Digital simulations diffusion

Axial and radial dispersion or non-ideal flow in tubular reactors is usually characterised by analogy to molecular diffusion, in which the molecular diffusivity is replaced by eddy dispersion coefficients, characterising both radial and longitudinal dispersion effects. In this text, however, the discussion will be limited to that of tubular reactors with axial dispersion only. Otherwise the model equations become too complicated and beyond the capability of a simple digital simulation language. 

The presence of a liquid layer on the surface of the filter cake will cause solute to diffuse from the top layer of cake into the liquid. Also if disturbed the layer of liquid will mix with the surface layer of filter cake. This effect can be incorporated into the digital simulation by assuming a given initial depth of liquid as an additional segment of the bed which mixes at time t=0 with the top cake segment. The initial concentrations in the liquid layer and top cake segment are then found by an initial mass balance. 

J. Heinze, Diffusion processes at finite (micro) disk electrodes solved by digital simulation. J. Electroanal. Chem. 124, 73-86 (1981). 

L6. Feldberg, S. W. Digital Simulation A General Method for Solving Electrochemical Diffusion-Kinetic Problems, in Electroanalytical Chemistry, Bard, A. J., Ed., Marcel Dekker New York, 1969, Vol. 3. 

If the diffusion process is coupled with other influences (chemical reactions, adsorption at an interface, convection in solution, etc.), additional concentration dependences will be added to the right side of Equation 2.11, often making it analytically insoluble. In such cases it is profitable to retreat to the finite difference representation and model the experiment on a digital computer. Modeling of this type, when done properly, is not unlike carrying out the experiment itself (provided that the discretization error is equal to or smaller than the accessible experimental error). The method is known as digital simulation, and the result obtained is the finite difference solution. This approach is described in more detail in Chapter 20. 

For the purposes of considering diffusion at microelectrodes, it is convenient to introduce two categories of electrodes those to which diffusion occurs in a linear fashion and those to which diffusion occurs in a nonlinear fashion. The former category consists of cylindrical and spherical electrodes. As shown schematically in Figure 12.2A, the lines of flux (i.e., the pathway followed by material diffusing to the electrode) are straight, and the current density is the same at all points on the electrode. Thus, the diffusion problem is one-dimensional (i.e., distance from the electrode surface) and involves solution of the appropriate form of Fick s second law, Equation 12.7 or 12.8, either by Laplace transform methods or by digital simulation (Chap. 20). 

Two general approaches have been used in low-temperature studies. In the first, the uncompensated resistance, electrode capacitance, diffusion coefficient, and kinetic and thermodynamic parameters describing the electrode reaction are incorporated in a master model, which is treated (usually by some form of digital simulation) to calculate the expected voltammetric response for comparison with experiment [7,49]. 

Any discussion of the digital simulation of problems involving diffusion begins with a consideration of the combined form of Fick s laws [5],... 

The real power of digital simulation techniques lies in their ability to predict current-potential-time relationships when the reactants or products of an electrode reaction participate in some intervening chemical reaction. These kinetic complications often result in a fairly difficult differential equation (when combined with the conditions for diffusion or convection encountered in electrochemical problems) that resists solution by ordinary means. Through simulation, however, the effect of any number of chemical steps may be predicted. In practice, it is best to limit these predictions to cases where the reactants and products participate in one or two rate-determining steps each independent step adds another dimensionless kinetics parameter that must be varied over the range of... 

Substituting in typical electrochemical values for L, C, and tk, one obtains kd = 108 L/(mol s). Comparison of this result with the equation of Osborne and Porter [11] shows that diffusion control in the simulation occurs at a rate two orders of magnitude less than predicted by accepted theory. In other words, one cannot distinguish any difference between the effects of second-order rate constants in the vicinity of 108 L/(mol s) from those in the vicinity of 1010 L/(mol s) by using digital simulation. 

No discussion of digital simulation would be complete without some mention of the methods employed in treating electrogenerated chemiluminescence (ECL). In the most common statement of the problem, an aromatic hydrocarbon (A) is alternately oxidized and reduced at a single electrode to produce its radical anion and cation species (B and D). These species, in turn, react within the diffusion layer to regenerate the hydrocarbon. 

Figure 20.8 Model used in the digital simulation of chronopotentiometric behavior. A constant amount (5) is removed from the contents of the first element following the diffusion step. The electrode surface concentration is determined by extrapolating the line of known slope through the point representing first element concentration to the electrode surface.

Many problems involving competitive reaction kinetics may be treated by invoking the steady-state assumption within the digital simulation this has been done in at least two instances [29-34]. The first of these involves the development of a model for enzyme catalysis in the amperometric enzyme electrode [29-31]. In this model, the enzyme E is considered to be immobilized in a diffusion medium covering an electrode that is operated at a fixed potential such that the product (P) of enzyme catalysis is electroactive under diffusion-controlled conditions. (This model has also served as the basis for the simulation of the voltammetric response of the enzyme electrode [35].) The substrate (S) diffuses through the medium that contains the immobilized enzyme and is catalyzed to form P by straightforward enzyme kinetics ... 

Attempts to develop a model for the digital simulation of the cyclic voltammetric behaviour of PVF films on platinum62 electrodes required inclusion of the following features (a) environmentally distinct oxidized and reduced sites within the film (b) interconversion of the above sites and interaction between them (c) rate of electrochemical reactions to depend on the rate of interconversion of redox sites, the rate of heterogeneous electron transfer between film and substrate, intrafilm electron transfer and the rate of diffusion of counter ions and (d) dependence on the nature of the supporting electrolyte and the spacing of electroactive groups within the film. 

This is a typical diffusion-reaction scheme, as discussed in Chapter 2 (2.13). The major difference is the unconfined heat flow, which makes the solution of the response equation, even by digital simulation, difficult. 

This is Fick s second diffusion equation [242], an adaptation to diffusion of the heat transfer equation of Fourier [253]. Technically, it is a second-order parabolic partial differential equation (pde). In fact, it will mostly be only the skeleton of the actual equation one needs to solve there will usually be such complications as convection (solution moving) and chemical reactions taking place in the solution, which will cause concentration changes in addition to diffusion itself. Numerical solution may then be the only way we can get numbers from such equations - hence digital simulation. 

In digital simulation, when discretising the diffusion equation, we have a first derivative with respect to time, and one or more second derivatives with respect to the space coordinates sometimes also spatial first derivatives. Efficient simuiation methods will always strive to maximise the orders. 

The Feldberg approach to digital simulation [229] uses a somewhat different method of discretisation, and the method is alive and well it is, for example, the basis for the commercial program DigiSim [482], It begins with Fick s first diffusion equation, using fluxes between boxes or finite volumes, rather than concentrations at points in the discretisation process (see below). 

Digital simulation — Data from electrochemical experiments such as cyclic voltammetry are rich in information on solution composition, diffusion processes, kinetics, and thermodynamics. Mathematical equations describing the corresponding parameter space can be written down but can be only very rarely solved analytically. Instead computer algorithms have been devised to ac-... 

The fast complexation rate of Zn(II) ion with 8-quinolinol (Hqn) or 5-octyloxymethyl-8-quinolinol (Hoeqn) at the 1-butanol/water interface was measured by the micro-two-phase sheath flow method [17], The formation of a fluorescence complex at the interface was measured within a period of less than 2 milliseconds after the contact of the two phases. The depth profile of the fluorescence intensity observed across the inner organic phase flow proved that the fluorescence complex was formed only at the interface and it increased in proportion to the contact time. The diflusion length of Hoeqn in the 1-butanol phase for 2 milliseconds was calculated as 0.8 pm, which is smaller than the experimental resolution depth of 2 pm in the microscopy used. Therefore, the observed rate constant was analysed by taking diffusion and reaction rates into account between Zn(II) and Hoeqn at the interfacial region by a digital simulation method. The digital simulation has been used in the analysis of electrode reactions,... 

In order to have theoretical relationships with which experimental data can be compared for analysis it is necessary to obtain solutions to the partial differential equations describing the diffusion-kinetic behaviour of the electrode process. Only a very brief account f the theoretical methods is given here and this is done merely to provide a basis for an appreciation of the problems involved and to point out where detailed treatments can be found. A very lucid introduction to the theoretical methods of dealing with transient electrochemical response has appeared (MacDonald, 1977) which is highly recommended in addition to the classic detailed treatment (Delahay, 1954). Analytical solutions of the partial differential equations are possible only in the most simple cases. In more complex cases either numerical methods are used to solve the equations or they are transformed into finite difference forms and solved by digital simulation. 

In cases where comparisons have been made, theoretical data obtained by digital simulations are always in agreement with those from analytical solutions of the diffusion-kinetic equations within the limit of experimental error of quantities which can be measured. A definite advantage of simulation over the other calculation techniques is that it does not require a strong mathematical background in order to learn and to use the technique. A very useful guide for the beginner has recently appeared (Britz, 1981). 

The solution of the diffusion equation is best obtained by digital simulation. Fortunately, one can understand the behavior of such ensembles qualitatively, and the conclusions reached in this way are in quite good agreement with the results of (rather tedious) numerical calculations. 

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