Research Papers

Algebraic Form and New Approximation of Butler–Volmer Equation to Calculate the Activation Overpotential

[+] Author and Article Information
H. Kazemi Esfeh

Process Systems Engineering
Centre (PROSPECT),
Faculty of Chemical Engineering,
Universiti Teknologi Malaysia, UTM,
Skudai 81310, Johor, Malaysia;
Department of Chemical Engineering,
Mahshahr Branch,
Islamic Azad University,
Mahshahr, Iran
e-mail: h.kazemi.esfeh@gmail.com

M. K. A. Hamid

Process Systems Engineering Centre (PROSPECT),
Faculty of Chemical Engineering,
Universiti Teknologi Malaysia, UTM,
Skudai 81310, Johor, Malaysia
e-mail: kamaruddin@cheme.utm.my

Manuscript received May 6, 2016; final manuscript received August 25, 2016; published online October 11, 2016. Assoc. Editor: William Mustain.

J. Electrochem. En. Conv. Stor. 13(2), 021003 (Oct 11, 2016) (10 pages) Paper No: JEECS-16-1059; doi: 10.1115/1.4034754 History: Received May 06, 2016; Revised August 25, 2016

The Butler–Volmer equation has been widely used to analyze the electron transfer for electrochemical simulation. Although it has been broadly employed with numerous successful applications, the Butler–Volmer equation needs to be solved numerically to find the activation overpotential, which results in the increase of the calculation difficulties. There are also some parameters in Butler–Volmer equation such as exchange current density and symmetry factor that are not always known parameters. In order to avoid the latest mentioned limitation and the numerical calculation which is time consuming and for simplification, there are some approximation equations such as Tafel, linear low polarization, and hyperbolic sine approximation. However, all these equations are only applicable in a specific range of current density or definite condition. The aim of this paper is to present a new form of Butler–Volmer equation using algebraic operation to calculate activation overpotential. The devised equation should be accurate, have a wide application range, able to remove any numerical calculation, and be useful to find exchange current density. In this research, a new form of Butler–Volmer equation and a new approximation equation (called K–J equation) have been successfully derived. The comparison result shows that the new derived form is exactly equal to the Butler–Volmer equation to calculate the activation overpotential, and it removed the necessity of numerical calculation to find the activation overpotential. In addition, the K–J approximation has a good agreement with Butler–Volmer equation over a wide range of current density and is applicable to predict the activation loss.

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Hoogers, G. , 2002, Fuel Cell Technology Handbook, CRC Press, Boca Raton, FL.
Shen, S. , and Ni, M. , 2015, “ 2D Segment Model for a Solid Oxide Fuel Cell With a Mixed Ionic and Electronic Conductor as Electrolyte,” Int. J. Hydrogen Energy, 40(15), pp. 5160–5168. [CrossRef]
Sierra, J. M. , Figueroa-Ramírez, S. J. , Díaz, S. E. , Vargas, J. , and Sebastian, P. J. , 2014, “ Numerical Evaluation of a PEM Fuel Cell With Conventional Flow Fields Adapted to Tubular Plates,” Int. J. Hydrogen Energy, 39(29), pp. 16694–16705. [CrossRef]
Chandesris, M. , Médeau, V. , Guillet, N. , Chelghoum, S. , Thoby, D. , and Fouda-Onana, F. , 2015, “ Membrane Degradation in PEM Water Electrolyzer: Numerical Modeling and Experimental Evidence of the Influence of Temperature and Current Density,” Int. J. Hydrogen Energy, 40(3), pp. 1353–1366. [CrossRef]
Kang, T. , Kim, M. , Kim, J. , and Sohn, Y.-J. , 2015, “ Numerical Modeling of the Degradation Rate for Membrane Electrode Assemblies in High Temperature Proton Exchange Membrane Fuel Cells and Analyzing Operational Effects of the Degradation,” Int. J. Hydrogen Energy, 40(15), pp. 5444–5455. [CrossRef]
Mehrpooya, M. , Akbarpour, S. , Vatani, A. , and Rosen, M. A. , 2014, “ Modeling and Optimum Design of Hybrid Solid Oxide Fuel Cell-Gas Turbine Power Plants,” Int. J. Hydrogen Energy, 39(36), pp. 21196–21214. [CrossRef]
Salomov, U. R. , Chiavazzo, E. , and Asinari, P. , 2015, “ Gas-Dynamic and Electro-Chemical Optimization of Catalyst Layers in High Temperature Polymeric Electrolyte Membrane Fuel Cells,” Int. J. Hydrogen Energy, 40(15), pp. 5425–5431. [CrossRef]
Misran, E. , Mat Hassan, N. S. , Wan Daud, W. R. , Majlan, E. H. , and Rosli, M. I. , 2013, “ Electrochemical Properties of a PEMFC Operating With Saturated Hydrogen and Dry Air,” Int. J. Hydrogen Energy, 38(22), pp. 9395–9400. [CrossRef]
Antunes, R. , and Skrzypkiewicz, M. , 2015, “ Chronoamperometric Investigations of Electro-Oxidation of Lignite in Direct Carbon Bed Solid Oxide Fuel Cell,” Int. J. Hydrogen Energy, 40(12), pp. 4357–4369. [CrossRef]
Kuzmin, R. , Maximov, D. , Savenkova, N. , and Shobukhov, A. , 2012, “ Mathematical Modeling of Hysteresis in Porous Electrodes,” J. Math. Chem., 50(9), pp. 2471–2477. [CrossRef]
Sørensen, B. , 2012, Hydrogen and Fuel Cells: Emerging Technologies and Applications, Academic Press, New York.
Henstridge, M. C. , Ward, K. R. , and Compton, R. G. , 2014, “ The Marcus-Hush Model of Electrode Kinetics at a Single Nanoparticle,” J. Electroanal. Chem., 712(1), pp. 14–18. [CrossRef]
Laborda, E. , Suwatchara, D. , Rees, N. V. , Henstridge, M. C. , Molina, A. , and Compton, R. G. , 2013, “ Variable Temperature Study of Electro-Reduction of 3-Nitrophenolate Via Cyclic and Square Wave Voltammetry: Molecular Insights Into Electron Transfer Processes Based on the Asymmetric Marcus–Hush Model,” Electrochim. Acta, 110(1), pp. 772–779. [CrossRef]
Bhat, M. A. , and Ingole, P. P. , 2012, “ Evidence for Formation of Ion Pair Stabilized Diiodomethane Radical Anion in 1-Butyl-3-Methylimidazolium Tetrafluoroborate Room Temperature Ionic Liquid,” Electrochim. Acta, 72(1), pp. 18–22. [CrossRef]
Mamedov, B. , 2013, “ Analytical Evaluation of the Marcus–Hush–Chidsey Function Using Binomial Expansion Theorem and Error Functions,” J. Math. Chem., 51(10), pp. 2699–2703. [CrossRef]
Henstridge, M. C. , Laborda, E. , Rees, N. V. , and Compton, R. G. , 2012, “ Marcus–Hush–Chidsey Theory of Electron Transfer Applied to Voltammetry: A Review,” Electrochim. Acta, 84(1), pp. 12–20. [CrossRef]
Mousa, G. , Golnaraghi, F. , DeVaal, J. , and Young, A. , 2014, “ Detecting Proton Exchange Membrane Fuel Cell Hydrogen Leak Using Electrochemical Impedance Spectroscopy Method,” J. Power Sources, 246(1), pp. 110–116. [CrossRef]
Orvananos, B. , Malik, R. , Yu, H.-C. , Abdellahi, A. , Grey, C. P. , Ceder, G. , and Thornton, K. , 2014, “ Architecture Dependence on the Dynamics of Nano-LiFePO4 Electrodes,” Electrochim. Acta, 137(1), pp. 245–257. [CrossRef]
He, Z. , Liu, J. , Han, H. , Chen, Y. , Zhou, Z. , Zheng, S. , Lu, W. , Liu, S. , and He, Z. , 2013, “ Effects of Organic Additives Containing NH2 and SO3H on Electrochemical Properties of Vanadium Redox Flow Battery,” Electrochim. Acta, 106(1), pp. 556–562. [CrossRef]
Allen, J. A. , Tulloch, J. , Wibberley, L. , and Donne, S. W. , 2014, “ Kinetic Analysis of the Anodic Carbon Oxidation Mechanism in a Molten Carbonate Medium,” Electrochim. Acta, 129(1), pp. 389–395. [CrossRef]
Chae, J. E. , Annaka, K. , Hong, K. , Lee, S.-I. , Munakata, H. , Kim, S.-S. , and Kanamura, K. , 2014, “ Electrochemical Characterization of Phosphorous-Doped Soft Carbon Using Single Particle for Lithium Battery Anode,” Electrochim. Acta, 130(1), pp. 60–65. [CrossRef]
Yang, X.-G. , Ye, Q. , and Cheng, P. , 2014, “ Oxygen Starvation Induced Cell Potential Decline and Corresponding Operating State Transitions of a Direct Methanol Fuel Cell in Galvanostatic Regime,” Electrochim. Acta, 117(1), pp. 179–191. [CrossRef]
Senarathna, K. G. C. , Mantilaka, M. M. M. G. P. G. , Peiris, T. A. N. , Pitawala, H. M. T. G. A. , Karunaratne, D. G. G. P. , and Rajapakse, R. M. G. , 2014, “ Convenient Routes to Synthesize Uncommon Vaterite Nanoparticles and the Nanocomposites of Alkyd Resin/Polyaniline/Vaterite: The Latter Possessing Superior Anticorrosive Performance on Mild Steel Surfaces,” Electrochim. Acta, 117(1), pp. 460–469. [CrossRef]
Schluckner, C. , Subotić, V. , Lawlor, V. , and Hochenauer, C. , 2014, “ Three-Dimensional Numerical and Experimental Investigation of an Industrial-Sized SOFC Fueled by Diesel Reformat—Part I: Creation of a Base Model for Further Carbon Deposition Modeling,” Int. J. Hydrogen Energy, 39(33), pp. 19102–19118. [CrossRef]
Bove, R. , and Ubertini, S. , 2008, Modeling Solid Oxide Fuel Cells, Springer, Berlin.
Kakaç, S. , Pramuanjaroenkij, A. , and Vasil'ev, L. L. , 2008, Mini-Micro Fuel Cells: Fundamentals and Applications, Springer, Dordrecht, The Netherlands.
Yang, P. , Zhang, H. , and Hu, Z. , 2016, “ Parametric Study of a Hybrid System Integrating a Phosphoric Acid Fuel Cell With an Absorption Refrigerator for Cooling Purposes,” Int. J. Hydrogen Energy, 41(5), pp. 3579–3590. [CrossRef]
Noren, D. , and Hoffman, M. , 2005, “ Clarifying the Butler–Volmer Equation and Related Approximations for Calculating Activation Losses in Solid Oxide Fuel Cell Models,” J. Power Sources, 152(1), pp. 175–181. [CrossRef]
Thanomjit, C. , Patcharavorachot, Y. , Ponpesh, P. , and Arpornwichanop, A. , 2015, “ Thermodynamic Analysis of Solid Oxide Fuel Cell System Using Different Ethanol Reforming Processes,” Int. J. Hydrogen Energy, 40(21), pp. 6950–6958. [CrossRef]
Singhal, S. , 2003, High-Temperature Solid Oxide Fuel Cells: Fundamentals, Design and Applications, Elsevier, Oxford, UK.
Zhu, H. , and Kee, R. J. , 2003, “ A General Mathematical Model for Analyzing the Performance of Fuel-Cell Membrane-Electrode Assemblies,” J. Power Sources, 117(1), pp. 61–74. [CrossRef]
Cardoso, D. S. P. , Amaral, L. , Santos, D. M. F. , Šljukić, B. , Sequeira, C. A. C. , Macciò, D. , and Saccone, A. , 2015, “ Enhancement of Hydrogen Evolution in Alkaline Water Electrolysis by Using Nickel-Rare Earth Alloys,” Int. J. Hydrogen Energy, 40(12), pp. 4295–4302. [CrossRef]
Minutillo, M. , Perna, A. , and Jannelli, E. , 2014, “ SOFC and MCFC System Level Modeling for Hybrid Plants Performance Prediction,” Int. J. Hydrogen Energy, 39(36), pp. 21688–21699. [CrossRef]
Hajimolana, S. A. , Hussain, M. A. , Daud, W. A. W. , Soroush, M. , and Shamiri, A. , 2011, “ Mathematical Modeling of Solid Oxide Fuel Cells: A Review,” Renewable Sustainable Energy Rev., 15(4), pp. 1893–1917. [CrossRef]
Scott, K. , and Mamlouk, M. , 2009, “ A Cell Voltage Equation for an Intermediate Temperature Proton Exchange Membrane Fuel Cell,” Int. J. Hydrogen Energy, 34(22), pp. 9195–9202. [CrossRef]
Ni, M. , Leung, M. K. , and Leung, D. Y. , 2007, “ Parametric Study of Solid Oxide Fuel Cell Performance,” Energy Convers. Manage., 48(5), pp. 1525–1535. [CrossRef]
Apostol, T. M. , 2007, Calculus, Vol. 1, Wiley, New York.
Artin, E. , 1964, The Gamma Function, Vol. 14, Dover Publications, New York.
Doddathimmaiah, A. , and Andrews, J. , 2009, “ Theory, Modelling and Performance Measurement of Unitised Regenerative Fuel Cells,” Int. J. Hydrogen Energy, 34(19), pp. 8157–8170. [CrossRef]
Song, C. , Tang, Y. , Zhang, J. L. , Zhang, J. , Wang, H. , Shen, J. , McDermid, S. , Li, J. , and Kozak, P. , 2007, “ PEM Fuel Cell Reaction Kinetics in the Temperature Range of 23–120 C,” Electrochim. Acta, 52(7), pp. 2552–2561. [CrossRef]
Ticianelli, E. , Derouin, C. , Redondo, A. , and Srinivasan, S. , 1988, “ Methods to Advance Technology of Proton Exchange Membrane Fuel Cells,” J. Electrochem. Soc., 135(9), pp. 2209–2214. [CrossRef]
Mendoza-Hernandez, O. S. , Ishikawa, H. , Nishikawa, Y. , Maruyama, Y. , Sone, Y. , and Umeda, M. , 2014, “ State of Charge Dependency of Graphitized-Carbon-Based Reactions in a Lithium-Ion Secondary Cell Studied by Electrochemical Impedance Spectroscopy,” Electrochim. Acta, 131(1), pp. 168–173. [CrossRef]
Tant, S. , Rosini, S. , Thivel, P. X. , Druart, F. , Rakotondrainibe, A. , Geneston, T. , and Butel, Y. , 2014, “ An Algorithm for Diagnosis of Proton Exchange Membrane Fuel Cells by Electrochemical Impedance Spectroscopy,” Electrochim. Acta, 135(1), pp. 368–379. [CrossRef]


Grahic Jump Location
Fig. 1

Dependence of the electrode current by Butler–Volmer equation on overpotential

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Fig. 2

Comparison of Tafel and low-polarization approximations with Butler–Volmer equation at specific condition (β = 0.5 and T = 800 °C) [28]

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Fig. 3

Comparison of Butler–Volmer equation and its new form at specific condition (β = 0.5 and T = 800 °C)

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Fig. 4

Polarization curves obtained from the model and the experiment data at different pressures

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Fig. 5

Comparison between Butler–Volmer and K–J equation with different symmetry factor in high temperature; (a) β=1/4, (b) β=1/3, (c) β=1/2, and (d) β=3/4

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Fig. 6

Comparison between Butler–Volmer and K–J equation with different symmetry factor in low temperature; (a) β=1/4, (b) β=1/3, (c) β=1/2, and (d) β=3/4

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Fig. 7

R-squared value of K–J and Butler–Volmer equation difference in wide range of symmetry factor

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Fig. 8

Standard deviation between K–J and Butler–Volmer equation in wide range of symmetry factor and temperature

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Fig. 9

The comparison between the K–J equation and other Butler–Volmer approximation in wide range of current density

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Fig. 10

Logarithmic plot of the activation loss against the current density. (a) Layout of the exchange current density prediction and (b) example of exchange current density prediction.

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Fig. 11

Change in activation overpotential by changing the symmetry factor at specific condition (β = 0.5 and T = 800 °C)

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Fig. 12

Change in first-order differential function of activation overpotential in different symmetry factor using new form equation

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Fig. 13

Change in second-order differential function of activation overpotential in different symmetry factor using new form equation




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