## Abstract

Asymptotic methods are used to analyse a recent numerical model for the electrochemical pickling of steel. Although the process is characterized by an excess of supporting electrolyte, the asymptotic structure of the solution turns out not to be the same as that given classically for such situations. In the original theory, the asymptotic expansions for the ionic concentrations and electric potential are regular and uniformly valid; here, singular perturbation theory is required to take account of the bulk electrolyte and concentration boundary layers adjacent to reacting surfaces. The reworked theory gives a leading-order problem that is solved numerically; the results are in excellent agreement with those of the earlier computations. Also, invoking the slenderness of the geometry yields, in a combined quintuple asymptotic limit, an analytical estimate for the current density that captures well the qualitative trends and that enables a rapid assessment of the effect of operating conditions on the process.

## 1. Introduction

Electrochemical pickling is a process that is used in the steel-making industry to remove a surface oxide layer that is formed during the manufacture of thin steel strips. In this process, an example of which is shown in figure 1, a steel strip is passed between pairs of anodic and cathodic electrodes in an electrochemically neutral electrolyte, usually sodium sulphate (Na_{2}SO_{4}). When a current is passed through the cell, the ‘pickling’ reaction, i.e. the removal of the oxide layer, thought to consist predominantly of chromium oxide (Cr_{2}O_{3}), occurs according to
1.1
as do other electrochemical reactions that result in the evolution of oxygen and hydrogen,
1.2
and
1.3
respectively. Further information on many aspects of the pickling of austenitic stainless steels can be found in the survey by Li & Celis (2003) and the thesis by Ipek (2006).

The realization that the industrial process is not particularly efficient (Henriet 1995) has led to attempts at mathematical modelling to optimize it. Following earlier work that focused on laboratory situations with a non-moving strip (Ipek *et al.*2002, 2006), Ipek *et al*. (2007) derived a model that accounts for the electrochemical reactions described above and for the translational motion of the steel strip. Three variations of the model were derived, which either included or omitted some or all of the leading chemical reactions that are thought to occur in the electrolyte bulk, and the model equations were solved numerically using finite-element methods. While this gave qualitatively reasonable results, the drawback of a purely numerical approach for this application is the long computation time. This is due both to the size of the system of partial differential equations that must be simultaneously solved for and the need to resolve the concentration boundary layers adjacent to reacting surfaces; in addition, artificial streamwise diffusion is necessary to maintain numerical stability, particularly as the strip velocity is increased. Furthermore, it should be noted that these drawbacks are already prevalent for the simplest of the three variants, which indicates that further meaningful extensions of this canonical model for electrolytic pickling, incorporating two-phase flow (Ipek *et al.* 2008) or further bulk reactions (Ipek 2006), will always be hindered by computational limitations.

More generally, electrolytic pickling is an example of a multi-ion electrochemical system in which ionic transport, by diffusion, migration and convection, and chemical bulk reactions occur simultaneously, and there is a growing literature in this topic devoted to developing efficient numerical schemes to tackle multi-ion problems (Georgiadou & Alkire 1994; Van den Bossche *et al*. 1995, 1996; Bortels *et al*. 1996; Georgiadou 1997; Dan *et al*. 2001; Volgin *et al*. 2001, 2002; Georgiadou & Veyret 2002; Nelissen *et al*. 2003, 2004; Volgin & Davydov 2007). In many systems of interest, either in commercial electrochemical cells or in studies of mass transfer or electrode kinetics (Newman & Thomas-Alyea 2004), the concentration of some of the ionic species is much greater than that of the others, e.g. CuSO_{4} with an excess of H_{2}SO_{4} as supporting electrolyte, where the concentration of SO_{4}^{2−} and H^{+} is much greater than that of Cu^{2+}. In such situations, a potentially attractive modelling alternative to large-scale numerics is an asymptotic approach, which can lead to a formulation requiring much lower computational cost; the earliest exposition of the asymptotics, which we shall refer to as supporting electrolyte theory, is due to Levich (1942), with a recent summary by Newman & Thomas-Alyea (2004). It is apparent that this theory ought also to apply for pickling since the bulk concentrations of Na^{+} and SO_{4}^{2−} ions are much greater than those of H^{+}, OH^{−} and Cr_{2}O_{7}^{2−} ions (Ipek *et al*. 2007). However, the case of pickling appears already to be more complex than many situations encountered in electrochemical modelling literature where supporting electrolyte theory has actually been applied, in that at least three minority ions need to be considered for a model of the complete cell; in situations encountered earlier, there was at most one minority ion, e.g. Cu^{2+} in the Cu^{2+}/H^{+}/SO_{4}^{2−} system considered by Bark (1990) and Bark & Vynnycky (2008).

The purpose of this paper is therefore to apply the idea of supporting electrolyte theory to the pickling model of Ipek *et al*. (2007). In §2, we recapitulate the model of Ipek *et al*. (2007), in the supplementary material we demonstrate why existing expositions of supporting electrolyte theory surprisingly *cannot* be applied to it. With a view to re-working the original theory, in §3, we non-dimensionalize the governing equations and propose a consistent asymptotic structure for the model, consisting of a bulk flow and boundary layers; in particular; this identifies a strategy for the numerical solution of the problem. In §4, we obtain analytically an estimate for the cell current density when both the electrolyte region and the strip are slender; in §5, we compare this, the numerical solution to the leading-order problem suggested by the analysis in §3 and the results of the earlier paper. Conclusions, in particular, regarding the generalization of our analysis to other electrochemical systems are drawn in §6.

## 2. Model

In this paper, we focus on the formulation of reduced model (1) in Ipek *et al*. (2007); since the equations were given there in the context of three models, here we recap the relevant equations, which are for the concentrations, *c*_{i}, of five ionic species (*i*=H^{+}, OH^{−}, SO_{4}^{2−}, Cr_{2}O_{7}^{2−}, Na^{+}), the electric potential in the electrolyte, *Φ*^{(e)}, and the electric potential in the steel strip, *Φ*^{(b)}. The model geometry is as given in figure 2 and consists of a two-dimensional region of length *L*^{(b)} and width *D*_{eb}+*d*; anode and cathode electrodes at *x*=0, each of length *L*, at a distance *D*_{ie} apart and held at electric potentials ±*U*/2, respectively; and a steel strip with electric conductivity *κ*^{(b)} that moves vertically upwards with speed *V*^{(b)} and is at a horizontal distance *D*_{eb} from the electrodes. Because of symmetry, only one-half of the electrode–steel strip section is modelled; thus, *d* denotes the half-width of the steel strip, so that the centreline of the strip lies at *x*=*D*_{eb}+*d*. The remainder of the boundary at *x*=0 is assumed to be impermeable and insulated. Electrolyte is assumed to enter at *y*=0 and exit at *y*=*L*^{(b)}. Characteristic values for *d*,*D*_{eb}, *D*_{ie}, *L*, *L*^{(b)}, *U*, *V*^{(b)} and *κ*^{(b)} are given in table 1.

### (a) Governing equations

Three mechanisms, diffusion, migration and convection, are assumed to contribute to the transport of ionic species in the electrolyte. Assuming the applicability of transport equations in dilute solutions (e.g. Newman & Thomas-Alyea 2004), the molar flux, **N**_{i}, of the ionic species *i* can be expressed via the Nernst–Planck equation as, for *i*=H^{+}, OH^{−}, SO_{4}^{2−}, Cr_{2}O_{7}^{2−}, Na^{+},
2.1
where **u** is the hydrodynamic velocity of the electrolyte, *D*_{i} the diffusion coefficient and *z*_{i} the charge number for species *i*. The quantities *F*(=96 485 C mol^{−1}), *R*(=8.314 J mol s^{−1}) and *T* are the Faraday constant, the universal gas constant and the absolute temperature, respectively. The differential material balance for species *i* is given, in steady state, by
2.2
so that
2.3
In addition, the solution is assumed to be electrically neutral, which is expressed by
2.4
Equations (2.3) and (2.4) then provide a consistent description of the transport processes in the dilute electrolyte. In addition, the current density, **i**, is calculated from the flux of charged species and is given by Faraday’s law as
2.5
Substituting equation (2.1) into equation (2.5), we can identify the electrolyte conductivity, *κ*^{(e)}, as
2.6
Also, **u** is taken to be the simplest possible profile that is consistent with the fixed electrodes and the translational motion of the steel strip; consequently, we take **u**=(*u*,*v*), where
2.7

For the potential field in the steel strip, conservation of charge gives 2.8

### (b) Boundary conditions

At reacting surfaces (see §2*b*(i)–(iii)), Faraday’s law is used to relate local ionic fluxes to the current density. At all of these interfaces, electrochemical reactions are assumed to proceed according to Tafel, rather than Butler–Volmer, kinetics, which seems reasonable in view of the high current densities (of the order of several kA m^{−2}) at which the process is typically operated. In the following, *i*_{O2}, *i*_{H2} and *i*_{Cr2O3} are the current densities (A m^{−2}), *i*_{0,O2} and *i*_{0,H2} are the exchange current densities (A m^{−2}) for reactions (1.2) and (1.3), respectively, α_{O2} and α_{H2} are the transfer coefficients and and are the equilibrium potentials. Values for α_{O2},α_{H2}, *i*_{0,O2} and *i*_{0,H2} are determined from experimental polarization curves (see table 2 and Ipek (2006) for details). Obtaining corresponding kinetic data to model reaction (1.1) explicitly is, however, much trickier: reactions (1.1) and (1.2) occur simultaneously, so that the measured current is due to both reactions; and, to the best of our knowledge, the kinetic data for reaction (1.1) is not documented in the electrochemical literature. Therefore, the approach used was to carry out extensive experiments using steel type 316 in a small test cell to determine what fraction, *Λ*, of the total current is attributable to reaction (1.1); that data then form the basis of a constitutive relation for *i*_{Cr2O3} that is applied pointwise at the surface of the steel strip in the model presented here. Experimental details and results are given elsewhere (Ipek 2006; Ipek *et al*. 2005). Those results indicate that, for the current density range that is of interest in industrial pickling, usually no more that 10–15% of the overall current is attributable to reaction (1.1); therefore, to enable direct comparison with computed results in Ipek *et al*. (2007), we set *Λ*=0.1.

#### (i) Anode

Taking into account the oxygen evolution, reaction (1.2) gives
2.9
2.10
2.11
2.12
and
2.13
where
2.14
with where pH^{eq} (pH^{eq}=7).

#### (ii) Cathode

From the hydrogen evolution reaction (1.3), we have
2.15
2.16
2.17
2.18
and
2.19
where
2.20
with where pH^{eq}.

#### (iii) Strip/electrolyte

From the oxygen and hydrogen evolution, as well as chromium oxide removal according to equation (1.1), we obtain 2.21 2.22 2.23 2.24 and 2.25 where 2.26 2.27 and 2.28 with Continuity of the normal current at the steel strip/electrolyte interface gives the boundary condition where 2.29

#### (iv) Inlet

At the inlet at *y*=0, the concentrations must be such that there is a local equilibrium; this will imply that
2.30
where and *k*^{b}_{H2O} are the forward and backward reaction constants, respectively, for the reaction
In practice, one would expect to be able to measure *c*_{Na2SO4} in the homogeneous electrolyte entering the domain, from which we obtain immediately
2.31
Moreover, we will take Also, the electroneutrality condition is
2.32
which reduces to giving
2.33
The values for and are as given in table 3, and the boundary condition is written as
2.34

#### (v) Outlet

At the outlet boundary, we assume that the convective flux in the axial direction of the channel, for all ionic species, is dominant. Consequently, we set the sum of the diffusion and migration fluxes to zero, i.e. for *i*=H^{+}, OH^{−}, SO_{4}^{2−}, Cr_{2}O_{7}^{2−}, Na^{+},
2.35
Combining this with the electroneutrality condition, we obtain in simpler form
2.36
and
2.37

#### (vi) Insulated and symmetry boundaries

At the boundary of the electrolyte domain that is assumed insulated, i.e. *x*=0, *L*≤*y*≤*L*+*D*_{ie}, we have, for *i*=H^{+}, OH^{−}, , , Na^{+},
2.38
Similarly at the boundaries of the steel strip, which are either symmetric planes or insulated, we have
2.39

## 3. Analysis

With a view to determining the correct asymptotic structure for this problem, we begin by non-dimensionalizing with
3.1
Note, however, that although the computations of Ipek *et al*. (2007) suggest that *ϕ*^{(e)} will be O(1) for the values of *U* used there, *ϕ*^{(b)}≪1.

### (a) Electrolyte

Substituting equation (3.1) into equation (2.3) gives
3.2
where
Noting from Ipek *et al*. (2007) that *D*_{i}∼10^{−9} m^{2} s^{−1}, and hence that we use table 1 to observe that and *δ*∼10^{−2}; this motivates investigating the asymptotic regime for which and *δ*≪1. In particular, since we can expect boundary layers and we will need to consider these and the bulk separately.

#### (i) Bulk

In the bulk, we expect
3.3
where, at leading order,
3.4
and
3.5
Consequently, equation (3.2) reduces to
3.6
In what follows, it will be convenient to denote the leading-order solution for *ϕ*^{(e)} in the bulk as *ϕ*^{bulk}.

#### (ii) Boundary layers

*Anode*. We rescale the boundary layer near *X*=0 with
3.7
There is no need to rescale any of the *C*_{i}, but for *ϕ*^{(e)} we write
3.8
For the cell potentials of interest in pickling, *Π*≫1; for example, for *U*=20 V and *T*=298 K, *Π*∼10^{3}. The leading-order governing equations are
3.9
At leading order, the boundary conditions at are
3.10
and
3.11
where
3.12
and
3.13
The significant simplification here is in equation (3.11), since the right-hand side contains only the bulk potential, not Note that, at this stage, we should have that
implying that
3.14
There is, however, little point trying to rescale *ϕ*^{bulk} further, since any such rescaling would be valid only at this electrode, whereas we will ultimately need to solve for *ϕ*^{bulk} everywhere. Also, since governing equation (3.9) for this layer is parabolic, the boundary conditions (2.34) are now treated as initial conditions; the relevant ones at *Y* =0 are
3.15

In the boundary layer, we thus have at this stage four unknowns () and there are four equations: three equations from (3.9) and the electroneutrality condition, which gives at leading order
3.16
We can manipulate these to give
which can then be integrated once with respect to to give
3.17
throughout the layer, which comes from the boundary condition at . Also, we note for later use that, as
3.18
where
3.19
Then, we observe that equation (3.9) for *i*=H^{+} can be reduced to
3.20
on using electroneutrality. Rearranging and integrating with respect to then gives
3.21
where
3.22
Hence
3.23
At this stage, we have arrived at
3.24
Eliminating leaves just two partial differential equations in two unknowns, and for *i*=Na^{+}, SO_{4}^{2−},
3.25
where
3.26
This gives a particularly compact form for the boundary conditions (3.10) at : we will have simply
3.27

Consequently, if *ϕ*^{bulk} can be determined *a**p**r**i**o**r**i*, it will be possible to decouple the boundary layer problem from the bulk problem. For the Tafel laws used in this model, which do not depend on ion concentration at the reacting surfaces, this indeed turns out to be possible. Returning to equation (3.17), we see that the left-hand side is the *x*-component of the current density vector; as we approach the edge of the boundary layer, this must tend to its value in the bulk, which is ϒ(∂*ϕ*^{bulk}/∂*X*)_{X=0}; combining this with equation (3.17) gives
3.28

*Cathode*. Here also, the scalings (3.7) and (3.8) are applicable, and we obtain
3.29
Thus, although only OH^{−} ions are produced in the electrochemical reaction, we can expect the H^{+} ions advected up from the anode to participate in the electroneutrality condition at leading order. The boundary conditions are, at
3.30
and
3.31
where
3.32
and
3.33
Analogous to equation (3.14), we expect at this electrode that
3.34

For the anode, we were able to eliminate two of the dependent variables, and to leave the problem formulated in and here, we can once again eliminate two of the dependent variables ( and ), but the remaining problem will be formulated in three dependent variables, , and Starting with equation (3.29) for *i*= OH^{−}, we obtain
3.35
Integrating once with respect to gives
3.36
where
3.37
Hence, for *i*=Na^{+}, , H^{+}, we arrive at
3.38
where
3.39
and the boundary conditions in equation (3.30) can be compactly expressed as
3.40
Analogous to equation (3.28), we have that
3.41

Strictly speaking, the above equations are valid for where meaning that an initial condition analogous to equation (3.15) is to be expected. We have already mentioned that the H^{+} ions produced at the anode would be advected up to the cathode; in addition, the Na^{+} and SO_{4}^{2−} ions will also be advected and we can expect the profiles for *C*_{i} for at to be those which result from numerical integration from *Y* =0 up to that point. For on the other hand, we will have, at leading order,
3.42

*Boundary layer at cathodic strip*. For the boundary layer at the strip, we set
3.43
and
3.44
The leading-order governing equations are
3.45
The boundary conditions at are
3.46
and
3.47
where and
3.48
The majority of the details of the elimination of and are similar to those of the preceding subsections, and so we omit them here. Finally, for we arrive at
where
3.49
and
3.50
Boundary condition (3.46) at will simply become
while matching to the bulk as gives
3.51

*Boundary layer at anodic strip*. Here, we use equations (3.43) and (3.44) to obtain
3.52
The boundary conditions at are
3.53
3.53
and
3.55
where
3.56
Starting with equation (3.52) for we obtain
3.57
which can then be integrated to give
3.58
where
3.59
and
3.60

Finally, we arrive at, for *i*=Na^{+}, , OH^{−},H^{+},
3.61
where
3.62
At we will have simply
3.63
Finally, matching to the bulk gives
3.64

### (b) Strip

Here, we have 3.65 The coupling boundary condition to the electrolyte, equation (2.29), becomes 3.66 where 3.67

### (c) Summary

A systematic strategy for solving the problem would therefore be summarized as follows.

**(A)** Solve first for *ϕ*^{bulk} and *ϕ*^{(b)}:
3.68
and
3.69
where *ϵ*=*d*/*D*_{eb}, subject to the boundary conditions: at *Y* =0,1,
3.70
and
3.71
at *X*=0,
3.72
at *X*=1,
3.73
at *X*=*δ*+*ϵ*,
3.74

**(B)** Solve for the boundary layer at *X*=0: in the region and 0≤*Y* ≤1, for *i*=Na^{+}, , H^{+},
3.75
where
3.76
subject to the boundary conditions
3.77
3.78
and the initial conditions
3.79

**(C)** Solve for the boundary layer at *X*=1: in the region and 0≤*Y* ≤1, for *i*=Na^{+}, , H^{+}, *O**H*^{−},
3.80
where
3.81
subject to the boundary conditions
3.82
3.83
3.84
and the initial conditions
3.85

In principle, all three subproblems can now be solved numerically. The numerical solution of (B) and (C) will be given elsewhere, but we focus in the remainder of this paper on obtaining a numerical solutions for (A) and an analytical estimate for the solution to (A) when the geometry is slender.

## 4. Slender geometry asymptotics

The computations carried out by Ipek *et al*. (2007) were for a geometry where *δ*=0.04 and *ϵ*=0.0008. For such slender geometries, one would expect comparatively little short-circuiting between the anode and cathode electrodes, so that most of the current that exits the anode should make its way through the strip to the cathode, rather than through the electrolyte; indeed, the computations of Ipek *et al*. (2007) indicate short-circuiting of between 9 and 15% for the values of *U* and *V*^{(b)} used. Furthermore, those computations suggest that the potential has a rather simple structure, although it was not transparent how to find it quantitatively. Thus, further simplifications are possible as regards computing the potential in the electrolyte bulk and the steel strip; for example, the current density at the steel strip appears to consist of two piecewise constant regions that are separated by a transition region. The simplicity of this structure is contributed by the fact that all the Tafel laws at reacting surfaces are independent of concentration. Here, we indicate how the foregoing analysis helps to identify the asymptotic structure of the solution; ultimately, this gives the cell performance characteristics.

Although non-dimensional variables were useful for the analysis in §3 for easily identifying orders of magnitudes of various terms, it turns out to be more useful to return to dimensional variables to consider the total potential drop over the cell. Assuming there is no short-circuiting, this will consist of seven contributions: (1) overpotential at the anode electrode; (2) ohmic losses between the anode electrode and cathodic part of the strip; (3) overpotential at the cathodic part of the strip; (4) ohmic losses between the cathodic part of the strip and the anodic part of the strip; (5) overpotential at the anodic part of the strip; (6) ohmic loss between the anodic part of the strip and the cathode electrode; and (7) overpotential at the cathode electrode. These are summarized as
4.1
where [*i*] is a current density scale and , and are overpotential scales that are all to be determined; note that , whereas , In equation (4.1), terms 2 and 6 come from the potential drop predicted by equation (3.68) for *δ*≪1. In term 4, denotes the current density in and parallel to the strip, and we expect
Also, [*l*^{(b)}] is a length scale that is to be associated with the potential drop in term 4. Plausible lower and upper limits on [*l*^{(b)}] are *L*^{(b)}−2*L* and *L*^{(b)}, respectively, but we find that even if we choose the upper limit, which renders term 4 as large as possible, it is still much smaller than the sum of terms 2 and 6, a conclusion that holds even before we have determined [*i*]. To see this, we simply observe that
Hence, we will neglect term 4. Now,
4.2
4.3
4.4
and
4.5
from equations (2.14), (2.20), (2.26) and (2.28), respectively. So,
4.6
where
Setting
4.7
we obtain
4.8
where *I*=ν[*i*]. An equation of this form has been obtained previously in the context of polymer electrolyte fuel cells by Birgersson *et al*. (2005) and Vynnycky *et al*. (2009) and appears to be generic in electrochemical problems where activation and ohmic losses are of the same order of magnitude; here, we have ϒν≫1, giving
4.9
With this result, we are able to quantify explicitly potential losses over different parts of the cell, even without doing full numerical computations.

## 5. Results

In figure 3, we plot −[*i*], as given by equation (4.9), as well as that given by the numerical solution to
5.1
The latter is a transcendental equation for *I*, which can be solved in straightforward fashion using the Newton–Raphson method. Figure 3 also includes the average current density at the strip obtained numerically by Ipek *et al*. (2007), as well as the average current density at the electrodes. In addition, we have modified the earlier code so as to solve just equations (3.68)–(3.74), i.e. subproblem (A) in §3*c*, and the results of this are also included. The agreement between the two sets of numerical results is excellent and fully justifies the analysis of §3; we also remark that since the modified code only needs to solve for the electric potential, for which there are no boundary layers, the computing time is negligible by comparison with that of the original code. While there is some discrepancy between the analytical and numerical solutions in figure 3, this appears to be of the order of magnitude of the short-circuiting effects mentioned earlier which the analytical model cannot capture; we can therefore be certain that the analysis of §4 is qualitatively correct. However, it may be difficult to find combinations of parameters for which the analysis is also quantitatively correct: for example, the degree of short-circuiting can be decreased if *D*_{eb} is decreased, but this will mean that term 4 in equation (4.1), which it was convenient to be able to neglect, may no longer be negligible by comparison with terms 2 and 6.

Lastly, the result for [*i*] can be used to write down a compact formula for the amount of chromium oxide scale removed from a particular point of the steel strip as it passes through the cell. In Ipek *et al*. (2007), this quantity was denoted by *δ*_{Cr2O3} and derived to be
5.2
where and *M*_{Cr2O3} (= 0.152 kg mol^{−1}) are the density and relative molecular weight, respectively, of chromium oxide, and
5.3
In the notation of this paper and in the zero short-circuit limit, equation (5.2) becomes
5.4
There is little point comparing the numerical results obtained by Ipek *et al*. (2007) with the prediction of equation (5.2), since it is evident that there will be a discrepancy between them that scales as that already shown in figure 3. Instead, in figure 4, we plot *δ*_{Cr2O3} as a function of *U* and *V*^{(b)} using equations (4.9) and (5.4). This constitutes, in a very compact way, an expression for the performance of the cell as a function of the main operating parameters; equally importantly, this bypasses the need for time-consuming computations.

## 6. Conclusions

This paper has used asymptotic methods to analyse a recent mathematical model for the electrochemical pickling of steel. A significant preliminary step was to rework the classical asymptotic theory for a supporting electrolyte. Although this holds in systems where one of the minority ions carries the current at all reacting surfaces, this is not the case in pickling; consequently, although the classical theory can be built up using regular perturbation expansions for the ionic species, a singular perturbation approach is necessary for the reworked theory. A self-consistent asymptotic structure consisting of a core region and two boundary layers was proposed; the analysis for this is based on three non-dimensional parameters: *ϵ*(≪1),*Π*(≫1) and In industrial pickling, the operating geometry is slender, leading to two further asymptotic parameters: *δ*≪1 and *ϵ*≪1. In this quintuple asymptotic limit, the average current density for the cell can be estimated analytically, which is an attractive alternative to the lengthy computation times encountered in earlier attempts to solve the full problem numerically (Ipek *et al*. 2007). Furthermore, the asymptotic analysis explains in hindsight why a Laplace model for only the electric potentials that was used in earlier work, in which model results were compared against experiment (Ipek *et al*. 2006), was appropriate: there, we found that such a model was able to reproduce the experimental results rather well, although we did not at that stage understand why.

Also, while the asymptotic structure obtained here has passing similarities with that considered by Newman (1967), there are also significant differences: there, a binary electrolyte was considered, and a boundary-layer/core decomposition was arrived at, whereas here this has been done for the more complicated case of a supporting electrolyte. The fact that the reaction laws used here were taken to be independent of local ion concentration makes any future numerical computation of the decomposed problem simpler than that of Newman (1967), although there is no doubt that it can be extended to cases where there is concentration dependence also. In addition to such numerical work, future analytical pathways include the retention of the bulk reactions; in Ipek *et al*. (2007), these appeared to give minor alterations to the local current density, although a qualitatively different behaviour for the ionic concentration profiles, and the asymptotic analysis presented here should help to interpret those numerical results also.

An interesting question is how the supporting electrolyte asymptotic theory derived here can be generalized. First of all, we recall that the principal reason why the conventional theory, which works in 3-ion models, failed for the pickling model was that different minority ions were carrying the current at the cell electrodes, meaning that the electric potential of the electrolyte could not be constant at leading order. Hence, our new theory comes into play in models having at least four ionic species: the two majority ions of the supporting electrolyte, which do not participate in reactions at electrodes and do not carry the current there, and the minority ions which do. In 4-ion models, the minority ions must necessarily be of opposite charge; however, as we have seen in our 5-ion pickling model, it is possible for some of the minority ions to have charge of the same sign. Based on these considerations, other examples where this theory could be applied are in models for the electrochemical treatment of tumours (Nilsson *et al*. 1999) and the chlorate process (Byrne *et al*. 2001).

## Acknowledgements

M.V. acknowledges the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland Mathematics Initiative Grant 06/MI/005.

## Footnotes

- Received May 31, 2009.
- Accepted August 18, 2009.

- © 2009 The Royal Society