Transport phenomena associated with capacity loss of all-vanadium redox flow battery

International Journal of Heat and Mass Transfer 148 (2020) 119040

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Transport phenomena associated with capacity loss of all-vanadium redox flow battery Dong Kyu Kim a, Sang Jun Yoon c,d, Sangwon Kim b,d,∗ a

School of Mechanical Engineering, Chung-Ang University, Seoul 06974, Korea KIST Europe, Korea Institute of Science and Technology, Campus E71, Saarbrücken 66123, Germany c Center for membranes, Korea Research Institute of Chemical Technology, Daejeon 34114, Korea d Transfercenter Sustainable Electrochemistry, Saarland University, Saarbrücken 66125, Germany b

a r t i c l e

i n f o

Article history: Received 5 April 2019 Revised 7 November 2019 Accepted 10 November 2019

Keywords: All-vanadium flow battery Capacity loss Species transport Osmosis Prediction of state of charge

a b s t r a c t This study investigates transport of different species through the Nafion® 115 membrane in an all vanadium redox flow battery to understand transport phenomena associate with capacity loss. We consider several driving forces related to transport of vanadium ions, proton, and water molecules and examine the variations in ion concentration during charge-discharge, and long-term operation. First, variations in ion concentration are analyzed during the 3rd and the 300th charge-discharge cycles to compare the ion transport process at between early stage and late stage. The capacity loss is closely related to the selfdischarge reaction caused by diffusion through the membrane, and proton transport is another important factor of capacity loss since the imbalance in proton concentration can accelerate water transport through the membrane. Furthermore, variations in solution volume and ion concentration are examined during long-term charge-discharge cycles. Since ions accumulate on the positive electrode due to self-discharge reaction and repeated electrochemical reaction, water molecules transport to the positive electrode side. Finally, the relationship between the changes in solution volume and capacity loss is examined, and an empirical equation is suggested for the prediction of capacity loss from changes in the solution volume during long-term operation. Through the results, we can predict the lifetime of VFB system by simple measurement. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Over the last few decades, large-scale energy-storage systems (ESSs) have gained much attention as an alternative energy source of power grids. Since the power generation from renewable energy sources such as wind turbines can be intermittent, many researchers and manufacturers started to develop large-scale ESSs to supplement power supply [1,2]. Among the large-scale ESSs, allvanadium redox flow batteries (VFBs) have garnered most interest because of their high energy efficiency, long cycle life, low maintenance cost, and flexibility for scale-up [3,4]. The VFB system was first proposed by the Skyllas-Kazacos group in 1985. Since then, many companies have succeeded in developing megawatt (MW)class VFB systems [5,6], and some companies even tried to develop large-scale commercial VFB systems to supply power to the grid [7,8]. Although there is much competition among these companies,

∗ Corresponding author at: KIST Europe, Korea Institute of Science and Technology, Campus E71, Saarbrücken 66123, Germany. E-mail address: [email protected] (S. Kim).

https://doi.org/10.1016/j.ijheatmasstransfer.2019.119040 0017-9310/© 2019 Elsevier Ltd. All rights reserved.

their understanding of the transport phenomena associated with capacity loss of VFBs remains unsatisfactory. Many experimental and numerical studies have been conducted to study the transport phenomena that are related to capacity loss of the VFB system. Some experimental studies investigated the variations in vanadium ions concentration and changes in solution volume during operation [9–11]; the results showed that the vanadium ion concentration and volume of the electrolyte vary during self-discharge and long-term charge-discharge cycling. Other studies developed three-dimensional transient models to show the distribution of vanadium ions in a VFB by considering the crossover of vanadium ions during change and discharge [12,13]; the results showed that the distribution of vanadium ions and the capacity loss of a VFB are affected by the crossover of vanadium ions. Other researchers developed dynamic models to understand the effect of side reactions on the capacity loss during repetitive cycling [14– 16]. Although most previous studies offer a basic understanding of the mechanism of capacity loss and help us to predict the amount of capacity loss during long-term operation of a VFB, they did not consider the effect of transport of protons and water molecules on

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D.K. Kim, S.J. Yoon and S. Kim / International Journal of Heat and Mass Transfer 148 (2020) 119040

vanadium ions, protons, and water molecules through the membrane. Specially, we considered self-discharge reaction, charge imbalance, osmotic drag force, and hydraulic pressure difference between electrodes to explain transport of proton and water accurately. Furthermore, we conducted experiments to obtain the cell voltages and changes in solution volume to validate the model. In addition, to ensure the reliability of the developed model, some of electrolytes were collected, and the concentration changes of vanadium ions were measured using UV–VIS spectroscopy during charge-discharge cycles.

Nomenclature ɛ Ve C Q j A w D nd

η

P

ρ μ κ

Porosity Volume of electrode (m3 ) Concentration (mol/m3 ) Flow rate (m3 /s) Current density (A/m2 ) Area (m2 ) Thickness (m) Diffusion coefficient (m2 /s) Electro-osmotic drag coefficient Overpotential (V) Pressure (bar) Density (kg/m3 ) Viscosity (Pa•s) Permeability (cm2 )

2.1. Modeling The assumptions of the developed model follow that of previous studies [6-9]. The assumptions are listed below: (1) The flow is regarded as incompressible and laminar. (2) Since the model is developed based on the lumped model, membrane and electrode have isotropic and homogeneous physical properties. (3) The cell is operated in moderate temperature range. Therefore, the effect of temperature changes on the performance is neglected. (4) The cell is operated in stable voltage range and the electrolyte is perfectly sealed. Therefore, hydrogen evolution, oxygen reaction, and oxidation of V2+ ions are also neglected.

Subscripts i Ions m Membrane e Electrode s Surface d Drag pore Pore of the membrane res Reservoir capacity loss. Therefore, it is still necessary to understand relationship between the various ions transport and their associated reactions to explain the capacity loss. The objective of this study is to examine the transport phenomena associated with capacity loss of an all-vanadium redox flow battery. The study began with the development of a transient model to understand the transport phenomena through the membrane during long-term charge-discharge cycling. Since capacity loss is closely related to the transport of ions through the membrane, the model described the transport of vanadium ions, protons, and water molecules affected by electrochemical reaction, diffusion, osmosis, hydraulic pressure difference, and self-discharge reaction. The investigation began with an examination of the variations in ion concentration and solution volume during the 3rd and the 300th charge–discharge cycles, respectively. Next, the changes in the capacity loss of the VFB was analyzed by examining the relationship between the variations in ion concentration and solution volume during long-term charge-discharge cycling. Finally, an empirical equation is suggested to predict the change in state of charge (SOC) using change in electrolyte volume. 2. Methods The all-vanadium redox flow battery is a type of flow battery that uses vanadium as the electroactive material at the negative electrode and positive electrode, and the electrolyte circulates between the cell and the reservoir. The redox reaction takes place at electrodes in the cell, while the energy is stored chemically in the external reservoir; therefore, the power and capacity are independently determined by the cell size and electrolyte volume, respectively. The main reactions at the positive electrode and negative electrode of the VFB are shown as follows:

VO2+ + H2 O−e−

charge



discharge

V3+ + e−

VO2 + + 2H+ charge



discharge

V2+

E0 = 1.00Vvs.SHE

(1)

E0 = −1.00Vvs.SHE

(2)

The following section describes the VFB model that we developed; it includes several driving forces to explain the transport of

When modeling the VFB, we focused on explanation of the transport of ions in the VFB as a result of different driving forces that are shown in Fig. 1(a). First, we calculated the variations in vanadium ion concentration by considering the electrochemical reaction, side reaction, diffusion, and convection at each electrode. The variations in the vanadium ion concentration are expressed using species conservation shown as follows [17].

εVe

dCi = n˙ i, f low ± n˙ i,electrochem − n˙ i,di f f dt

(3)

where ɛ is the porosity, Ve is the volume of the electrode, Ci is the concentration of each ion, and n is number of moles.

n˙ i, f low = Q (Cires − Ci ) Q is the flow rate, and

n˙ i,electrochem = Ae

(4) Cires

is concentration of ion in the tank.

j F

(5)

Ae is surface area of electrode, j is the current density, and F is Faraday’s number. The plus sign is used for V2+ , and VO2 + ions, and minus sign is used for V3+ , and VO2+ ions. The third term in Eq. (3) is expressed as below.

Am (D C + 2DV (V )CV (V ) + DV (IV )CV (IV ) ) wm V ( I I ) V ( I I ) A n˙ V (III ),di f f = m (DV (III )CV (III ) − 3DV (V )CV (V ) − 2DV (IV )CV (IV ) ) wm A n˙ V (IV ),di f f = m (DV (IV )CV (IV ) − 3DV (II )CV (II ) − 2DV (III )CV (III ) ) wm A n˙ V (V ),di f f = m (DV (V )CV (V ) + 2DV (II )CV (II ) + DV (III )CV (III ) ) wm

n˙ V (II ),di f f =

(6) (7) (8) (9)

Here, Am is the area of the membrane, wm is the thickness of the membrane, and Di is the diffusion coefficient of each ion. The diffusion coefficient is 5.261 × 10−6 cm2 /min for V2+ , 1.933 × 10−6 cm2 /min for V3+ , 4.095 × 10−6 cm2 /min for VO2+ , and 3.538 × 10−6 cm2 /min for VO2 + [18]. In Eq. (3), the first term on the right-hand side is the change in vanadium concentration resulting from the flow rate, and the second term is that resulting

D.K. Kim, S.J. Yoon and S. Kim / International Journal of Heat and Mass Transfer 148 (2020) 119040

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at the positive electrode is represented by the equation

εVe

dCH + (+) = n˙ i, f low + n˙ i,H + ,(+) + n˙ H + ,di f f dt −n˙ H + ,sel f (+) − n˙ H + ,balance

(16)

while the transport of protons at the negative electrode is represented by the equation

εVe

dCH + (−) = n˙ i, f low + n˙ i,H + ,(−) − n˙ H + ,di f f dt −n˙ H + ,sel f (−) + n˙ H + ,balance

(17)

The first and the second terms represent the changes in proton concentration due to convection and the electrochemical reaction, respectively. Since the electrochemical reactions at positive and negative electrodes involve one electron and 2 mol of hydrogen are generated at positive electrode during charge process, variation in proton is expressed as follow [11].

n˙ i,H+ ,(+) = −(Am − 2Ae ) n˙ i,H+ ,(−) = Am

j F

(18)

j F

(19)

The third term in Eqs. (16) and (17), n˙ H + ,di f f , is the proton transport due to diffusion induced by the difference between the concentration of the negative electrode and positive electrode given by

n˙ H+ ,di f f =

 Am DH +  CH + (−) − CH + (+) wm

(20)

The fourth term in Eqs. (16) and (17), n˙ H + ,sel f , on the right-hand side shows the consumption of protons due to self-discharge reaction described by Eqs. (10) to (15):

n˙ H+ ,sel f (+) = 2 wAmm DV (II )CV (II )  n˙ H+ ,sel f (−) = wAmm 2DV (IV )CV (IV ) + 4DV (V )CV (V )

(+ )electrode (− )electrode (21)

Fig. 1. Schematic diagram of species transports in all-vanadium redox flow battery: (a) transport of vanadium ions and protons; (b) transport of water molecules.

The last term in Eqs. (16) and (17), n˙ H + ,balance , shows the transport of protons due to charge imbalance. The transport of vanadium ions with different charge numbers causes the charge imbalance between the negative electrode and positive electrode. Therefore, the proton transports through the membrane to match the charge balance that is shown as follows:

n˙ H+ ,balance = from the electrochemical reaction. The third term represents the variation in ion concentration resulting from diffusion and selfdischarge reactions. The self-discharge reaction at the positive electrode is represented by the equations

V2+ + 2VO2 + + 2H+ → 3VO2+ + H2 O

(10)

V3+ + VO2 + → 2VO2+

(11)

V

2+

+ VO

2+

+

+ 2H → 2V

3+

+ H2 O

(12)

+

VO2 + 2V +

VO2 + V

2+

3+

+

+ 4H → 3V

→ 2VO

2+

3+

+ 2H2 O

(13) (14) (15)

These self-discharge reactions are induced by the crossover of vanadium ions that are calculated at the third term in Eq. (3). In the case of proton transport, we considered several driving forces associated with the transport of protons. The transport of protons

(22)

This study also examined the water transport through the membrane as a result of several driving forces that are shown in Fig. 1(b). The variations in water concentration on the side of the positive electrode and the side of the negative electrode are expressed as follows:

εVe

while the self-discharge reaction at the negative electrode is represented by the equations

VO2+ + V2+ + 2H+ → 2V3+ + H2 O

Am (2DV (II )CV (II ) + 3DV (III )CV (III ) wm −2DV (IV )CV (IV ) − DV (V )CV (V ) )

εVe

dCH2 O(+) = n˙ i, f low + n˙ water,(+) + n˙ water,di f f dt +n˙ water,sel f (+) + n˙ water,osmos + n˙ water,press

(23)

dCH2 O(−) = n˙ i, f low + n˙ water,(−) − n˙ water,di f f dt +n˙ water,sel f (−) − n˙ water,osmos − n˙ water,press

(24)

The first and the second terms on the right-hand side of Eqs. (23) and (24) show the changes in water concentration due to convection and the electro-osmotic drag force, respectively.

Cwater,(+) = −(Ae nd,H+ + Am )

j F

(25)

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D.K. Kim, S.J. Yoon and S. Kim / International Journal of Heat and Mass Transfer 148 (2020) 119040

Cwater,(−) = Ae nd,H+

j F

(26)

The third term, n˙ water,di f f , expresses the water transport through the membrane as a result of water molecules being dragged along by the diffused vanadium ions and protons:

n˙ water,di f f =

Em =

RT ln F



CH + ,ca CH + ,an



(37)

 Am  n D C + nd,V (III ) DV (III )CV (III ) − nd,V (IV ) DV (IV )CV (IV ) − nd,V (V ) DV (V )CV (V ) wm d,V (II ) V (II ) V (II )  Am DH +  −nd,H + CH + (−) − CH + (+) (27) wm

where nd, i is the osmotic drag coefficient, which is equal to 6 for V2+ and V3+ , 5 for VO2+ , 3 for VO2 + , and 2.5 for H+ [19–22]. Because VFB cell is filled with liquid, inherent interfacial resistance of the individual membrane can be negligible [23]. Therefore, we used intrinsic drag coefficient for each vanadium ion as an osmotic drag coefficient regardless of types of membrane. Water generation due to the self-discharge reaction, n˙ water,sel f , is considered in the fourth term, and the water generations at the positive electrode and negative electrode are expressed as follows:

Am D2CV (II ) wm  Am  n˙ water,sel f (−) = D4CV (IV ) + 2D5CV (V ) wm

n˙ water,sel f (+) =

(28) (29)

In addition, we considered the water transport by osmosis, n˙ water,osmos , i.e., the force to equalize the ion concentration on both sides. The variation in water concentration by osmosis is expressed as

n˙ water,osmos = κ Am RT (CCa,total − CAn,total )

(30)

Here, κ is permeability of water through Nafion® 115 that is 4 × 10−16 cm2 [22]. Ci, total is the total sum of ion concentration at each electrode. Lastly, we included water transport due to the difference of hydraulic pressure, n˙ water,press , between negative electrode and positive electrode that is expressed by Darcy’s law derived from Navier-Stokes equation shown as below [24].

κπ r pore 2 ρ P n˙ water,press = · μ wm

(31)

where ρ is the density of electrolyte, rpore is the radius of pores in the membrane (2.07 nm) [25], and μ is the viscosity of the electrolyte (20.1 × 10−3 Pa•s) [26]. The pressure difference, P, that is resulted from difference of the electrolyte volume in the external tank is calculated shown as follow.

P = ρ gh

(32)

his difference of electrolyte height in negative and positive side external tank. The concentration of each ion at the external tank is expressed as follow.

dCires Q =ε (C res − Ci ) dt Vres i

(33)

The obtained water concentration is used to calculate volume of electrolyte in reservoir which is expressed as follow. res dVwater Q Mwater ρwater = res dt Cwater

(34)

Cell voltage is then calculated as: rev Ecell = Ecell −

E(+ ) and E(− ) are open circuit potential for positive and negative electrode that are expressed as follow. Em is a Donnan potential across the membrane due to the differences in proton activities



(IR )k −

k



|η|

(35)

k

rev is the reversible open circuit voltage that is shown as where Ecell follow. rev Ecell = E ( + ) − E ( − ) + Em

(36)



(IR )k is total ohmic loss containing the resistance of the

k

membrane, electrode, and current collector; and η is the overpotential as expressed by the Butler–Volmer equation.



2RT η (+ ) = a sinh F



2F kca



CV (IV )CV (V )

 2RT η (− ) = − a sinh F

j

2F kan



(38)



j

(39)

CV (III )CV (II )

Here, k(+ ) and k(− ) are the reaction rate constant for each electrode that are shown follow.



k(+) = k(+),re f exp



0 F Eca Tre f



R

 k(−) = k(−),re f exp −



0 Tre f F Ean

R

1 1 − Tre f T





1 1 − Tre f T

(40)



(41)

SOC is determined by the portion of VO2 + ions in the catholyte at 1.6 V because the VO2 + ions limit the use of the VFB.

SOC =

CV (V ) CV (V ) + CV (IV )

(42)

Using the above expressions, we developed a lumped transient model for VFB using the program Simulink (MathWorks, USA) to express the detailed physicochemical processes inside a unit cell [11]. The operating conditions and physical properties for model are the same with the values used in the experiment that are shown in next section. 2.2. Experiment Experimental setups which is shown in Fig. 2(a) were prepared to examine the variations in capacity loss by measuring the changes in solution volume and voltage during charge and discharge, and long-term charge and discharge cycling. The VFB single cell was assembled by sandwiching the membrane (Nafion® 115, Sigma-Aldrich, USA) between two graphite bipolar plates. The active area of the single cell was 49 cm2 , and the electrochemical reaction took place in the carbon felt (GFD4.6 EA, SGL Group, Germany) that was prepared by acid and thermal treatments for 30 h at 400 °C under the atmosphere. Electrons transported to the potentiostat (SP-150, BioLogic, France) through the current collector. 1.6 M V3+ in 3 M H2 SO4 solution (80 mL) was prepared as the negative electrolyte and 1.6 M VO2+ in 3 M H2 SO4 solution (80 mL) was prepared as the positive electrolyte, and the electrolyte was stored in a mass cylinder to measure the changes in solution volume. The electrolyte was circulated through the mass cylinder and the cell by a pump (labV1, Shenchen, China). Physical properties of cell components are listed in Table 1. Using this experimental setup, the charge and discharge reaction continued until the 300th cycle at a current density of 80 mA/cm2 .

D.K. Kim, S.J. Yoon and S. Kim / International Journal of Heat and Mass Transfer 148 (2020) 119040

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The measurement of SOC during charge-discharge cycle is conducted using UV–VIS spectroscopy (Agilent Technologies, Cary 8454 UV–vis). We selected VO2+ ions in catholyte to analyze the concentration of ions because V2+ ions in anolyte is easily oxidized in the air. During the cell operation, catholyte of 0.125 ml is collected to measure the concentration of VO2+ ions concentration. We also collected anolyte of the same amount with catholyte to match the balance of electrolyte. Using the collected catholyte, 0.02 M VO2+ in 3 M H2 SO4 solution is prepared to analyze changes in concentration of vanadium ions. The prepared samples are shown in Fig. 2(b). Here, absorbance of UV–VIS spectroscopy was measured at 760 nm. 3. Results and discussion In this section, we will validate the VFB model by comparing the output of the model with experimental results obtained during a charge–discharge reaction. Using this model, we analyzed the variations in ion concentration of the VFB for long-term operation. 3.1. Model validation

Fig. 2. Images of experimental setup for examination of changes in vanadium ion concentration: (a) Experimental setup of the VFB; (b) Samples of catholyte at different SOC. Table 1 Physical properties of cell components. Parameters

Value

Thickness of membrane Conductivity of membrane Water uptake of membrane Nominal thickness of electrode Open porosity of electrode Area-specific resistance of electrode Density of electrolyte Molecular weight of electrolyte

127 μm [27] 0.1 S/cm [27] 38% water [27] 4.6 mm [28] 94 [28] <0.15  · cm2 [28] 1.35 kg/l [29] 163.01 g/mol [29]

Table 2 Operating conditions of the experiment. Parameters

Value

Area of membrane Upper / lower voltage limit Current density Volume of electrolyte Flow rate Maximum number of cycles

49 cm2 1.6 V/1.0 V 80 mA/cm 80 mL 65 mlpm 300

The upper and lower limits of the voltage were 1.6 V for charging and 1.0 V for discharging to avoid hydrogen evolution, respectively, and the mass cylinder was covered by rubber cap to prevent air contact. Temperature of laboratory was maintained at 20 °C, and the operating voltage was in moderate condition, so we assume that the operating temperature of the cell is in reasonable range. We measured the solution volume at intervals of five cycles. The experimental conditions are listed in Table 2.

The model was validated by comparisons of the voltage and changes in solution volume in the 3rd and the 300th cycle that is shown in Fig. 3(a), and (b). Moreover, the adequacy of the considered driving forces was convinced by comparing the experimental results of the relationship between SOC and voltage with the model results that is shown in Fig. 3(c). The calculated results are represented by lines, and the experimental results are represented by symbols in Fig. 3. The model shows good agreement (within 8% of the average error) with the experimental data. Even though there are small deviations between the experimental and numerical results at the beginning of the discharge process, this model is reasonably accurate for analysis of the species transport in the VFB. 3.2. Species transport during charge-discharge reaction in the 3rd and the 300th cycles In this section, we will compare the transport of various species in the VFB during the charge and discharge reaction in the 3rd and the 300th cycles. Different from previous studies that examined only the transport of vanadium ions, this study includes an examination of the transport of vanadium ions, protons, and water molecules. The variations in voltage and solution volume are shown in Fig. 3. It can be seen that the charge and discharge time of the 300th cycle (2799 s) is lower than that at the 3rd cycle (4400 s), indicating capacity loss of the VFB; the energy efficiency reduced by 1.75%. Meanwhile, the solution volume at negative electrode increases during charging and decreases during discharging. The variation in solution volume at positive electrode varied in opposite direction compared with negative electrode. After the 300th cycle, the trend is the same as that at the 3rd cycle, but total volume of electrolyte at positive electrode increases, and that at negative electrode decreases. The variations in solution volume and capacity loss can be explained by examining the transport of ions in the VFB. The variations in the concentration of vanadium ions at the positive electrode are shown in Fig. 4(a); it can be seen that the VO2 + ion concentration increases and the VO2+ ion concentration decreases during the charging process at the 3rd and the 300th cycles, as explained in a previous study [16]. The variations in vanadium concentration are the result of the self-discharge reaction and diffusion as shown in Fig 4(b) and (c). Here, the loss of ion concentration due to diffusion is described by negative value. When charge process starts, the amount of VO2 +

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D.K. Kim, S.J. Yoon and S. Kim / International Journal of Heat and Mass Transfer 148 (2020) 119040

Fig. 4. Changes in vanadium ion concentration at positive electrode as a result of different driving forces during the charge and discharge reaction: (a) variations in vanadium ion concentration during the charge and discharge reaction; (b) changes in VO2+ ion concentration due to self-discharge reaction and diffusion; (c) changes in VO2 + ion concentration due to self-discharge reaction and diffusion.

Fig. 3. Model validation by comparing the numerical (lines) and experimental (symbols) data for the 3rd and 300th charge–discharge cycles: (a) variations in voltage; (b) changes in solution volume; (c) variations in SOC with respect to voltage.

transported across the membrane is small, 0.01 mol/m. At the end of charge, however, the amount of VO2 + transported to negative electrode become large 0.03 mol/ m. Since the number of VO2 + ions is large at the end of charge process, the amount of diffusion is also large. The diffused ions result in the self-discharge reaction; VO2 + ions are consumed and VO2+ ions are generated at the positive electrode, which lead to changes in ion concentration and reduces the capacity. The variations in ion concentration by self-discharge reaction are similar to that by diffusion, but the amount of concentration variation by self-discharge is larger than

that by diffusion. Therefore, we can expect that VO2+ ion increases and VO2 + ion decreases as cycle proceed. Changes in vanadium ion concentration at negative electrode are shown in Fig. 5. The trend is similar to the phenomena occurred in positive electrode, except direction of reaction is reverse. During the self-discharge reaction, V2+ ions are consumed, and V3+ ions are generated. Therefore, the concentration changes of V2+ ion have negative values, and that of V3+ ion have positive values. The same as positive electrode, changes due to diffusion shows ion loss. In addition, this model describes the transport of protons in the VFB in response to the driving forces. Fig 6(a) shows variation of proton concentration considering all of driving forces. The total amount of proton ion increases during charge and decreases during discharge, and the maximum value of proton ions increases as the reaction progress. Since the charge time is longer than discharge time, the proton ions are continuously accumulated. The dominant driving forces to determine the proton concentration are

D.K. Kim, S.J. Yoon and S. Kim / International Journal of Heat and Mass Transfer 148 (2020) 119040

Fig. 5. Changes in vanadium ion concentration at negative electrode as a result of different driving forces during the charge and discharge reaction: (a) variations in vanadium ion concentration during the charge and discharge reaction; (b) changes in V2+ ion concentration due to self-discharge reaction and diffusion; (c) changes in V3+ ion concentration due to self-discharge reaction and diffusion.

flow rate and the main reaction, because the amount of changes in proton concentration by flow rate and the main reaction is 100 times larger than that by other driving forces. Fig 6(b) shows changes in proton ion concentration during the 3rd charge and discharge reaction. During charge process, about 8 mol/m3 of hydrogen are continuously generated, and about 7 mol/m3 of hydrogen are emitted by circulated flow between reservoir and the cell (right side axis in Fig 6(b)). Therefore, we can expect that the proton concentration increases during the charging process. The same phenomena occur in opposite direction during discharge reaction, so the amount of proton ions decreases. Furthermore, protons are consumed continuously because of the self-discharge reaction. The amount of self-discharge reaction, however, is changed during charge and discharge reaction. The amount of proton consumption is 0.018 mol/m3 at the end of charge process, however, that at the end of discharge process is 0.002 mol/m3 . Since the amount of required vanadium ions for self-discharge reaction are changed during the reaction, the amount of proton consumption due to self-discharge reaction is varied. Diffusion is another impor-

7

Fig. 6. Changes in proton concentration at positive electrode as a result of different driving forces during the charge and discharge reaction: (a) variations in proton concentration during charge and discharge reaction; (b) comparison of driving forces acting during the 3rd cycle; (c) comparison of driving forces acting during the 300th cycle.

tant driving force of the transport of protons during the charge and discharge cycles. The amount of proton transport across the membrane is negligible during the 3rd cycle. Because proton ion concentration at positive and negative electrode is almost the same at early stage. Next, charge imbalance should be considered as another cause of the proton transport. Since the transported vanadium ions have different charge numbers, protons are transported to match the charge balance between the two electrodes; the protons move from the positive electrode to the negative electrode because the charge number of vanadium ions on the positive electrode side is higher than that on the negative electrode side. However, the amount of proton transport across the membrane due to charge imbalance is negligible compared with other driving forces. It only affects proton transport of 0.001 mol/m3 . In Fig. 6(c), variation of proton concentration during the 300th cycle is presented. One noticeable change compared with early stage is the proton transport due to diffusion. As the reaction progressed, the proton

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D.K. Kim, S.J. Yoon and S. Kim / International Journal of Heat and Mass Transfer 148 (2020) 119040

Fig. 8. Changes in solution volume at positive electrode resulting from different driving forces during the charge and discharge reaction: (a) comparison of driving forces acting during the 3rd cycle; (b) comparison of driving forces acting during the 300th cycle.

Fig. 7. Changes in proton concentration at negative electrode as a result of different driving forces during the charge and discharge reaction: (a) variations in proton concentration during charge and discharge reaction; (b) comparison of driving forces acting during the 3rd cycle; (c) comparison of driving forces acting during the 300th cycle.

concentration at positive electrode is largely accumulated, so concentration difference between two electrodes become large. Therefore, proton transport due to diffusion increases significantly. The amount of proton transport is 0.0 0 02 mol m−3 during the 3rd cycle, but 0.018 mol/m3 during the 300th cycle. Changes in proton concentration at negative electrode are shown in Fig. 7(a). The trend is similar to the phenomena occurred in positive electrode. During the 3rd cycle, proton concentration increases during charge and decreases during discharge process. The proton concentration at the end of charge at negative electrode is relatively small compared with that at positive electrode. Since the number of protons associated with self-discharge reaction at negative electrode is large shown in Fig. 7(b), total amount of proton concentration at negative electrode is smaller than that at positive electrode. In addition, during the 300th cycles, the maximum value

of the concentration of protons increased significantly at the positive electrode, but slightly at the negative electrode: 1550 mol/m3 increase at positive electrode, and 400 mol/m3 increase at negative electrode. This is resulted from the proton transfer due to diffusion that is shown in Fig. 7(c). With charge-discharge cycles proceed, the difference of proton concentration become large. It is resulted in large amount of proton transfer across the membrane and complement the loss of proton due to self-discharge reaction. Therefore, the amount of proton increase is relatively small at negative electrode. Finally, we analyze the amount of water transported through the membrane in response to different driving forces at the positive electrode; the results are shown in Fig. 8. First, we know that the dominant driving force for water transport is osmosis in both the 3rd and 300th cycles. About 70% of the total volume of water is transported through the membrane. Furthermore, water molecules can be dragged by the diffusion of vanadium ions and protons. During the 3rd cycle, the amount of water transported by diffusion of vanadium ions is larger than that transported by the diffusion of proton. During the 300th cycle, however, the effect of proton transport is more significant than that of the transport of vanadium ions. Since the drag coefficient of vanadium ions is larger than that of protons, vanadium ions can carry a large amount of water at the beginning. However, during the 300th cycle, the difference between the concentration of protons at the negative electrode and positive electrode is higher than that the difference in

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Fig. 10. Changes in solution volume during long-term cycles of charge and discharge; the symbols represent the experimental results, and the lines represent the numerical results.

Fig. 9. Changes in solution volume at negative electrode resulting from different driving forces during the charge and discharge reaction: (a) comparison of driving forces acting during the 3rd cycle; (b) comparison of driving forces acting during the 300th cycle.

vanadium ion concentration. Therefore, a larger volume of water is transported by the protons than vanadium ions, which is the reason why the proton transport should be considered as a driving force of water transport. On the other hand, the amount of water generated by the self-discharge reaction is similar during the 3rd and the 300th cycle. Because the variations in ion concentration due to the self-discharge reaction are similar during the processes. Meanwhile, the amount of water transported due to the difference between the hydraulic pressure difference at the negative electrode and positive electrode is negligible. Changes in solution volume resulting from different driving forces at negative electrode are shown in Fig. 9. The effect of driving forces on the water transport at negative electrode is similar to that at positive electrode. At the 3rd cycle, the dominant driving forces of the water transport is osmosis and diffusion of vanadium ions, while at the 300th cycle, the dominant driving forces are the osmosis and diffusion of proton. The direction of water transport is determined by the difference of ions concentration accumulated in each electrode that is explained in the next section. 3.3. Species transport during long-term cycles of charge and discharge reaction Capacity loss is an important issue for long-term operation of the VFB. In this section, we present the results of analysis of the changes in solution volume and ion concentration during the longterm operation.

Fig. 11. Changes in ion concentration during the long-term cycles of charge and discharge at 1.0 V; the symbols represent the experimental results, and the lines represent the numerical results.

The changes in the solution volume at the negative electrode and the positive electrode during long-term operation are shown in Fig. 10, where the symbols represent the experimental results and the lines represent the numerical results. The solution volume at the positive electrode increases from 46 to 51.4 mL, and that at the negative electrode decreases from 46 to 38.2 mL, i.e., the decrease in solution volume at the negative electrode is larger than the increase in solution volume at the positive electrode. These results can be understood by examining the variation in ion concentration that are shown in Figs. 11 and 12. The Fig. 11 shows the variations in ion concentration at 1.0 V during long-term operation of the VFB. As the VFB continues to operate, the VO2+ concentration increases, but the V3+ concentration decreases. Since a large amount of V2+ ions are transported to the positive electrode due to crossover, the self-discharge reaction on the positive electrode side actively proceeds, and a large amount of VO2+ ions is produced and accumulated at the positive electrode. The consumption of V2+ ions at negative electrode leads to decrease of the maximum concentration of V3+ ions. Moreover, the amount of VO2 + ions decreases because reduced V2+ ion concentration limits the conversion of VO2 + ions during charging. These variations in vana-

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D.K. Kim, S.J. Yoon and S. Kim / International Journal of Heat and Mass Transfer 148 (2020) 119040

Fig. 13. Relationship between volume change at negative electrode, capacity loss, and SOC in the long-term cycles of charge and discharge reaction.

creases of the solution at the negative electrode, and an imbalance in the volume of the solution between the two electrodes occurs. 3.4. Prediction of capacity loss from the change in solution volume

Fig. 12. Changes in solution volume in response to different driving forces during the long-term cycles of the charge and discharge reaction: (a) changes in solution volume at the positive electrode; (b) changes in solution volume at the negative electrode.

dium ions result in capacity loss during long-term operation of the VFB. On the other hand, we already know that the concentration of protons increases as VFB continues to operate due to proton generation during charging process, as explained in the previous section. Therefore, the total amount of ions in the positive electrode halfcell increases with cycles. The changes in ion concentration lead to water transport in the long-term operation of the VFB shown in Fig. 12. Since the total ion concentration at the positive electrode side increases more than that at the negative electrode side, the water transport by osmosis is headed toward the positive electrode side. Water molecules that are dragged by diffused ions move towards the opposite direction as that caused by osmosis. On the other hand, the amount of water generated by the selfdischarge reaction (see Eqs. (10) to (15)) is higher at negative electrode than that at the positive electrode and the effect of the difference in hydraulic pressure is negligible. Except for the self-discharge reaction, the amount of water transport due to each reaction occurs symmetrically. Nevertheless, the imbalance between volume increase of the solution at the positive electrode and volume decrease of the solution at the negative electrode is due to the water decomposition reaction occurring during the charging process at the positive electrode. Once water is transported from negative electrode to positive electrode due to osmosis, some of water transported to the positive electrode is consumed during charge process. Thus, the volume increase of the solution at the positive electrode is smaller than the volume de-

In the previous section, we discussed the changes in solution volume caused by ion transport. In this section, we will show how the SOC and capacity loss can be predicted by measuring the changes in solution volume at the negative electrode. The relationship between the ratio of volume change at the negative electrode, capacity loss, and SOC are shown in Fig. 13. We used the anolyte to measure the changes in solution volume because the changes are large at the negative electrode. The measurement showed that the variations in capacity and SOC were inversely proportional to the change in solution volume. While the solution volume was changed by 4%, the capacity decreased by 30%, and the SOC dropped from 84% to 76%. Moreover, the SOC and capacity loss decreased continuously as the solution volume changed. Using the relationship between change in solution volume and SOC, we suggested the following empirical equations to predict the variation in SOC when the temperature is about 20 °C:

SOC = −3 ln(rvol,an ) + 80.4 rvol,an

V = nth Vinitial

(43) (44)

where, rvol, an is the volume ratio during the initial cycle, Vinitial is the volume during the initial cycle and Vnth is the volume during the nth cycle. Using these equations, we can easily predict the changes in SOC of the VFB during long-term operation. 4. Conclusions The transport phenomena in an all vanadium redox flow battery were analyzed to understand the relationship between changes in solution volume and capacity loss. We first examined the driving forces associated with the transport of vanadium ions, protons, and water molecules. To explain the water transport through the membrane, we developed model considering diffusion, self-discharge reaction, osmosis, and the difference in hydraulic pressure at the negative electrode and positive electrode. Using this model, we examined the variation in ion concentration during the 3rd and 300th cycles. As the reaction progresses, the charge and discharge

D.K. Kim, S.J. Yoon and S. Kim / International Journal of Heat and Mass Transfer 148 (2020) 119040

time decreases owing to the imbalance in vanadium ion concentration. The concentration of VO2+ ions and protons at the positive electrode increase with continued operation, but the respective concentration of the other ions decreases. Furthermore, we examined the changes in solution volume in the long-term cycles of charge and discharge. The solution volume increases on the positive electrode side. Since large number of protons accumulated on the positive electrode side during a charge-discharge reaction, water molecules head toward the positive electrode side. Finally, we examined the relationship between capacity loss and changes in the solution volume. As the solution volume decreases, the charge capacity decrease; the capacity and SOC decrease drastically until the volume decrease reaches 4%. Using this relationship, an empirical equation is suggested for predicting the changes in SOC during long-term operation of the VFB by simple measurement. These results can contribute to accelerate commercialization of VFB system. Declaration of Competing Interest None. Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Science and ICT (NRF-2019R1F1A1058036). Additional support was provided by basic research project funded by Korea Institute of Science and Technology Europe and ‘GO-KRICT’ project funded by Korea Research Institute of Chemical Technology. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijheatmasstransfer. 2019.119040. References [1] A.Z. Weber, M.M. Mench, J.P. Meyers, P.N. Ross, J.T. Gostick, Q. Liu, Redox flow batteries: a review, J. Appl. Electrochem. 41 (2011) 1137–1164. [2] X. Ma, H. Zhang, C. Sun, Y. Zou, T. Zhang, An optimal strategy of electrolyte flow rate for vanadium redox flow battery., J. Power Sources 203 (2012) 153–158. [3] A. Tang, S. Ting, J. Bao, M. Skyllas-Kazacos, Thermal modelling and simulation of the all-vanadium redox flow battery, J. Power Sources 203 (2012) 165–176. [4] J. Marschewski, P. Ruch, N. Ebejer, O.H. Kanan, G. Lhermitte, Q. Cabrol, B. Michel, D. Poulikakos, On the mass transfer performance enhancement of membraneless redox flow cells with mixing promoters, Int. J. Heat Mass Trans 106 (2017) 884–894. [5] B.W. Zhang, Y. Lei, B.F. Bai, T.S. Zhao, A two dimensional model for the design of low fields in vanadium redox flow batteries, Int. J. Heat Mass Trans 135 (2019) 460–469. [6] S.J. Yoon, S. Kim, D.K. Kim, Optimization of local porosity in the electrode as an advance channel for all-vanadium redox flow battery, Energy 172 (2019) 26–35.

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