Mathematical modeling electrochemical

Weber, A.Z. and Newman, J. (2004) Transport in polymer-electrolyte membranes II. Mathematical model./. Electrochem. Soc., 151 (2), A311-A325. 

Scale- Up of Electrochemical Reactors. The intermediate scale of the pilot plant is frequendy used in the scale-up of an electrochemical reactor or process to full scale. Dimensional analysis (qv) has been used in chemical engineering scale-up to simplify and generalize a multivariant system, and may be appHed to electrochemical systems, but has shown limitations. It is best used in conjunction with mathematical models. Scale-up often involves seeking a few critical parameters. Eor electrochemical cells, these parameters are generally current distribution and cell resistance. The characteristics of electrolytic process scale-up have been described (63—65). 

In maldug electrochemical impedance measurements, one vec tor is examined, using the others as the frame of reference. The voltage vector is divided by the current vec tor, as in Ohm s law. Electrochemical impedance measures the impedance of an electrochemical system and then mathematically models the response using simple circuit elements such as resistors, capacitors, and inductors. In some cases, the circuit elements are used to yield information about the kinetics of the corrosion process. 

In recent years the mechanism of crevice has been mathematically modelled and a more thorough understanding of the corrosion processes has been evolved . From such mathematical modelling it is feasible to predict critical crevice dimensions to avoid crevice corrosion determined with relatively simple electrochemical measurements on any particular stainless steel. 

Very little work has been done in this area. Even electrolyte transport has not been well characterized for multicomponent electrolyte systems. Multicomponent electrochemical transport theory [36] has not been applied to transport in lithium-ion electrolytes, even though these electrolytes consist of a blend of solvents. It is easy to imagine that ions are preferentially solvated and ion transport causes changes in solvent composition near the electrodes. Still, even the most sophisticated mathematical models [37] model transport as a binary salt. 

These two amphoteric rules play an important role both in classical and in electrochemical promotion as further discussed at the end of this Chapter and in the mathematical modeling of Chapter 6. 

C.G. Vayenas, S. Brosda, and C. Pliangos, Rules and Mathematical Modeling of Electrochemical and Chemical Promotion 1. Reaction Classification and Promotional Rules,/. Catal., in press (2001). 

C.G. Vayenas, and S. Brosda, Electrochemical promotion Experiment, rules and mathematical modeling, Solid State Ionics, submitted (2001). 

C.G. Vayenas, and G. Pitselis, Mathematical Modeling of Electrochemical Promotion and of Metal-Support Interactions, I EC Research 40(20), 4209-4215 (2001). 

MATHEMATICAL MODELLING OF ELECTROCHEMICAL PROMOTION AND CLASSICAL PROMOTION... 

The mathematical model of equations (6.63) to (6.65) is in excellent qualitative agreement with experiment as shown in Figures 6.18 to 6.25. It describes in a semiquantitative manner all electrochemical promotion studies up to date and predicts all the local and global electrochemical and classical promotion rules LI, L2 and G1 to G7. 

MATHEMATICAL MODELING DIMENSIONLESS NUMBERS GOVERNING ELECTROCHEMICAL PROMOTION AND METAL-SUPPORT INTERACTIONS... 

The parameter a in Equation (11.6) is positive for electrophobic reactions (5r/5O>0, A>1) and negative for electrophilic ones (3r/0Oelectrochemical promotion behaviour is frequently encountered, leading to volcano-type or inverted volcano-type behaviour. However, even then equation (11.6) is satisfied over relatively wide (0.2-0.3 eV) AO regions, so we limit the present analysis to this type of promotional kinetics. It should be remembered thatEq. (11.6), originally found as an experimental observation, can be rationalized by rigorous mathematical models which account explicitly for the electrostatic dipole interactions between the adsorbates and the backspillover-formed effective double layer, as discussed in Chapter 6. 

Since electrochemical processes involve coupled complex phenomena, their behavior is complex. Mathematical modeling of such processes improves our scientific understanding of them and provides a basis for design scale-up and optimization. The validity and utility of such large-scale models is expected to improve as physically correct descriptions of elementary processes are used. 

Palusinski, OA Allgyer, TT Mosher, RA Bier, M Saville, DA, Mathematical Modeling and Computer Simulation of Isoelectric Focusing with Electrochemically Defined Ampholytes, Biophysical Chemistry 13, 193, 1981. 

Verbrugge MW, Tobias CW (1985) A mathematical model for the periodic electrodeposition of multicomponent alloys. J Electrochem Soc 132 1298-1307... 

Mathematical models have been developed [1144—1146,1623]. The scale formation of iron carbonate and iron monosulfide has been simulated by thermodynamic and electrochemical models [49,1144,1154,1893]. 

Darling RM, Meyers JP. 2005. Mathematical model of platinum movement in PEM fuel cells. J Electrochem Soc 152 A242-A247. 

Meyers, J. P. Villwock, R. D. Darling, R. M. Newman, J. In Advances In Mathematical Modeling and Simulation of Electrochemical Processes and Oxygen Depolarized Cathodes and Activated Cathodes for Chlor-Alkall Processes, Van Zee, J. W., Puller, T. P., Poller, P. C., Hine, P., Eds. The Electrcohemical Society Proceedings Series Pennington, NJ, 1998 Vol. PV 98-10. 

M.E. Coltrin, RJ. Kee, and G. H. Evans. A Mathematical Model of the Fluid Mechanics and Gas-Phase Chemistry in a Rotating Disk Chemical Vapor Deposition Reactor. J. Electrochem. Soc., 136(3) 819-829,1989. 

Bessette N.F., Wepfer W.J., Winnick J. (1995) A mathematical-model of a solid oxide fuelcell. Journal of the Electrochemical Society 142, 3792-3800. 

Nonlinear problems frequently arise in engineering, but many texts are oriented towards linear problems due to the difficulty of non-linearity. For chemical systems as well as electrochemical systems, the mathematical models are typically nonlinear and even strongly nonlinear due to the nature of kinetics influenced by transport phenomena. 

A non-linear mathematical model, which is a set of ordinary differential equations, for the process in the SPBER was developed.19 The model accounts for the heterogeneous electrochemical reaction and homogeneous reaction in the bulk solution. The lateral distributions of potential, current density and concentration in the reactor were studied. The potential distribution in the lateral dimension, x, of the packed bed was described by a one dimensional Poisson equation as ... 

If a chemical reaction is operated in a flow reactor under fixed external conditions (temperature, partial pressures, flow rate etc.), usually also a steady-state (i.e., time-independent) rate of reaction will result. Quite frequently, however, a different response may result The rate varies more or less periodically with time. Oscillatory kinetics have been reported for quite different types of reactions, such as with the famous Belousov-Zha-botinsky reaction in homogeneous solutions (/) or with a series of electrochemical reactions (2). In heterogeneous catalysis, phenomena of this type were observed for the first time about 20 years ago by Wicke and coworkers (3, 4) with the oxidation of carbon monoxide at supported platinum catalysts, and have since then been investigated quite extensively with various reactions and catalysts (5-7). Parallel to these experimental studies, a number of mathematical models were also developed these were intended to describe the kinetics of the underlying elementary processes and their solutions revealed indeed quite often oscillatory behavior. In view of the fact that these models usually consist of a set of coupled nonlinear differential equations, this result is, however, by no means surprising, as will become evident later, and in particular it cannot be considered as a proof for the assumed underlying reaction mechanism. 

Verbrugge M, Liu P. Microstructural analysis and mathematical modeling of electric double-layer supercapacitors. Journal of the Electrochemical Society 2005 152(5) D79-D87. 

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