The use of channel flow cells for electrochemical kinetic studies in high temperature aqueous solutions

Electrochimica Acta 52 (2007) 4124–4131

The use of channel flow cells for electrochemical kinetic studies in high temperature aqueous solutions George R. Engelhardt a , Ritwik Biswas b , Zaki Ahmed c , Serguei N. Lvov d , Digby D. Macdonald e,∗ a

OLI Systems Inc, 108 The American Road, Morris Plains, NJ 07950, USA Siemens Power Generation, 4400 Alafaya Trail, Orlando, FL 32826, USA c Research Institute, King Faud University of Petroleum and Minerals, Dhahran, Saudi Arabia d Department of Energy and Geo-Environmental Engineering, The Pennsylvania State University, University Park, PA 16802, USA e Center for Electrochemical Science and Technology, Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA b

Received 13 November 2006; received in revised form 22 November 2006; accepted 22 November 2006 Available online 17 January 2007

Abstract It has been shown that, for the case of one-step reactions of arbitrary order, the relationship between the average current density and the limiting current density on a working electrode mounted on the inner radius of an annular flow channel of arbitrary length obeys, with great accuracy, the same relations as does a reaction on a uniformly accessible surface. This allows us to combine the advantages of non-uniformly accessible surfaces (high sensitivity, and no need to use rotating contacts) with the advantages of uniformly accessible surface systems (simple treatment of experimental data). This feature can be very important when investigating systems at high temperatures and pressures, where RDEs are difficult to employ. Using this approach, and by employing previously measured polarization data, the kinetic parameters (exchange current density and anodic transfer coefficient) for the oxidation of hydrogen on platinized nickel in 0.1 M NaOH + 0.7 × 10−3 m H2 at temperatures between 25 ◦ C and 300 ◦ C have been derived. The anodic transfer coefficient is found to be almost temperature independent with a value of 0.43. The exchange current density displays Arrhenius behavior with temperature, increasing from 1.9 × 10−4 A cm−2 at 25 ◦ C to 3.9 × 10−3 A cm−2 at 300 ◦ C. © 2007 Elsevier Ltd. All rights reserved. Keywords: Annular flow channel; Uniformly accessible electrode; Hydrogen oxidation; Elevated temperature

1. Introduction The quantitative description of many corrosion and electrochemical processes at elevated temperatures is often restricted by the lack of accurate experimental data for transport coefficients and charge transfer kinetic parameters. Only a few data for reaction rate constants, exchange current densities, and transfer coefficients for temperatures higher than 100 ◦ C are available for even the most commonly studied reactions. For this reason, unknown parameters are often set equal to the same parameters for different electrochemical reactions (that are known), or values are simply assumed without explanation or justification. For example, it is sometimes assumed that the kinetic param-

∗

Corresponding author. E-mail address: [email protected] (D.D. Macdonald).

0013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2006.11.027

eters (rate constant and transfer coefficient) for hydrogen ion reduction coincide with those for the reduction of water [1]. Also, it should be noted that, even in cases where direct experimental data for the exchange currents and transfer coefficients are available, the accuracy of the data are often questionable, because they are not obtained from cells possessing well-defined hydrodynamic and mass transfer properties. To our knowledge, the sole laboratory system having well-controlled hydrodynamic characteristics that has been used for measuring kinetic parameters of redox reactions at elevated temperatures (T > 200 ◦ C) is the annular flow channel [2,3]. For elevated temperature work, this system has some advantages over the rotating disk electrode (RDE), because of the difficulty of devising rotating electrical contacts for use at high temperatures and pressures. A second issue concerns the uniform accessibility of the reaction surface. While the RDE represents a uniformly accessible surface when operating under complete mass transfer control,

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most other electrode configurations depart from this condition to greater or lesser extents. This issue is important, because a non-uniformly accessible surface implies a distribution of electrochemical kinetic states, with only some averaged response being detected in the external circuit. Thus, previous studies [2,3] employing annular flow channels did not explicitly recognize the non-uniform accessibility of the surface, and hence the calculated kinetic parameters might only be approximate in nature. The primary goal of this work is to show that, in many important cases, it is possible to interpret the experimental data obtained from channel electrodes using the theory for charge transfer at uniformly accessible surfaces, without possessing detailed information about the hydrodynamic conditions in the cell. Using this approximation, kinetic parameters for the anodic oxidation of hydrogen in 0.1 M NaOH + 0.7 × 10−3 m H2 aqueous solutions at temperatures between 25 ◦ C and 300 ◦ C have been extracted from measured polarization curves. 2. Mass transport in a channel electrode

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Eliminating cs and σ from Eqs. (1)–(3) yields   jav jav = Fc∞ 1 − jlim,av

(4)

Eq. (4) clearly shows, that, in some simple cases, the kinetic parameters can be obtained by measuring only the current densities (including the limiting current density) on the uniformly accessible electrode (UAE) without possessing information about diffusion coefficients and the hydrodynamic conditions. For example, let us suppose that an electrochemical reaction of order μ takes place at the electrode surface, so that the diffusion flux at the electrode surface can be expressed as j = kcsμ

(5)

where the rate constant, k, depends on the temperature at the electrode surface and on the electrode potential. For this case, Eq. (4) takes the form  μ jav μ jav = kc∞ 1 − , (6) jlim,av and the reaction order, μ, can be found from the expression [4,5]

The advantage of a uniformly accessible electrode (UAE) (e.g. a RDE under complete mass transfer control) over nonuniformly accessible electrodes (e.g. channel electrode) is that the local current density, i, at the uniformly accessible surface coincides with the experimentally determined average current density, iav [4,5]. This condition simplifies the quantitative description of mass transport processes and, in some cases, it allows one to obtain kinetic parameters for heterogeneous reactions in the absence of reliable information on the hydrodynamic conditions in the system. Below, we review some general relations that describe transport processes at uniformly accessible surfaces. Let us suppose that the molar flux density of species, j, to a uniformly accessible surface (which coincides with the averaged flux density, jav ) is described by the kinetic equation jav = j = F (cs )

(1)

where c is the concentration of species. We will denote all values at the electrode surface by “s” and in the bulk by the “∞”. On the other hand, the flux density to a uniformly accessible reactive surface can be expressed as jav = j = σ(c∞ − cs )

(2)

where σ is the mass transfer coefficient of the system, which depends on the hydrodynamic conditions and on the diffusion coefficient of the species. Accordingly, the average surface concentration, cs,av , which, on the uniformly accessible surface, coincides with cs , can be expressed as   jav (3) cs,av = cs = c∞ 1 − jlim,av where the limiting current, jlim , exists when cs  c∞ , i.e. jlim,av = jlim = σc∞ .

(1)

μ=

(2)

ln(jav /jav ) (1)

(1)

(2)

(2)

ln[(1 − jav /jlim,av )/(1 − jav /jlim,av )]

(7)

where (1) and (2) denote values corresponding to different (and possibly unknown) hydrodynamic conditions. If the reaction order, μ, is known, the rate constant, k, can be found from Eq. (6). The second advantage of a UAE is that the electrode allows us to reduce the problem of defining the average flux density on the surface to the solution of the algebraic Eq. (4), if the average limiting flux (current) density, jlim,av is known. The latter value can be readily determined experimentally (for example, by applying a high potential to the electrode, such that the reaction occurs under complete mass transfer control) or by analytical or numerical calculation, as has been done for many systems. Accordingly, it would be remarkable if Eq. (6) (or Eq. (4), in the general case) can be used with reasonable accuracy for a channel electrode, recognizing that they strictly apply to a uniformly accessible electrode. Below we will investigate this possibility by the direct comparison of values of average flux densities obtained by numerical solution of the corresponding boundary problem and the value of jav , obtained from Eq. (6), where jlim,av is also obtained by numerical solution of the mass transfer equation. In this article, we consider the case of hydrodynamically fully developed flow (turbulent or laminar) with constant physical properties, negligible axial diffusion, and axially symmetric mass exchange in circular annular channels and tubes under steady-state conditions. Under these conditions, the mass transfer equation can be written as: u(r)

∂c 1 ∂ ∂c = [r(D + Dt (r))] ∂x r ∂r ∂r

(8)

Here x and r are the axial and radial coordinates of the annulus, respectively (x is measured along the surface from the edge of the

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electrode), u the local axial velocity, D the diffusion coefficient, and Dt is the eddy diffusivity, which is assumed to be rationally defined (Dt = 0 in the case of laminar flow). The boundary conditions for this equation become, −D

∂c = kcμ ∂r

∂c =0 ∂r

at r = ri and x > 0

(9)

at r = ro and x > 0

(10)

at x ≤ 0

(11)

and c = c∞

Here, to be specific, we assume that the inner tube (with radius ri ) is electrochemically active with respect to the test species and that the outer tube (with radius ro ) is insulated. For the case of laminar flow, the local axial velocity is described by the following equation [7]:     2 r 2V r (12) + B ln u= 1− M ro ro where V is the mean velocity, a = ri /ro is the annulus radius ratio, B = (a2 − 1)/ln(a) and M = 1 + a2 − B. It is most convenient to solve Eqs. (9)–(12) by introducing the following dimensionless variables and parameters r , ro − r i

C =

c , c∞

X=

xD 2x = , 2 Sc Re de 2V (ro − ri )

R=

K=

μ−1

k(ro − ri )c∞ D

(13)

where Sc = ν/D is the Schmidt number, Re = Vde /ν is the Reynolds number and de = 2(ro − ri ) is the hydraulic diameter. For our purposes, the most important quantity is the dimensionless average flux density at the electrode surface, defined as    jav (ro − ro ) 1 X ∂C Jav ≡ = dX. (14) Dc∞ X 0 ∂R R=a

approaches. Firstly, we calculate, Jav , by direct numerical solution of the boundary value problem, Eqs. (8)–(11), by using the fully implicit up-wind scheme [8]. On the other hand, we solved the same boundary problem for limiting conditions (when Boundary condition (9) was replaced by condition (15)), and calculated the value of Jlim,av (the step for the rectangular grid [a/(1 − a) ≤ R ≤ 1/(1 − a), 0 ≤ X ≤ 10] was chosen, so that the calculated dimensionless limiting current density, Jav,lim , differs from the analytical solution for the L´evˆeque region by no more than 0.1%). After that, Jav , was calculated by using Eq. (6) in it dimensionless form: μ  jav Jav = K 1 − (16) jlim,av It is evident that Jav from Eq. (16) satisfies the limiting conditions at K → 0 and K → ∞. In the general case, the numerical solution of Eq. (16) can be easily obtained for arbitrary values of the reaction order, μ; however, for some values of μ (e.g. at μ = 1/2, 1, 2), analytical solutions can also be obtained. Thus, for μ = 1, we have Jav =

K , 1 + K/Jav,lim

(17)

Figs. 1–3 show that Eq. (16) yields a reasonable approximation for arbitrary values of X and for all reasonable values of the dimensionless parameters (K, μ and a = ri /ro ) upon which the solution depends, in the case of an annulus cell. Fig. 3 also shows the results for a tubular channel, when the inner surface of the tube is electrochemically active. The relative error, ε, between Jav , which is obtained from Eq. (16), and the same quantity that is derived by numerical solution of the boundary problem (Eqs. (8)–(11)) is practically invisible on the scales employed in Figs. 1–3 at X ≤ 0.5 (ε < 5% for this region). The condition X ≤ 0.5 corresponds to the condition that x/de ≤ 0.25ReSc. At a temperature of 25 ◦ C, for example, for Re = 500 and Sc = 1000, we have x/de ≤ 125,000. Even at elevated temperatures (e.g. 300 ◦ C), when Sc is of the order of 1, we have x/de ≤ 125. This condition is usually satisfied in experiments. Moreover, the dis-

For the case of a very slow reaction (K → 0), we have Jav → K μ μ (jav → Kc∞ ), i.e. the constant kinetic current kc∞ flows in the cell. In the opposite case (for K → ∞), the reactant that reaches the surface immediately reacts to form the product, so that Boundary condition (9) can be replaced by the condition for the limiting current: c=0

at r = ri

(15)

The average limiting current can be easily found analytically, in this case, for a sufficiently short electrode in the so called L´evˆeque region (i.e. for the region where the variation of u with distance from the wall, y, is approximately linear) [6] or numerically, as in the corresponding Graetz problem, by using, for example, the method of separation of variables [6,7]. For intermediate values of the dimensionless reaction rate, K, we calculate the average value of the flux, Jav , using two

Fig. 1. Averaged dimensionless fluxes on the electrode surface under laminar flow conditions for different values of the dimensionless rate constant, K, for μ = 1 and a = 0.5. The solid lines are the numerical solutions of the boundary value problem and the dashed lines are obtained by using Eq. (16).

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where τ o is the shear stress on the outer tube and ρ is the density. In this work we assume the following expressions for the eddy diffusivity and axial velocity [9]: v+ = y+ − A1 (y+ ) + A2 (y+ ) 4

5

for y+ ≤ 20,

v+ = 2.5 ln y+ , for y+ ≥ 20

(19)

and Dt 4A1 (y+ ) − 5A2 (y+ ) = ν 1 − 4A1 (y+ )3 + 5A2 (y+ )4 3

Dt y+ = −1 ν 2.5 Fig. 2. Averaged dimensionless fluxes on the electrode surface under laminar flow conditions for different values of the reaction order, μ, at K = 10 and for a = 0.5. The solid lines are the numerical solutions of the boundary value problem and the dashed lines are obtained by using Eq. (16).

crepancy that we see for X > 0.5 (Fig. 1 for K = 1 and in Fig. 2 for μ = 2) does not exceed 16% and the discrepancy disappears at sufficiently high values of X. It follows from the fact that, due to depletion of the reactant at the surface, Jav → Jav,lim for X → ∞. As can be seen, Eq. (16) assures this transition. It must also be noted that the relatively large value of ε at X > 0.5 is observed only for low value of K, i.e. for relatively slow reactions that can be investigated even under stagnant conditions. Because Eq. (16) is more applicable for smaller values of X, corresponding, among other things, to higher values of the average hydrodynamic velocity, V (see Eq. (13)), we can expect that Eq. (6) will be satisfied more closely for turbulent flow than for laminar flow conditions. For the case of turbulent flow, it is convenient to use the following dimensionless variables:   x+ = x τo /ρ/ν, y+ = (r − ri ) τo /ρ/ν, 

u+ = u/

τo /ρ,

K+ =

μ−1

kνc∞ √ D τo /ρ

(18)

Fig. 3. Averaged dimensionless fluxes on the electrode surface under laminar flow conditions for different values of annulus radius ratio, a, at K = 10 and μ = 1. The solid lines are the numerical solutions of the boundary value problem and the dashed lines are obtained by using Eq. (16).

for y+ ≥ 20

4

for y+ ≤ 20, (20)

where A1 = 1.0972 × 10−4 , A2 = 3.295 × 10−6 , and y+ is the dimensionless distance from the nearest surface. The dimensionless averaged flux density for turbulent flow is defined, as:   L+  ∂C 1 + Jav = + dx+ (21) L 0 ∂y+ y+ =0 + be the corresponding averaged limiting flux. The Also let Jav,lim equation  μ + Jav + + (22) Jav = K 1 − + Jav,lim

is written in analogy with Eq. (6). Fig. 4 shows that, for the case of turbulent flow for Sc = 1, Approximation (6) holds even more closely than for the laminar case. The results of our calculations (see Fig. 4) show no + obtained from significant difference between the values of Jav a numerical solution of Eq. (8) and the corresponding values obtained from Eq. (22) (the relative error always being less than 2%). Fig. 5 also shows that Approximation (6) yields acceptable results for arbitrary values of Sc. More sophisticated expressions for u+ and Dt were developed in Ref. [10] by breaking the flow area into four sections: (1) A sublayer adjacent to the inner surface, (2) a sublayer adjacent to

Fig. 4. Averaged dimensionless fluxes on the electrode surface under turbulent flow conditions for different values of the dimensionless rate constant K+ . The solid lines are the numerical solutions of the boundary value problem and the dashed lines are obtained b using Eq. (22).

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Fig. 5. Averaged dimensionless fluxes on the electrode surface under turbulent flow conditions for different values of the Schmidt number, Sc. The solid lines are the numerical solutions of the boundary value problem and dashed lines are obtained b using Eq. (22).

the outer surface, (3) a fully developed turbulent region from the inner sublayer to the point of maximum velocity in the annulus, and (4) a fully developed turbulent region from the outer sublayer to the point of maximum velocity. Our calculations show that, if we use these expressions instead of Eqs. (19) and (20), only a + small difference is observed between calculated values of Jlim,av (maximum error is within 5%) and an even smaller difference is + . Furthermore, no visible difference is observed observed for Jav + obtained from a numerical solution between the values of Jav of Eq. (8) and the corresponding values obtained from Eq. (22). This means that the accuracy of Eq. (22) very slightly depends on the expressions employed for the eddy diffusivity and turbulent velocity. This finding is very important, because, in contrast with the case of laminar flow, the expressions for Dt and u for turbulent flow are still subject to considerable uncertainty and controversy. We would like to emphasize that the results of calculations (Figs. 1–5) on no account mean that the channel electrode can be considered as to be approximately uniformly accessible surface. Of course, the flux density, j, and limiting flux density, jlim change with the distance, x, from the electrode leading edge. Accordingly, the average values jav and jlim,av depend on the length of the electrode, L (see Figs. 1–5). The only conclusion that has been demonstrated above is that the relation between jav and jlim,av (Eq. (6)) for the case of a channel electrode is approximately the same as for a uniformly accessible surface. In particular, we can conclude that, for the case of a single irreversible reaction under normal experimental conditions in a channel, the dependence of the average current density on overvoltage (which is assumed to be constant along the electrode surface) and the average limiting current density itself can be described with a great accuracy by the same functions as for a uniformly accessible electrode. Thus, for the case of a first order reaction obeying Tafel kinetics (for example, an anodic reaction, such as the oxidation of hydrogen) we have: iav =

i0 exp(αFη/RT ) 1 + i0 exp(αFη/RT )/ ia,lim

(23)

where iav is the average current density, i0 the exchange current density, αa the anodic transfer coefficients, η the overvoltage of the electrochemical reaction that is assumed to be constant along the electrode surface, T the Kelvin temperature, F Faraday’s constant, and R is the gas constant. Accordingly, the method yields rather simple analytical expressions for the averaged mass transfer rate in the channel, which can be readily used for obtaining kinetic parameters by optimizing the expression on the experimental polarization data. It must be emphasized that, in non-linear cases (μ = 1), no exact analytical solutions can be obtained. Moreover, even in the linear case (μ = 1), for example, for a first-order irreversible reaction, the exact analytical solution that was obtained in Ref. [11] is so complicated that it has only ever been treated using a computational method. Thus, it was previously shown [11] that the current-potential curve of a totally irreversible, first order reaction for the region 0.1 ≤ iav /iav,lim ≤ 0.9 at 25 ◦ C for the L´evˆeque region at a tubular electrode can be accurately approximated by the linear relation:   ilim,av 0.0618 E = E1/2,irrev − log10 −1 (24) αa iav where E and E1/2,irrev are the electrode potential and the halfwave potential, respectively. As follows from Eq. (23), the current–potential curve for a totally irreversible, first order reaction can be presented in the form   iav,lim B E = E1/2,irrev − log10 −1 (25) αa iav where 2.303RT B= (26) F For a temperature of 25 ◦ C, Eq. (26) yields B = 0.0592, which is close to the corresponding value from Eq. (24). However, it must be emphasized that, contrary to Eq. (24), Eq. (25) describes the current–potential curve for any temperature and is applicable not only for the L´evˆeque region of the tubular electrode, but is applicable to the whole length of an annular electrode. The last circumstance can be very important for elevated temperatures, because, in this case, the length of the L´evˆeque region can be less than the length of the working electrode, L, especially for turbulent flow. Thus, for the case of the laminar flow in a tubular electrode, the length of the L´evˆeque region can be estimated from the condition [12] L < 0.005ReScde

(27)

In our dimensionless variables, Condition (27) can be rewritten as X < 0.01

(28)

Strictly speaking, Condition (28) was derived for the case of circular tubes, but it also can be applied to flat channels and circular annuli. For comparison, the case of mass transfer to the inner surface of a circular tube was also considered (see Fig. 2), where Reaction (5) takes place on the inner surface. For the usual Schmidt number (Sc ∼ 103 ), this condition (28) is usually

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satisfied in an experiment. However, at elevated temperatures, when Sc can be of order of 1, the situation changes. Thus, for Re = 500 and Sc = 1, we have L/de = 2.5. It is clear that such a restriction has to be taken into account when designing the apparatus. For the case of turbulent flow, the length of the L´evˆeque region for low Schmidt numbers can be estimated from the condition [12]: L+ < 64Sc

(29)

which yields that L+ < 64 if Sc has the order of 1 (strictly speaking, turbulent transport will not be important provided that the length of the electrode is chosen to satisfy the more restrictive conditions L+ < 64 Sc and L+ < 700 [12]). On the other hand, the value of L+ can be of the order of 102 to 103 , for example in the cells described in Refs. [2,3]. This means that, under turbulent flow conditions, the L´evˆeque region is usually much less in length than is the working electrode. It must also be noted that, generally, at elevated temperatures turbulent (but not laminar) flow occurs, due to the very low value of the viscosity and hence to the high value of Re. These relationships mean that, if we would like to calculate the average flux density, jav , in the annular electrode by using Eq. (6), we have to clearly understand which limiting average flux density, jlim,av , must be substituted into this equation. Thus, the usual analytical relations for jlim ,av , for example those from Ref. [13], cannot be used in the general case at elevated temperatures, because they are obtained only for the L´evˆeque region. For this purpose, the values of jlim,av obtained by the analytical or numerical solution of the mass transfer equation for the whole cell must be used (as is done in this article) or simply the experimentally measured values of jlim ,av must be adopted. Furthermore, the usual analytical relations from Ref. [13] must be used with great caution when calibrating the experimental setup at elevated temperatures. However, it must be noted that, if our aim is only to determine the kinetic parameters of the electrochemical reaction, it can be done without any calibration of the experimental setup. As follows from the above, only measured values for iav and ilim ,av are necessary for this purpose (we do not need any information about the hydrodynamic conditions and the transport coefficients), as is demonstrated below in this paper. Of course, if we need to measure the transport properties in the system (for example diffusion coefficients), the experimentally measured values of ilim ,av must be compared with analytically or numerically calculated values of ilim ,av . It is also important to note that the method employed in this study can be compared with the approach of Frank-Kamenetskii [14] for describing mass transfer to arbitrarily shaped bodies, which has been criticized by many authors (see, for example, Ref. [15]). The approach is reduced to the calculation of the local value of flux density, j, via the local value jlim from Eq. (6) that is used in it local form and subsequent averaging of resulting values. It is evident that this procedure is much more complicated than that described above and also that it leads to relations that are different from those that can be obtained by

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using Eq. (6) directly. The application of the Frank-Kamenetskii method to different systems in the L´evˆeque region shows that the accuracy of this method is about 10–15%, whereas the accuracy of the method described above in the same region is within 2%. 3. Experimental An annular flow channel, similar to the apparatus described by Macdonald et al. [2], was constructed to perform electrochemical and corrosion studies. The flow channel was placed within a pressure vessel comprising a 300 ml Type 316 SS PARR bench top reactor fitted with a PARR magnetic drive. The remainder of the test system comprised a solution storage tank, a high-pressure metering pump, a pre-heater, a regenerative heat exchanger, a pressure control value, and a cooler. The operating solution was accelerated through the flow channel by the impeller resulting in high flow velocities. The tandem cylindrical working electrodes (outer diameter of 1.270 cm and length of 0.635 cm) were located on the central column with the electrical leads being routed through the bottom of the autoclave. The thermocouple, counter electrode, external reference electrode, internal Pt reference electrode, and solution inlet and outlet were fitted through the autoclave head. The platinized nickel working electrode was platinum plated to a thickness of 0.5–1.3 ␮m. The counter electrode was a Pt mesh electrode located around the base of the inner electrode housing and the external reference electrode used in this work was a Ag/AgCl external pressure balanced reference electrode (EPBRE) operating with a saturated KCl internal solution. A more detailed description of the apparatus and the experimental procedure can be found in Refs. [2,3]. 4. Results and discussion We assume that the hydrogen oxidation reaction H2 → 2H+ + 2e−

(30)

takes place on the working electrode. Examples of polarization measurements at a temperature of 300 ◦ C for different values of the rotation speed of the impeller (and hence velocity of the solution in the channel) are presented in Fig. 6. In this paper, we present results only for sufficiently large values of the overvoltage, η = E − Ee , where E and Ee are the potential of electrode and equilibrium potential that the reaction can be considered to be irreversible and ultimately under complete mass transfer control. As noted above, we used a Ag/AgCl, saturated KCl external pressure balanced reference electrode (EPBRE). The equilibrium hydrogen electrode potential at 300 ◦ C for mH2 = 0.7 × 10−3 m and mNaOH = 0.1 m on the EPBRE scale was calculated abd e measured to be EPt/Ni−Ag/AgCl = −0.9402 V versus SHE. The average current density, iav for this case is described by Eq. (23) for sufficiently large values of η. Fig. 6 shows that the anodic limiting currents arise from mass transport limitations for the transfer of hydrogen to the electrode surface. Of course, the limiting currents increase as the fluid velocity increases, i.e. with increasing rotation speed of

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Fig. 7. Temperature dependence of anodic transfer coefficient for the hydrogen oxidation reaction in 0.1 M NaOH.

Fig. 6. Anodic polarization curves for the oxidation of H2 (0.7 × 10−3 m) on platinized nickel in 0.1 M NaOH aqueous solution at 300 ◦ C for different rotation speeds of the impeller. The solid lines correspond to the best fits of Eq. (23).

the impeller. The values of ia,lim are listed in Table 1. The values of the transfer coefficient, αa , and exchange current density i0, which were obtained from optimization of Eq. (23) on the experimental polarization data, are presented in Table 1 (see also Fig. 6). We see that the values of αa , and i0 do not depend on the hydrodynamic condition, which confirms the applicability of the method. Using the experimental data given in Ref. [3], we also calculated the values of αa and i0 for temperatures of 25, 100 and 200 ◦ C. Figs. 7 and 8 show the temperature dependencies (between 25 ◦ C and 300 ◦ C) of the transfer coefficient, αa , and the exchange current density, i0 , respectively. As can be seen from the figures, the anodic transfer coefficient is essentially independent of temperature and is close to a value of 0.45. The exchange current density demonstrates Arrhenius behavior, yielding the apparent activation energy of 16.0 kJ/mol. It should be noted that the overall Reaction (30) has been considered as a one-stage reaction in this work; in reality, the hydrogen electrode is an example of a multi-stage electrochemical reacTable 1 Results of a non-linear regression analysis of the anodic polarization curve for the oxidation of H2 (aq) (0.7 × 10−3 m) on platinized nickel in 0.1 M NaOH aqueous solution at 300 ◦ C Rotation speed (rpm)

ia,lim (A/cm2 )

αa

i0 (A/cm2 )

200 400 600

0.0145 0.0230 0.0354

0.45 0.46 0.43

4.0 × 10−3 4.0 × 10−3 3.6 × 10−3

Average

–

0.45

3.9 × 10−3

Fig. 8. Temperature dependence of the exchange current density for the hydrogen electrode reaction in 0.1 M NaOH.

tion [16]. More accurate determination of the kinetic parameters will be reported in future publications. 5. Summary and conclusions It has been shown that for the case of one-step reactions of arbitrary order the relationship between average current density and the limiting current density in an annular channel with the electrode mounted on the inner cylinder obeys, with great accuracy, the same relationship as for a uniformly accessible surface (e.g., a rotating disk electrode (RDE) under complete mass transfer control). This allows us to combine the advantages of inherently non-uniformly accessible surfaces (high sensitivity and no need to use rotating contacts) with the advantages of uniformly accessible surface systems (simple treatment of experimental data). This feature can be very important when investigating systems at high temperatures and pressures, where RDEs are difficult to employ. In particular, this finding allows us to determine the kinetic parameters for heterogeneous charge transfer reactions without having any information about the hydrodynamic conditions in the channel or of the transport coefficients of the species in the solution. To illustrate the method,

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we describe the determination of the anodic transfer coefficient and the exchange current density for the anodic oxidation of hydrogen on platinized nickel in 0.1 M NaOH for temperatures between 25 ◦ C and 300 ◦ C. It is important to emphasize that the method described above is applicable not only to the L´evˆeque region of the channel, but practically to electrodes of any arbitrary lengths. This circumstance is very important for studies at elevated temperatures, because of the difficulty in engineering channels to strict hydrodynamic and mass transport specifications. It is also important to emphasize that, in this work, only the case of a single species that is involved in an electrochemical reaction has been considered. In future publications, we will generalize the method to the case that involves more than one species. Our preliminary studies show that this method can be successfully used in the cases of the first-order reversible reactions, consecutive first order reactions, and second-order irreversible reactions. These cases will also be reported in future publications. Acknowledgments The authors gratefully acknowledge the support of this work by the Empire State Electrical Energy Research Corporation under Contract EP93-33 and by the Department of Energy through Award No. DE-FC36-04GO14043.

K+ R Re Sc u+ y+ x+ X

4131

non-dimensional rate constant, K+ = kνc∞ / √ D τo /ρ non-dimensional radial coordinate, r/(ro − ri ) Reynolds number, dh V/ν Schmidt number, ν/D √ non-dimensional axial velocity, u/ (τ/ρ) non-dimensional distance in radial direction, √ y (τ/ρ)/ν √ non-dimensional distance in axial direction, y (τ/ρ)/ν axial coordinate, 2x/ScRede μ−1

Greek letters α anodic transfer coefficient η overvoltage ν kinematic viscosity ρ density σ mass transfer coefficient τ shear stress Subscripts ∞ either in the bulk or entrance of the channel av average i inner surface lim limiting o outer surface s electrode surface

Appendix A. Nomenclature References a c de D Dt F i j r R T u V x

ratio of radii of inner to outer cylinder concentration equivalent hydraulic diameter, 2(ro − ri ) diffusion coefficient turbulent diffusion coefficient Faraday number current density molar flux density radial coordinate of annulus geometry, measured from axis gas constant temperature local axial velocity mean velocity axial coordinate of annulus geometry

Non-dimensional groupings C non-dimensional concentration, c/c∞ Jav non-dimensional average flux density, see Eq. (14) μ−1 K non-dimensional rate constant, k(ro − ri )c∞ /D

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