Modeling the Electrical Behavior Of An Alkaline Hydrogen Peroxide

1494

J. Phys. Chem. B 2001, 105, 1494-1502

Modeling the Electrical Behavior Of An Alkaline Hydrogen Peroxide Generating Flow-Through Electrochemical Reactor Piotr Piela†,‡ and Piotr K. Wrona*,†,§ Industrial Chemistry Research Institute, Rydygiera 8, 01-793 Warsaw, Poland, and Department of Chemistry, UniVersity of Warsaw, Pasteura 1, 02-093 Warsaw, Poland ReceiVed: March 1, 2000; In Final Form: NoVember 29, 2000

Two limiting-case simplified models of mass- and charge-transport phenomena in a flow-through electrochemical reactor were developed. It is postulated that the true experimental system’s behavior falls in between. The models account for energy loss due to the ohmic drop, forced convection, diffusion, and migration. The calculation results were compared against experimental data.

Introduction In the electrochemical generation of alkaline hydrogen peroxide for the pulp and paper industry, obtaining a solution with a low hydroxide-to-peroxide ratio (less than 2:1 by weight NaOH to H2O2) is a prerequisite because of the bleaching process’ requirements. One way to achieve this is to generate the product in a fuel-cell-type flow-through electrochemical reactor fed with a zero ionic strength catholyte1 and a high ionic strength anolyte (Figure 1). The use of zero ionic strength solutions in electrosynthesis is a new and uninvestigated concept. To our knowledge, all commercial electrolytic processes to date operate on concentrated electrolyte solutions to reduce electricity expenditure. Here we report on a theoretical study of a large laboratory-scale project with promising economy which exploits the new concept. In our experimental work (to be published), the reactor uses two thin, planar and porous gas-diffusion electrodes such as those described in the review.1 The electrodes contact current collectors in the form of expanded nickel mesh. There are two independently fed electrolyte compartments separated by a Nafion 117 (Du Pont) ionomeric membrane. The compartments are filled with expanded PVC turbulence promoters, which simultaneously serve for the compression of the electrodes to the current collectors. The catholyte is pumped with a peristaltic pump, and the anolyte is circulated with a diaphragm pump at a speed of about 10 times that of the catholyte. On the reactor’s cathode, oxygen is reversibly reduced to the peroxide anion

O2 + H2O + 2e a HO2- + OH-

(1)

and on the anode hydrogen is reversibly oxidized

H2 + 2OH- a 2H2O + 2e

(2)

so the net reaction (without considering side reactions) is

H2 + O2 + OH- a HO2- + H2O

(3)

* To whom correspondence should be addressed. Fax: +48 22 8225996. E-mail: [email protected]. † Industrial Chemistry Research Institute. ‡ Fax: +48 22 6338295. E-mail: [email protected]. § University of Warsaw.

Figure 1. Schematic of the fuel cell type flow-through reactor for alkaline hydrogen peroxide generation.

For every two electrons passing in the external circuit, two sodium ions traverse the high-conductivity cation-exchange membrane from the high NaOH concentration anolyte area to the low ionic strength catholyte area. One can easily operate such a system on pure water as the catholyte and 20% NaOH as the anolyte, achieving a current density on the order of 0.1 A cm-2 at a potential difference of 0 V (at 323 K). With the principle of electroneutrality in mind, it is not (should not be) straightforward for an electrochemist how this is possible. The only way to do classical (low-voltage) electrochemical experiments without a supporting electrolyte has been shown to be the use of microelectrodes.2 There, the ohmic drop is kept sufficiently low because the flowing currents are minuscule. In the case of a large-area electrode, one would expect the voltage drop on the low-conductivity electrolyte to prevent any significant current from flowing. Although the theory of transport and ionics is a classical topic, we were not able to find any treatment of the combined phenomena for a system with large electrodes and a low ionic strength electrolyte. In this modeling work, we tried to show that it is sufficient to have, initially, a low concentration of the supporting electrolyte in the catholyte for the modeled current density to develop to considerable values. As a matter of fact, such a small

10.1021/jp0007989 CCC: $20.00 © 2001 American Chemical Society Published on Web 01/26/2001

Electrical Behavior Of A Reactor

Figure 2. Experimental average current density vs time curves recorded in the flow-through reactor. Conditions: voltage 0.000 V, 25 °C, 1 atm O2, 1 atm H2. Anolyte: 20% NaOH. Catholyte: 0.002 M EDTA. Linear catholyte flow velocity: O, 0.7 cm min-1; 0, start at 5.7 cm min-1 (A) then switch to 2.5 (B), 1.2 (C), and 0.7 cm min-1 (D).

amount of electrolyte is present in the catholyte prior to polarizing the cell because of the leaking properties of the membrane and the occasional addition of sodium ethylenediaminetetraacetate (EDTA, typically 0.002 M). Our aim was also to explain the shape of the experimental average current density vs time curves (Figure 2) in terms of basic laws of electrochemistry: the principle of electroneutrality, the Faraday’s law of mass and charge equivalency, and the laws of transport of charged particles in viscous media (for diffusion, migration, and convection). In these quite unusual chronoamperometry curves, the current at steady operation parameters exhibits an induction time, a maximum, and finally a plateau. It is also typical that lowering the velocity of the catholyte flow raises the current, which is exactly the opposite of how a cell with a high supporting electrolyte concentration would react. Unlike in the usual situation, when a high concentration of the supporting electrolyte is present, here the current is obviously carried to a considerable extent by the electrogenerated ions and, in the initial period of electrolysis, is resistance-controlled. With this modeling, we also tried to create some insight into the local operating conditions across the electrochemical reactor at steady state. Such information might be useful in cell design, where the need to eliminate any local strain of the reactor’s components is crucial. When the resistance at the electrolyte inlet is much higher than that at the outlet, the problem of local wear is expected to be particularly severe. Even at the large laboratory scale (∼100 cm2 electrode area and an operation time of weeks) we observe such effects. They are not fatal but can be expected to when constant operation is to last for over a year. Also, the product stability is at risk. The ceiling temperature of operation for alkaline hydrogen peroxide is 55-60 °C. With variations in current distribution, it locally goes higher, hampering the current efficiency. Models From Figure 2 (squares) showing that the experimental current rises upon lowering the velocity of low-conductivity catholyte pumping and from the fact that there is no response of the

J. Phys. Chem. B, Vol. 105, No. 7, 2001 1495 current in our reactor to changes of anolyte pumping (curve not shown), it is apparent that the electrical current flowing across the actual reactor is mostly determined by the conductivity of the catholyte chamber. This conductivity is a function of the charge carriers concentration distributions within the chamber. Therefore, the main objective of the model must be the calculation of these distributions. This is not an easy task, however, because the distributions result obviously from (i) charge carriers injection on the cathode and on the membrane and (ii) complex transport phenomena within the solution involving diffusion, migration, and convection (both natural and forced). Especially, the mathematical treatment of convection in its full is an extremely difficult task. In the case of laminar flow, one could consider the Navier-Stokes equation3 to calculate the spatial velocity distribution of liquid flow in the spacer-filled catholyte compartment. However, in our minds, it is difficult to accept a priori that the conditions of laminar flow are operative in our electrochemical reactor. In the general case of a liquid flowing between two ideally smooth, parallel plates, on the basis of the condition for laminarity which requires that the work done by an external pressure differential on the liquid be larger than the kinetic energy of the liquid,4 one obtains, with the geometric parameters of our reactor, a critical value of the Reynolds number equal to 260. For average linear velocities resulting in the actual Reynolds number below this value, the laminar flow is to be expected. Taking the velocities used in this paper, we obtain Reynolds numbers below 1 and the condition for laminarity seems to be well fulfilled. Two things, however, are not taken into account in this reasoning. First, turbulence promoters present in our system and the fact that the walls of the reactor are porous will decrease significantly the critical value of the Reynolds number. Second, the peristaltic pump we used had a periodic output pressure characteristic which excludes laminarity by definition (laminarity is when the local velocity vectors are time-invariant). These findings discouraged us from using the laminar approach. Perhaps the most adequate mathematical description, the physics of turbulent flow, is a challenge in itself and was also very quickly abandoned by the present authors. Instead, to cope somehow with the convection problem, we decided to consider two limiting-case models and say, that the reality must be somewhere between them. In the first model (referred to as model I), we assume that the liquid performs a layered, laminar movement between the parallel walls of the reactor and there is no natural convection of any kind, so the result is a parabolic distribution of the flow velocity in the direction along the x axis (see Figure 3). In model I, the transport of charge carriers in the direction perpendicular to the electrodes (x axis, see Figure 3) was taken to be of a combined diffusional and migrational nature. In the second model (referred to as model II), we assumed that there was no mechanical interaction of the liquid with the walls of the reactor and, also, no natural convection in the direction of flow, so the liquid can be thought of as moving with a uniform velocity in every point of the solution. In the x direction, we assumed a convection fast enough to even the concentrations of charge carriers, so that the concentration of a given ion was the same everywhere along the x axis (Figure 4). Detailed Assumptions for Model I. (i) The hydrogen anode, the anolyte compartment, and the membrane are treated altogether as an ideally nonpolarizable electrode. The potential of this ensemble is assumed to be constant regardless of the magnitude of current and close to the reversible hydrogen potential in base (-0.828 V vs NHE). Therefore, we only need to model the concentration distributions in the catholyte

1496 J. Phys. Chem. B, Vol. 105, No. 7, 2001

Piela and Wrona TABLE 1. Parameters for Models I and II of a Flow-Through, Alkaline Hydrogen Peroxide Generating, Fuel Cell-type Electrochemical Reactor parameter

symbol

cathode-membrane distance length of the electrolyzer (direction of flow) formal potential of the cathode reaction standard rate constant of the cathode reaction absolute temperature oxygen solubility

Figure 3. Space discretization scheme for model I. b, locations where the concentrations of ions are calculated; O, locations where the electrical potential gradient and diffusion-migration fluxes of ions are calculated; 0, locations where the flow fluxes of ions are calculated.

cation of the supporting electrolyte charge anion of the supporting electrolyte charge number of cations in the molecule of the supporting electrolyte diffusion coefficient of OHdiffusion coefficient of Na+ diffusion coefficient of HO2diffusion coefficient of Cat diffusion coefficient of An a

0.5 cm 6.5 cm

Ef0 ks

0.838 Va 104 cm s-1 M-1

T sO2 ZCat ZAn nCat

298.15 K 1.2685 × 10-3 M atm-1 1 -4 4

DOHDNa+ DHO2DCat DAn

5.28 × 10-5 cm2 s-1 1.33 × 10-5 cm2 s-1 2.0 × 10-5 cm2 s-1 1.33 × 10-5 cm2 s-1 0.8 × 10-5 cm2 s-1

See the text.

layer of the gas-diffusion cathode, CHO2-(0) and COH-(0) are the surface concentrations of the respective ions, and Ef0 is the formal potential (standard conditions: CO2 ) CHO2-(0) ) COH(0) ) 1 M) and is given, for convenience, vs the potential described in (i) (Ef0 ) -0.076 V - (-0.828 V) - RT/2F ln sO2 ) 0.838 V; sO2 denotes the solubility of oxygen in water at 298.15 K; see Table 1). (iii) The local potential of the cathode is determined by the externally applied voltage U and the local ohmic drop in the catholyte compartment (i denotes local current density in current divided by area units and Rint is the local internal resistance of the catholyte in resistance times area units; Ecath given again vs the potential described in (i)):

Ecath ) U - iRint

Figure 4. Space discretization scheme for model II. b, locations where the concentrations of ions are calculated; 0, locations where the flow fluxes of ions are calculated.

compartment. This assumption may hold strong taking that the hydrogen anode was a platinum microparticle-on-carbon-type with a dense nickel mesh current collector, the anolyte was NaOH at its maximum conductivity, and the membrane was a Nafion 117 ionomeric membrane (du Pont) known for its high sodium ion conductivity. (ii) The oxygen reduction on the cathode is reversible (transport-controlled) and proceeds entirely via pathway 1 (100% current efficiency for the peroxide ion). This assumption is well fulfilled because oxygen was reduced on carbon and the medium was alkaline.5,6 We also assume that the equilibrium potential for the cathode is equal to the thermodynamic potential of the two-electron oxygen reduction in base (-0.076 V vs NHE). The assumption is expressed by the Nernst equation

Ecath ) Ef0 +

C O2 RT ln 2F CHO -(0)COH-(0) 2

(4)

where CO2 is the dissolved oxygen concentration in the reaction

value

d l

(5)

(iv) The dissolved oxygen concentration in the reaction layer of the gas-diffusion cathode is constant and equal to the equilibrium oxygen concentration in water under a given oxygen pressure at 298.15 K. The transport of oxygen in the reaction layer is not modeled. It is also assumed that the concentration of oxygen in the bulk catholyte is zero. These assumptions hold weak and, in the model, yield currents that are somewhat greater than they should be, but we found that putting even a very small oxygen concentration still gave reasonable current densities. Some support for this assumption is found later in the results of the simulation. (v) The model does not comprise any double layer effects. It deals only with bulk ion transport phenomena. Therefore, the principle of electroneutrality holds in every point of the modeled space:

ZionCion ) 0 ∑ ions

(6)

Zion is the ion’s charge, and Cion is its local concentration. The sum is taken over all ions in solution, i.e., HO2-, OH-, Na+, Cat, and An (the latter two being the cation and the anion of the supporting electrolyte). (vi) We neglect all ion-pairing effects. The catholyte is treated as an ideal solution regardless of the concentrations of ions. This allows us to use ion concentrations instead of activities as well as estimate the conductivity of the solution as the simple sum of contributions of all ions. This assumption creates a


J. Phys. Chem. B, Vol. 105, No. 7, 2001 1497

discrepancy between the model and the reality especially at longer times when the concentrations become high. However, this only makes the model less quantitative and, in our minds, does not put the qualitative-like objective of this work into doubt. (vii) In this limiting case, the ions diffuse and migrate in the x direction (no convection) but collectively move in the y direction with a linear velocity V(x), and this kind of convective movement predominates in this direction over the migration and diffusion phenomena. From the assumption of layered, laminar flow, we obtain the following parabolic distribution of V(x) along the x axis:

V V(x) ) 6 2x(d - x) d

(7)

where V is the average linear velocity, a parameter to the simulation, and d is the distance between the electrode and the membrane. Each ion’s flux has, therefore, two components:

fion(x) ) -Dion

∂Cion ZionF ∂φ D C ∂x RT ion ion ∂x

fion(y) ) V(x) Cion

(8) (9)

Dion is the diffusion coefficient for the ion and φ is the local electrical potential. The expressions 8 and 9 for the fluxes are used in the following transport-governing equation (continuity equation):

∂fion(x) ∂fion(y) ∂Cion )∂t ∂x ∂y

(10)

Detailed Assumptions for Model II. (i)-(vi) are the same as those in model I, except for the way in which the cathodic reaction is ascertained to be transport-controlled (assumption (ii)). Here, we use the electrode kinetics theory

(

kfh ) ks exp kbh ) ks exp

(

)

0.5F(Ecath - Ef0) RT

)

1.5F(Ecath - Ef0) RT

(11)

(12)

and put a high value for ks (104 cm s-1 M-1). The formal potential, Ef0 has the same meaning as that in model I. The forward and the back transfer coefficients (0.5 and 1.5) correspond to reaction 1 being limited by the transfer of the first electron. The need to use kinetic expressions in model II comes from the fact that, now, we only have one concentration value for every species in solution at a given y coordinate and cannot calculate the local current from the flux of products at the electrode surface, as we do in model I (for this we need two concentration values per species at a given y, see eqs 19 and 20). Although formally different, both approaches result in reversibility at the cathode. (vii) In this case, the flow in the y direction is dealt with as in model I (eq 9) with the exception that the parabolic velocity distribution is not taken into account, so V(x) is replaced by V in eq 9. In the x direction, we assume uniform concentrations of all species. Therefore, the governing equation for model II incorporates simply Faraday’s law of electrolysis and takes the form

∂fion(y) ∂Cion i )∂t nionFd ∂y

(13)

where nion is the number of electrons passing per molecule of generated ion (nHO2- ) nOH- ) 2 and nNa+ ) 1). Algorithm of the Simulation The method to include diffusion, migration, and resistance effects in an electrochemical simulation has been described in the literature.3,7-9 Here we adopt these ideas to our flow-through reactor, which only necessitates the incorporation of an additional step in the calculations responsible for the concentration changes because of the catholyte flow. To solve the governing eqs 10 and 13, we used the explicit finite differences method.10-12 The problem is two-dimensional for model I and one-dimensional for model II. The space between the cathode and the membrane was discretized for model I and model II, as is shown in Figures 3 and 4, respectively. In model I, the space was divided into M(N + 1) cells with dimensions ∆x and ∆y, with the total number of concentrations to calculate being (M + 1)(N + 1). In model II, there are N + 1 cells with dimensions d and ∆y, and the number of concentrations is N + 1. (M and N are the number of mesh spaces in the x and y directions, respectively.) On the basis of these discretization schemes, the simulation involved the following steps. Step 0. Implementation of the starting conditions. Model I: CHO2-(m, n) ) 0 M, COH-(m, n) ) 0 M, CNa+(m, n) ) 0 M, CCat(m, n) ) CCat0, CAn(m, n) ) CAn0,

∂φ (m, n) ) 0 V cm-1 ∂x m ) 0, 1, ..., M for the concentrations, and m ) 0, 1, ..., M 1 for the electrical field gradient; n ) 0, 1, ..., N. Model II: CHO2-(n) ) 0 M, COH-(n) ) 0 M, CNa+(n) ) 0 M, CCat(n) ) CCat0, CAn(n) ) CAn0, n ) 0, 1, ..., N. CCat0 and CAn0 denote the initial concentrations of the cation and the anion of supporting electrolyte. Step 1. Calculation of the local cathode-membrane internal resistances Rint(n) and the internal resistance of the entire cell Rint. The local internal resistances can be obtained from the specific conductivity (κ) distribution along the x axis:

Rint(n) )

∫0d dxκ

(14)

with the conductivity being the following function of the local concentrations of all ions:

κ)

F2

Zion2DionCion ∑ RTions

(15)

In model I, using the trapezoidal rule, we arrive at the following numerical formula for the local resistance:

Rint(n) )

RT∆x F2

M

2

∑ m)1 Zion2Dion[Cion(m - 1, n) + Cion(m, n)] ∑ ions

(16)

In model II, the formula takes a simpler form:


Rint(n) )

RTd

1

F2

Zion2DionCion(n) ∑ ions

Piela and Wrona

(17)

i(n) ) 2FfO2(n)

Then, the overall internal resistance (Rint(n)s in parallel connection) can be calculated:

N+1

Rint )

N

∑ n)0R

Model II: Here we use an expression equivalent to eq 19:

The flux of oxygen consumed at the electrode fO2(n) is now given by a kinetic expression

fO2(n) ) -kfhCO2 + kbhCHO2-(n)COH-(n)

(18)

1 int(n)

Step 2. Calculation of the local current densities, i(n) and the average current density ˆı. Model I: From Faraday’s law of electrolysis we have

i(n) ) -2FfHO2-(0, n)

(19)

fHO2-(0, n) ) -DHO2F D RT HO2-

∆x

i)F

2

∂x

∂φ

1

∂x

(0, n) (20)

The necessary surface concentrations of HO2-, CHO2-(0, n), can be expressed by the first, adjacent to the electrode concentrations of HO2- and OH-, CHO2-(1, n) and COH-(1, n) using eqs 4, 5, and 8 (in numerical form) and stoichiometry (fHO2-(0, n) ) fOH(0, n)). Because eq 5 contains the local current density, eq 19 becomes nonexplicit with respect to the local current density (and nonlinear). The root of this equation was extracted numerically with a procedure similar to the Mu¨ller’s procedure13 (bisection + parabolic interpolation). The average current density is given by

ˆı )

N

∑i(n)

N + 1n)0

(21)

The membrane surface concentrations of sodium ions CNa+(M, n) are simultaneously obtained from eq 8 (in numerical form), and the assumption that the membrane surface flux of Na+ compensates the electrode surface fluxes of HO2- and OH(fNa+(M - 1, n) ) -fHO2-(0, n) - fOH-(0, n)). The concentrations of Na+ on the electrode, CNa+(0, n), and the concentrations of HO2- and OH- on the membrane, CHO2(M, n) and COH-(M, n), are calculated in the same step from the relationships expressing the impermeability of the electrodesolution and membrane-solution boundaries to the respective ions:

fHO2-(M, n) ) fOH-(M, n) ) fNa+(0, n) ) 0

Zionfion ∑ ions

(27)

and eq 8 we arrive at the general expression for the gradient of electrical potential in an electrolyte solution

+

CHO2-(0, n) + CHO2-(1, n) ∂φ

(26)

Using eqs 5, 11, and 12, we obtain, again, a nonexplicit equation for the local current density which is solved numerically. The expression for the average current density is the same as that in model I (eq 21). Step 3. Recalculation of the electrical potential gradients on the basis of the calculated local current densities (model I only). Using the Faraday’s law of mass and charge equivalency

where fHO2-(0, n) is the electrode surface flux of HO2- ions given by eq 8. This equation takes the numerical form

CHO2-(1, n) - CHO2-(0, n)

(25)

(22)

Once the surface concentrations of the electroactive species are calculated, we can calculate the electrode and membrane surface concentrations of ions of the supporting electrolyte CCat(0, n), CAn(0, n), CCat(M, n), and CAn(M, n) from the principle of electroneutrality (eq 6) written for the boundaries and

fCat(0, n) + fAn(0, n) ) 0

(23)

fCat(M - 1, n) + fAn(M - 1, n) ) 0

(24)

)-

(

1 κ

i+F

ZionDion ∑ ions

)

∂Cion ∂x

(28)

which takes the numerical form

∂φ

-2RT

(m, n) )

∂x

F2

[

Zion2Dion[Cion(m, n) + Cion(m + 1, n)] ∑ ions

i(n) + F

ZionDion ∑ ions

]

Cion(m + 1, n) - Cion(m, n) ∆x

(29)

Step 4. Recalculation of the bulk concentrations of all species changing because of diffusion and migration (model I) or flowing local current (model II). Finite differences versions of eqs 10 (combined with eq 8) and 13 are used respectively to calculate the temporal changes of concentrations

C′ion(m, n) ) Cion(m, n) -

∆t [f (m, n) - fion(x)(m - 1, n)] ∆x ion(x) (30)

C′ion(n) ) Cion(n) -

i(n)∆t nionFd

(31)

m ) 1, 2, ..., M - 1 and n ) 0, 1, ..., N. Naturally, in eq 31, only the electrogenerated ions are considered. Step 5. Recalculation of concentrations of all species changing because of the catholyte flow. The finite differences version of eq 10 (combined with eq 9) is used:

C′′ion(m, n) ) C′ion(m, n) -

∆t [f (m, n) - fion(y)(m, n - 1)] ∆y ion(y) (32)

C′′ion(n) ) C′ion(n) -

∆t [f (n) - fion(n - 1)] ∆y ion

(33)

m ) 0, 1, ..., M and n ) 0, 1, ..., N. Step 6. Correction of the bulk concentrations of all species for the electroneutrality condition (model I only). Each ion’s concentration is adjusted proportionally to its magnitude to


Figure 5. Average current density vs electrolysis time plots calculated with model I (solid lines) and model II (dotted lines). Voltage, 0.000 V (0.838 V cathode overvoltage); oxygen pressure, 1 atm; base electrolyte concentration, 0.002 M; catholyte linear velocity, curves (A) 0.5 cm min-1, (B) 0.7 cm min-1, (C) 1.5 cm min-1.


Figure 6. Total ohmic drop vs electrolysis time plots calculated with model I (solid lines) and model II (dotted lines). Voltage, 0.000 V (0.838 V cathode overvoltage); oxygen pressure, 1 atm; base electrolyte concentration, 0.002 M; catholyte linear velocity, curves (A) 0.5 cm min-1, (B) 0.7 cm min-1, (C) 1.5 cm min-1.

achieve local electroneutrality. This operation had almost no impact on the simulated ˆı-t curves. Step 7. Return to Step 1, hence, advancing in time by ∆t. The computer program allowed changing the average linear velocity of the catholyte flow V, the concentration of the inflowing supporting electrolyte, the pressure of oxygen, and the electrode-membrane voltage U at run time. Table 1 summarizes all of the other simulation parameters which were kept constant during the calculations. The values are typical for one of the experimental setups studied by the authors. The values of diffusion coefficients of OH- and Na+ were taken from reference14 and are the zero-concentration extrapolated values. The diffusion coefficient of HO2- was estimated from DOH- (a similar transport mechanism is possible). We used EDTA as the supporting electrolyte, so Cat is Na+, and the diffusion coefficient of An (EDTA4-) was given a value similar to those found in the literature13 for organic anions of comparable mass and charge. Results and Discussion Figure 5 shows a comparison of the average current density vs the electrolysis time plots obtained with models I and II. The simulation conditions for curves labeled with B (see figure captions) match those for the experimental curve from Figure 2 (circles). The models reasonably explain (i) the magnitude of the current density, model I yields values smaller and model II values greater than the experimental ones (model II is closer to the experiment which indicates that convection is the predominating way of transport in the real system), (ii) the current induction time, and (iii) the relationship between current density and catholyte linear velocity, the faster the flow the smaller currents are attained; however they fail at explaining the shape of the curve with a distinct maximum, at this time. Figure 6 presents the evolutions of the total ohmic drop in the catholyte compartment with electrolysis time for the same calculations. Closely, the entire cathode overvoltage (0.838 V) is lost in the ohmic drop. This could explain the experimental fact that we could never record the limiting current in a potentiodynamic

Figure 7. Steady-state (on current plateau) ion concentrations and electrical potential gradient profiles calculated with model I. Voltage, 0.000 V (0.838 V cathode overvoltage); catholyte linear velocity, 0.7 cm min-1; oxygen pressure, 1 atm; base electrolyte concentration, 0.002 M.

experiment with our cell, simply because the electrode was not sufficiently cathodically polarized for the surface oxygen concentration to fall close to zero. Figure 7 shows steady-state (on current plateau) ion concentration profiles and electrical potential gradient distribution in the catholyte compartment obtained from model I under the same conditions as those for curve B, solid line, in Figure 5. Apparently, the current raises in the following way. At an early


Figure 8. Local current density (solid lines) and local cathode potential (dotted lines) distribution along the direction of flow from model I (O) and model II (0) at steady state (on current plateau). Voltage, 0.000 V (0.838 V cathode overvoltage); catholyte linear velocity, 0.7 cm min-1; oxygen pressure, 1 atm; base electrolyte concentration, 0.002 M.

stage after applying voltage, the ions of the supporting electrolyte accumulate in the vicinity of the cathode (Na+ from EDTA) and the membrane (EDTA4-), allowing a small but growing amount of the HO2-, OH-, and Na+ to be generated into the catholyte. Soon, because of diffusion and migration (and convection, in reality), the concentration profiles of the electrogenerated ions start to overlap, and then those ions become the principal charge carriers. The resistance of the solution is lowered, and the current builds up. Figure 8 shows how models I and II predict the local current density and local cathode potential distribution along the direction of flow (y) at steady state under the same conditions as those for curves B in Figure 5. The electrode is nonequipotential, and the current density is several times greater at the catholyte exit than at the entrance. This could cause severe problems in industrial practice and should be addressed through special electrolyzer design. Figures 9-12 present calculation results for parameters V (catholyte average linear velocity) and U (cathode-membrane voltage) outside of the experimental range. U was set to 0.738 V (0.100 V cathode overvoltage), and V was lowered almost 2 orders of magnitude (complete description in the captions). Figure 9 shows the average current density vs time plots. The curves labeled with B are similar in shape to the experimental curves (Figure 2, circles), having a maximum and a plateau. The position of the maximum on the time scale is inconsistent with that of the experiment (up to 1 h in the experiment against approximately 3 h from calculations). In the calculations, this position is a function of the geometric parameters (d, l) and the diffusion coefficients of ions but not the catholyte average linear velocity V. Also, the difference in the attained current densities from models I and II is no longer big. Figure 10 shows the respective total ohmic drop vs time curves. In this case, the ohmic drop can fall below 20% of the total cathode overvoltage. Figure 11 presents the steady-state ion concentration profiles and the electrical potential gradient distribution in the catholyte space obtained from model I with parameters the same as those for curve B, solid line, on Figure 9. One can see that the maximum on the average current density vs time curve

Piela and Wrona

Figure 9. Average current density vs electrolysis time plots calculated with model I (solid lines) and model II (dotted lines). Voltage, 0.738 V (0.100 V cathode overvoltage); oxygen pressure, 1 atm; base electrolyte concentration, 0.002 M; catholyte linear velocity, curves (A) 0.0 cm min-1, (B) 0.02 cm min-1, (C) 0.04 cm min-1.

Figure 10. Total ohmic drop vs electrolysis time plots calculated with model I (solid lines) and model II (dotted lines). Voltage, 0.738 V (0.100 V cathode overvoltage); oxygen pressure, 1 atm; base electrolyte concentration, 0.002 M; catholyte linear velocity, curves (A) 0.0 cm min-1, (B) 0.02 cm min-1, (C) 0.04 cm min-1.

originates from the fact that the catholyte becomes saturated with the products of the cathodic reaction and that the cathodic reaction is reversible (accumulation of products extinguishes the reaction). In fact, the current falls to zero when there is no catholyte flow (curves A, Figure 9). It also becomes obvious why the location of the maximum on the time scale depends on the volume of the catholyte compartment and the mobility of ions (these parameters determine the time it takes to saturate the compartment with ions). To be more accurate, one should say that in reality the volume in which the ions strongly accumulate, not the volume of the whole catholyte compartment, is decisive. In the studied generator, which utilizes a gasdiffusion cathode, this volume could correspond to the volume



Figure 12. Local current density (solid lines) and local cathode potential (dotted lines) distribution along the direction of flow from model I (O) and model II (0) at steady state (on current plateau). Voltage, 0.738 V (0.100 V cathode overvoltage); catholyte linear velocity, 0.02 cm min-1; oxygen pressure, 1 atm; base electrolyte concentration, 0.002 M.

Figure 11. Steady-state (on current plateau) ions concentrations and electrical potential gradient profiles calculated with model I. Voltage, 0.738 V (0.100 V cathode overvoltage); catholyte linear velocity, 0.02 cm min-1; oxygen pressure, 1 atm; base electrolyte concentration, 0.002 M.

of the liquid-filled pores in the structure of the electrode. Another fact in support for this is that when the experiment is conducted at a higher temperature (meaning faster transport of product ions away from the pores) the recorded current density vs time curves no longer have a maximum and are just like the curves in Figure 5. The present simple models do not encompass this structural feature of the system and cannot, therefore, produce the maximum with the real system’s catholyte velocity. This sets clear prospects for future work on a single physical model of the reactor. A model similar to model I but taking into account a layer of liquid-filled pores in the electrode should be appropriate. In that layer, there would be no forced convection, so the products would accumulate there faster, and this process would be virtually insensitive to the average linear velocity of catholyte. Second, the unitized model should comprise a convection treatment in the bulk of the catholyte assuming either laminarity or turbulence and taking into account the geometry of turbulence promoters, to simulate the resistance changes of the reactor properly. Figure 12 presents the steady-state local current density and local cathode potential distributions along the y direction for curves B in Figure 9. Relationships calculated with models I and II are now very similar. The cathode is still nonequipotential. The current is now carried mostly by the center of the electrolyzer. At the catholyte inlet, the resistance is too high for the current to develop, and at the outlet, the product ions are accumulated, such that the reverse electrode reaction takes place to a considerable extent and the resulting current is low again. In Figure 13, there is a comparison of the experimental average current density vs time curve, shown also in Figure 2: squares, recorded with changes of catholyte flow velocity during

Figure 13. Average current density vs electrolysis time plots. O, curve simulated with model II; 0, experimental curve; ], curve simulated with modified model II (see text). Voltage, 0.000 V; 1 atm O2; base electrolyte, O, 0, 0.002 M EDTA; ], 0.01 M EDTA; catholyte linear velocity, start at 5.7 cm min-1 (A), then switch to 2.5 (B), 1.2 (C), and 0.7 cm min-1 (D).

experiment, with a model II-simulated curve; circles, also with changes of flow velocity during simulation. Although the curves do not overlap, they follow the same trends. The substantial differences can originate from several facts such as the use of concentrations instead of activities in the models to calculate the conductivity, the fact that model II assumes an infinitely fast transport in the x direction which is certainly not the case in the experiment, or the fact that the initial conductivity of the catholyte chamber is probably higher than the one calculated from 0.002 M EDTA concentration because of leaking properties of the membrane. To obtain a better fit of the experimental curve in Figure 13 with the simulated one, we have slightly modified model II. We have added a constant component into the local internal resistance given by eq 17. This resistance could

1502 J. Phys. Chem. B, Vol. 105, No. 7, 2001 correspond to the sum of all constant resistances in the real system, such as the resistance of the electrode-current collector assemblies or the resistance of the ionomeric membrane. The curve labeled with diamonds in Figure 13 is the new modeled curve. We used a constant internal resistance component of 14 Ω cm2 and a supporting electrolyte concentration of 0.01 M (both are realistic values). One can see that, despite the differences in shape still present, at least the attained average current density is now correctly modeled. We have also tested the effect of this modification on previously calculated results (both models). Although some changes in the shape of the ˆı-t curves originated, the conclusions reached remained valid. Summary Two simplified models of mass and charge transport in the catholyte chamber of a flow-through electrochemical reactor for the generation of alkaline hydrogen peroxide were presented. In the first model, we assumed that the transport in the direction perpendicular to the catholyte flow is of a diffusionalmigrational nature, solely (no convection). In the second model, contrary to the first, we assumed a convection in the direction perpendicular to the flow fast enough to even the concentrations of all ions (in that direction). Both models dealt with the transport in the direction of flow in the same way: forced convection was predominating (migration and diffusion were neglected). In the first model, we used a parabolic velocity distribution in the direction perpendicular to the flow corresponding to layered, laminar movement of the liquid. In the second model, a uniform velocity was assumed in every point of the solution. The catholyte had initially a low ionic strength (0.002 M EDTA), so the ohmic drop was addressed in the models. Both models were able to (i) explain the magnitude of current density recorded in the experimental system (however, the model assuming a fast convection in the direction normal to the electrode did it better), (ii) explain the induction time for the

Piela and Wrona current to develop, and (iii) simulate the behavior of the system when the catholyte velocity was changed (decrease in the velocity causes the current to increase). The shape of the chronoamperometric curves, possessing a maximum and a subsequent plateau, could only be simulated correctly using cell operation parameters outside of the experimental range. This discrepancy was discussed upon the differences between the models and the real system, and an outline of a unitized model of the reactor was proposed. Acknowledgment. The authors greatly acknowledge the help of Dr. Jerzy Chlistunoff and Dr. Marek Orlik in the elaboration of the models. This work was supported by the Polish Committee for Scientific Research through Grant 3 T09A 046 18. References and Notes (1) Foller, P. C.; Bombard, R. T. J. Appl. Electrochem. 1995, 25, 613. (2) Bond, A. M.; Fleischmann, M.; Robinson, J. J. Electroanal. Chem. 1984, 168, 299. (3) Orlik, M. J. Phys. Chem. B 1999, 103, 6629. (4) Szczeniowski, S. Experimental Physics, Part I, Mechanics and Acoustics; PWN: Warsaw, Poland, 1972; Chapters 86-96. (5) Tarasevich, M. R.; Sadkowski, A.; Yeager, E. Oxygen Electrochemistry. In ComprehensiVe Treatise of Elctrochemistry; Conway, B. E., Bockris, J. O’M., Yeager, E., Khan, S. U. M., White, R. E., Eds.; Plenum Press: New York, 1983; Vol. 7, Chapter 6, p 353. (6) Hoare, J. P. Oxygen. In Encyclopedia of Electrochemistry of the Elements; Bard, A. J., Ed.; Marcel Dekker: New York, 1974; Vol. II, pp 191-382. (7) Orlik, M.; Doblhofer, K.; Ertl, G. J. Phys. Chem. B 1998, 102, 6367. (8) Laser, D.; Bard, A. J. J. Electrochem. Soc. 1976, 123, 1834. (9) Jaworski, A.; Donten, M.; Stojek, Z. J. Electroanal. Chem. 1996, 407, 75. (10) Feldberg, S. W. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1986; Vol. 3, p 199. (11) Britz, D. Digital Simulation in Electrochemistry, 2nd ed.; Springer: Berlin, Germany, 1988. (12) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; Wiley: New York, 1980. (13) Rodman, R. D. Commun. ACM 1963, 6, 442. (14) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworth: London, 1959.

Modeling the Electrical Behavior Of An Alkaline Hydrogen Peroxide

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