Theoretical and Experimental Study of the Primary Current Distribution

LABORATORY EXPERIMENT pubs.acs.org/jchemeduc

Theoretical and Experimental Study of the Primary Current Distribution in Parallel-Plate Electrochemical Reactors Armando I. Vazquez Aranda,† Eduardo R. Henquín,‡ Israel Rodríguez Torres,† and Jose M. Bisang*,‡ †

Facultad de Ingeniería, Instituto de Metalurgia, Universidad Autonoma de San Luis Potosí, Av. Sierra Leona 550, Lomas 2a seccion, San Luis Potosí, S.L.P., Mexico CP 78210 ‡ Programa de Electroquímica Aplicada e Ingeniería Electroquímica (PRELINE), Facultad de Ingeniería Química, Universidad Nacional del Litoral, Santiago del Estero 2829, S3000AOM Santa Fe, Argentina

bS Supporting Information ABSTRACT: A laboratory experiment is described to determine the primary current distribution in parallel-plate electrochemical reactors. The electrolyte is simulated by conductive paper and the electrodes are segmented to measure the current distribution. Experiments are reported with the electrolyte confined to the interelectrode gap, where the current distribution is uniform, and the case of unconfined electrolyte is treated by raising some segments. Experimental data are compared with theoretical results, and a close agreement is achieved. This experiment enables the student to gain a useful knowledge of the primary current distribution in electrochemical reactors. KEYWORDS: Upper-Division Undergraduate, Chemical Engineering, Laboratory Instruction, Hands-On Learning/Manipulatives, Electrochemistry

everal authors14 have analyzed the potential distribution and have claimed the necessity to design electrochemical reactors with uniform current distribution at the electrode surfaces to obtain better quality products, improve the utilization of electrode materials, avoid explosion risk, increase current efficiency, and achieve a better use of electrical energy. Current distribution depends on the geometric factors of the electrochemical reactor, the conductivity of the electrolyte, and the kinetics of the electrochemical reactions. According to these variables, current distribution is classified as primary when only geometric factors are considered. Taking into account kinetics, current distribution is called secondary when the reaction is under charge transfer control, and tertiary when mass transfer is included in the analysis. The primary current distribution represents the simplest situation for mathematical treatment and in monopolar reactors it shows the most pronounced case. Thus, for a given electrochemical system, if the primary current distribution is acceptable, the secondary and tertiary ones are still better. For these reasons, the primary current distribution is frequently analyzed. Electrochemical reactors with parallel-plate electrodes are the more common cell arrangement used in the industrial practice,5 that is, electrowinning and electrorefining of metals, electrosynthesis of inorganic and organic compounds, fuel cells, and batteries. These reactors present, as an advantage, simplified constructive features; the interelectrode gap is uniform and can be defined by a single and easily adjustable geometric variable. Moreover, the primary and secondary current distributions are substantially uniform over most of the central portion of the electrode. However, a considerable edge effect occurs near the

S

Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.

inlet and outlet of the electrolyte to the electrode region, where the current density tends toward very high values. Furthermore, Frias-Ferrer et al.6 planned a laboratory experiment to examine the fluidodynamic aspects of the entrance and exit effects in parallel-plate electrochemical reactors. The aim of the present contribution is to describe a laboratory experiment for a course of electrochemistry to introduce students to the study of current distribution in electrochemical systems and to compare experimental data with theoretical results from the numerical and analytical solution of the Laplace equation. This laboratory experiment was carried out by advanced students of chemical engineering during the last four years. It was performed in approximately 2 h by groups of two students and it provided the students with an overall understanding of primary current and potential distribution in electrochemical systems. The students presented a full report about the experiment a week later at the end of the class. The report contains the experimental procedure, a brief summary of the theory, discussion of the results, and a comparison between experimental and theoretical results.

’ EXPERIMENTAL DETAILS Description of the Experimental Setup

The experimental arrangement is shown in Figure 1 and the symbols are defined in Table 1. The electrolyte was simulated by a sheet of conductive paper (Pasco Scientific, PK 9025)7 Published: October 17, 2011 163

dx.doi.org/10.1021/ed200170v | J. Chem. Educ. 2012, 89, 163–167

Journal of Chemical Education

LABORATORY EXPERIMENT

mounted on a nonconducting board and the electrodes were formed by copper segments, 6.1 103 m wide, at opposite sides of the conductive paper. The segments were insulated from one another by an approximately 5 104 m thick Teflon slide and were connected to the conductive paper by colloidal silver paint (Pasco Scientific, PK9031B) to minimize the contact resistance. The thickness of the conductive paper was 1.3 104 m and the resistivity was 3.48 Ω m. Calibrated resistors, approximately 30 Ω resistance, were intercalated between each segment and the current feeder of the electrode. The effect of the calibrated resistors on the current distribution can be neglected due to the small value of their resistance in comparison to that of the conductive paper. A dc power supply was used to apply a constant current to the feeders. The electric connection was made in the middle point of the current feeder of each electrode. Two electrochemical systems were examined. In the first one, the conductive paper and 15 copper segments were trimmed so that they were the same length, 0.1 m with 0.02 m interelectrode gap. This case, called confined electrolyte and represented in Figure 1A, permits for, by pressing the segments, the adjustment of the contact resistance with the conductive paper to obtain a uniform current distribution. In the second system, some terminal segments were raised. Thus, the paper length is larger than the electrode length, represented as unconfined electrolyte in Figure 1B.

The experimental current distribution at each electrode is determined by measuring the ohmic drops in the calibrated resistors. The data acquisition is performed using a computer controlled analogue multiplexer. An alternative procedure is to connect the calibrated resistors to a selector switch with 15 positions coupled to a recorder. For a given value of the total current, several experiments are carried out to verify the reproducibility of the results. Typical values of the applied potential to the system are in the range from 5 to 40 V to obtain a total current between 1 and 5 mA. The next step is to compare the experimental results with the theoretical calculations obtained by solving the Laplace equation.

’ THEORETICAL CONCEPTS The primary current distribution in electrochemical reactors is obtained by solving the Laplace eq 1 in the solution phase ∇2 ϕ ¼ 0

ð1Þ

with the following boundary conditions at the terminal anode, ϕ ¼ UA

ð2Þ

at the terminal cathode, ϕ ¼ UC

Experimental Procedure

Before each laboratory session, to facilitate the student work and shorten the duration of the experiments, instructors should calibrate the resistors. The instructors should also verify that the contact resistance between each segment and the conductive paper is minimal and that it approximately has the same value for all segments, following the guidelines given in the Supporting Information.

ð3Þ

and at the insulating walls ∂ϕ ∂ϕ ¼ ∂x ∂y inslulating walls

¼0

ð4Þ

insulating walls

The current density at any point at the electrode surface is given by 1∂ϕ ji ¼ with i ¼ A or C ð5Þ F ∂x at electrode i

Numerical Resolution of the Laplace Equation

The numerical resolution of the Laplace equation for the unconfined system shown in Figure 1B is outlined in the following paragraphs. The geometric parameters are defined in Figure 2. Therefore, the solution of eq 1, by the finite difference method with an equidistant grid,8 yields the potential for the bulk of the electrolyte: Figure 1. Schematic view of the experimental arrangement for (A) confined system and (B) unconfined system: (1) conductive paper, (2) segmented electrodes, (3) calibrated resistors, (4) current feeder of the electrode.

ϕðx, yÞ ¼

ϕðx þ h, yÞ þ ϕðx h, yÞ þ ϕðx, y þ hÞ þ ϕðx, y hÞ 4

ð6Þ

Table 1. Description of the Symbols Symbol

Description

Symbol

Description

ai

Constants in eq 15

U

Potential of the metal phase (V)

e

Interelectrode gap (m)

x

Axial coordinate (m)

h

Distance between two nodes in the potential grid (m)

y

Axial coordinate (m)

H

Distance from the electrode end to the reactor bottom (m)

ε

Dimensionless geometric parameter given by eq 19

j jmean

Current density (A m2) Mean current density (A m2)

F ϕ

Electrolyte resistivity (Ω m) Potential in the solution phase (V)

L

Electrode length (m)

A

Anode

r

Electrolyte thickness outer the interelectrode gap (m)

C

Cathode

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dx.doi.org/10.1021/ed200170v |J. Chem. Educ. 2012, 89, 163–167



Figure 2. Geometric representation of an electrochemical reactor with unconfined electrolyte to the interelectrode gap. Thick black lack lines, electrodes; light gray shading, electrolyte; A, anode; C, cathode. Left normal arrangement. Right inverted arrangement.

The finite difference method applied to the boundary condition given by eq 4 produces For x = 0 r and L < y < L + H is ϕðx, yÞ ¼

2ϕðx þ h, yÞ þ ϕðx, y þ hÞ þ ϕðx, y hÞ 4

Figure 3. Contour plots of potential distribution and current lines for an unconfined system. L = H, e = L + H, r = 0. Full lines: potential distribution. Dashed lines: current lines. Thick black lines: electrodes. UA = 1 V. UC = 0 V.

ð7Þ

ϕ(2h, y), ϕ(3h, y), and ϕ(4h, y), were fitted with the polynomial:9

For x = e + r and L < y < L + H is ϕðx, yÞ ¼

2ϕðx h, yÞ þ ϕðx, y þ hÞ þ ϕðx, y hÞ 4

ϕðx, yÞ ¼ a0 þ a1 x þ a2 x2 þ a3 x3

ð8Þ

Introducing the first derivative of eq 15, evaluated at the electrode surface, in eq 5 the current density is given by 1 ∂ϕðx, yÞ a1 ðyÞ ð16Þ jðyÞ ¼ ¼ F ∂x F

For y = 0 and 0 < x < e is ϕðx, yÞ ¼

ϕðx þ h, yÞ þ ϕðx h, yÞ þ 2ϕðx, y þ hÞ 4

ð9Þ

at electrode

For y = L + H, 0 r < x < e + r is ϕðx þ h, yÞ þ ϕðx h, yÞ þ 2ϕðx, y hÞ ϕðx, yÞ ¼ 4

Thus, the current density distribution results in jðyÞ a1 ðyÞ ¼ Z L jmean 1 a1 ðyÞ dy L 0

ð10Þ

Likewise, the potential in the corners of the electrolyte is given by For y = L + H and x = 0 r is ϕðx, yÞ ¼

2ϕðx þ h, yÞ þ 2ϕðx, y hÞ 4 2ϕðx h, yÞ þ 2ϕðx, y hÞ 4

ð11Þ

ð12Þ

Moreover, the electrodes are represented by For x e 0 and 0 e y e L is ϕðx, yÞ ¼ UA

Analytical Solution of the Laplace Equation

Two plane electrodes placed opposite to each other in the walls of a channel flow of infinite length has been theoretically considered by Parrish and Newman.11 In this case, the following equation represents the primary current distribution

ð13Þ

and For x g e and 0 e y e L is ϕðx, yÞ ¼ UC

ð17Þ

The current lines are obtained solving the potential distribution for the inverted arrangement,10 sketched on the right-hand side of Figure 2, where the electrodes are replaced by insulating walls, whereas the insulating walls of the normal configuration are considered electrodes. Thus, the equipotential lines of the inverted cell are equivalent to the current flow lines of the normal cell. The students can calculate the theoretical primary potential and current density distributions with an Excel spreadsheet included in the Supporting Information.

For y = L + H and x = e + r is ϕðx, yÞ ¼

ð15Þ

jðyÞ ε coshðεÞ=K½tanh2 ðεÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jmean sinh2 ðεÞ sinh2 ðyε=LÞ

ð14Þ

ð18Þ

where K(m) is the complete elliptic integral of the first kind and ε is a characteristic geometric parameter given by eq 19. In eq 18, the y coordinate is measured from the center of the electrode

To calculate the current density for a given axial position y along the electrode length, the four potential points in the solution phase nearest to each electrode surface, that is, ϕ(h, y), 165




Figure 4. Current distribution: (0) confined system, (b) unconfined system with 10 segments. Vertical bars: standard error of the mean. Full line: numerical solution of the Laplace equation, r = 5 mm. Dashed line: current distribution according to eq 18.

Experimental results of unconfined systems with 10 and 8 segments, respectively, are shown in Figures 4 and 5. Each point represents the mean value at both electrodes for three experiments with different currents; the standard error of the mean is given by the vertical bars, and the full line was obtained by numerical solution with the finite difference method of the Laplace equation. The results of the confined system are also included. As expected, the unconfined system presents a pronounced current distribution and there is a close agreement between experimental and theoretical results. The dashed lines in Figures 4 and 5 correspond to the behavior according to eq 18, where a more pronounced current distribution is shown because the unconfined electrolyte is larger in this theoretical treatment. From Figures 4 and 5, it is concluded that the presence of electrolyte outside the interelectrode gap produces a pronounced current distribution that may compromise the performance of the electrochemical reactor. The solution to obtain a uniform current density distribution is to enclose the electrodes with perpendicular insulating plates as shown the confined system.

’ HAZARDS Equipment and materials used in this work do not present risks to people or the environment. ’ CONCLUSION This laboratory experiment allowed students to obtain an improvement in understanding of the primary current and potential distribution in electrochemical systems, which is very important in electrochemical technology and also in basic electrochemistry. During the exercise, students clearly observed that the presence of unconfined electrolyte in an electrochemical reactor produces current distribution at the electrode surfaces. Students corroborated the theoretical results, according to the numerical solution of the Laplace equation, with the experimental ones and they also compared theoretical and experimental results with the current distribution according to a model of electrochemical reactor. The final student laboratory reports were of high quality. ’ ASSOCIATED CONTENT

bS

Figure 5. Current distribution: (0) confined system, (b) unconfined system with 8 segments. Vertical bars: standard error of the mean. Full line: numerical solution of the Laplace equation, r = 7 mm. Dashed line: current distribution according to eq 18.

Supporting Information Instructions for students, notes for the instructor; and an Excel spreadsheet to calculate the theoretical primary potential and current density distributions. This material is available via the Internet at http://pubs.acs.org.

with a total electrode length of 2L. ε¼

πL e

ð19Þ

The experimental results are compared with the theoretical calculations obtained solving the Laplace equation with the numerical procedure previously explained or given by eq 18.

’ RESULTS AND DISCUSSION The potential distribution and the current lines for an unconfined system, obtained by numerical solution of the Laplace equation, are shown in Figure 3. A symmetrical potential distribution is observed and the current lines are concentrated on the lower edge of the electrodes because of the excess of electrolyte in this region.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: jbisang@fiq.unl.edu.ar.

’ ACKNOWLEDGMENT Israel Rodriguez thanks the SEP-CONACyT-2003-C02 project for the financial support to do a stay of research at the PRELINE Laboratory. ’ REFERENCES (1) Newman, J.; Thomas-Alyea, K. E. Electrochemical Systems, 3rd ed.; Wiley: Hoboken, NJ, 2004; Ch 18, pp 419458. 166




(2) Deconinck, J. Current distributions and Electrode Shape Changes in Electrochemical Systems, Lecture Notes in Engineering 75; Springer Verlag: Berlin, 1992. (3) Ibl, N. Current distribution. In Comprehensive Treatise of Electrochemistry; Yeager, E., Bockris, J. O’M., Conway, B. E., Sarangapani, S., Eds.; Plenum Press: New York, 1983; Vol. 6, Ch 4, pp 239315. (4) Heitz, E.; Kreysa, G. Principles of Electrochemical Engineering; VCH Verlagsgesellschaft mbH: Weinheim, 1986; Ch. 3, pp 5374. (5) Walsh, F.C. A First Course in Electrochemical Engineering; Alresford Press, Alresford, 1993, p 137. (6) Frias-Ferrer, A.; Gonzalez-García, J.; Saez, V.; Exposito, E.; Sanchez-Sanchez, C. M.; Montiel, V.; Aldaz, A.; Walsh, F. C. J. Chem. Educ. 2005, 82, 1395–1398. (7) Pasco Home Page. http://www.pasco.com/. (8) Chapra, S. C.; Canale, R. P. Numerical Method for Engineers, 5th ed.; Mc Graw Hill: New York, 2006; Ch. 29, pp 820824. (9) Prentice, G. A.; Tobias, C. W. AIChE J. 1982, 28, 486–492. (10) Hall, D. E.; Taylor, E. J.; Kerstanski, D. J. Plat. Surf. Finish 1985, 72, 60–63. (11) Parrish, W. R.; Newman, J. J. Electrochem. Soc. 1970, 117, 43–48.

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Theoretical and Experimental Study of the Primary Current Distribution

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