A vanadium redox flow battery model incorporating the effect of ion concentrations on ion mobility

Applied Energy 158 (2015) 157–166

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

A vanadium redox flow battery model incorporating the effect of ion concentrations on ion mobility X.L. Zhou, T.S. Zhao ⇑, L. An, Y.K. Zeng, X.H. Yan Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Special Administrative Region

h i g h l i g h t s
An ion concentration-dependent mobility is proposed.
A VRFB model incorporating the proposed mobility is developed.
The model enables a more realistic simulation of cell performance.
Numerical results show that thinning conventional electrodes improves performance.

a r t i c l e

i n f o

Article history: Received 18 May 2015 Received in revised form 14 July 2015 Accepted 12 August 2015

Keywords: Flow battery Ion mobility Numerical modeling

a b s t r a c t Previous vanadium redox flow battery models (VRFB) use the ion mobility deduced from the ion diffusivity measured at low ion concentrations, resulting in an overestimation of the ionic conductivity in VRFBs that virtually operate at much higher ion concentrations. To address this issue, we propose to use the Stokes–Einstein relationship to determine an ion concentration-dependent ion mobility. A twodimensional, transient model that incorporates the effect of ion concentrations on ion mobility is developed for VRFBs. It is shown that the present model results in: (i) a more accurate estimation of ionic conductivity, (ii) a more accurate prediction of cell voltage particularly at high current densities, and (iii) a more realistic simulation of the concentration distributions and local current density distributions in the electrodes. Finally, the model is applied to the study of the effects of important electrode design parameters and operating conditions on cell performance. It is found that the local current density, being distributed across the electrode in a manner opposite to that predicted by previous models, is much lower at the current collector side than that at the membrane side. This fact suggests that the region away from the membrane is not well utilized in conventional electrodes, thus a thinner electrode is preferred. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Redox flow batteries are energy storage systems that are designed for use in large-scale applications such as peak load leveling and effective utilization of intermittent renewable energy sources [1–4]. In particular, the all-vanadium redox flow battery (VRFB) has been regarded as the most promising technology, primarily because it uses the same element in both half-cells, which avoids cross-contamination between the two half-cell electrolytes [5,6]. Although promising, issues with VRFBs, including a low power density on a system level, ion crossover through the polymer membrane, and corrosion of current collectors, must be addressed before the widespread commercialization of this type of technology. To address these issues, efforts have been devoted ⇑ Corresponding author. Tel.: +852 2358 8647. E-mail address: [email protected] (T.S. Zhao). http://dx.doi.org/10.1016/j.apenergy.2015.08.028 0306-2619/Ó 2015 Elsevier Ltd. All rights reserved.

to developing optimal materials for electrodes, ion exchange membranes and current collectors [7–11]. Moreover, electrode optimization was found to be critical in improving cell performance. Recently, Aaron et al. [12,13] experimentally demonstrated that the optimization of electrode thickness heavily relies on making a tread-off between the reaction surface area and charge transport resistances that result from the ion transport and electron transport processes. In addition to experimental investigations, numerical modeling also plays an important role in improving and optimizing the performance of VRFBs. Previous numerical investigations on VRFBs mainly focused on developing robust formulations to describe the transport phenomena occurring in VRFBs [14–29], and little attention was focused on determining accurate transport properties. However, to ensure that numerical models provide meaningful insight into understanding these complex phenomena and optimizing engineering design and operation of a VRFB, accurate

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Nomenclature a c d D F Hcell I j k± km N Nm p R S t T u ~ v U0 U 00 V wcell z

specific surface area of porous electrode, m1 concentration, mol m3 thickness, m diffusivity, m2 s1 Faraday’s constant, C mol1 length of the electrode applied current density, A m2 local current density, A m2 reaction rate constant, m s1 mass transfer coefficient, s1 superficial molar flux, mol m2 s1 MacMullin number pressure, Pa universal gas constant, J mol1 K1 source term time, s temperature, K ion mobility, mol s1 kg1 superficial velocity, m s1 equilibrium potential, V standard equilibrium potential, V electrolyte volume in half-cell tank, m3 width of the electrode valence

transport properties are needed, in addition to a robust formulation. One of the key transport properties is the mobility that is required to model the ion transport process in the framework of dilute solution theory. To date, most models for VRFBs simply adopt ion mobility deduced from ion diffusivity measured at low ion concentration solution and assume that ion mobility is independent of ion concentration. However, there is an intricate relationship between the ion mobility in the electrolyte of VRFBs and the ion concentration. With an increase in the ion concentrations, the ion mobility declines dramatically as a result of higher viscosity. A critical issue seen in previous models is the effects of this relationship are missing, resulting in an overestimation in the ionic conductivity of VRFBs that operate at much higher ion concentrations. Specifically, the ionic conductivity predicted by the previous model is around 200 S m1 [23], while the actual value is closer to 10–40 S m1 [30]. The margin of errors in the estimation is over 500%. As a consequence, the error associated with the prediction of the ionic conductivity affects the accuracy of the predicted cell voltage and distribution of concentrations and local current density, especially at high current densities. Thus, to obtain more meaningful insight in the underlying physical processes and predicting cell performance, an accurate correlation between ion mobility and ion concentration is necessary. Since the electrolytes of VRFBs contain 4 or 5 charge ions, determining the link between these two properties by experimental measurements is a grueling task. A more straightforward method must be developed in securing more accurate and meaningful information on the relationship between ion concentration and ion mobility in the electrolyte of VRFBs. The Stokes–Einstein relationship, which describes the relationship between ion diffusivity, dynamic viscosity and temperature, is typically used to characterize the temperature dependence of ion diffusivity and ion mobility in dilute solution theory [31]. It can also be adopted to estimate the concentration dependence of ion diffusivity and mobility and has been applied into models of electrochemical systems such as lithium ion batteries [32]. Based on the dilute solution assumption, we propose to use the

Greek

q l e j r

/

g a x

density, kg m3 viscosity, kg m1 s1 porosity permeability, m2 conductivity, S m1 potential, V overpotential, V charge transfer coefficient volumetric flow rate, m3 s1

Superscripts and subscripts e electrolyte eff effective i species in inlet m membrane out outlet s solid V(II) V2+ V(III) V3+ V(IV) VO2+ V(V) VOþ 2

Stokes–Einstein relationship to determine the ion concentrationdependent ion mobility in electrolytes of VRFBs. A twodimensional, transient model that incorporates the proposed ion motility is developed. In addition, the effects of applied current densities and electrode thickness on the cell performance are examined using the proposed model. The formulation of the model is detailed in Sections 2 and 3, followed by the results and discussion in Section 4 and the summary of main findings in Section 5. 2. The ion concentration dependence of ion mobility and diffusivity Based on the framework of dilute solution theory, mobility, which is usually deduced form diffusivity measured by experiment, is a key transport property that describes the ion transport process. The ion mobility heavily depends on the electrolyte components and their respective concentrations. Namely, the mobility of a certain ion is the function of the concentrations of ions in the electrolyte. However, the composition of the electrolyte in VRFBs is very complex as it contains 4 or 5 charged ions. Thus, it is difficult to correlate the ion mobility directly with the concentrations of the charged ions. Thus, we propose to correlate the ion mobility indirectly with ion concentrations by using the Stokes–Einstein relationship, as depicted in Fig. 1a. The Stokes–Einstein relationship can be expressed as [31]:

ri ¼

RT 6pDi g

ð1Þ

where ri is the radius of the charged ion i, Di is the diffusivity of ion i,

g is the dynamic viscosity, R is the universal gas constant, and T is the operating temperature. According to Nernst–Einstein equation, the ionic mobility of species i can be calculated by:

ui ¼

Di RT

where ui is the ion mobility.

ð2Þ

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Fig. 1. (a) Illustration of the approach for relating the ion mobility to ion concentrations; (b) schematic of the VRFB system and the computational domain.

Combining Eqs. (1) and (2), the relationship between mobility and dynamic viscosity can be expressed as:

ri ¼

1 6pui g

ð3Þ

where ri is the radius of the charged ion i. Based on the dilute solution assumption, Eqs. (1) and (2) are valid for electrolytes in VRFBs. In this model, we assumed that the radii of the ions are kept constant during change in ion concentrations, neglecting the metal ion complexing in the strong acid solution, which can be partially validated in [32]. Thus, Eqs. (2) and (3) give:

ui ¼

u0i g0

g

¼

D0i g0 RT g

ð4Þ

Table 1 Electrolyte properties. Parameter

Value

Density of the electrolyte Dynamic viscosity of the electrolyte [33] Initial V(II) diffusion coefficient in the electrode, DV2 [15] Initial V(III) diffusion coefficient in the electrode, DV3 [15] Initial V(IV) diffusion coefficient in the electrode, DV4 [15] Initial V(V) diffusion coefficient in the electrode, DV5 [15] Initial H+ diffusion coefficient in the electrode, D+H [15] Initial SO2 4 diffusion coefficient in the electrode, DSO2 [15]

1400 kg m3 8.876 Pa s 2.4  1010 m2 s1

Initial dynamic viscosity, g0

1 Pa s

2.4  1010 m2 s1 3.9  1010 m2 s1 3.9  1010 m2 s1 9.312  1010 m2 s1 1.065  1010 m2 s1

4

where g0 is the initial experimental determined values of dynamic viscosity and D0i is the initial experimental determined values of dynamic viscosity and ion mobility, which are provided in Table 1. For electrolytes with given ion concentrations, the dynamic viscosity g can be obtained easily by experimental measurements or numerical simulations. Combining the obtained dynamic viscosity

g, the initial experimental determined values of dynamic viscosity g0 , and ion diffusivity D0i , the mobility of a certain ion ui in the electrolyte with given ion concentrations can be calculated by Eq. (4).

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In this model, the dynamic viscosity used is obtained from the experimental data [33,34]. The modeling of the viscosity of the electrolytes can be found in [35].

Table 3 Geometric parameters, material properties and operating conditions.

3. Mathematical model Fig. 1b shows the schematic of the computational domain, which includes two porous electrodes separated by a membrane and two current collectors. In the present model, several simplifications and assumptions are made: (1) Variations in z-direction (the direction perpendicular to x–y plane in Fig. 1b are neglected. (2) The fluid flow in the electrodes is assumed to be incompressible. (3) The non-isothermal condition is not taken into account. (4) Side reactions, such as hydrogen evolution and oxygen evolution, are not taken into consideration. (5) The membrane is impermeable to all ions and water, except for protons. (6) Both the first and second steps of the dissociation of sulfuric acid are fully complete. 3.1. Governing equations 3.1.1. In the porous electrode In each half-cell, the electrolyte which contains water and charged ions (including VO+2, VO2+, V2+, V3+, H+ and SO2 4 ) flows through the porous electrode. The mass conservation of charged ions, except for SO2 4 , can be expressed as:

! @ eci þ r  N i ¼ Si @t

ð5Þ

! where ci is the concentration of species i, N i is the flux of the species, Si is the source term, which is given in Table 2, and e represents the electrode porosity, which is given in Table 3 along with other geometric parameters, material properties and operating conditions. The concentration of SO2 can be calculated according to 4 the electro-neutrality condition:

X zi c i ¼ 0

ð6Þ

i

where zi is the valence of species i. The concentration flux of each charged species in Eq. (5) can be calculated via the Nernst–Planck equation:

! eff ~ N i ¼ Deff i rc i  zi ui c i F r/e þ v c i

ð7Þ

where F is the Faradaic constant, /e is the ionic potential in the eff electrolyte, and Deff i and ui are the effective diffusivity and effective diffusivity of species i, which can be calculated by:

Deff i ¼

Di NM

ð8aÞ

ueff i ¼

ui NM

ð8bÞ

*

Parameter

Value

Length of cell, Hcell Width of cell, wcell Thickness of the electrode, Le Thickness of the membrane, Lm Porosity of the electrode [36], e Specific surface of electrode*, a Hydraulic permeability of electrode [37], j Electronic conductivity of electrode [38], rs Operating temperature, T Ionic conductivity of membrane [39], rm Total contact resistance [38], Rcon

0.026 m 0.026 m 0.0036 m 0.000125 m 0.883 1.75  105 m2/m3 6.0  1011 220 S m1 300 K 5.5 S m1 74 mX cm2

Fitted parameter.

where Di is the diffusivity of species i in the bulk electrolyte, ui is the mobility of species i in the bulk electrolyte, and NM is the MacMullin number. The bulk velocity ~ v can be calculated using Darcy’s law:

~ v¼

j rp l

ð9Þ

where j represents the permeability of the porous electrode, l is the dynamic viscosity of the electrolyte, and p is the liquid pressure, which can be determined from an overall mass balance as:

r  ðq~ vÞ ¼ 0

ð10Þ

where q denotes the density of the electrolyte, which is given in Table 3. The conservation of charge can be expressed as:

r ~ie ¼ r ~is ¼ j

ð11Þ

where j donates the local current density, ~ ie and ~is represent the ionic current density and electronic current density, respectively, and can be given as follows:

~ie ¼ F

X ! zi N i

ð12Þ

i

~is ¼ rs r/ s

ð13Þ

where /s is the electronic potential in the solid phase, and rs is the electronic conductivity. The local current density is defined by the Butler–Volmer equations, which are given in Eq. (11) for both the negative (‘‘”) and positive (‘‘+”) electrodes:

j ¼ aFk ðcVðIIÞ Þð1a Þ ðcVðIIIÞ Þa

 s  cVðIIÞ

cVðIIÞ  s    cVðIIIÞ a F g exp   cVðIIIÞ RT

jþ ¼ aFkþ ðcVðIVÞ Þð1aþ Þ ðcVðVÞ Þaþ

exp

 s  cVðIVÞ

 s    cVðVÞ aþ F gþ exp   cVðVÞ RT

cVðIVÞ

  ð1  a ÞF g RT ð14aÞ

exp

  ð1  aþ ÞF gþ RT ð14bÞ

where a represents the specific surface area of the porous electrode; k is the reaction rate constant; a denotes the charge transfer coefficient, and g is the overpotential expressed as:

Table 2 Source terms of the governing equations. Term

Positive electrode

Negative electrode

V(II) concentration equation, SV(II) V(III) concentration equation, SV(III) V(IV) concentration equation, SV(IV) V(V) concentration equation, SV(V) Proton concentration equation, S+H

– – j/F j/F 2j/F

j/F j/F – – –

gj ¼ /s  /e  U 0;j

ð15Þ

where U0,j is the equilibrium potential in each half-cell given as:

U 0; ¼ U 00; þ

  cVðIIIÞ RT ln F cVðIVÞ

ð16aÞ

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X.L. Zhou et al. / Applied Energy 158 (2015) 157–166

U 0;þ ¼ U 00;þ þ

cVðVÞ  ðcþ;Hþ Þ RT ln F cVðIVÞ

! 2 ð16bÞ

where U 00; and U 00;þ are the standard equilibrium potentials for the negative and positive redox couple. The electro-kinetic parameters are listed in Table 4. In the above Butler–Volmer equations (Eq. (14)), the term csi represents the concentration of vanadium species i at the liquid– solid interface, which can be correlated to the bulk concentrations ci by balancing the electrochemical reaction rate with the rate of mass transfer of the reactants to (or from) the electrode surface. For the positive half-cell, the balance is:

 s    cVðIVÞ km cVðIVÞ  csVðIVÞ ¼ Fkþ ðcVðIVÞ Þð1aþ Þ ðcVðVÞ Þaþ cVðIVÞ    s    cVðVÞ ð1  aþ ÞF gþ aþ F gþ   exp exp RT cVðVÞ RT ð17aÞ km ðcVðVÞ  csVðVÞ Þ ¼ Fkþ ðcVðIVÞ Þð1aþ Þ ðcVðVÞ Þaþ



csVðVÞ



! Ni ~ n ¼ 0;

i ¼ VðIIÞ; VðIIIÞ; Hþ ; HSO4

ð22Þ

At x ¼ x1 , the Donna effects are considered for the electrolyte potential.

/m ¼ /e þ

  cm;Hþ RT ln F ce;Hþ

! n ¼ 0; Ni ~

ð23Þ

i ¼ VðIIÞ; VðIIIÞ; HSO4

ð24Þ

~ie  ~ n ¼ ~im  ~ n

ð25Þ

At x ¼ x2 , the interfacial conditions are similar to those at x ¼ x1 ,

/m ¼ /e þ

  cm;Hþ RT ln F ce;Hþ

! n ¼ 0; Ni ~

ð26Þ

i ¼ VðIVÞ; VðVÞ; HSO4

ð27Þ

~ie  ~ n ¼ ~im  ~ n

cVðVÞ    s    cVðIVÞ aþ F gþ ð1  aþ ÞF gþ exp  exp  RT cVðIVÞ RT ð17bÞ

where km is the pore level mass transport coefficient, which can be defined as [36]:

km ¼ 1:6  104~ v 0:4

ð18Þ

ð28Þ

At x ¼ x3 ,

@/e ¼ 0; @x

rs

@p ¼ 0; @x

@p ¼0 @y

! Ni ~ n ¼ 0;

@/s ¼I @x

ð29Þ ð30Þ

i ¼ VðIVÞ; VðVÞ; Hþ ; HSO4

ð31Þ

At y ¼ 0, 3.1.2. In the membrane Since protons are the only charge carrier in the membrane, the conservation of current gives:

@/e ¼ 0; @y

! rm N Hþ ¼  r/m F

ci ¼ cin i ;

ð19Þ

where rm is the conductivity of the membrane and /m is the ionic potential of the membrane. In addition, the Donna effects are considered in the interfacial boundary (the membrane/electrolyte interface) conditions [23]. 3.2. Boundary conditions The boundary conditions can be specified by referring to Fig. 1b as following: At x ¼ 0, the non-slip boundary condition is adopted for the velocity.

@/e ¼ 0; @x @p ¼ 0; @x

/s ¼ 0

ð20Þ

ð32Þ

i ¼ VðIIÞ; VðIIIÞ; Hþ ; HSO4

i ¼ VðIVÞ; VðVÞ; Hþ ; HSO4

when 0 < x < x1

when x2 < x < x3

~ v  ~n ¼ v in

ð33Þ ð34Þ ð35Þ

At y ¼ Hcell ,

@/e ¼ 0; @y @ci ¼ 0; @x

@/s ¼0 @y

ð36Þ

i ¼ VðIIÞ; VðIIIÞ; Hþ ; HSO4

i ¼ VðIVÞ; VðVÞ; Hþ ; HSO4

when 0 < x < x1

when x2 < x < x3

p ¼ pout @p ¼0 @y

ð37Þ ð38Þ ð39Þ

ð21Þ 3.3. Inlet concentrations, electrolyte tanks and initial values

Table 4 Kinetic parameters. Parameter

Value

Reaction constant for negative reaction, k [19] Reaction constant for positive reaction, k+ [19] Negative charge transfer coefficient, a [23] Positive charge transfer coefficient, a+ [23] Standard reduction potential at negative electrode, U 00;

6.8  107 m s1 1.7  107 m s1 0.45 0.55 0.255 V

[15] Standard reduction potential at positive electrode, U 00;þ

1.004 V

[15]

@/s ¼0 @y

During the VRFB operation, the concentrations of the species in the electrolyte tanks change due to the electrochemical reactions in both the positive and negative half-cells. In order to capture changes in species concentrations, the inlet concentration for each species can be calculated based on the mass conservation:

@cin ewcell i ¼ @t V tank

Z

v out cout i dl 

Z

v in cini dl

 0 cin i ð0Þ ¼ c i

ð40Þ

where Vtank is the volume of tank. The superscript in and out refer to the value at the inlet or outlet of the electrode. The initial value c0i is given in Table 5. The state of charge (SOC) is defined as:

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X.L. Zhou et al. / Applied Energy 158 (2015) 157–166 Table 5 Initial species concentrations. Parameter

Value at 10% SOC (mol m3)

Initial V2+ concentration in negative electrolyte, c0VðIIÞ

150

250

Initial V3+ concentration in negative electrolyte, c0VðIIIÞ

1350

1250

Initial VO2+ concentration in positive electrolyte, c0VðIVÞ

1350

1250

Initial VO+ concentration in positive electrolyte, c0VðVÞ

150

250

Initial H+ concentration in positive electrolyte, c0Hþ ;pos

6150

6250

Initial H+ concentration in negative electrolyte, c0Hþ ;neg Initial

SO2 4

Initial

SO2 4

Negative electrolyte : Positive electrolyte :

4650

4750

concentration in positive electrolyte,

c0SO2 ;pos 4

4500

4500

concentration in negative electrolyte,

c0SO2 ;neg 4

4500

4500

SOC ¼ SOC ¼

cVðIIÞ cVðIIÞ þ cVðIIIÞ

cVðVÞ cVðIVÞ þ cVðVÞ

ð41aÞ ð41bÞ

3.4. Cell voltage With a given applied current density I, the cell voltage can be derived from:

R

V cell ¼

Value at 16.7% SOC (mol m3)

/s dy þ I  Rcon hcell

ðx ¼ x3 Þ

ð42Þ

where the Rcon is the contact resistance at the interface between the current collector and the porous electrode. 4. Results and discussion The variables in the conservation equations were iteratively solved based on the finite-element method. The relative error tolerance was set to 1.0E6. 4.1. Model validation 4.1.1. Ionic conductivity The first part of validation is to compare the simulated ionic conductivity with the experimental data. For choices, we validate the bulk ionic conductivity of the positive electrolyte, which is composed of 2 M vanadium salt and 3 M sulfuric acid. With the dynamic viscosity data from [33] and [34], we correlated the viscosity with state of charge (SOC) linearly:

l ¼ 14:08  4:81  SOC ðmPa sÞ

4.1.2. Polarization curve and charge–discharge curve The second part of validation is to compare the simulated polarization curve and charge–discharge curve with the experimental data reported in [40], as shown in Fig. 3. The simulation was conducted under the same conditions as the experiment: the initial electrolyte concentrations of vanadium salt and sulfuric acid is 1.5 M and 3 M, corresponding to a dynamic viscosity of 8.876 mPa s [33]; the electrodes of both positive and negative electrodes are carbon felts with an original thickness of 1 mm and the compression ratio 20%. Fig. 3a shows that the result of the model with ion concentration-dependent mobility agrees fairly well with the experimental data (less than 2%) while the result of the model without ion concentration-dependent mobility shows a large discrepancy with the experimental data, which is more pronounced at high current densities. It should be noted that the specific area and reaction rate constant are fitted parameters due to the lack of experiment data. However, these parameters affect the predicted cell voltage mainly at low current density regions. It is still reasonable to conclude that the present model captures charge transport in the electrode accurately, which governs the performance at higher current density regions. In addition, the charge– discharge performance at the applied current density of 80 mA cm2 was also simulated and compared with experimental data, as shown in Fig. 3b. It can also be seen that the model result shows a good agreement with the experimental data. It should be pointed out that the starting and ending SOC of the simulation were fitted to match the starting and ending charge–discharge voltage.

ð43Þ

The conductivity of the bulk electrolyte can be calculated by:

re ¼

F2 X 2 z Di c i RT i i

ð44Þ

A comparison between the simulated ionic conductivity of positive electrolyte by the present model and the experimental data from [30] is shown in Fig. 2. With the changes of state of charge, only an average margin of error of 5% is observed for the model with ion concentration-dependent mobility, while the model without ion concentration-dependent mobility gives an average margin of error of around 1000%. The agreement in predicted conductivities by the present model and the experimental data indicates that the proposed approach offers a substantially more accurate capture of the effects of ion concentrations on the ion mobility. It should be pointed out that the proposed ion concentrationdependent ion mobility can be also applicable to other flow batteries, such as Fe–Cr flow batteries and Fe–V flow batteries.

Fig. 2. Comparison between the simulated and measured ionic conductivities for the positive electrolyte (the electrolyte contains 2 M vanadium ions and 3 M sulfuric acid).

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X.L. Zhou et al. / Applied Energy 158 (2015) 157–166 1.8

1.6 [39]

Experiment results Model with ion concentration-dependent mobility Model without ion concentration-dependent mobility

1.6

1.5

Voltage (V)

Volatge (V)

1.4

1.2

1.4

1.3

1.0 1.2 0.8 Model without ion concentration-dependent mobility Model with ion concentration-dependent mobility

1.1 0.6 0

2000

4000

6000

8000

10000

12000

0

Current density (A m-2)

2000

4000

6000

8000

10000

12000

14000

Time (s)

(a)

(a)

1.6

1.5 1.6

1.3

Voltage (V)

Voltage (V)

1.4

1.2

1.4

1.2

1.1 Model with ion concentration-dependent mobility Experiment - discharge [39] Experiment- charge [39]

1.0 0.0

0.2

0.4

0.6

0.8

1.0

Model without ion concentration-dependent mobility Model with ion concentration-dependent mobility

1.0

Normalized capacity

(b) Fig. 3. (a) Comparison between the simulated and measured polarization curves (the inlet SOC for both positive and negative electrolytes are set to be 0.33% and the flow rate is 15 ml min1); (b) comparison between the simulated and measured charge–discharge curves at the current density of 80 mA cm2.

4.2. Effect of applied current density In this section, the effects of applied current density on battery voltage are investigated as well as distributions of concentration and local current density, at the initial vanadium concentration in each electrode of 1.5 M and sulfuric acid concentration of 3 M, corresponding to a dynamic viscosity of 8.876 mPa s. A comparison of models with and without ion concentration-dependent mobility is also presented. Fig. 4 shows the simulated charge–discharge curves predicted by models with and without ion concentration-dependent mobility at the current densities of 100 mA cm2 and 200 mA cm2. The initial SOC for both positive and negative electrolyte is set to be 10%. A noticeable discrepancy in the predicted cell voltage by these two models can be observed. Moreover, this discrepancy increases with an increase in applied current density and a similar trend can be observed in the polarization curve. Specifically, the difference in predicted cell voltages at 100 mA cm2 should be around 50 mV and this value increases to around 100 mV at 200 mA cm2, which is nearly 8% of the total voltage. The discrepancy is attributed to the difference in estimations of the electrolyte conductivity by the two models, which results in a difference in the estimation of the ohmic overpotential as well as the cell voltage.

0

1000

2000

3000

4000

5000

6000

7000

Time (s)

(b) Fig. 4. (a) The simulated charge–discharge curves at the current density of (a) 100 mA cm2 and (b) 200 mA cm2 (the inlet SOC for both positive and negative electrolytes are set to be 10%).

Figs. 5 and 6 show the comparisons of concentration distribution and local current density distribution at the current densities of 100 mA cm2 and 200 mA cm2, respectively. The distribution of local current density predicted by the present model shows that the electrochemical reaction rates at the area close to the membrane are much higher than elsewhere. A likely explanation is that the electronic conductivity in this model is 220 S m1, which is substantially higher than the effective ionic conductivity (20 S m1), which means that the ion transport process, rather than electron transport, is the limiting step of the electrochemical reaction. More importantly, the local current density distribution shows that only about 1/4 of the electrode has been utilized as a result of low ionic conductivity, indicating that the remains contribute little to the electrochemical reactions but introduce large ohmic resistance. These results suggest that a thinner electrode design will improve the cell performance, especially at high current densities. 4.3. Effect of the electrode thickness In this section, the effects of electrode thickness on cell voltage as well as the distributions of concentration and local current density will be investigated at the current density of 200 mA cm2. The

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X.L. Zhou et al. / Applied Energy 158 (2015) 157–166

0

-2

Local current density (A m-2)

Local current density (A m-2)

0

-4 -6 -8 -10 Model without ion concentration-dependent mobility Model with ion concentration-dependent mobility

-12 0.0

0.2

0.4

0.6

0.8

-2 -4 -6 -8 -10 Model with ion concentration-dependent mobility Model without ion concentration-dependent mobility

-12 0.0

1.0

0.2

0.4

0.8

1.0

(a)

(a) 0

810

-2

Concentration (mol m-3)

Local current density (A m-2)

0.6

X/Le

X/Le

-4 -6 -8 -10

Model with ion concentration-dependent mobility Model without ion concentration-dependent mobility

800 790 780 770 760

Model without ion concentration-dependent mobility Model with ion concentration-dependent mobility

-12 0.0

0.2

0.4

0.6

0.8

750 1.0

0.0

0.2

0.4

X/Le

(b)

initial vanadium concentration and sulfuric acid concentration in each electrode are set to be 1.5 M and 3 M, respectively. Fig. 7 shows the simulated charge–discharge curves of the VRFB with different electrode thicknesses. When the electrode thickness is reduced from 3.6 mm to 0.8 mm (around 1/4 of the initial electrode thickness), an observed drop in overpotential (around 50 mV) can be found. To this end, the improvement in performance caused by a reduction in the electrode thickness is comparable to that of systems involving high-end catalysts. This can be explained by the fact that a thinner electrode reduces ohmic resistance and therefore less ohmic loss, as shown in Table 6. However, if the electrode thickness is further reduced to 0.4 mm, the battery performance is compromised as an increase in activation overpotential is seen. Therefore, by comparing the charge–discharge curves of the battery with different electrode thicknesses, an optimized electrode thickness of 0.8 mm is obtained in this study, a result which agrees well with previous experimental data [12,13,40]. To reveal the effect of electrode thickness, the concentration distributions and the local current density distributions of the VRFB with different electrode thicknesses are plotted in Fig. 8. It is seen that the uniformity of local current density distributions increases with a decrease in the electrode thickness due to a reduced charge transport pathway, leading to better electrode utilization. However, the magnitude of the local current density

0.8

1.0

(b) Fig. 6. Distributions of (a) V(II) concentration and (b) local current density during charging at 50% SOC at y = 0.5 Hcell. Data taken from simulation at the current density of 200 mA cm2.

1.6

1.5

Voltage (V)

Fig. 5. Distributions of (a) V(II) concentration and (b) magnitude of local current density during charging at 50% SOC at y = 0.5 Hcell. Data taken from simulation at the current density of 100 mA cm2.

0.6

X/Le

1.4

1.3

1.2 Electrode thickness: 0.4mm Electrode thickness: 0.8mm Electrode thickness: 3.6mm

1.1

1.0 0

1000

2000

3000

4000

5000

Time (s) Fig. 7. Simulated charge–discharge curves for the electrodes with different thicknesses at the current density of 200 mA cm2. The inlet SOC for both the positive and negative electrolytes are set to be 16.7%.

increases with decreased electrode thickness, which indicates higher activation overpotential. In addition, the concentration distributions display a similar trend with the local current density

X.L. Zhou et al. / Applied Energy 158 (2015) 157–166 Table 6 Overpotential analysis of VRFBs with different electrode thicknesses at SOC of 50% at the current density of 200 mA cm2. Electrode thickness (mm)

Activation overpotential (mV)

Ohmic overpotential (mV)

Concentration overpotential (mV)

3.6 0.8 0.4

38.2 51.3 68.4

120.5 67.5 58.7

1.3 1.8 2.5

Local current density (A m-2 )

-4

-8

-12

-16

Acknowledgement

Electrode thickness: 0.8mm Electrode thickness: 0.4mm Electrode thickness: 3.6mm -20 0.2

0.4

densities, and (iii) a more realistic simulation of the distributions of concentration and local current density in the electrodes. In addition, the effects of the applied current density and the electrode thickness on cell performance are examined with the present model. The main findings are summarized as follows: (1) The differences in predicted cell voltages and distributions of local current density and concentrations by models with and without ion concentration-dependent ion mobility increase with the increased applied current density, indicating that the model without ion concentration-dependent mobility shows limited ability in the prediction of battery behavior at high current densities. (2) It is found that the local current density, being distributed across the electrode in a manner opposite to that predicted by previous models, is much lower at the current collector side than that at the membrane side. This finding suggests the region away from the membrane cannot be wellutilized in conventional electrodes made of thick graphite felt, thus a thinner electrode is preferred.

0

0.0

165

0.6

0.8

1.0

X/Le

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. 623313).

(a)

References 810

Electrode thickness: 0.8mm Electrode thickness: 0.4mm Electrode thickness: 3.6mm

Concentration (mol m-3)

800 790 780 770 760 750 0.0

0.2

0.4

0.6

0.8

1.0

X/Le

(b) Fig. 8. Distributions of (a) V(II) concentration and (b) local current density of VRFBs with different electrode thicknesses during charging at 50% SOC at y = 0.5 Hcell. Data taken from simulation at the current density of 200 mA cm2.

distribution since the local current density determines the rate of consumption or production of active species.

5. Concluding remarks In this work, we have proposed ion concentration-dependent ion mobility for VRFBs. A two-dimensional, transient model that incorporates the proposed ion motility is developed. The comparisons between the models with and without ion concentrationdependent ion mobility indicate that the present model enables: (i) a more accurate estimation of ionic conductivity, (ii) a more accurate prediction of cell voltage, especially at high current

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