Fahidy, Approaches to the Study of Electrochemical Process Instability - A Review, 2006

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ISSN 1023-1935, Russian Journal of Electrochemistry, 2006, Vol. 42, No. 5, pp. 506511. MAIK Nauka/ Interperiodica (Russia), 2006.Published in Russian in Elektrokhimiya, 2006, Vol. 42, No. 5, pp. 567573.

506

INTRODUCTION

Instability is a word of many meanings. It can beinterpreted as permanent motion never reaching a finalsteady state, or irregular trajectories generated by prop-agation patterns, etc., and, in general, any state ofinconstancy. In this paper, instability is confined tooscillatory behavior and waves, liquid flow, and elec-trochemical reactor dynamics due to internal or exter-nal generating factors. The objective is to provide anoverview of intensive experimental and theoreticalprojects carried out during the fourth quarter of 20thcentury by the authors research group. During theseyears, important new horizons of modern electrochem-istry were opened up in this domain also by researchteams operating elsewhere.

The group conducted its investigation of instabilityphenomena via a two-prong program. Firstly, flowvisualization techniques based on dyes and laser illumi-nation were developed to study electrode reaction-induced spatiotemporal convective flow propagation inliquid media. The study of the effect of externallyimposed magnetic fields on such patterns has specifi-cally enhanced current understanding of magnetohy-drodynamic (MHD) interactions with electrolytic pro-cesses with particular respect to the operation of mag-netoelectrolytic reactors. In the second avenue ofresearch, pathways of theoretical analysis were openedby an extensive pursuit of the application of (classical)Lyapunov stability theory, and more recent techniquesof nonlinear dynamics, e.g., phase-plane methods, frac-tional Brownian motion (FBM) theory, and bifurcationtheory. A major, and by no means unexpected, resultthat electrochemical instability phenomena are not

fully explained by purely theoretical models, attests tothe challenging complexities of the subject matter, andthe need for continuing research into it [1].

EXPERIMENTAL INVESTIGATIONS

Oscillations and fluctuations with amplitudes con-fined in space do not represent instability in a strictmathematical sense, but they usually indicate nonad-ventitious physical influences on an otherwise stableprocess. Oscillatory response to regular periodic excita-tion may signify instability, e.g., in the case of magne-toelectrolysis with rough cathode surfaces underimposed periodic magnetic fields [2].

Convective flow instability generated at an electrodesurface by changing the interfacial pH due to passage ofcurrent was extensively examined by means ofchemoanalytical indicators tagging the convectingelements due to their characteristically dark color atalkaline pH values [3]. A subsequent series of investi-gations [35], including several cathode geometries [6]and inclined cylindrical electrodes [7], confirmed theenhancing effect of coupled electric and magnetic fieldson vortex formation and convective flow propagation.Computer-based image processing techniques com-bined with layer illumination via low power (about 250

mW) laser sheets provided an excellent tool for themonitoring of vortex and solid-particle propagationaccompanying anodic dissolution and precipitate for-mation [8].

Oscillatory anodic currents, observed during thedissolution of copper into cyanide ion-carrying electro-lytes, exhibit characteristic patterns and are highly sen-sitive to magnetic field imposition [9]. Time series, e.g.,FBM methods [10] employing Pox diagrams [11] wereemployed in the evaluation of numerous experimen-

Approaches to the Study of Electrochemical Process Instability:A Review*

T. Z. Fahidy

z

Department of Chemical Engineering, University of Waterloo, Waterloo, ON N2L 3G1 Canada

Received May 11, 2005

Abstract

This paper summarizes experimental and theoretical investigations conducted by the authorsresearch group, prior to his official retirement, on various aspects of electrochemically induced convective-flowand oscillatory instability and the stability of electrochemical reactors.

DOI: 10.1134/S1023193506050089

Key words

: electrochemical instability, convective instability, oscillatory instability, instability in electrochem-ical reactors

* The text was submitted by the author in English.

z

Authors email: [email protected].


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APPROACHES TO THE STUDY OF ELECTROCHEMICAL PROCESS 507

tally observed oscillation patterns [1218] accountingalso for the nature of the electrolyte [19]. Significantimpetus to oscillation analysis was gained by the adap-tation of confocal image processing [20] and laser bom-bardment [21, 22] of electrodes. The spilling of electro-lyte over a partitioning wall is perhaps the most dra-matic manifestation of instability generated by surfacewaves in a coaxial cylindrical magnetoelectrolytic cell

[23].

THEORETICAL INVESTIGATIONS ANDMODELS

Convective flow and parametric instability.

Onaccount of its intrinsic complexity, MHD-induced con-vective flows can be traced mathematically only to alimited extent by means of fundamental physical laws[24, 25]. Arguments based on the nonzero nature of the

rot

(

j B

)/

term (

, fluid density) in the vorticity equa-tion [26, 27] applied to the electrode/electrolyte inter-face layer predict qualitatively the existence of unusualpropagation patterns. The cross product of the currentdensity vectorj

and the magnetic flux density vectorB

,called the MHD body force density, and a fundamentalquantity in MHD theory, is nonzero due to the nonuni-form spatial distribution of its component vectors in theinterface layer. Their progressively increasing nonuni-formity during the passage of electric charges isresponsible for the snowballing nature of vortex for-mation and propagation.

Oscillatory instability. The nature of FBM-basedanalysis was briefly explained in [17, Section 3.6] andin Appendices A and B of [15]. The Hurst exponentH

[11], determined from the slope of a logarithmic plot ofrescaled rangeR/S

of time series data versus time lag s

is a quantitative measure of long-term correlation in afluctuating time series. It is also a fundamental param-eter in the kernel of the definition integral of a randomfunctionB

H

(

t

)

of time [28]:

(1)

where

is the gamma function, and

(2)

Within the 1/2


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RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 42 No. 5 2006

FAHIDY

diagram, illustrated in Fig. 2, yieldsH0.78, indicat-ing mediumstrong persistence.

Experimental data [9, 1222] provide considerableevidence forHbeing confined to the domain of persis-tence in the investigated systems. The cyclic forma-tiondissolution process involving CuO, Cu2O, CuCl orCuBr, and CuSCN species is responsible for oscilla-tions, but their exact role in determining the magnitudeof the Hurst exponent is not known. Analyses based onthe phase portrait/Poincar map approach [18, 2931],low-order Fourier expansions [18, 32], and BoxJen-kins differencing of time series [13, 33] suffer from thesame limitations.

A study of morphological instability of a planarmetal electrode under potentiostatic electrodepositionand electrodissolution [34] underlines the predominant

role of kinetic effects throughout the entire potentialrange, including the limiting plateau. Mapping ofparameter regions with a specific number of solutions[35] and different types of bifurcation diagrams demon-strate the usefulness of singularity theory for the analy-sis of complex electrodeposition of metals, e.g., zinc[36].

Instability in electrochemical reactors. The subjectmatter was intensively studied in the early 1980s bymeans of the celebrated stability theorems of Lyapunov

[37, 38]. The gist of the method is illustrated in theAppendix using a relatively simple example of alumped-parameter autonomous two-dimensional sys-tem.

In an electrochemical reactor with a single-electrodeprocess, the two state variables may be taken as activeion concentration and electrolyte temperature (a ratio-nal and convenient, but not an obligatory choice). Since

the reactor is driven by current, it is nonautonomous,and its Lyapunov functions require a more involvedform with respect to the Appendix. Stability conditionsfor the continuous-flow stirred tank electrolytic reactor(CSTER) model were studied [3941] via theLyapunov function

(6)

with positive definite matrices Q and M. This and amore incisive analysis [42] including electrolytic alloydeposition [43] and codeposition of two metal species

[44] underline the necessity of knowing the dependenceof electrolyte parameters and heat transport character-istics on temperature and concentration for a full utili-zation of Lyapunov analysis.

Considerably more challenging is the stability anal-ysis of plug flow electrolytic reactors (PFER) and elec-trochemical reactors with axial dispersion (ERAD).The challenge arises from their distributed parameternature requiring spatiotemporal governing equationsfor the concentration and temperature profile along thedominant space coordinate [4547]. Following thepath-finding work of Zhubov [48], Berger and Lapidus[49], and Weigand [50], Lyapunov functionals of the

form

, (7)

were employed to explore the size of AS regions, andregions of indeterminate stability, in terms of the statevariable vector w and positive-definite symmetricmatrix S. They are functions of the dimensionless axialcoordinatez/L, wherezis the axial coordinate andListhe active length of the reactor [51, 52].

Figure 3 demonstrates the involved nature of stabil-ity conditions in the case of an electrochemical reactorobeying the ERAD model. A (sudden) perturbation inelectrolyte concentration and/or temperature in regionsmarked Swith respect to its steady state will eventuallylead the reactor back to its steady-state position. If theperturbation point is located, however, in regionsmarkedI, the Lyapunov function in equation (7) is inca-pable of predicting stability or lack of it. Stabilitybehavior is expectably affected by the direction of per-turbation in temperature: the corresponding stabilitycondition expressed by equation (30) in [52] incorpo-

V x t,( ) xTQx TM ( ) dt

+=

V wTS ( )w ; z/Ld

0

1

=

log(R/S)

1.0

0.8

0.6

0.4

0.2

Slope:H0.78

0.6 0.8 1.0 1.2 1.4log(s)

Fig. 2.Quantitative illustration of the second phase of con-structing a Pox diagram. Oscillogram readings for varioustime instants T are collected for specified lag values andplotted on a logarithmic scale in term of the rescaled range

R/S (equations (3), (4)). Least-squares properties of theregression line: coefficient of determination, 0.94; RMS ofdeviations, 0.0094; slope, 0.7774; intersection, 0.25486.


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rates the exothermic or endothermic nature of the elec-trode reaction.

Electrochemical reactor instability due to reso-nance. The PFER can also be described in terms oftransfer function models written for any arbitrary axialposition [53]. The practically more interesting Gc(s,L)transfer function, related to the PFER exit, predicts anoscillatory frequency response of both gain amplitude

and phase angle. The gain-magnitude resonant frequen-cies are given by the implicit relationship

(8)

where = Tdfis the angular frequency related to phys-ical frequencyf,and is a parameter carrying physicalcharacteristics of the reactor and its transportation lagTd(the ratio of reactor length to the linear velocity ofthe electrolyte). Equation (8) applies to the specificcase of a constant current across the reactor axis atsteady-state conditions. In a practical PFER, isexpected to be very small; hence, 1, andequation (8) predicts the first three resonance positionsat excitation frequencies 1.71/Td, 9/Td, 15.45/Td,approximately. The distance between increasing reso-nance frequencies becomes progressively smaller.

Resonance-related local instability was shown to beapproximately predictable by the Guillemin frequencytransfer function approach [54] on the basis of experi-mental system step responses subjected to a number ofdifferentiation steps to obtain an aperiodic modulatedimpulse train. Various ramifications of the technique[5557] make it, at least in principle, attractive for thestudy of reactors which do not fit physically any partic-ular a priorimathematical model.

Electrochemical reactors subjected to random per-turbations.The instability problem may be defined asthe widening of the variance of a performance measure(e.g., active ion concentration and electrolyte tempera-ture) in the exit stream from a reactor, with respect tothe variance of a random input (e.g., inlet active ionconcentration and current) [58]. Using the systemdynamics framework [59, 60] provided by fundamentalprinciples [6165] of random function theory, electro-lyzer behavior was analyzed in terms of autocovariancefunctions for batch, CSTER, and two-element CSTERcascade in the case of white noise, Markovian, andquasi-Markovian fluctuations [66]. The approach indi-cates that tank electrolyzers are asymptotically wide-sense stationary when subjected to bounded stationaryrandom perturbations.

CONCLUSIONS

The success achieved in the two major avenues ofinstability studies, discussed in the preceding, bearswitness to the dedication and competence of numerousgraduate students and researchers of post-PhD status,with whom the author had the privilege to collaborate

( )sinh ( )cos ( )cos

-------------------------------------------2

2 2+-----------------=

h ( )cos

over many years. Their fresh ideas and experimentalskills were highly instrumental in contributing, even ifin a modest manner, to modern electrochemical scienceand engineering.

ACKNOWLEDGMENTS

Authors work was supported by the Natural Sci-ences and Engineering Research Council of Canada

(NSERC) and the University of Waterloo. The author isgrateful to Prof. A.D. Davydov of the Frumkin Instituteof Electrochemistry, Russian Academy of Sciences, forhis kind invitation to submit this paper.

APPENDIX

The dynamic equations for an autonomous two-dimensional (2D) system

(A.1)

possess critical points defined by the conditionf1=f2= 0in 2D state space (phase plane), in terms of state vectorxwith its elementsx1andx2being the state variables. If,at a critical point x0, a positive-definite function V(x)satisfies the conditions (i) V(x0) = 0; (2) V(x) > 0,xx0,at all times; and (iii) dV/dt=xgrad(V) < 0,xx0,xR,then the system is asymptotically stable (AS) withinspaceR, with reference to the critical pointx0. IfRis the

dx1dt

-------- f1 x1, x2( ),=

dx2

dt-------- f2 x1, x2( )=

S

I

T TS

C

S

I

TS

Fig. 3.Phase-plane portrait for the ERAD model. C, dimen-

sionless concentration; , electrolyte conductivity; T,dimensionless electrolyte temperature; Ts, dimensionlesssteady-state temperature;P, critical point.


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entire vector space (i.e., limV(x) , asx ),the system is asymptotically stable in the large ASL.

For instance, the Lyapunov function V= +

predicts that the system dx1/dt=x2, dx2/dt= x1 2x2with single critical point (0, 0) is ASL, inasmuch as Vispositive in the entire space, it grows to infinity as either

x1orx2reach infinity, and dV/dt= =4 is negative

everywhere. This result is borne out also by therepeated negative eigenvalues of 1, or equivalently,due to the negative trace and positive determinant of thesystem matrixAwith elements a11= 0, a12= 1, a21= 1,a22= 2.

Indeterminacy arises when the dV/dt< 0 condition(or its violation) cannot be established. For instance,

the stability of the system dx1/dt= x2, dx2/dt=

with critical point (0, 0) is impossible to ascertain with

V= + , because dV/dt= 0 in the entire state space.

However, the generalized function family V= +

, n = 1, 2, 3, , would predict widely different

regions of asymptotic stability. In this situation, thedV/dt < 0 condition is satisfied by the inequality

x2 > 0.

Among an infinite, in principle, number of possibleLyapunov function candidates, the quadratic forms V=xTPxand V=f(x)TIf(x)(the latter is widely known as theKrasovskii function [67]) were particularly successful;Pis an a prioriundefined symmetric positive definitematrix, and Iis the identity matrix (the simplest posi-tive definite symmetric matrix). The Krasovskii-typeLyapunov function tends to predict an RAS of conser-

vative size.

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Fahidy, Approaches to the Study of Electrochemical Process Instability - A Review, 2006