Digital simulation limitations

This analysis is limited, since it is based on a steady-state criterion. The linearisation approach, outlined above, also fails in that its analysis is restricted to variations, which are very close to the steady state. While this provides excellent information on the dynamic stability, it cannot predict the actual trajectory of the reaction, once this departs from the near steady state. A full dynamic analysis is, therefore, best considered in terms of the full dynamic model equations and this is easily effected, using digital simulation. The above case of the single CSTR, with a single exothermic reaction, is covered by the simulation examples, THERMPLOT and THERM. Other simulation examples, covering aspects of stirred-tank reactor stability are COOL, OSCIL, REFRIG and STABIL. 

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. 

Digital simulation [37] has shown that the approximations made by Levich are valid under most conditions limits to his assumptions are given, particularly with regard to potential scan rates. 

Not the least of these results from the absence of an analytical solution to almost any problem of consequence, except in the limits of the boundary value problem. An appreciation of this limitation and how one may minimize its impact is essential to the utilization of digital simulations in any application. 

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... 

The X value in the fast scan limit may be used to estimate the value of Keq, and its increases at slower sweep rates may be used to estimate the monomeri-zation/dimerization rate constants [16]. Full digital simulations may also be used... 

The conditions where Eq. (4.236) provides good results have been examined by comparison with those obtained from digital simulation in [79] and it is concluded that this solution gives rise to accurate results in RPV for (k + k2)1 2 > 5 (with t2 being the duration of the second pulse), with the error decreasing as K increases and always less than 5 % for the value of the oxidative limiting current. 

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). 

Note that this equation is the same as (12.3.29). This limiting current holds only in the region of small (o. When A (= b C lcS) becomes small, the behavior approaches the mass-transfer-controlled limiting current. Results of a digital simulation of the catalytic case (83) are shown in Figure 12.4.3. Other treatments of the ErCj case at the RDE, as well as variations of this mechanism, have also appeared (84-86). The treatment of the Ej-CJ case for the RRDE by digital simulation techniques showed that the results (i.e., plots of Nk vs. XKT) are indistinguishable from those of the Ej-Ci case for first- or pseudo-first-order reactions (83). 

Since the usual transmission experiment directly monitors the electrolytic product, it offers many of the diagnostic features of reversal chronoamperometry or reversal chrono-coulometry. In effect, is a continuous index of the total amount of the monitored species still remaining in solution at the time of observation. Equation 17.1.2 describes the limiting case in which the product is completely stable. If homogeneous chemistry tends to deplete the concentration of R, different absorbance-time relations will be seen. They can be predicted (e.g., by digital simulations see Appendix B), and curves for many mechanistic cases have been reported (17). 

The above discussion assumes the condition [S] Ks- When the dependence of [S] on the catalytic current is to be studied, the bulk concentration of S, [S], must be lowered. Under such conditions, however, no steady-state current is observed on CVs as shown by curve (b). This is because the substrate depression occurs in the vicinity of the electrode surface, and no steady state is attained. A digital simulation technique [12] would be the most straightforward way to analyze such nonsteady-state currents or [Sj dependence of the catalytic current [13-15]. Substrate depression can be avoided when two enzyme reactions are coupled in mediated bioelectrocatalysis in such a way that S in the first enzyme reaction is regenerated from the product P by the second enzyme reaction to keep the S/P ratio constant [16]. Under such conditions, the steady-state limiting current can be given by [16,17]... 

The shapes of the polarization curves depend strongly on the /oM ratios. It is obvious that the continuous change of the exchange current density to the limiting diffusion current density ratio for processes of the metal electrodeposition is only possible by the digital simulation. In this way, the relations between ratios, the shape of polarization curve, and the electrodeposition process control can be established. 

Enzyme-mediated feedback can be used to image enzyme patterns. To successfully image enzymatic features, lip fouling from oxide formation or adsorption from solution constituents must be avoided. The enzyme reaction at the substrate must not be inhibited by solution species. It must also be able to sustain a level of regeneration activity of the mediator that can compete with its mass transport from the bulk electrode to the tip. In the case of a glucose oxidase catalyzed reaction, a digital simulation of the positive feedback observed from this enzyme quantitatively expresses this limitation (143). 

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