Fick equation

UNSAT-H uses the Richards equation, Fick s law, and the Fourier equation to estimate the flow of soil-water, vapor, and heat. This may be the strongest part of the model because these are the most rigorous, currently known, theoretical methods for estimating these parameters. 

Permeability of an FML is evaluated using the Water Vapor Transmission test.28 A sample of the membrane is placed on top of a small aluminum cup containing a small amount of water. The cup is then placed in a controlled humidity and temperature chamber. The humidity in the chamber is typically 20% relative humidity, while the humidity in the cup is 100%. Thus, a concentration gradient is set up across the membrane. Moisture diffuses through the membrane, and with time the liquid level in the cup is reduced. The rate at which moisture is moving through the membrane is measured. From that rate, the permeability of the membrane is calculated with the simple diffusion equation (Fick s first law). It is important to remember that even if a liner is installed correctly with no holes, penetrations, punctures, or defects, liquid will still diffuse through the membrane. 

The reaction scheme at and near the phase boundary during the phase transformation is depicted in Figure 10-14. The width of the defect relaxation zone around the moving boundary is AifR, it designates the region in which the relaxation processes take place. The boundary moves with velocity ub(f) and establishes the boundary conditions for diffusion in the adjacent phases a and p. The conservation of mass couples the various processes. This is shown schematically in Figure 10-14b where the thermodynamic conditions illustrated in Figure 10-12 are also taken into account. The transport equations (Fick s second laws) have to be solved in both the a and p... 

If we attach a new reference frame (z) to the moving (stable planar) boundary, z = ,-V t. The transport equation (Fick s second law) reads in the z-system... 

Finally, let us briefly point out some essential features of the stability analysis for a more general transport problem. It can be exemplified by the moving a//9 phase boundary in the ternary system of Figure 11-12. Referring to Figure 11-7 and Eqn. (11.10), it was a single independent (vacancy) flux that caused the motion of the boundary. In the case of two or more independent components, we have to formulate the transport equation (Fick s second law) for each component, both in the a- and /9-phase. Each of the fluxes jf couples at the boundary b with jf, i = A,B,... (see, for example, Eqn. (11.2)). Furthermore, in the bulk, the fluxes are also coupled (e.g., by electroneutrality or site conservation). 

Sorption data were used to obtain values for A" L. As pointed out by Paul and Paciotti, the data in Figure 2.17 show that reverse osmosis and pervaporation obey one unique transport equation—Fick s law. In other words, transport follows the solution-diffusion model. The slope of the curve decreases at the higher concentration differences, that is, at smaller values for c,eimi because of decreases in the diffusion coefficient, as the swelling of the membrane decreases. 

The Diffusivity or Diffusion Coefficient (D) is the amount of matter (m) passing per second through a unit area, under the influence of a unit gradient of concentration (the "driving force"), as expressed in the equation (Fick s Law) ... 

Our example in polarography illustrates how a spreadsheet can be used to simulate a rather complex curve, in this case reflecting the interplay between the Nernst equation, Fick s law of diffusion, and drop growth. The first two factors also play a role in cyclic voltammetry, where we introduce semi-integration as an example of deconvolution. 

Diffusion Equations. Fick s second law can be solved for the special case of a layer of substance of constant (with time) concentration diffusing into a semi-infinite medium which initially contains no solute. Using the method of the Laplace transform (8) the solution yields an expression for the concentration in the medium ... 

Minimum mass transport occurs even in the absence of water filtration. We will review uni dimensional (linear) mass transport. If V = 0 and q, = 0, advective-dispersive equation acquires the format of a linear equation Fick s second law (equation 3.12). 

In the aforementioned step 1, the solvent diffuses into polymer networks (case-I diffusion). At this time, the solvent absorption behavior of the gel can be obtained by solving the diffusion equation (Fick s second law). In... 

As In the case of the isobaric experiment, it would also be possible to interpret these flux measurements in terms of a Fick equation with an effective... 

Wicke-Kallenbach experiment would incorrectly predict the flux in the second experiment if used in a simple Fick equation of the form (10.31). However, if the isobaric flux measurements had been interpreted in terms of... 

In contrast to the cell experiments of Gibilaro et al., it is now seen from equation (10.45) that measurement of the delay time gives no information about diffusion within the pellets this can be obtained only through equation (10.46) from measurements of the second moment. As in the case of the cell experiment, the results can also be Interpreted in terms of an "effective diffusion coefficient" associated with a Fick equation for the... 

A few simple equations describe most appHcations of barrier polymers. Equation 1 is an adaptation for films of Fick s First Law. 

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. 

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) ... 

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 Dunwald—Wagner equation, based on the application of Ficks second law of diffusion into or out of a sphere (radius r) [477], can be written... 

Hence, the current (at any time) is proportional to the concentration gradient of the electroactive species. As indicated by the above equations, the dififusional flux is time dependent. Such dependence is described by Fick s second law (for linear diffusion) ... 

This equation reflects the rate of change with time of the concentration between parallel planes at points x and (x + dx) (which is equal to the difference in flux at the two planes). Fick s second law is vahd for the conditions assmned, namely planes parallel to one another and perpendicular to the direction of diffusion, i.e., conditions of linear diffusion. In contrast, for the case of diffusion toward a spherical electrode (where the lines of flux are not parallel but are perpendicular to segments of the sphere), Fick s second law has the form... 

Derive the Cottrell equation by combining Fick s first law of diffusion with the tune-dependent change of the concentration gradient during a potential-step experiment. 

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). 

Liquid phase diffusivities are strongly dependent on the concentration of the diffusing component which is in strong contrast to gas phase diffusivities which are substantially independent of concentration. Values of liquid phase diffusivities which are normally quoted apply to very dilute concentrations of the diffusing component, the only condition under which analytical solutions can be produced for the diffusion equations. For this reason, only dilute solutions are considered here, and in these circumstances no serious error is involved in using Fick s first and second laws expressed in molar units. 

As reversible ion transfer reactions are diffusion controlled, the mass transport to the interface is given by Fick s second law, which may be directly integrated with the Nernst equation as a boundary condition (see, for instance. Ref. 230 232). A solution for the interfacial concentrations may be obtained, and the maximum forward peak may then be expressed as a function of the interfacial area A, of the potential scan rate v, of the bulk concentration of the ion under study Cj and of its diffusion coefficient D". This leads to the Randles Sevcik equation [233] ... 

Many extensions have been derived for the Ilkovic equation from the consideration that the dme does not behave as a flat electrode but in fact shows a spherical growth. For instance, Fick s second law of diffusion (cf., eqn. 3.2) becomes12... 

This equation can be derived by means of Laplace transformation of Fick s second law for a planar microelectrode ... 

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