This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 10841--10848 10841

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

Determination of the mass-transport properties ofvanadium ions through the porous electrodes ofvanadium 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

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,

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

proposed based on the experimental data.

1. Introduction

With growing interest in the deployment of renewable inter-mittent energy sources, such as solar and wind power, the needfor electrical energy storage (EES) in both mobile and stationaryapplications becomes exigent.1 Among various EES systems,the vanadium redox flow battery (VRFB) is one of the mostpromising candidates, since apart from having the commonfeature of the system scalability of all redox flow batteriesassociated with the separation of energy storage tanks andpower packs, it offers additional advantages including theability to fully charge and discharge without damaging thecells, no crossover contamination in the electrolytes, andmoderate cost.2–4 Nevertheless, issues with VRFBs, includinga low power density and energy density on a system level, ioncrossover through the polymer membrane, and corrosion, needto be addressed.5,6 To this end, efforts including improving theelectrode electrochemical kinetics,7–11 adding additives into theelectrolyte to improve its solubility,12–14 as well as modifyingexisting membranes and searching for alternatives15–20 havebeen made over the past decades. Moreover, minimizing the

mass transport polarization is also one of the crucial issues forimproving cell performance. Recently, Aaron et al. introduced aso-called zero-gap cell architecture with a serpentine flow field,enabling the peak power density to be 550 mW cm�2, which issignificantly higher than conventional cells; the enhancementof the cell performance was attributed to the enhanced masstransport and reduced internal resistance.21,22

In addition to experimental investigations, numerical modelingcan also play an important role in improving and optimizingthe performance of VRFBs. To enable numerical models toprovide meaningful insight into the operating characteristics ofa VRFB, accurate transport properties are needed, in addition toa robust formulation. One of the key transport properties is theeffective diffusivity that is required to model the mass transportat the representative elementary volume (REV) level of porouselectrodes in the framework of Darcy’s law. The most widelyused effective diffusivity23–25 is a simple correlation equationwith the intrinsic diffusivity and the porosity of the porousmaterial using a Bruggemann correction;26 an issue with thiscorrelation, however, is that the effect of the pore morphologyof different porous structures is missing. Attempts have alsobeen made to measure the effective diffusivity. A commonapproach is to place a porous structure sample between tworeservoirs, one of which contains an electrolyte solution, whilethe other contains DI water, respectively. UV-Vis spectroscopic

Department of Mechanical Engineering, The Hong Kong University of Science and

Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China.

E-mail: metzhao@ust.hk; Fax: +86 (852) 2358-1543; Tel: +86 (852) 2358-8647

Received 8th May 2013,Accepted 8th May 2013

DOI: 10.1039/c3cp51944a

www.rsc.org/pccp

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10842 Phys. Chem. Chem. Phys., 2013, 15, 10841--10848 This journal is c the Owner Societies 2013

measurements are carried out to determine the amount of ionsin the DI water over a period of time. After determining the ionconcentration of different aliquots, the effective diffusivity ofthe ions through the porous media can be obtained based onthe porous structure area, thickness and the overall solutionvolume.27–29 Nevertheless, a major problem associated withthis method is that the dispersion effect which exists in anoperating VRFB cannot be taken into account. Recently, Pantet al. proposed a new configuration to determine the effectivediffusivity through the electrodes in flow batteries.30 The flowingelectrolytes in two channels were separated by a porous diffusionlayer. The amount of ions diffused through the diffusion layerwas determined by the ion-chromatography measurements. Thedispersion effect is included in their method, but the experi-mental setup is rather complex and the measurements are time-consuming.

Another key mass-transfer property is the pore-level mass-transfer coefficient, which is related to the morphology of poresurfaces, electrolyte properties and the local velocity of electro-lyte. A common method to obtain the mass-transfer coefficientis to measure the non-aqueous phase liquids (NAPL)–waterinterfacial area in water-saturated columns by tracer studies.31

However, this approach suffers from troublesome setup andrelatively large measurement errors. Another way is to electro-chemically determine the mass-transfer coefficient by measur-ing the limiting current densities when the electrode operatesunder mass-transport control.32–35 Nevertheless, as the porouselectrode usually has a finite thickness (several millimeters),this approach can only be used to obtain the average mass-transfer coefficient over the electrode surface.

In this study, the effective diffusivity and the pore-levelmass-transfer coefficient in porous VRFB electrode are electro-chemically determined. By devising and using the uniquesetups, the effect of dispersion on the effective diffusivity canbe included. The limiting current densities caused by specificions under various flow rates are measured. Finally, based onthe experimental results, the correlation equations for theeffective diffusivity and the pore-level mass-transfer coefficient,respectively, are proposed.

2. Theoretical

Consider the transport of ions in an electrolyte flow through aporous electrode as illustrated in Fig. 1. The flux of a specificion as the result of diffusion, electrical migration and convectioncan be expressed by the modified Nernst–Planck equation as:24

Ni = � Deffi rci � Fzicimirfs + uci (1)

where u represents the Darcy’s velocity, c is the ion concen-tration at the representative elementary volume (REV) level,Deff is the effective diffusivity, F is Faradaic constant, z and m arethe charge number of the ion and ionic mobility, and fs is theionic potential in electrolyte. With eqn (1), the conservation ofspecies can be written as:

r�Ni =:Ri (2)

where:R represents the consumption rate of the ion as the

result of electrochemical reactions at the solid surfaces in thepores of the electrode and is related to the transport flux fromthe REV level, represented by the REV concentration c, to thesolid surfaces in pores, represented by the concentration at thesolid surfaces cs, as modeled by:

:Ri = kmAV(ci � ci

s) (3)

where km represents the pore-level mass-transfer coefficient andAV is the specific surface area of the porous electrode. Thereaction current density j is related to the local concentration cs

at the solid surfaces of the pores in the porous electrode andcan be determined from the Butler–Volmer equation:25

j ¼ eAVj0;icsicrefi

� �exp

aþFZRT

� �� exp �a�FZ

RT

� �(4)

where e is the porosity of the porous electrode, j0 is theexchange current density, Z is the overpotential, cref is thereference concentration, a+ and a� are the anodic and cathodictransfer coefficients. Combining eqn (3) and (4), the variable cs

in the Butler–Volmer equation can be eliminated by introducinga pore-level mass-transfer coefficient km in Section 2.2.

The flow of ions gives rise to the current density in theelectrolyte solution:

ii = ziFNi (5)

The electrolyte is considered to be electrically neutral:Xi

zici ¼ 0 (6)

By combining eqn (1), (5) and (6), we can express the totalcurrent density in the electrolyte as:

i ¼Xi

ii ¼ �FXi

ziDeffi rci � F2

Xi

zi2cimirfs (7)

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

Fig. 1 Schematic of the mass transport process through a porous electrode.

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current density induced by the electrochemical reaction of theactive ions:

r�i = j (8)

By solving eqn (2) and (8), the distributions of ion concentrationand overpotential within the porous electrode can be determined.

The above analysis suggests that the two transport proper-ties, the effective diffusivity Deff and the pore-level mass-transfercoefficient km, are needed to solve the coupled electrochemicaland mass transport process in porous electrodes.

2.1. Determination of the effective diffusivities of vanadium ions

This section describes the theoretical framework to determinethe effective diffusivity through a porous electrode. The basicidea is to measure the mass flux under a given concentrationdifference across the porous electrode. The direct determina-tion of the mass flux has been proven to be rather challenging.In this work, an electrochemical method to determine theeffective diffusivity through a porous electrode is proposed.Fig. 2 illustrates the experimental setup, which is essentially aVRFB. The porous sample to be tested and a membrane-electrode assembly (MEA) are separated by a stainless steelring serving as a current collector. A flow field is used to supplyand distribute liquid electrolyte onto the porous surface, wherethe concentration of vanadium ions, cf,i, could be approximatedas the average value of the concentrations at the inlet (cin) andthe outlet (cout) of the flow field if (i) the surface area of theelectrode is small; and (ii) the electrolyte flow rate is large. Thevanadium ion is then transported through the porous sampleand the gap between the porous sample and the catalyst layer,where it is reacted at a concentration, cs,i. As such, the mass fluxfrom the flow channel to the catalyst layer can be expressed as:

N i ¼i

F¼ cf ;i � cs;i

L

Deffi

þ l

Di

(9)

where L and l represent the thicknesses of the porous sampleand stainless steel square ring, and Di is the ion intrinsicdiffusivity in the electrolyte.23 When the current density reachesthe limiting current density, the concentration in the catalystlayer becomes zero. Consequently, eqn (9) is reduced to:

Deffi ¼

L

Fcf ;i

ilim� l

Di

(10)

where ilim is the limiting current density. Clearly, with ameasured limiting current density and a given concentrationcf,i, the effective diffusivity through the porous sample can bedetermined. The measured effective diffusivity through theporous electrode at different electrolyte flow rates can becorrelated in terms of the intrinsic diffusivity D, the porositye, and the flow velocity as:36–38

Deff

D¼ eað1þ bPe2Þ (11)

where Pe (= udp/D) is the Peclet number, with u representing theflow velocity and dp the pore diameter. The empirical constantsa and b will be determined based on the experimental data.It should be noted that Deff/D depends on the geometricproperties (e.g. porosity) of the porous electrodes and flowcharacteristics through the electrode, regardless of the specificions. 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 determinethe pore-level mass-transfer coefficient. The mass flux from theREV level to solid surfaces through pores needs to be measuredunder the condition that the concentration at the REV level cf

and that at solid pore surfaces cs are known. An electrochemicalapproach is conceived in this work. The idea is illustrated inFig. 3, where a porous layer (graphite felt) is placed between theflow field and membrane. The electrolyte with concentration

Fig. 2 Schematic of the setup used for the determination of the effectivediffusivity.

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

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cf in the flow field is transported through the pores to the solidsurface, where a redox reaction takes place. If the porous layeris sufficiently thin (with several pores), the mass-transferresistance from the flow field to the porous layer becomesnegligible. Hence, the concentration at the REV level can beapproximated by that in the flow field. The mass flux from theREV level to the pore surfaces can be modeled by:

N0 = km(cf � cs) (12)

Note that when the current density reaches the limitingcurrent density, the concentration at solid pore surfaces ofthe porous layer becomes zero. Consequently, eqn (12) isreduced to:

km ¼ilim

Fcf(13)

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

3. Experimental3.1. VRFB configuration and flow circuit

The fabricated VRFB consisted of two endplates and two flowfields with parallel flow channels and the negative and positiveelectrodes separated by a Nafions 117 membrane. Graphite felt(GFA Series, SGLs, Germany), whose physical properties can befound elsewhere,39 was used as the material for the negativeand positive electrodes. Each electrode had a geometric area of10 mm � 10 mm. The PTFE gaskets were placed between theparts to ensure no leakage of the electrolyte. The volume ofeach electrolyte compartment was 20 ml. Unless otherwiseindicated, the positive and negative electrolytes contained3.0 M sulfuric acid and the vanadium ions of different con-centrations for the performance tests. The electrolytes werekept at room temperature and both positive and negativeelectrolytes were circulated using a peristaltic pump (Glorys,BT100-1F) using corrosion-resistant tubing (Masterflexs,Cole-Parmer).

For the determination of the effective diffusivities of thevanadium ions, the following steps were taken in the designand fabrication of the experimental system: (i) to eliminate theelectric field effect on ion transport through the porous sample,the porous sample was electrically isolated from the catalystlayer by a separator (see Fig. 2); (ii) instead of measuring thevoltage of the cell, a reference electrode was used to measurethe respective potentials of the positive and negative electrodes;the measured potentials at the positive/negative electrode candirectly indicate whether the limiting current density is causedby the transport of a specific ion on the positive or negativeelectrode; and (iii) to maintain an even distribution of electro-lyte concentration onto the porous sample, a parallel flow fieldwas designed and utilized.

For the determination of the pore-level mass-transfer coeffi-cient, we ensure the porous layer is sufficiently thin (with severalpores), and a parallel flow field is applied to keep an even distri-bution of electrolyte concentration along the porous layer.

3.2. Electrochemical test rig

An Arbin BT2000 (Arbin Instrument Inc.) was employed tomeasure the polarization curves. The internal resistance ofthe cell was measured by the built-in function of the ArbinBT2000. Anode polarization data for the VRFB were obtainedemploying a saturated silver–silver chloride electrode, Ag–AgCl(ABB, Series 1400, 1.0 M KNO3), equipped with a liquid junctionprotection tube, placed about 3 cm away from the exit of theflow channel in line with the electrolyte circuit. Cathodepolarization data were derived by subtracting anode polariza-tion values from the respective cell polarization data.

4. Results and discussion4.1. 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 thepositive and negative sides at a flow rate of 40 ml min�1. It canbe seen that with a lower electrolyte concentration of 0.125 M,increasing the current density resulted in a sharp drop in thecell voltage, meaning that the electrolyte at the electrode activesites was depleted and the limiting current density reached.When the electrolyte concentration was increased to 1.0 M, thecell performance was greatly improved at moderate currentdensities, but the voltage dropped steeply at high currentdensities. However, with a higher electrolyte concentration of1.5 M, with increasing the current density the cell voltagedropped almost linearly towards zero and there was no sharpdrop in the cell voltage. This polarization behavior indicatesthat at higher electrolyte concentrations, the voltage loss ispredominately caused by the large internal resistance ratherthan the mass transport limitation. Therefore, no limitingcurrent density exists at sufficiently high electrolyte concentra-tions, unless the flow rate is extremely low. The objectivesof this work were to determine the effective diffusivities ofvanadium 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 currentdensities. Therefore, all the experimental data reported here-after were collected at an electrolyte concentration lowerthan or equal to 1.0 M and at a flow rate lower than or equalto 40 ml min�1, at which the limiting current density occurred.

4.2. Measured effective diffusivities of vanadium ions throughthe porous electrode

To measure the effective diffusivities of VO2+ and VO2+, it should

be ensured that the limiting current density occurs on thepositive side. Accordingly, the total concentration of vanadiumions in the positive electrolyte was set to be 0.25 M while the totalconcentration in the negative electrolyte was 0.4 M. The state ofcharge (SOC) of the positive electrolyte was kept at 0.9 at thebeginning of each test. Fig. 5a shows the polarization curves atdifferent electrolyte flow rates. As can be seen, the limitingcurrent density increases with the flow rate. When the velocityof the electrolyte solution is zero, the limiting current density is22 mA cm�2, representing the mass transport limitation of purediffusion through the porous layer. With an increase in the flowrate, the limiting current density increases: to 47 mA cm�2 at

5 ml min�1, and to 73 mA cm�2 at 40 ml min�1. The fact that thelimiting current density increases with the flow rate means thereis an enhanced mass transport as a result of the increased flowrate. Fig. 5b shows the respective positive and negative potentialsat different flow rates. Clearly, at each flow rate the masstransport limitation occurred at the positive electrode, suggest-ing that the limiting current density was determined by thetransport flux of VO2

+ ion. Limiting current densities were thenmeasured by varying the flow rate from 5 to 40 ml min�1. Themeasured data were then transformed to the effective diffusivityaccording to eqn (10). The following correlation was obtainedbased on a least-square fit of the experimental data:

Deff

D¼ e1:1ð1þ 1:46� 10�3Pe2Þ ð4:3oPeo 34:5Þ (14)

It should be noted that eqn (14) was obtained based on thetransport of VO2

+ ions. To verify whether eqn (14) is suitable forthe transport of other ions or not, additional experiments wereperformed to measure the limiting current densities induced bythe transport of V2+ ions. To this end, the total concentration ofvanadium 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 SOCof the negative electrolyte was kept at 0.9 at the beginning of eachtest. The polarization curves at different electrolyte flow rates areshown in Fig. 6a. When the flow rate is 5 ml min�1, the limiting

Fig. 5 Polarization curves for the VRFB using VO2+ ions at various electrolyte

flow rates (positive electrolyte concentration 0.25 M; negative electrolyte concen-tration 0.4 M). (a) Cell voltage vs. current density. (b) Positive and negative half-cell potentials vs. Ag–AgCl reference electrode.

Fig. 6 Polarization curves for the VRFB using V2+ ions at various electrolyte flowrates (positive electrolyte concentration 0.4 M; negative electrolyte concentration0.25 M). (a) Cell voltage vs. current density. (b) Positive and negative half-cellpotentials vs. Ag–AgCl reference electrode.

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current density is 37 mA cm�2. As the flow rate increases to40 ml min�1 the limiting current density becomes 55 mA cm�2.Fig. 6b shows the respective positive and negative potentials atdifferent flow rates. It is clear that the mass transport limitationoccurred at the negative electrode for each flow rate, indicatingthe limiting current density was determined by the transport ofV2+ ions. The measured Deff at different flow rates for thetransport of V2+ ions (6.9 o Pe o 55.3) is compared witheqn (14) in Fig. 7. It can be found that the experimental data forV2+ ions fit well with the correlation. Hence, eqn (14) representsthe 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-transfercoefficient illustrated in Fig. 3, the polarization curves of theVRFB can be obtained and are shown with vanadium ionconcentration in the positive side of 0.125 M (Fig. 8a) and0.25 M (Fig. 8b), respectively. The vanadium ion concentrationin the negative side was kept at 1.0 M to ensure that the masstransport limit occurs in the positive electrode. The SOC of thepositive electrolyte was kept at 0.9 at the beginning of each test.It is seen that for a given electrolyte concentration, the limitingcurrent density increased with increasing the flow rate as aresult of the improved mass transport of the electrolyte. For thevanadium ion concentrations of 0.125 M and 0.25 M, thelimiting current density increased from 46.2 to 81.1 mA cm�2

and from 63.3 to 107.8 mA cm�2, respectively, as the flow ratewas increased from 5 to 40 ml min�1.

The variations in the limiting current density with electro-lyte concentration at different flow rates are shown in Fig. 9.It can be found that at each given flow rate the limiting currentdensity increased almost linearly with the electrolyte concen-tration, due to the reduced concentration polarization.

The measured pore-level mass-transfer coefficients againstcurrent density are shown in Fig. 10. It can be seen that theeffect of the electrolyte flow rate on km is significant. Forinstance, when the flow rate was increased from 5 to 40 ml min�1,km doubled. On the other hand, if the electrolyte flow rateis reduced to zero, the transport within a pore space is inducedby pure diffusion, where km is simply expressed as 2D/dp.21

Therefore, the pore-level mass-transfer coefficient can be

related to D and the electrolyte flow rate.40,41 Fig. 11 presentsthe variation in ln(kmdp/D � 2) with ln Re based on the sameexperimental data shown in Fig. 10. It is seen that althoughscattered 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 s�1) was obtained based on aleast-square fit of the experimental data:

kmdp

D¼ 2þ 1:534Re0:912 ð0:3oReo 2:4Þ (15)

Fig. 7 Mass-transfer correlation, Deff/D vs. Pe.

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 electrolyteconcentration: 1.0 M.

Fig. 9 Limiting current density vs. electrolyte concentration at different flowrates.

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5. Concluding remarks

Numerical modeling of VRFBs can not only provide insight intotheir operation characteristics, but also enables rapid testing ofhypotheses aimed at improving cell performance. The accuracyof numerical simulation depends not only on the robustness ofthe mathematical formulation, but also on the accuracy ofconstitutive mass-transport properties through the porouselectrodes. In this work, we measured two critical constitutiveproperties for mass transport of ions through porous electrodessaturated with a liquid electrolyte solution. One is the effectivediffusivity that is required to model the mass transport at theREV level of porous electrodes in the framework of Darcy’s law,while the other is the pore-level mass-transfer coefficient formodeling the mass transport from the REV level to the solidsurfaces of pores. Based on the theoretical framework of masstransport through the electrodes of VRFBs, unique experi-mental setups for electrochemically determining the two trans-port properties were devised. The effective diffusivity and thepore-level mass-transfer coefficient through the porous elec-trode made of graphite felt, a typical material for VRFB electro-des, were measured at different electrolyte flow rates. Theobtained correlation equation for the effective diffusivity ofvanadium ions through the porous electrode includes theeffects of both the porous electrode structure and flow disper-sion. It is found that the effect of flow dispersion becomes more

significant with an increase in flow rates. The pore-level mass-transfer coefficient is found to be independent of currentdensity and the correlation equation in terms of Reynoldsnumber and the intrinsic diffusivity has been proposed.

Acknowledgements

The work described in this paper was fully supported by a grantfrom the Research Grants Council of the Hong Kong SpecialAdministrative Region, China (Project No. 622712).

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