Determination of the mass-transport properties of vanadium ions

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Cite this: Phys. Chem. Chem. Phys., 2013, 15, 10841

Determination of the mass-transport properties of vanadium ions through the porous electrodes of vanadium redox flow batteries Qian Xu and T. S. Zhao* This work is concerned with the determination of two critical constitutive properties for mass transport of ions through porous electrodes saturated with a liquid electrolyte solution. One is the effective diffusivity that is required to model the mass transport at the representative element volume (REV) level of porous electrodes in the framework of Darcy’s law, while the other is the pore-level mass-transfer coefficient for modeling the mass transport from the REV level to the solid surfaces of pores induced by redox reactions. Based on the theoretical framework of mass transport through the electrodes of vanadium redox flow batteries (VRFBs), unique experimental setups for electrochemically determining

Received 8th May 2013, Accepted 8th May 2013

the two transport properties by measuring limiting current densities are devised. The effective diffusivity and the pore-level mass-transfer coefficient through the porous electrode made of graphite felt,

DOI: 10.1039/c3cp51944a

a typical material for VRFB electrodes, are measured at different electrolyte flow rates. The correlation equations, respectively, for the effective diffusivity and the pore-level mass-transfer coefficient are finally

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proposed based on the experimental data.

1. Introduction With growing interest in the deployment of renewable intermittent energy sources, such as solar and wind power, the need for electrical energy storage (EES) in both mobile and stationary applications becomes exigent.1 Among various EES systems, the vanadium redox flow battery (VRFB) is one of the most promising candidates, since apart from having the common feature of the system scalability of all redox flow batteries associated with the separation of energy storage tanks and power packs, it offers additional advantages including the ability to fully charge and discharge without damaging the cells, no crossover contamination in the electrolytes, and moderate cost.2–4 Nevertheless, issues with VRFBs, including a low power density and energy density on a system level, ion crossover through the polymer membrane, and corrosion, need to be addressed.5,6 To this end, efforts including improving the electrode electrochemical kinetics,7–11 adding additives into the electrolyte to improve its solubility,12–14 as well as modifying existing membranes and searching for alternatives15–20 have been made over the past decades. Moreover, minimizing the Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China. E-mail: [email protected]; Fax: +86 (852) 2358-1543; Tel: +86 (852) 2358-8647

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mass transport polarization is also one of the crucial issues for improving cell performance. Recently, Aaron et al. introduced a so-called zero-gap cell architecture with a serpentine flow field, enabling the peak power density to be 550 mW cm2, which is significantly higher than conventional cells; the enhancement of the cell performance was attributed to the enhanced mass transport and reduced internal resistance.21,22 In addition to experimental investigations, numerical modeling can also play an important role in improving and optimizing the performance of VRFBs. To enable numerical models to provide meaningful insight into the operating characteristics of a VRFB, accurate transport properties are needed, in addition to a robust formulation. One of the key transport properties is the effective diffusivity that is required to model the mass transport at the representative elementary volume (REV) level of porous electrodes in the framework of Darcy’s law. The most widely used effective diffusivity23–25 is a simple correlation equation with the intrinsic diffusivity and the porosity of the porous material using a Bruggemann correction;26 an issue with this correlation, however, is that the effect of the pore morphology of different porous structures is missing. Attempts have also been made to measure the effective diffusivity. A common approach is to place a porous structure sample between two reservoirs, one of which contains an electrolyte solution, while the other contains DI water, respectively. UV-Vis spectroscopic

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measurements are carried out to determine the amount of ions in the DI water over a period of time. After determining the ion concentration of different aliquots, the effective diffusivity of the ions through the porous media can be obtained based on the porous structure area, thickness and the overall solution volume.27–29 Nevertheless, a major problem associated with this method is that the dispersion effect which exists in an operating VRFB cannot be taken into account. Recently, Pant et al. proposed a new configuration to determine the effective diffusivity through the electrodes in flow batteries.30 The flowing electrolytes in two channels were separated by a porous diffusion layer. The amount of ions diffused through the diffusion layer was determined by the ion-chromatography measurements. The dispersion effect is included in their method, but the experimental setup is rather complex and the measurements are timeconsuming. Another key mass-transfer property is the pore-level masstransfer coefficient, which is related to the morphology of pore surfaces, electrolyte properties and the local velocity of electrolyte. A common method to obtain the mass-transfer coefficient is to measure the non-aqueous phase liquids (NAPL)–water interfacial area in water-saturated columns by tracer studies.31 However, this approach suffers from troublesome setup and relatively large measurement errors. Another way is to electrochemically determine the mass-transfer coefficient by measuring the limiting current densities when the electrode operates under mass-transport control.32–35 Nevertheless, as the porous electrode usually has a finite thickness (several millimeters), this approach can only be used to obtain the average masstransfer coefficient over the electrode surface. In this study, the effective diffusivity and the pore-level mass-transfer coefficient in porous VRFB electrode are electrochemically determined. By devising and using the unique setups, the effect of dispersion on the effective diffusivity can be included. The limiting current densities caused by specific ions under various flow rates are measured. Finally, based on the experimental results, the correlation equations for the effective diffusivity and the pore-level mass-transfer coefficient, respectively, are proposed.

2. Theoretical Consider the transport of ions in an electrolyte flow through a porous electrode as illustrated in Fig. 1. The flux of a specific ion as the result of diffusion, electrical migration and convection can be expressed by the modified Nernst–Planck equation as:24 Ni = Deff i rci Fzicimirfs + uci

(1)

where u represents the Darcy’s velocity, c is the ion concentration at the representative elementary volume (REV) level, Deff is the effective diffusivity, F is Faradaic constant, z and m are the charge number of the ion and ionic mobility, and fs is the ionic potential in electrolyte. With eqn (1), the conservation of species can be written as: : rNi = Ri (2)

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Fig. 1

Schematic of the mass transport process through a porous electrode.

: where R represents the consumption rate of the ion as the result of electrochemical reactions at the solid surfaces in the pores of the electrode and is related to the transport flux from the REV level, represented by the REV concentration c, to the solid surfaces in pores, represented by the concentration at the solid surfaces cs, as modeled by: : Ri = kmAV(ci cis) (3) where km represents the pore-level mass-transfer coefficient and AV is the specific surface area of the porous electrode. The reaction current density j is related to the local concentration cs at the solid surfaces of the pores in the porous electrode and can be determined from the Butler–Volmer equation:25 s ci aþ FZ a FZ exp (4) exp j ¼ eAV j0;i ref RT RT ci where e is the porosity of the porous electrode, j0 is the exchange current density, Z is the overpotential, cref is the reference concentration, a+ and a are the anodic and cathodic transfer coefficients. Combining eqn (3) and (4), the variable cs in the Butler–Volmer equation can be eliminated by introducing a pore-level mass-transfer coefficient km in Section 2.2. The flow of ions gives rise to the current density in the electrolyte solution: ii = ziFNi

(5)

The electrolyte is considered to be electrically neutral: X zi ci ¼ 0 (6) i

By combining eqn (1), (5) and (6), we can express the total current density in the electrolyte as: X X X 2 i i ¼ F zi Deff zi2 ci mi rfs (7) i¼ i rci F i

i

i

The charge entering the electrolyte is balanced by the charge leaving the solid phase, which is essentially the

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current density induced by the electrochemical reaction of the active ions: ri = j

(8)

By solving eqn (2) and (8), the distributions of ion concentration and overpotential within the porous electrode can be determined. The above analysis suggests that the two transport properties, the effective diffusivity Deff and the pore-level mass-transfer coefficient km, are needed to solve the coupled electrochemical and mass transport process in porous electrodes. 2.1. Determination of the effective diffusivities of vanadium ions This section describes the theoretical framework to determine the effective diffusivity through a porous electrode. The basic idea is to measure the mass flux under a given concentration difference across the porous electrode. The direct determination of the mass flux has been proven to be rather challenging. In this work, an electrochemical method to determine the effective diffusivity through a porous electrode is proposed. Fig. 2 illustrates the experimental setup, which is essentially a VRFB. The porous sample to be tested and a membraneelectrode assembly (MEA) are separated by a stainless steel ring serving as a current collector. A flow field is used to supply and distribute liquid electrolyte onto the porous surface, where the concentration of vanadium ions, cf,i, could be approximated as the average value of the concentrations at the inlet (cin) and the outlet (cout) of the flow field if (i) the surface area of the electrode is small; and (ii) the electrolyte flow rate is large. The vanadium ion is then transported through the porous sample and the gap between the porous sample and the catalyst layer, where it is reacted at a concentration, cs,i. As such, the mass flux from the flow channel to the catalyst layer can be expressed as: Ni ¼

i cf;i cs;i ¼ L l F þ eff Di Di

(9)

Fig. 2 Schematic of the setup used for the determination of the effective diffusivity.

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where L and l represent the thicknesses of the porous sample and stainless steel square ring, and Di is the ion intrinsic diffusivity in the electrolyte.23 When the current density reaches the limiting current density, the concentration in the catalyst layer becomes zero. Consequently, eqn (9) is reduced to: Deff i ¼

L Fcf;i l ilim Di

(10)

where ilim is the limiting current density. Clearly, with a measured limiting current density and a given concentration cf,i, the effective diffusivity through the porous sample can be determined. The measured effective diffusivity through the porous electrode at different electrolyte flow rates can be correlated in terms of the intrinsic diffusivity D, the porosity e, and the flow velocity as:36–38 Deff ¼ ea ð1 þ bPe2 Þ D

(11)

where Pe (= udp/D) is the Peclet number, with u representing the flow velocity and dp the pore diameter. The empirical constants a and b will be determined based on the experimental data. It should be noted that Deff/D depends on the geometric properties (e.g. porosity) of the porous electrodes and flow characteristics through the electrode, regardless of the specific ions. For this reason, the subscript i can be removed. 2.2.

Determination of the pore-level mass-transfer coefficient

This section presents the theoretical framework to determine the pore-level mass-transfer coefficient. The mass flux from the REV level to solid surfaces through pores needs to be measured under the condition that the concentration at the REV level cf and that at solid pore surfaces cs are known. An electrochemical approach is conceived in this work. The idea is illustrated in Fig. 3, where a porous layer (graphite felt) is placed between the flow field and membrane. The electrolyte with concentration

Fig. 3 Schematic of the setup used for the determination of the mass-transfer coefficient.


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cf in the flow field is transported through the pores to the solid surface, where a redox reaction takes place. If the porous layer is sufficiently thin (with several pores), the mass-transfer resistance from the flow field to the porous layer becomes negligible. Hence, the concentration at the REV level can be approximated by that in the flow field. The mass flux from the REV level to the pore surfaces can be modeled by: 0

N = km(cf cs)

(12)

Note that when the current density reaches the limiting current density, the concentration at solid pore surfaces of the porous layer becomes zero. Consequently, eqn (12) is reduced to: km ¼

i lim Fcf

(13)

Eqn (13) indicates that the mass-transfer coefficient at the pore level can be determined with a measured limiting current density and a given concentration cf. Caution needs to be taken to ensure that the limiting current density is caused by the mass transport from the flow field to the thin porous layer.

3. Experimental 3.1.

VRFB configuration and flow circuit

The fabricated VRFB consisted of two endplates and two flow fields with parallel flow channels and the negative and positive electrodes separated by a Nafions 117 membrane. Graphite felt (GFA Series, SGLs, Germany), whose physical properties can be found elsewhere,39 was used as the material for the negative and positive electrodes. Each electrode had a geometric area of 10 mm 10 mm. The PTFE gaskets were placed between the parts to ensure no leakage of the electrolyte. The volume of each electrolyte compartment was 20 ml. Unless otherwise indicated, the positive and negative electrolytes contained 3.0 M sulfuric acid and the vanadium ions of different concentrations for the performance tests. The electrolytes were kept at room temperature and both positive and negative electrolytes were circulated using a peristaltic pump (Glorys, BT100-1F) using corrosion-resistant tubing (Masterflexs, Cole-Parmer). For the determination of the effective diffusivities of the vanadium ions, the following steps were taken in the design and fabrication of the experimental system: (i) to eliminate the electric field effect on ion transport through the porous sample, the porous sample was electrically isolated from the catalyst layer by a separator (see Fig. 2); (ii) instead of measuring the voltage of the cell, a reference electrode was used to measure the respective potentials of the positive and negative electrodes; the measured potentials at the positive/negative electrode can directly indicate whether the limiting current density is caused by the transport of a specific ion on the positive or negative electrode; and (iii) to maintain an even distribution of electrolyte concentration onto the porous sample, a parallel flow field was designed and utilized.

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For the determination of the pore-level mass-transfer coefficient, we ensure the porous layer is sufficiently thin (with several pores), and a parallel flow field is applied to keep an even distribution of electrolyte concentration along the porous layer. 3.2.

Electrochemical test rig

An Arbin BT2000 (Arbin Instrument Inc.) was employed to measure the polarization curves. The internal resistance of the cell was measured by the built-in function of the Arbin BT2000. Anode polarization data for the VRFB were obtained employing a saturated silver–silver chloride electrode, Ag–AgCl (ABB, Series 1400, 1.0 M KNO3), equipped with a liquid junction protection tube, placed about 3 cm away from the exit of the flow channel in line with the electrolyte circuit. Cathode polarization data were derived by subtracting anode polarization values from the respective cell polarization data.

4. Results and discussion 4.1.

Measured limiting current density

Fig. 4 shows the polarization curves of the VRFB with 0.125 M, 1.0 M and 1.5 M vanadium ion concentrations in both the positive and negative sides at a flow rate of 40 ml min1. It can be seen that with a lower electrolyte concentration of 0.125 M, increasing the current density resulted in a sharp drop in the cell voltage, meaning that the electrolyte at the electrode active sites was depleted and the limiting current density reached. When the electrolyte concentration was increased to 1.0 M, the cell performance was greatly improved at moderate current densities, but the voltage dropped steeply at high current densities. However, with a higher electrolyte concentration of 1.5 M, with increasing the current density the cell voltage dropped almost linearly towards zero and there was no sharp drop in the cell voltage. This polarization behavior indicates that at higher electrolyte concentrations, the voltage loss is predominately caused by the large internal resistance rather than the mass transport limitation. Therefore, no limiting current density exists at sufficiently high electrolyte concentrations, unless the flow rate is extremely low. The objectives of this work were to determine the effective diffusivities of vanadium ions with eqn (10) and the pore-level mass-transfer

Fig. 4

Polarization curves for different electrolyte concentrations in the VRFB.

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coefficient with eqn (13) by measuring the limiting current densities. Therefore, all the experimental data reported hereafter were collected at an electrolyte concentration lower than or equal to 1.0 M and at a flow rate lower than or equal to 40 ml min1, at which the limiting current density occurred. 4.2. Measured effective diffusivities of vanadium ions through the porous electrode To measure the effective diffusivities of VO2+ and VO2+, it should be ensured that the limiting current density occurs on the positive side. Accordingly, the total concentration of vanadium ions in the positive electrolyte was set to be 0.25 M while the total concentration in the negative electrolyte was 0.4 M. The state of charge (SOC) of the positive electrolyte was kept at 0.9 at the beginning of each test. Fig. 5a shows the polarization curves at different electrolyte flow rates. As can be seen, the limiting current density increases with the flow rate. When the velocity of the electrolyte solution is zero, the limiting current density is 22 mA cm2, representing the mass transport limitation of pure diffusion through the porous layer. With an increase in the flow rate, the limiting current density increases: to 47 mA cm2 at

Fig. 5 Polarization curves for the VRFB using VO2+ ions at various electrolyte flow rates (positive electrolyte concentration 0.25 M; negative electrolyte concentration 0.4 M). (a) Cell voltage vs. current density. (b) Positive and negative halfcell potentials vs. Ag–AgCl reference electrode.

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5 ml min1, and to 73 mA cm2 at 40 ml min1. The fact that the limiting current density increases with the flow rate means there is an enhanced mass transport as a result of the increased flow rate. Fig. 5b shows the respective positive and negative potentials at different flow rates. Clearly, at each flow rate the mass transport limitation occurred at the positive electrode, suggesting that the limiting current density was determined by the transport flux of VO2+ ion. Limiting current densities were then measured by varying the flow rate from 5 to 40 ml min1. The measured data were then transformed to the effective diffusivity according to eqn (10). The following correlation was obtained based on a least-square fit of the experimental data: Deff ¼ e1:1 ð1 þ 1:46 103 Pe2 Þ ð4:3 o Pe o 34:5Þ D

(14)

It should be noted that eqn (14) was obtained based on the transport of VO2+ ions. To verify whether eqn (14) is suitable for the transport of other ions or not, additional experiments were performed to measure the limiting current densities induced by the transport of V2+ ions. To this end, the total concentration of vanadium ions in the negative electrolyte was set to be 0.25 M, while that in the positive electrolyte was set to be 0.4 M. The SOC of the negative electrolyte was kept at 0.9 at the beginning of each test. The polarization curves at different electrolyte flow rates are shown in Fig. 6a. When the flow rate is 5 ml min1, the limiting

Fig. 6 Polarization curves for the VRFB using V2+ ions at various electrolyte flow rates (positive electrolyte concentration 0.4 M; negative electrolyte concentration 0.25 M). (a) Cell voltage vs. current density. (b) Positive and negative half-cell potentials vs. Ag–AgCl reference electrode.


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Fig. 7

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Mass-transfer correlation, Deff/D vs. Pe.

current density is 37 mA cm2. As the flow rate increases to 40 ml min1 the limiting current density becomes 55 mA cm2. Fig. 6b shows the respective positive and negative potentials at different flow rates. It is clear that the mass transport limitation occurred at the negative electrode for each flow rate, indicating the limiting current density was determined by the transport of V2+ ions. The measured Deff at different flow rates for the transport of V2+ ions (6.9 o Pe o 55.3) is compared with eqn (14) in Fig. 7. It can be found that the experimental data for V2+ ions fit well with the correlation. Hence, eqn (14) represents the effective diffusivity in this type of porous electrode. 4.3.

Measured pore-level mass-transfer coefficient

Using the setup for determining the pore-level mass-transfer coefficient illustrated in Fig. 3, the polarization curves of the VRFB can be obtained and are shown with vanadium ion concentration in the positive side of 0.125 M (Fig. 8a) and 0.25 M (Fig. 8b), respectively. The vanadium ion concentration in the negative side was kept at 1.0 M to ensure that the mass transport limit occurs in the positive electrode. The SOC of the positive electrolyte was kept at 0.9 at the beginning of each test. It is seen that for a given electrolyte concentration, the limiting current density increased with increasing the flow rate as a result of the improved mass transport of the electrolyte. For the vanadium ion concentrations of 0.125 M and 0.25 M, the limiting current density increased from 46.2 to 81.1 mA cm2 and from 63.3 to 107.8 mA cm2, respectively, as the flow rate was increased from 5 to 40 ml min1. The variations in the limiting current density with electrolyte concentration at different flow rates are shown in Fig. 9. It can be found that at each given flow rate the limiting current density increased almost linearly with the electrolyte concentration, due to the reduced concentration polarization. The measured pore-level mass-transfer coefficients against current density are shown in Fig. 10. It can be seen that the effect of the electrolyte flow rate on km is significant. For instance, when the flow rate was increased from 5 to 40 ml min1, km doubled. On the other hand, if the electrolyte flow rate is reduced to zero, the transport within a pore space is induced by pure diffusion, where km is simply expressed as 2D/dp.21 Therefore, the pore-level mass-transfer coefficient can be

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Fig. 8 Polarization curves for the VRFB at different electrolyte concentrations. (a) 0.125 M in positive side. (b) 0.25 M in positive side. Negative electrolyte concentration: 1.0 M.

Fig. 9 rates.

Limiting current density vs. electrolyte concentration at different flow

related to D and the electrolyte flow rate.40,41 Fig. 11 presents the variation in ln(kmdp/D 2) with ln Re based on the same experimental data shown in Fig. 10. It is seen that although scattered slightly due to the influence of current density, ln(kmdp/D 2) generally increases in a linear fashion with ln Re. The following correlation of km (m s1) was obtained based on a least-square fit of the experimental data: km dp ¼ 2 þ 1:534Re0:912 D

ð0:3 o Re o 2:4Þ

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Paper significant with an increase in flow rates. The pore-level masstransfer coefficient is found to be independent of current density and the correlation equation in terms of Reynolds number and the intrinsic diffusivity has been proposed.

Acknowledgements The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 622712).

Notes and references Fig. 10

Fig. 11

Mass-transfer coefficient vs. current density at different flow rates.

Mass-transfer correlation for km.

5. Concluding remarks Numerical modeling of VRFBs can not only provide insight into their operation characteristics, but also enables rapid testing of hypotheses aimed at improving cell performance. The accuracy of numerical simulation depends not only on the robustness of the mathematical formulation, but also on the accuracy of constitutive mass-transport properties through the porous electrodes. In this work, we measured two critical constitutive properties for mass transport of ions through porous electrodes saturated with a liquid electrolyte solution. One is the effective diffusivity that is required to model the mass transport at the REV level of porous electrodes in the framework of Darcy’s law, while the other is the pore-level mass-transfer coefficient for modeling the mass transport from the REV level to the solid surfaces of pores. Based on the theoretical framework of mass transport through the electrodes of VRFBs, unique experimental setups for electrochemically determining the two transport properties were devised. The effective diffusivity and the pore-level mass-transfer coefficient through the porous electrode made of graphite felt, a typical material for VRFB electrodes, were measured at different electrolyte flow rates. The obtained correlation equation for the effective diffusivity of vanadium ions through the porous electrode includes the effects of both the porous electrode structure and flow dispersion. It is found that the effect of flow dispersion becomes more

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1 G. L. Soloveichik, Annu. Rev. Chem. Biomol. Eng., 2011, 2, 503. 2 C. P. de Leon, A. F. Ferrer, J. G. Garcia, J. G. Garcia, D. A. Szanto and F. C. Walsh, J. Power Sources, 2006, 160, 716. 3 T. Nguyen and R. F. Savinell, Electrochem. Soc. Interface, 2010, 54. 4 M. Schreiber, M. Harrer and A. Whitehead, J. Power Sources, 2012, 206, 483. 5 N. Armaroli and V. Balzani, Energy Environ. Sci., 2011, 4, 3193. 6 C. M. Araujo, D. L. Simone, S. J. Konezny, A. Shim and G. L. Soloveichik, Energy Environ. Sci., 2012, 5, 9534. 7 Y. Y. Shao, X. Q. Wang, M. Engealhard, C. M. Wang, S. Dai, J. Liu, Z. G. Yang and Y. H. Lin, J. Power Sources, 2010, 195, 4375. 8 H. M. Tsai, S. J. Yang, C. C. Ma and X. F. Xie, Electrochim. Acta, 2012, 77, 232. 9 K.-J. Kim, M.-S. Park, J.-H. Kim, U. K. Hwang, N.-J. Lee, G. Jeong and Y.-J. Kim, Chem. Commun., 2012, 48, 5455. 10 H.-M. Tsai, S.-Y. Yang, C.-C. Ma and X. Xie, Electroanalysis, 2011, 23, 2139. 11 C. Yao, H. Zhang, T. Liu, X. Li and Z. Liu, J. Power Sources, 2012, 218, 455. 12 X. Wu, S. Liu, N. Wang, S. Peng and Z. He, Electrochim. Acta, 2012, 78, 475. 13 Z. Jia, B. Wang, S. Song and X. Chen, J. Electrochem. Soc., 2012, 159, A843. 14 L. Li, S. Kim, W. Wang, M. Vijayakumar, Z. Nie, B. Chen, J. Zhang, G. Xia, J. Hu, G. Graff, J. Liu and Z. Yang, Adv. Energy Mater., 2011, 1, 394. 15 X. Teng, Y. Zhao, J. Xi, Z. Wu, X. Qiu and L. Chen, J. Power Sources, 2009, 189, 1240. 16 M. Vijayakumar, B. Schwenzer, S. Kim, Z. Yang, S. Thevuthasan, J. Liu, G. L. Graff and J. Hu, Solid State Nucl. Magn. Reson., 2012, 42, 71. 17 P. Han, Y. Yue, Z. Liu, W. Xu, L. Zhang, H. Xu, S. Dong and G. Cui, Energy Environ. Sci., 2011, 4, 4710. 18 W. Wei, H. Zhang, X. Li, Z. Mai and H. Zhang, J. Power Sources, 2012, 208, 421. 19 X. Zhao, Y. Fu, W. Li and A. Manthiram, RSC Adv., 2012, 2, 5554. 20 C. Jia, J. Liu and C. Yan, J. Power Sources, 2012, 203, 190.


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21 D. S. Aaron, Z. Tang, A. B. Papandrew and T. A. Zawodzinski, J. Appl. Electrochem., 2011, 41, 1175. 22 D. S. Aaron, Q. Liu, Z. Tang, G. M. Grim, A. B. Papandrew, A. Turhan, T. A. Zawodzinski and M. M. Mench, J. Power Sources, 2012, 206, 450. 23 A. A. Shah, M. J. Watt and F. C. Walsh, Electrochim. Acta, 2008, 53, 8087. 24 D. You, H. Zhang and J. Chen, Electrochim. Acta, 2009, 54, 6827. 25 M. Verbrugge, J. Electrochem. Soc., 1990, 137, 886. 26 M. Vynnycky, Energy, 2011, 36, 2242. 27 X. Luo, Z. Lu, L. Chen and X. Qiu, J. Phys. Chem. B, 2005, 109, 20310. 28 J. Qiu, M. Li, M. Zhai and G. Wei, J. Membr. Sci., 2007, 297, 174. 29 C. Jia, J. Liu and C. Yan, J. Power Sources, 2012, 203, 190. 30 L. M. Pant, MPhil Thesis, University of Alberta, 2011. 31 J. Cho, M. Annable and P. S. Rao, Environ. Sci. Technol., 2005, 39, 7883.

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32 M. Matlosz and J. Newman, J. Electrochem. Soc., 1986, 133, 1850. 33 B. Delanghe, S. Tellier and M. Astruc, Electrochim. Acta, 1990, 35, 1369. 34 S. N. Yang and Z. Lewandowski, Biotechnol. Bioeng., 1995, 48, 737. 35 J. L. Nava and A. Recendiz, et al., Port. Electrochim. Acta, 2009, 27, 381. 36 R. Aris, Proc. R. Soc. London, Ser. A, 1956, 235, 67. 37 I. Frankel and H. Brenner, J. Fluid Mech., 1989, 204, 97. 38 S. E. Jarlgaard, Microfluidics Theory and Simulation Course Report, Technical University of Denmark, Denmark, 2007. 39 http://www.sglgroup.com/cms/_common/downloads/products/ productgroups/gs/carbon-and-graphite-for-continuouscasting/ Ringsdorff_marken_strangguss_e.pdf. 40 N. Wakao, S. Kaguei and T. Fumazkri, Chem. Eng. Sci., 1979, 34, 325. 41 A. Amiri and K. Vafai, Int. J. Heat Mass Transfer, 1998, 41, 4259.

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