model based on Pick s diffusion law has been developed to describe trichloroethylene volatilization from quiescent water, and the rate constant was found to be inversely proportional to the square of the water depth (Peng et al. 1994). 

Consider the specific example of a spherical electrode having the radius a. We shall assume that diffusion to the spherical surface occurs uniformly from all sides (spherical symmetry). Under these conditions it will be convenient to use a spherical coordinate system having its origin in the center of the sphere. Because of this synunetry, then, aU parameters have distributions that are independent of the angle in space and can be described in terms of the single coordinate r (i.e., the distance from the center of the sphere). In this coordinate system. Pick s second diffusion law becomes... 

The Stem-Volmer equations discussed so far apply to solutions of the luminophore and the quencher, where both species are homogeneously distributed and Fick diffusion laws in a 3-D space apply. Nevertheless, this is a quite unusual situation in fluorescent dye-based chemical sensors where a number of factors provoke strong departure from the linearity given by equation 2. A detailed discussion of such situations is beyond the scope of this chapter however, the optosensor researcher must take into account the following effects (where applicable) ... 

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. 

Cyclic voltammetry at spherical electrodes. As discussed in Chapter 1, Section 4.2.3, diffusion laws at a spherical electrode must take into account the curvature r0 of the electrode. The mathematical treatment of diffusion at a spherical electrode becomes somewhat more complicated6 with respect to the preceding one for planar diffusion and we will not dwell on it. On the basis of what we will see in Chapter 11, Section 2, it is important to consider that, under radial diffusion, the cyclic voltammogram loses its peak-shaped profile to assume a sigmoidal profile, see Figure 6. 

The diffusional process can be treated by applying the first Pick s diffusion law to the two diffusional processes occurring on both sides of inter face, under the simplifying assumptions of steady-state and linear concentration gradients shown in Fig. 5.8. The diffusional fluxes through the aqueous, and organic, diffusion films will then be... 

Basically, whenever isotopic exchanges occur between different phases (i.e., heterogeneous equilibria), isotopic fractionations are more appropriately described in terms of differential reaction rates. Simple diffusion laws are nevertheless appropriate in discussions of compositional gradients within a single phase— induced, for instance, by vacancy migration mechanisms, such as those treated in section 4.10—or whenever the isotopic exchange process does not affect the extrinsic stability of the phase. 

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 non-steady-state electrolysis, the concentration of the reactant Ox is a function of the distance x from the electrode (cathode) and the time f, [Ox] = Concurrently, concentration of the reaction product Red increases with time. For simplicity, the concentrations will be used instead of activities. Weber (19) and Sand (20) solved the differential equation expressing Pick 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. 

Usually, experimentalists quantify step fluctuations by averaging the data to find the correlation function G(t) = 0.5 < (h(x,i) - h(x,0)Y >, where h x,t) specifies the step position at time t and the average is over many sample points, x. G(f) measures how far a position on a step wanders with time. If that position were completely free to wander, it would obey a diffusive law G(t) t. However, its motion is restricted by the fact that it is connected to the other parts of the step. For that reason G(t) is sub-diffusive. The detailed law which G(f) obeys is dependent on the atomic processes which mediate step motion. For example, if the step edge is able to freely exchange... 

The surface oxidation of a metal such as copper is accompanied by the growth of an oxide layer, the thickness of which may be measured by the method of colour interference when due allowance is made for the refractive index of the oxide formed, or by the decrease in electrical resistance of a thin wire or tube of the metal as oxidation ensues. Investigations have been made on the rate of such oxidations by Tammann (Zeit.f anorg. GJiem. cxi. 78 cxxiii. 196 cxxiv. 196), Hinshelwood (Proc. Mog. Soc. A, cii. 318), Palmer Proo. Roy. Soc. A, cm. 444) and Dunn (unpublished, see also Pilling and Bedworth, Jour. Inst. Metals, xxix. 629, 1923). It is found that the rate of increase in thickness of the oxide film oc) obeys under ideal conditions the ordinary diffusion law or CG = kt. ... 

A similar departure from the diffusion law to the logarithmic is likewise to be noted when the oxide formed is not of uniform composition but consists of a solid solution of two oxides such as obtains in the case of iron. 

Pick s law is an empirical diffusion law for binary systems. For multicomponent systems. Pick s law must be generalized. There are several ways to generalize Pick s law to multicomponent systems. One simple treatment is called the effective binary treatment, in which Pick s law is generalized to a multicomponent system in the simplest way ... 

In the literature there are several, mostly just slightly different, equations that describe the rate coefficient of the diffusion controlled reactions these equations are usually based on the solutions of Pick II diffusion law assuming that the reaction probability at contact distance is 1. Andre et al. [131] used the following equation to describe the time dependence of excited molecule concentration [RH ] produced by an infinite excitation pulse ... 

Between samples, the residual solute in the polymer is removed by continued exposure to back extractant. This process requires about 2 min for 1 mM sample solutions. As expected from diffusion laws, this time period for residual removal varies with the square of the sample concentrations. Higher sampling rates require keeping analyte concentrations below about 1 mM. 

Molecular rotation in a solution is described quantitatively by diffusion laws (analogous to Fick s laws)... 

It is worth noting that the derivation outlined above is more generally applicable since, for many types of diffusional mass transfer (spherical, cylindrical, bounded, etc.), it is possible to rewrite the original differential equations in the time domain in terms of new variables in such a way that the second diffusion law is of the same form as eqn. (19b), with appropriate formulation of the boundary conditions [22, 75]. However, in finding the inverse transforms, difficulties may arise because of the more complex meaning of the time domain variables. 

The latter formulation is more convenient to account for mass transfer, since in the case that equilibria between the complexes are fast, the diffusion laws have to be applied to their total concentrations. The major consequence for our mathematical expressions is that the quantity exp (f) should be replaced by 1/2... 

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