Experimental systems circuit model

The Kramers-Kronig relations have been applied to electrochemical systems by direct integration of the equations, by experimental observation of stability and linearity, by regression of specific electrical circuit models, and by regression of generalized measurement models. 

Experimental studies undertaken by Schutte (61) to examine the partitioning of polar constituents of the bitumen to various process streams in the extraction circuit indicated that froth is enriched in asphaltenes. Schutte postulated that asphaltenes play a key role in promoting air attachment to bitumen. Further observations also indicated that the water and solids content of froth is directly related to the asphaltene content of the bitumen. Findings were rationalized in terms of the effect of asphaltenes on the rate of coalescence of aerated bitumen droplets at a froth interface. Schutte believed that rapid coalescence would inhibit the entrainment of water and solids and thereby favor good product quality. Experimentation with a model system using paraffin oil and process water indicated that the presence of asphaltenes substantially increases the time required for coalescence. 

The goal of ac impedance measurements is to determine the values of elements in the equivalent circuit. It is necessary to experimentally study the response of corrosion system to an ac excitation and to fit the data to an equivalent circuit model that accurately describes the corrosion process at the metal-electrolyte interface. 

In the theoretical analysis of electrically driven pattern formation in nematics one deals only with the theoretical AC voltage Utheo, which drops over the nematic layer. Utheo differs, however, from the experimental voltage Uexp applied to the whole LC cell and recorded in experiments. Thus a quantitative comparison between experiments and theory is far from trivial as has been emphasized for instance by Krekhov et Typical liquid crystal cells consist of a nematic layer confined between ITO- or Sn02-coated glass plates covered with a thin film of an aligning polymer. As the polymer is a quite good insulator this sandwich has fairly complicated electric properties. In particular, at low frequencies the whole system has to be represented by a complex equivalent electric circuit model. 

EIS changed the ways electrochemists interpret the electrode-solution interface. With impedance analysis, a complete description of an electrochemical system can be achieved using equivalent circuits as the data contains aU necessary electrochemical information. The technique offers the most powerful analysis on the status of electrodes, monitors, and probes in many different processes that occur during electrochemical experiments, such as adsorption, charge and mass transport, and homogeneous reactions. EIS offers huge experimental efficiency, and the results that can be interpreted in terms of Linear Systems Theory, modeled as equivalent circuits, and checked for discrepancies by the Kramers-Kronig transformations [1]. 

Experimentally determined frequency characteristics were used to map the corrosion processes using models based on suitable equivalent circuits. Each element of such a circuit models the specific process or phenomenon occurring in the corrosion system imder investigation. 

Equivalent electrical circuits obtained by minimizing the mean square error were further used for the analysis of experimentally identified frequency characteristics and a description of corrosive processes in the systems under investigation. A simple electric circuit consisting of three elements of type R and C with a single time constant was adopted as system model for Al. Figs. 6 and 7 show the circuits modeling respectively the... 

Owing to the complexity of the actual experimental system, irregularities may occur in both Bode and Nyquist plots. Therefore the plots should be inspected carefully in deriving the components of the equivalent circuit model. The Bode plot is more sensitive than the Nyquist plot for identifying the presence of plot shape irregularities. If clear separation into two time constants does not occur, then the plots may require more sophisticated and complicated data analysis to extract values of all components. 

It is almost impossible to cover the entire range of models in Figure 25.1, and in this chapter we will limit ourselves to the different modeling approaches at the continuum level (micro-macroscopic and system-level simulations). In summary, there are computational models that are developed primarily for the lower-length scales (atomistic and mesoscopic) which do not scale to the system-level. The existing models at the macroscopic or system-level are primarily based on electrical circuit models or simple lD/pseudo-2D models [17-24]. The ID models are limited in their ability to capture spatial variations in permeability or conductivity or to handle the multidimensional structure of recent electrode and solid electrolyte materials. There have been some recent extensions to 2D [29-31], and this is still an active area of development As mentioned in a recent Materials Research Society (MRS) bulletin [6], errors arising from over-simplified macroscopic models are corrected for when the parameters in the model are fitted to real experimental data, and these models have to be improved if they are to be integrated with atomistic... 

Equivalent circuits (EC) should always be selected on the basis of an intuitive understanding of the electrochemical system, as long as they are based on the chemical and physical properties of the system and do not contain arbitrarily chosen circuit elements [1]. The condition of "the best fit" between the model and the experimental data still applies, but producing the best fit does not necessarily mean that the developed equivalent circuit model has physical meaning. One needs to apply knowledge of the physical processes... 

FIGURE 6-1 An equivalent circuit impedance model for a realistic experimental system... 

FTEIS is an experimentally convenient technique for such potentiodynam-ic impedance measurements. In this approach, a series of Nyquist (or Bode) spectra is obtained in parallel with the recording of CV data. DC voltage-dependent electrode equivalent-circuit models can be developed through CNLLS analysis of the impedance spectra. However, in FTEIS postexperiment processing is often cumbersome and inconvenient, and FTEIS limits the amplitude of the excitation signal to imder 10 mV (for aqueous solutions) because of the nonlinearity of the electrochemical system. 

An important aspect of the use of the computer here comes about in the following way. Few of the components of the various surface elements are known accurately. On the other hand, arough idea of these quantities is known from experiment on other systems and from theory. A computer can be programmed with a range of reasonable numbers for the R s and the Cs of each of the circuit elements concerned and asked to find those values which, for the given model, fit the experimental impedance curves. 

The interpretation of measured data for Z(oi) is carried out by their comparison with predictions of a theoretical model based either on the (analytical or numerical) integration of coupled charge-transport equations in bulk phases, relations for the interfacial charging and the charge transfer across interfaces, balance equations, etc. Another way of interpretation is to use an -> equivalent circuit, whose choice is mostly heuristic. Then, its parameters are determined from the best fitting of theoretically calculated impedance plots to experimental ones and the results of this analysis are accepted if the deviation is sufficiently small. This analysis is performed for each set of impedance data, Z(co), measured for different values of external parameters of the system bias potentials, bulk concentrations, temperature... The equivalent circuit is considered as appropriate for this system if the parameters of the elements of the circuit show the expected dependencies on the external parameters. 

In an early work by Kottke and Niiler (1988), a cellular model was used to simulate the combustion wave initiation and propagation for the TH-C model system. The interactions between neighboring cells were described by the electrical circuit analogy to heat conduction. At the reaction initiation temperature (i.e., melting point of titanium), the cell is instantly converted to the product, TiC, at the adiabatic combustion temperature. The cell size was chosen to be twice as large as the Ti particles (44 /xm). Experimentally determined values for the green mixture thermal conductivity as a function of density were used in the simulations. As a result, the effects of thermal conductivity of the reactant mixture on combustion wave velocity were determined (see Fig. 21). Advani et al. (1991) used the same model, and also computed the effects of adding TiC as a diluent on the combustion velocity. 

Such circuits are constructed on the basis of three elementary units a spring, a dashpot and a slider, which are sketched in fig. 3.50. Following computer language, we call these pictures icons. Icon (a) mimics a purely elastic spring, icon (b) the purely viscous movement of a piston in a viscous liquid. The slider (c) represents a system with a yield stress, i.e. where a minimum force is required to achieve flow. Here, we shall only consider icons (a) and (b). In mechanical models we construct circuits consisting of a number of springs and a number of dashpots, arranged in such a way that the experimental observations are optimally accounted for. The two simplest circuits are sketched in fig. 3.51a and b. 

FRA systems are versatile, and they can be controlled to acquire and analyse the data required to construct Mott-Schottky plots, for example. Unfortunately, the ease of use of FRA-fitting software can lead to errors of interpretation that arise from a failure to relate fitting elements to the physical system. Several equivalent circuits may give the same frequency-dependent impedance response. No a priori distinction between degenerate circuits is possible, ft is necessary to study the system response as a function of additional experimental variables (DC voltage, concentration, mass transport conditions etc.) in order to establish whether the circuit elements are related in a predictable way to a model of the physical system. 

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