Fick number

Generally, the solids are not stracturally homogeneous, but the solid and liquid nevertheless wiU be called phases and leaching will be treated as a two-phase, mass transfer process. The solid consists of a matrix of insoluble solids, the marc, and the occluded solution. It may also contain undissolved solute and a nonextractable secondaiy phase, for example, coffee oil in water-soaked coffee grounds. This secondary phase is treated as part of the marc. Dimensionless parameters that can affect solute transfer include the solute equilibrium distribution coefficients, m and M the Fick number, r the stripping factor, a the Biot number, Bi and the Peclet number, Pe. These parameters are defined more precisely in the Notation section. 

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 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. 

Equation 10.66 is referred to as Fick s Second Law. This also applies when up is small, corresponding to conditions where C, is always low. This equation can be solved for a number of important boundary conditions, and it should be compared with the corresponding equation for unsteady state heat transfer (equation 9.29). 

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

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) ... 

V, where the plateau is reached and a quantitative determination can be done. The column effluent is pumped through the detector, and if this contains compounds that can be oxidized at the set potential the current through the electrode increases. The current is equivalent to the amount of compounds, but not necessarily the same for all kinds of compounds. This depends on the number of electrons that are involved in the electrode reaction, on the electrode kinetics, and on the thickness of the diffusion layer, d. This is expressed in Fick s first law ... 

Figures 7(a)-(c) show a comparison between the numerically computed absorption flux and the absorption flux obtained from expression (31), using eqs (24), (30) and (34)-(37). From these figures it can be concluded that for both equal and different binary mass transfer coefficients absorption without reaction can be described well with eq. (24), whereas absorption with instantaneous reaction can be described well with eq. (30). If the Maxwell-Stefan theory is used to describe the mass transfer process, the enhancement factor obeys the same expression as the one obtained on the basis of Fick s law [eq. (35)]. Finally, from Figs 7(b) and 7(c) it appears that the use of an effective mass transfer coefficient m the Hatta number again produces satisfactory results.

J is the flux of particles crossing a 1-cm2 plane in 1 s (i.e., number cm 2 s ). The constant D is known as the diffusion coefficient and is simply the proportionality constant relating the flux to the concentration gradient. (Fick s first law applies, of course, not only to particles but also to gas and liquid molecules.)... 

Two N0 sensor cells in series were used to measure n, the number of electrons involved in the oxidation reaction. The following relationship was derived from Fick s Law, and was used to calculate n ... 

A number of unsteady state problems in diffusion have been considered, in which chemical reactions are occurring (F6, G8). Sherwood and Pigford (S9, pp. 332-337) have studied the unsteady state absorption of a substance A which diffuses into the solvent S and undergoes an infinitely fast, irreversible, second-order reaction with a solute B (that is A + B— AB). It is assumed that Fick s second law adequately describes the diffusion process and that because of the infinitely fast reaction of A and B there will be a plane parallel to the liquid surface at a distance a from it, which separates the region containing no A from that containing no B. The distance s is a function of t, inasmuch as the boundary between A and B retreats as B is used up in the chemical reactions. 

Schmidt numbers vary widely depending on the value of the Fick diffusion coefficient, which in the case of gases is strongly dependent on the concentration of the diffusing component. It should be noted that in a mixture the Schmidt number is not a property solely of either the component or... 

The following, well-acceptable assumptions are applied in the presented models of automobile exhaust gas converters Ideal gas behavior and constant pressure are considered (system open to ambient atmosphere, very low pressure drop). Relatively low concentration of key reactants enables to approximate diffusion processes by the Fick s law and to assume negligible change in the number of moles caused by the reactions. Axial dispersion and heat conduction effects in the flowing gas can be neglected due to short residence times ( 0.1 s). The description of heat and mass transfer between bulk of flowing gas and catalytic washcoat is approximated by distributed transfer coefficients, calculated from suitable correlations (cf. Section III.C). All physical properties of gas (cp, p, p, X, Z>k) and solid phase heat capacity are evaluated in dependence on temperature. Effective heat conductivity, density and heat capacity are used for the entire solid phase, which consists of catalytic washcoat layer and monolith substrate (wall). 

As mentioned above, in 1917 M. Smoluchowski applied the theory of diffusion to this situation to evaluate the rate of doublet formation. According to Fick s first law (Equation (2.22)) J, the number of particles crossing a unit area toward the reference particle per unit of time is given by... 

This approach of subdividing space into an increasing number of discrete pieces provides the basis for many numerical computer models (e.g., the so-called finite difference models) an example will be discussed in Chapter 23. Although these models are extremely powerful and convenient for the analysis of field data, they often conceal the basic principles which are responsible for a given result. Therefore, in the next chapter we will discuss models which are not only continuous in time, but also continuous along one or several space axes. In this context continuous in space means that the concentrations are given not only as steadily varying functions in time [QY)], but also as functions in space [C,(r,x) or C,(t,x,y,z)]. Such models lead to partial differential equations. A prominent example is Fick s second law (Eq. 18-14). 

The flux of particles is in the opposite sense to the direction of the concentration gradient increase. Equation (6) is Fick s first law, which has been experimentally confirmed by many workers. D is the mutual diffusion coefficient (units of m2 s 1), equal to the sum of diffusion coefficients for both reactants, and for mobile solvents D 10 9 m2 s D = DA + jDb. The diffusion coefficient is approximately inversely dependent upon viscosity and is discussed in Sect. 6.9. As spherical symmetry is appropriate for the diffusion of B towards a spherically symmetric A reactant, the flux of B crossing a spherical surface of radius r is given by eqn. (6) where r is the radial coordinate. The total number of reactant B molecules crossing this surface, of area 4jrr2, per second is the particle current I... 

The diffusion coefficient as defined by Fick s law, Eqn. (3.4-3), is a molecular parameter and is usually reported as an infinite-dilution, binary-diffusion coefficient. In mass-transfer work, it appears in the Schmidt- and in the Sherwood numbers. These two quantities, Sc and Sh, are strongly affected by pressure and whether the conditions are near the critical state of the solvent or not. As we saw before, the Schmidt and Prandtl numbers theoretically take large values as the critical point of the solvent is approached. Mass-transfer in high-pressure operations is done by extraction or leaching with a dense gas, neat or modified with an entrainer. In dense-gas extraction, the fluid of choice is carbon dioxide, hence many diffusional data relate to carbon dioxide at conditions above its critical point (73.8 bar, 31°C) In general, the order of magnitude of the diffusivity depends on the type of solvent in which diffusion occurs. Middleman [18] reports some of the following data for diffusion. 

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. 

Let us assume that the oxide sample exists in the form of a parallelepiped. The solution to Fick s second law under the given boundary conditions gives for the change in time of the integral number of vacancies wv... 

The laws of Fick are concerned with the concentration and the change of concentration with respect to space and time. In this paper we are interested in the number of molecular displacements. If we calculate the change of concentration we must use the laws of Fide. The diffusion constant in all cases is given by... 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

In boundary condition (iv), /B(i c,t) is the flow of B molecules across the surface of a sphere with radius Rc, the distance between the reactants, where a reaction may take place. This flow is expressed by the flux density, JB(i c,f) (be., the number of molecules per second that are passing through an area of one square meter), as given by Fick s first law... 

The process, which involves diffusion, is initiated by a temperature rise. Under these conditions, a diffusion flux (/diff) will emerge symmetrically from both sides of the sheet and the particle number density (n) will vary with time according to Fick s law ... 

Diffusion is the random migration of molecules or small particles arising from motion due to thermal energy. A very simple derivation of Fick s first law, based on the random walk problem, can be obtained in one dimension. In this case, Jx(x, t), that is, the number of particles, N, that move across unit area, A, in unit time, x, can be defined as... 

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