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- State estimation in Volmer–Heyrovsky reactions coupled with sorption processes: Application to the hydrogen reaction

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<ul><li><p>yio</p><p>R.</p><p>, U</p><p>Elec</p><p>300</p><p>rme 14</p><p>Abstract</p><p>systems, the determination of H concentration dependencewith respect to position and time is important. In the case</p><p>directly related to the state of charge (SOC) of the elec-</p><p>approximated by ordinary linear dierential equations withconstant parameters. In this way, having previous knowl-edge of initial and boundary conditions, the state variablessuch as surface concentration of intermediate species andbulk concentration of H may be well predicted. However,even when knowing the initial conditions, the presence of</p><p>* Corresponding author. Tel./fax: +54 299 4488305.E-mail addresses: bcastro@inifta.unlp.edu.ar (B.E. Castro), milocco@</p><p>uncoma.edu.ar (R.H. Milocco).</p><p>Journal of Electroanalytical Chem</p><p>Journal ofThe study of hydrogen evolution, sorption, and diu-sion on metals and alloys has been the subject of numerousscientic publications. These processes are related to theoperation of hydrogen storage materials in rechargeablebatteries such as Ni/metal hydride (MH) [13]. They arealso related to the ingress of H into ferrous alloys, whichis a major cause of embrittlement and damage coupled tometallic corrosion in many technological processes [4]. Inboth cases, the mechanism for hydrogen evolution andadsorption may be described by the VolmerHeyrovskyTafel scheme coupled to subsequent absorption and diu-sion of H atoms into the metallic substrate [57]. For both</p><p>trode. In the case of systems undergoing corrosion pro-cesses, the hydrogen content is related to the failure ofthe material. Typically the SOC of the battery is desiredto be kept within appropriate limits, for example20% 6 SOC 6 95%, so the estimation of the SOC is essen-tial for the battery to operate within these safe limits.</p><p>In this work we study the modeling and state observa-tion of hydrogen evolution and absorption reactions cou-pled to H diusion. The reactions shall be described interms of the mechanism associated with absorption anddiusion processes. Using spatial discretization of themetallic substrate, the diusion dierential equations areIn this work, a procedure to estimate variables of electrochemical systems involving adsorption and absorption of intermediate speciescoupled to diusion is presented. The method is model-based and needs potential and current measurements. Since the system is linearand time variant, the well known Kalmans lter theory is used to estimate the variables. An analysis of theoretical observability showsthe procedure success even in the case of constant potential/current. The error estimation converge asymptotically to zero for slow timevariations. The procedure is useful to estimate H concentrations in metallic substrates during H evolution and insertion. To illustrate theproposed procedure, state observations of the hydrogen concentration on steel and metal hydride electrode are presented. In the last case,the procedure is used to estimate the state of charge of the electrode. 2007 Elsevier B.V. All rights reserved.</p><p>Keywords: State of charge; Observability; Metal hydride; Hydrogen reaction; Kalman lter</p><p>1. Introduction of metal hydride electrodes, the H concentration prole isState estimation in VolmerHesorption processes: Applicat</p><p>B.E. Castro a,a Instituto de Investigaciones Fisicoqumicas Teoricas y Aplicadas (INIFTA)</p><p>b Grupo Control Automatico y Sistemas (GCAyS), Departamento de</p><p>Buenos Aires 1400, 8</p><p>Received 7 March 2006; received in revised foAvailable onlin0022-0728/$ - see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jelechem.2007.01.002rovsky reactions coupled withn to the hydrogen reaction</p><p>H. Milocco b,*</p><p>niversidad Nacional de La Plata. Suc 4, CC16 (1900), La Plata, Argentina</p><p>trotecnia, Facultad de Ingeniera, Universidad Nacional del Comahue,</p><p>Neuquen, Argentina</p><p>30 December 2006; accepted 10 January 2007January 2007</p><p>www.elsevier.com/locate/jelechem</p><p>istry 604 (2007) 18</p><p>ElectroanalyticalChemistry</p></li><li><p>lecdisturbances may produce dierences between predictedand real values. In a more realistic scenario where initialconditions and disturbances are unknown, state estimationby direct model simulation is imprecise. Instead, using themodel and ltering theory, it is possible to estimate thestate evolution by just measuring output variables likecurrent and voltage [8,9]. In this paper the state observa-tion problem is solved by using the Kalmans lter fornon-stationary systems [10]. For this purpose, the modelwas written as a lineal time/dependent set of equations.</p><p>The paper is organized as follows: in Section 2, electro-chemical mechanisms are mathematically modeled as a setof nonlinear dierential equations in terms of mass andcharge balances. Subsequently, the model equations arespatially discretized and represented as ordinary nonlineardierential equations with time varying parameters. After-ward, reformulation using the measured current allow us toset up the model as linear and time-varying, which is usedproperly in the Kalmans lter. In Section 3, the observ-ability condition is analyzed in the context of the Kalmanslter. In Section 4, the state estimation procedure is per-formed in two dierent electrochemical systems by usingmodel-based simulations of real electrodes whose parame-ters were reported in the literature. The rst one deals withH2 evolution on an AISI 1045 planar steel electrode in0.1 M NaOH [11], and the second one, with potentiostaticvariations of a metal hydride electrode constituted byMmNi3.6Co0.8Mn0.4Al0.3 quasi-spherical alloy particles, in6 M KOH [12]. Finally, conclusions are presented in Sec-tion 5.</p><p>2. Model formulation</p><p>Hydrogen evolution coupled to insertion in metallic sub-strates constitutes an electrochemical intercalation processinvolving charge transfer steps, adsorption and absorptionof electroactive species, and mass transfer. The process canbe modeled as follows [5,7]: in the rst step, water reactswith a surface metal atom (M) producing an adsorbedintermediate (MHad) on the surface. This electroreductionstep is described by the Volmer reaction and can be repre-sented by the following electron-transfer equation:</p><p>H2OM e ()k1</p><p>k1MHad OH; 1</p><p>where ki k0i ebaigt and ki k0ieb1aigt with b = F/RT,ai the symmetry factor being a number in the interval[0,1], g(t) = E(t) Eeq is the overpotential Eeq is theHydrogen reaction equilibrium potential. In a second elec-troreduction step, adsorbed species react with water pro-ducing hydrogen evolution (Heyrovskys reaction) whichcan be described as</p><p>MHad H2O e ()k2</p><p>k2MH2 OH: 2</p><p>2 B.E. Castro, R.H. Milocco / Journal of EThe equilibrium kinetic constants related to Volmer andHeyrovsky steps are related in terms of [17]k01k02</p><p>k01k02 1: 3</p><p>In a third step, the hydrogen adatoms, MHad, are trans-ferred to a free interstitial site in the metal (S) just beneaththe metal surface. This process is parallel to the Heyrov-skys step. Calling (SHab) the absorbed atom of hydrogen,the absorption reaction may be written as follows:</p><p>MHad S ()k03</p><p>k03M SHab H sorption reaction; 4</p><p>where k03; k03 are both potential-independent rate con-</p><p>stants. Then, the equations for mass and charge balancesrepresenting the kinetics of the system are the following:</p><p>dhdt k1~h k1h k2h k2~h J0; tC ; 5</p><p>J0; t Ck03h~x k03x~h; 6I f FACk1~h k1h k2h k2~h; 7followed by diusion of MHad. J(0, t) is the ux of hydro-gen diusing from the surface to the interior of the metal.Let us assume that h(t) 2 [0,1] is the surface coverage ofthe intermediate species MHad and ~ht 1 ht thecorresponding free metal surface. Let us call x(z,t) the frac-tional concentration of SHab species, being xz; t cz; t=c, where c is the maximum SHab concentration,being z the spatial position and t, the time. x(z,t) isexpressed adimensionally in the interval [0, 1]. The comple-mentary concentration is given by ~xz; t 1 xz; t,which represents the fractional concentration of availablevacant sites for H in the bulk; J(z,t) is the ux of hydrogendiusing from the surface to the interior of the metal atspatial position z and time t; C is the MHad maximum sur-face concentration; If is the faradaic current; F is theFaraday constant; and A is the active electrode area.</p><p>To complete the model given by Eqs. (5)(7) we need toinclude the equations describing hydrogen diusionaltransport in the metal substrate. This may be expressedby Ficks rst and second laws, which in the case of spher-ical geometry corresponds to [13]</p><p>Jz; t Dc oxz; toz</p><p>; 8oxz; t</p><p>ot D o</p><p>2xz; toz2</p><p> 2zoxz; toz</p><p> ; 9</p><p>where D is the diusion coecient. Using Eq. (8) in (9) weget</p><p>oxz; tot</p><p> 1cz2</p><p>oz2Jz; t oz</p><p>: 10</p><p>The analytical solution of Eq. (10) is complex [13]. Then, inorder to simplify, it can be approximated into a set of or-dinary dierential equations by using a spatial discretiza-tion. Spatial discretization is a very well known method</p><p>troanalytical Chemistry 604 (2007) 18to approximate partial dierential equations in ordinarydierential equations, for details see [6,7]. Eq. (10) can be</p></li><li><p>lecdiscretized along the space variable z by considering Nslices of the metal with thickness Dz, as illustrated in Fig. 1.</p><p>If each cell is small enough, the concentration x(zi,t) inthe ith cell (0 6 i 6 N 1) can be considered constant withinput and output hydrogen ow given by J(zi1,t) andJ(zi,t), respectively. In Fig. 1, z = 0 corresponds to the elec-trode surface. Using this approximation, Eqs. (8) and (10)can be written as</p><p>Jzi; t DcDz xzi1; t xzi; t; 11dxzi; t</p><p>dt 1</p><p>cz2iDzz2i1Jzi1; t z2i Jzi; t: 12</p><p>By replacing (11) in (12) and considering a boundary con-dition for the ux J(zN, t) = 0, the following set of ordinarydierential equations for the hydrogen concentration pro-le is fullled:</p><p>dxz0;tdt a11d0xz0; ta11d0xz1; t Jz0; t=cDz</p><p>dxz1;tdt a11xz0; ta111d1xz1; ta11d1xz2; t... ...</p><p>dxzi;tdt a11xzi1; ta111dixzi; ta11dixzi1; t... ...</p><p>dxzN1;tdt a11xzN2; ta111dN1xzN1; ta11dN1xzN ; t</p><p>dxzN ;tdt a11xzN1; ta11xzN ; t</p><p>13</p><p>where a11 D=Dz2; C cDz, and di z2i1=z2i i1=i2. Notice that in the case of spherical diusion at theparticle centre the condition is always J(zN,t) = 0, whichis equivalent to dx(zN,t)/dz = 0. Thus, the approximationx(zN,t) = x(zN+1,t) holds and the last equation of (13) fullsby direct inspection. Latter we will give the conditions forthe case of planar diusion. If the interfacial area of theelectrode is large, as in battery electrodes, a substantialcontribution of the double layer capacitive current Ic(t) isexpected, which is modeled in parallel with the faradaiccurrent, If(t), according to the following dynamics:</p><p>Ict Cdl dEdt</p><p>; 14</p><p>Fig. 1. Spatial discretization of the active lm.</p><p>B.E. Castro, R.H. Milocco / Journal of Ewhere Cdl is the double layer capacity and E(t) correspondsto the potential at the electrode interface. The total mea-sured current I(t) is the sum of both the faradaic and thecapacitive currents, which leads to</p><p>It V t EtRe</p><p>; 15where V(t) is the applied potential and Re corresponds tothe ohmic resistance between the working and the referenceelectrodes. Accordingly, the dynamics of the interfacialpotential may be expressed as</p><p>dEdt</p><p> V t EtReCdl</p><p> I ftCdl</p><p> V tReCdl</p><p> EtReCdl</p><p> b21tCdl</p><p>ht b22tCdl</p><p>; 16</p><p>where V(t) is the measured potential and</p><p>b21 FACk1t k1t k2t k2t;b22t FACk1t k2t:</p><p>The complete model is then given by Eqs. (5)(7), (13),(15), and (16). It is useful to note that for slow time varia-tions where dE(t)/dt 0, the Eq. (16) become If(t) = I(t).This condition will be used latter. The resistance Re doesnot aect the derivative of the potential since the rst termof (16) is governed by (15).</p><p>The model obtained above corresponds to the more gen-eral case in which the geometry is spherical. In the casewhere a planar geometry is considered, Eqs. (9) and (10)are replaced by</p><p>oxz; tot</p><p> D o2xz; toz2</p><p>17</p><p>and</p><p>oxz; tot</p><p> 1coJz; t</p><p>oz; 18</p><p>respectively. The model equations in this case are given byEqs. (5)(7), (13) considering di = 1 "i and (15). In thecase of linear diusion, at the limit, there is eitherJ(zN,t) = 0 or dx(zN,t)/dt = 0. The rst case coincides withEq. (13) derived for spherical diusion, in the second casethe last equation of (13) should be replaced by dx(zN,t)/dt = 0. The same approximation as above is made with re-spect to the potential variations dE(t)/dt 0.</p><p>Proper simulation of the state variables can be done bysolving numerically the set of model equations with giveninitial conditions. However, the initial conditions are oftenunknown and there are also disturbances that aect thestate evolutions which are not taken into account in themodel. Thus, it is expected for simulated-evolutions tobehave dierently from the real ones. In order to overcomethese diculties, we propose to use a model based observeremploying the measured potential V(t) and current I(t).The observer theory for linear systems was a very activeresearch area around the sixties and up to day is includedas a basic contents of graduated courses in engineering.However, it is an active research area for nonlinear sys-tems, [8,9]. In the following section we propose to use the</p><p>troanalytical Chemistry 604 (2007) 18 3well known Kalmans lter theory for estimation ofvariables.</p></li><li><p>0 0 0 0 0 0 0 0 0 0 0 0 </p><p>b21Cdl</p><p>0 0 0 0 0 </p><p>664 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0. .</p><p>. . .. . .</p><p>. . .. ..</p><p>.</p><p> a11 a111dN1 a11dN1 0 0 a11 a11 0 0 0 0 1ReCdl</p><p>377777777777775</p><p>;</p><p>Bt a2; 0; . . . ; 0; V tReCdl b22Cdl</p><p> T;</p><p>lec3. Estimation of state variables</p><p>Consider a linear time variant system described by thefollowing state space formulation:</p><p>_nt Atnt Bt wt;It Ctnt Dt rt; 19</p><p>where n(t) is a column vector of n state variables, A(t), is amatrix of dimension n n, B(t) is a column vector of lengthn, C(t) is a row vector of dimension n, and D(t) is a scalar.Consider also that the entries of matrices and vectors A(t),B(t), C(t) and D(t) are known nonlinear functions of thepotential E(t) as we will see later in Eqs. (22). Lets assumeunknown initial conditions, disturbances w(t), and mea-surement noise r(t). We call disturbances to all possibleundesirable environmental stochastic variations aectingthe system.</p><p>Consider the system to be observable. Observable meansthat it is possible to estimate the states from some mea-sured variables, which in our case are potential and cur-rent. In the next subsection the observability of thesystem is analyzed in more detail. Based on current andpotential measurement the goal is to obtain estimationsof the state vector n(t) such that the error et nt n^t between real, n(t), and estimated, n^t valuesbe minimum. From the Kalman lter theory it is wellknown that the optimal estimation is given by the followingpair of dierential equations:</p><p>_^nt Atn^t Bt S1tCTtR1It Ctn^t Dt;20</p><p>_St WSt ATtSt StAt CTtR1Ct; 21These couple of equations are solved numerically by start-ing with an arbitrary pair of initial values (n(0),S(0)). Thematrix S(t) is positive semidenite at time t and R and Ware given positive semi-denite matrices. In the ideal casewhere the disturbances does not aect the system,w(t) = 0, using (20) and (21) the error estimation, due tothe...</p></li></ul>