Numerical Investigations of Flow Field Designs for VRFB 2013

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Numerical investigations of flow field designs for vanadium redox flowbatteries

Q. Xu, T.S. Zhao ⇑, P.K. LeungDepartment of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

h i g h l i g h t s

" The performance of VRFBs with different flow fields is numerically simulated." A power-based efficiency is defined and calculated for different flow fields." An optimal flow rate exists for each type of flow field." The serpentine flow field appears to be more suitable for VRFBs.

a r t i c l e i n f o

Article history:Received 16 October 2012Received in revised form 5 December 2012Accepted 13 December 2012

Keywords:Flow batteryVanadium redox flow battery (VRFB)Numerical modelingFlow fieldSystem efficiency

a b s t r a c t

As a key component of flow batteries, the flow field is to distribute electrolytes and to apply/collect elec-tric current to/from cells. The critical issue of the flow field design is how to minimize the mass transportpolarization at a minimum pressure drop. In this work a three-dimensional numerical model is proposedand applied to the study of flow field designs for a vanadium redox flow battery (VRFB). The performanceof three VRFBs with no flow field and with serpentine and parallel flow fields is numerically tested.Results show that when a flow field is included a reduction in overpotentials depends not only onwhether a flow field can ensure a more even distribution of electrolytes over the electrode surface, butalso on whether the flow field can facilitate the transport of electrolytes from the flow field towardsthe membrane, improving the distribution uniformity in the through-plane direction. It is also shownthat the pumping power varies with the selection of flow fields at a given flow rate. To assess the suit-ability of flow fields, a power-based efficiency, which takes account of both the cell performance andpumping power, is defined and calculated for different flow fields at different electrolyte flow rates.Results indicate that there is an optimal flow rate for each type of flow field at which the maximum effi-ciency can be achieved. As the cell with the serpentine flow field at the optimal flow rate shows the high-est energy-based efficiency and round-trip efficiency (RTE), this type of flow field appears to be moresuitable for VRFBs than the parallel flow field does.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Renewable energies like solar and wind are among a few of thecentral topics of our time. However, the random and intermittentnature of renewable energy affects the final quality of power out-put. Energy storage is the key technology to solve this problem [1].Among various existing energy-storage techniques, the all-vana-dium redox flow battery (VRFB) offers the promise for large scaleenergy storage due to its unique features: tolerance to deep dis-charge without any risk of damage, long cycle life, active thermalmanagement and independence of energy and power ratings [2–6].

Although progress has been made over the past decades, signif-icant technical challenges, including slow electrochemical kineticsin the positive electrode, low solubility of active species in electro-lytes and ions crossover through the polymer membrane, are thebarriers that prevent VRFBs from widespread commercialization.To address these issues, previous efforts include decorating elec-trodes with metal or inorganic elements [7–9] or find alternativeelectrode materials [10–12], adding additives to electrolyte to im-prove its solubility [13–15], as well as modifying existing mem-branes and searching for alternatives [16–20], have been widelyreported. The design of flow field of VRFBs is also closely relatedto the above mentioned technical issues. However, the studies onthe flow field design for VRFBs remain limited.

The flow field serves four functions [21–24]: to distribute elec-trolytes on the electrode surface, to apply/collect electric current

0306-2619/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.apenergy.2012.12.041

⇑ Corresponding author. Tel.: +852 2358 8647; fax: +852 2358 1543.E-mail address: [email protected] (T.S. Zhao).

Applied Energy 105 (2013) 47–56

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to/from the cells, to provide a structural support for the electrodematerial and to facilitate heat management. Among these func-tions, the requirements for the latter three are relatively readilymet. However, to design a flow field that meets the requirementfor the first function is much more challenging. To make a VRFBefficient, it is essential to minimize the mass transfer polarizationover the entire electrode. To this end, a uniform distribution ofelectrolytes on the electrode surface is required. However, the uni-formity of electrolytes on the electrode surface is usually achievedat the cost of high flow rates, requiring a higher pumping power,which will reduce the overall efficiency of flow batteries. Hence,the key issue associated with the flow field design is how to min-imize the mass transport polarization at a minimum pressure dropthrough the flow field.

A few experimental investigations into flow fields of VRFBs havebeen reported [25–27]. Zhu et al. [25] investigated the effects oftwo different flow fields, one with a flow-pass pattern while theother with a flow-through pattern, on the performance of a VRFB.The experimental results suggested that the flow-through patterntended to increase the electrode effective active area and to en-hance the uniformity of liquid electrolyte, resulting in an improve-ment in the energy efficiency by up to 5%. Recently, Zawodzinski

et al. [26–28] introduced a so-called zero-gap cell architecture witha serpentine flow field, enabling the peak power density to be767 mW cm�2, which is significantly higher than the conventionalcell configuration; the enhancement of the cell performance wasattributed to the enhanced mass transport and reduced internalresistance.

Numerical modeling and simulation can provide insight to theflow field design. To the best of our knowledge, no numericalinvestigations into the effect of flow fields on the performance ofVRFBs have been reported. In this work, a three-dimensional modelbased on computational fluid dynamics and electrochemical reac-tions is developed. With this model, the effects of flow field design(flow-through with no flow field, with serpentine and parallel flowfields, as shown in Fig. 1) on the distributions of ion concentrations,local overpotential and local current density at various flow rates,as well as on the system efficiency are investigated and the flowfield design that is more suitable for VRFBs is identified.

2. Mathematical model

Consider a typical single VRFB consisting of a Nafion membrane,two graphite electrodes separated by the membrane and two

Nomenclature

AV specific surface area of the porous electrode (m2 m�3)c concentration (mol m�3)D diffusivity (m2 s�1)df mean fibre diameter (m)E1 negative open circuit potential (V)E2 positive open circuit potential (V)Ecell cell voltage (V)E0 equilibrium potential (V)F Faraday constant (C mol�1)I current density (A m�2)~im current density in the solid matrix (A m�2)~is current density in the electrolyte solution (A m�2)j1 negative electrode current density (A m�3)j2 positive electrode current density (A m�3)K permeability of porous material (m2)KCK Carman–Kozeny constantk1 standard reaction rate constant for negative reaction

(m s�1)k2 standard reaction rate constants for positive reaction

(m s�1)km local mass transfer coefficient (m s�1)lmem membrane thickness (m)

N!

molar flux (mol m�2 s�1)~n unit normal vector at a domain boundaryP power (W)p pressure (Pa)Q volumetric flow rate (ml s�1)R gas constant, 8.314 (J mol�1 K�1)_R source term in species conservation equation

(mol m�3 s�1)Rcell cell resistance (X m2)S surface area of the electrodeT temperature (K)U uniformity factorW work (J)~u superficial velocity (m s�1)x coordinate (m)y coordinate (m)z valence of ion

Greeka+ anodic transfer coefficientsa� cathodic transfer coefficientse porosity of porous electrodew efficiencyg over-potential (V)l ionic mobility (m2 V�1 s�1)m kinematic viscosity (m2 s�1)q density (kg m�3)rm conductivity of the solid matrix (S m�1)rmem conductivity of polymer electrolyte membrane (S m�1)/m potential in the solid matrix (V)/mem potential in the membrane (V)/s potential in the electrolyte solution (V)

Superscripts0 initial valuee ending valueeff effective values value at the pore surface of porous electrode

Subscripts1 negative electrode2 positive electrodechar chargedisch dischargeH+ protonH2O waterin inletm solid matrixmem membraneout outlets solutionHSO�4 bisulfate ionV(II) V2+ ionV(III) V3+ ionV(IV) VO2+ ionV(V) VOþ2 ion

48 Q. Xu et al. / Applied Energy 105 (2013) 47–56


current collectors outside the electrodes. Positive and negativeelectrolytes are stored in respective reservoirs and circulatedthrough respective half-cell compartments by pumps. The electro-lytes are made of sulfuric acid solution containing vanadium ions.Chemical and electrical energy can be interchanged by reversibleelectrochemical reactions that take place in the electrodes whenthe battery is charged or discharged. The reactions that occur inthe electrodes of the VRFB can be expressed as:

At negative electrode : V3þ þ e� ¢Charge

DischargeV2þ ð1Þ

At positive electrode : VO2þ þ H2O� e� ¢Charge

DischargeVOþ2 þ 2Hþ ð2Þ

During operation, protons transfer through the membrane whileelectrons transfer through an external load to form an electrical cir-cuit. In the electrochemical processes, the concentration of eachvanadium ion with specific valence keeps changing, while the totalvanadium ions concentration is constant in both the positive andnegative electrodes. To describe the relative amount of vanadiumions with different valences, we define the SOC as:

SOC ¼ cVðIIÞ

cVðtotalÞ

� �1

¼ cVðVÞ

cVðtotalÞ

� �2

ð3Þ

where cV(II) and cV(V) are the molar concentrations of V2þ; VOþ2 ;cvðtotalÞ is the total vanadium ions concentration in a certain electro-lyte, and the subscripts 1 and 2 represent the negative and positiveelectrode, respectively. Alternatively, SOC can also be defined as:

SOC ¼ 1� cVðIIIÞ

cVðtotalÞ

� �1

¼ 1� cVðIVÞ

cVðtotalÞ

� �2

ð4Þ

where cV(III) and cV(IV) are the molar concentrations of V3+ and VO2+.

2.1. Simplifications and assumptions

We consider a three-dimension system with the x–y face lo-cated in the membrane middle face and the z-axis located at thenormal line through the middle point of the membrane surface.The simplifications and assumptions used in the present workare as follows:

(1) As indicated in Eqs. (3) and (4), SOC varies with time. How-ever, when the reservoir is sufficiently big, the change in SOCis relatively small. As a result, the transient charging and dis-charging processes can be simplified as a steady-stateprocess.

(2) An isothermal condition is assumed in the entire domain.(3) The membrane is impermeable to all ions and species,

except for protons.(4) Possible side reactions, such as hydrogen and oxygen evolu-

tions, are not considered.

2.2. Governing equations

2.2.1. Transport in the flow channelSeveral flow field designs are considered: flow-through with no

flow field, a serpentine flow field, and a parallel flow field. For theVRFB with the serpentine and parallel flow fields, the flow of elec-trolyte in the channel can be expressed as:

ðq~u � rÞ~u ¼ �rpþ qmr2~u ð5Þ

where q is the electrolyte density, ~u is the electrolyte velocity, p isthe pressure and m is the kinematic viscosity.

2.2.2. Transport through the porous electrodeIn the flow through the porous electrode, the molar flux of each

species, N!

i (with i representing V2þ; V3þ; VO2þ; VOþ2 ; Hþ; HSO�4 ),can be expressed in terms of the modified Nernst–Planck equationas:

N!

i ¼ �Deffi rci � Fzicilir/s þ~uci ð6Þ

where ci represents the concentration of each species, F is the Fara-daic constant, zi and li are the valence and ionic mobility of eachspecies, /s is the ionic potential in the electrolyte solution, and

Deffi ¼ e1:5Di ð7Þ

is the effective diffusion coefficient of each species, with e and Di

representing, respectively, the porosity of the electrode and thefree-space diffusivity of each species. The superficial electrolytevelocity ~u can be expressed using Darcy’s law as:

qmK~u ¼ �rp ð8Þ

where K is the permeability coefficient. The permeability of a por-ous medium can be described by the Kozeny–Carman equation as[29]:

K ¼d2

f e3

KCKð1� eÞ2ð9Þ

where df is the fiber diameter, KCK is the Carman–Kozeny constant,which characterizes the shape and orientation of the fibrous material.

The flow of charged species results in the current in electrolytesolution as:

~ii ¼ ziF N!

i ð10Þ

The electrolyte is considered to be electrically neutral:Xi

zici ¼ 0 ð11Þ

Combining Eqs. (6), (10), and (11), we can express the total currentdensity in the electrolyte as:

Fig. 1. Schematic of the three types of flow fields in the VRFB.

Q. Xu et al. / Applied Energy 105 (2013) 47–56 49


~i ¼X

i

~ii ¼ �FX

i

ziDeffi rci � F2

Xi

z2i cilir/s ð12Þ

The conservation of species can be written as:

r � N!

i ¼ _Ri ð13Þ

where _Ri is the generation rate for species i [30].Combining Eqs. (6) and (13) gives:

r � ð~uciÞ ¼ r � ðDeffi rciÞ þ r � ðFzicilir/sÞ þ _Ri ð14Þ

Due to the conservation of charge, the charge entering the electro-lyte solution is balanced by the charge leaving the solid phase:

r �~i ¼ r �~is ¼ �r �~im ð15Þ

where~is is the current density in the electrolyte solution and~im isthe current density in the solid matrix.

The electronic potential in the solid matrix is given by Ohm’slaw:

�reffm r2/m ¼ r �~im ð16Þ

where reffm and /m are the effective conductivity and electronic po-

tential of the solid matrix.

2.2.3. Transport through the membraneAs proton is the only mobile ion, the current conservation equa-

tion is:

N!

Hþ ¼ �rmem

Fr/mem ð17Þ

where rmem and /mem are the conductivity and electronic potentialof the ion-exchange membrane. The values of the constants relatedto mass and charge transport can be found in Table 1.

2.2.4. Reaction kineticsThe electrochemical reactions taking place on the solid surfaces

of the porous electrode can be expressed using the Butler–VolmerEquation as:

j1 ¼ eAV Fk1ðcsVðIIIÞÞ

a�;1 ðcsVðIIÞÞ

aþ;1 expaþ;1Fg1

RT

� �� exp �a�;1Fg1

RT

� �� ð18Þ

and

j2 ¼ eAV Fk2ðcsVðVÞÞ

a�;2 ðcsVðIVÞÞ

aþ;2 expaþ;2Fg2

RT

� �� exp �a�;2Fg2

RT

� �� ð19Þ

Eqs. (18) and (19) are for the negative and positive electrodes,respectively, where AV is specific surface area of the porous elec-trode, k1 and k2 are the standard rate constants for negative and po-sitive electrochemical reactions, cs

i ; i 2 fV2þ;V3þ;VO2þ;VOþ2 g arethe vanadium-ion concentrations at the liquid–solid interfaces of

the porous region, a+ and a� are the anodic and cathodic transfercoefficients, g1 and g2 are the overpotentials in the negative and po-sitive electrodes, respectively.

The interface concentrations csi can be related to the bulk con-

centrations ci by balancing the electrochemical reaction rate withthe rate of mass transfer of the vanadium ions to (or from) the elec-trode surface. For the negative electrode during discharge, the bal-ance is:

km1ðcVðIIÞ � csVðIIÞÞ ¼ ek1 cs

VðIIÞ expaþ;1Fg1

RT

� �� cs

VðIIIÞ exp �a�;1Fg1

RT

� �� ð20Þ

and

km1ðcVðIIIÞ � csVðIIIÞÞ ¼ ek1 cs

VðIIIÞ expa�;1Fg1

RT

� �� cs

VðIIÞ exp �aþ;1Fg1

RT

� �� ð21Þ

where km is the mass transfer coefficient and can be approximatedby [31]:

km ¼ 1:6� 10�4~u0:4 ð22Þ

Combining Eqs. (20) and (21), we can express the concentrations ofV(II) and V(III) at the liquid–solid interface as:

csVðIIÞ ¼

kmcVðIIÞ þ ek1kmcVðIIIÞ exp � a�;1Fg1RT

� =A1

B1 � C1=A1ð23Þ

and

csVðIIIÞ ¼

kmcVðIIIÞ þ ek1kmcVðIIÞ exp � aþ;1Fg1RT

� =B1

A1 � C1=B1ð24Þ

where

A1 ¼ km þ ek1 expa�;1Fg1

RT

� �ð25Þ

B1 ¼ km þ ek1 expaþ;1Fg1

RT

� �ð26Þ

and

C1 ¼ ðek1Þ2 exp�aþ;1 � a�;1

RTFg1

� ð27Þ

A similar approach can be used to solve the concentrations of V(IV)and V(V) at the liquid/solid interface of positive electrode.

The overpotentials in Eqs. (18) and (19) are defined as follows:g1 ¼ /m;1 � /s;1 � E1 ð28Þ

and

g2 ¼ /m;2 � /s;2 � E2 ð29Þ

where E1 and E2 are open circuit potentials for reactions (1) and (2),respectively, and can be approximated using the relevant Nernstequation as:

E1 ¼ E01 þ

RTF

lncVðIIIÞ

cVðIIÞð30Þ

and

E2 ¼ E02 þ

RTF

lncVðVÞc2

Hþ

cVðIVÞð31Þ

where the equilibrium potentials E01 and E0

2 are given in Table 2.Then the cell voltage can be written as:

Ecell ¼ E02 � E0

1 þRTF

lncVðVÞcVðIIÞc2

Hþ

cVðIVÞcVðIIIÞ� g1 þ g2 � IRcell ðdischargingÞ

ð32Þ

Table 1Transport properties.

Parameters Symbols Value Unit Ref.

V2+ ion diffusivity in electrolyte DV(II) 2.4 � 10�10 m2 s�1 [38]V3+ ion diffusivity in electrolyte DV(III) 2.4 � 10�10 m2 s�1 [38]VO2+ ion diffusivity in electrolyte DV(IV) 3.9 � 10�10 m2 s�1 [38]VOþ2 ion diffusivity in electrolyte DV(V) 3.9 � 10�10 m2 s�1 [38]HSO�4 ion diffusivity in electrolyte DHSO�4 1.33 � 10�9 m2 s�1 [39]Proton diffusivity in membrane Deff

Hþ3.5 � 10�10 m2 s�1 [40]

Carman–Kozeny constant KCK 5.55 [29]Electrolyte kinematic viscosity m 1.07 � 10�6 m2 s�1

Electrolyte density q 1500 kg m�3

Conductivity of the solid phase rm 1000 S m�1

Conductivity of the membrane rmem 10 S m�1



and

Ecell ¼ E02 � E0

1 �RTF

lncVðVÞcVðIIÞc2

Hþ

cVðIVÞcVðIIIÞþ g1 � g2

þ IRcell ðchargingÞ ð33Þ

where I is the current density and Rcell is the cell electricalresistance.

2.3. Boundary conditions

The boundary conditions can be specified by referring to Fig. 2as follows:

At z = z1 (the positive electrolyte/endplate interface),

@ci

@z¼ 0; i ¼ VO2þ;VOþ2 ;H

þ;HSO�4 : ð34Þ

At z = z2 (the electrolyte/electrode interface),

cijþ ¼ cij�; i ¼ VO2þ;VOþ2 ;Hþ;HSO�4 ; ð35Þ

@/s;2

@z¼ 0; ð36Þ

pjþ ¼ pj�: ð37Þ

At z = z2 (the rib/electrode interface),

@/s;2

@z¼ 0; /m;2 ¼ Ecell; ð38Þ

N!

i �~n ¼ 0; i ¼ VO2þ;VOþ2 ;Hþ;HSO�4 : ð39Þ

At z = z3 (the positive electrolyte/membrane interface),

/s;2 ¼ /mem; ð40Þ

N!

i �~n ¼ 0; i ¼ VO2þ;VOþ2 ;HSO�4 ; ð41Þ~imem �~n ¼~is �~n ðfor protonÞ: ð42Þ

At z = z4 (the negative electrolyte/membrane interface),

/s;1 ¼ /mem; ð43Þ

N!

i �~n ¼ 0; i ¼ V2þ;V3þ;HSO�4 ; ð44Þ~imem �~n ¼~is �~n ðfor protonÞ: ð45Þ

At z = z5 (the electrolyte/electrode interface),

cijþ ¼ cij�; i ¼ V2þ;V3þ;Hþ;HSO�4 ; ð46Þ@/s;1

@z¼ 0 ð47Þ

pjþ ¼ pj�: ð48Þ

At z = z5 (the rib/electrode interface),

@/s;1

@z¼ 0; /m;1 ¼ 0; ð49Þ

N!

i �~n ¼ 0; i ¼ V2þ;V3þ;Hþ;HSO�4 : ð50Þ

At z = z6 (the negative electrolyte/endplate interface),

@ci

@z¼ 0; i ¼ V2þ;V3þ;Hþ;HSO�4 : ð51Þ

At x = 0 and x = L,

@/s

@x¼ 0;

@/m

@x¼ 0; ð52Þ

@ci

@x¼ 0; i ¼ V2þ;V3þ;HþandHSO�4 for negative side;

i ¼ VO2þ;VOþ2 ;Hþand HSO�4 for positive side: ð53Þ

At y = 0 and y = H,

@/s

@y¼ 0;

@/m

@y¼ 0; ð54Þ

@ci

@y¼ 0; i ¼ V2þ;V3þ;Hþand HSO�4 for negative side;

i ¼ VO2þ;VOþ2 ;Hþand HSO�4 for positive side: ð55Þ

At the inlet,

~u ¼~u0 ð56Þ

At the outlet,

p ¼ pout ð57Þ

At all the boundaries except the inlet/outlet, the pressure satisfiesthe Neumann condition:

rp �~n ¼ 0 ðexcept inlet=outletÞ ð58Þ

The internal interfacial conditions (42) and (45) are based on thecurrent balance to associate the variables in both electrodes for iter-ative solution.

3. Results and discussion

The variables in the conservation equations were iterativelysolved using the ANSYS� 13.0 package with a combination of theself-written source terms. The package is based on the finite-volume method. The relative error tolerance was set to 1.0E�6.

Fig. 2. Computational domain.

Table 2Geometric and operating parameters.


Cathodic transfer coefficient forreaction (1)

a-,1 0.5 Assumed

Anodic transfer coefficient forreaction (1)

a+,1 0.5 Assumed

Cathodic transfer coefficient forreaction (2)

a-,2 0.5 Assumed

Anodic transfer coefficient forreaction (2)

a+,2 0.5 Assumed

Standard rate constant forreaction (1)

k1 1.75 � 10�7 m s�1 [30]

Standard rate constant forreaction (2)

k2 3.0 � 10�9 m s�1 [30]

Equilibrium potential forreaction (1)

E01

�0.26 V [42]

Equilibrium potential forreaction (2)

E02

1.004 V [42]



3.1. Model validation

The electrode area used here is 100 mm � 100 mm, with a thick-ness of 3 mm. For the serpentine and parallel flow fields, both theheight and width of the channel are 3 mm, and the rib width is3 mm. The detailed geometric and operating parameters are shownin Table 3. The numerically predicted performance of the VRFBwith no flow field is validated against the experimental data[32]. The operating condition was set to be the same as that inthe experiment. Since the cell voltage was measured as a functionof time, the time-dependant experimental data need to be trans-formed to the SOC-dependant data for the purpose of comparison.The transformation method can be found elsewhere [12]. The com-parison between the numerical and experimental data is shown inFig. 3. It is seen that the three-dimensional model well captures thetrend. The slight discrepancies in the cell voltage between the re-sults may be caused by ions crossover through the membraneand the side reactions that occurred in the experiment but arenot taken account in numerical simulation.

3.2. Species concentration distribution

In order to quantify the uniformity of species concentration inthe in-plane surface, we introduce a uniformity factor as [33]:

U ¼ 1� 1ci;m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1S

ZZðci � ci;mÞ2ds

sð59Þ

where S is the in-plane surface area of the electrode, ci,m is the meanspecies concentration in that surface.

Taking the concentration of VOþ2 ion as an example, Fig. 4 showsthe concentration uniformity factors at the outer surface of theelectrode (at the interface between the current collector and theelectrode) for the VRFB with no flow field and with two types offlow field at the current density of 40 mA/cm2. The ion concentra-tion at the inlet was set to 1200 mol/m3. When the flow rate is5 ml/s, it can be seen that the differences in concentration unifor-mity between three cases are large: the porous electrode with noflow field results in a uniformity factor of 0.684, but the parallelflow field exhibits a uniformity factor of 0.836. This result impliesthat adding a flow field can significantly improve the uniformity,which tends to improve the cell performance. When the flow rateincreases, the uniformity factors associated with all the three casesare improved and the difference in the uniformity factor betweendifferent flow fields becomes smaller. For instance, at 20 ml/s,

the uniformity factors for that with no flow field, with theserpentine and parallel flow fields become 0.929, 0.962 and0.978, respectively. It should be noted that although increasingthe flow rate can effectively improve the in-plane distribution,

Table 3Electrochemistry properties.


Specific surface area AV 2 � 105 m�1 [41]Electrode thickness de 3 � 10�3 mPolymer electrolyte membrane dmem 1.8 � 10�4 mElectrode height h 1 � 10�1 mElectrode width w 1 � 10�1 mOperating temperature T 300 KElectrode porosity e0 0.7 1Electrode fiber diameter df 17.6 lm [41]Initial V2+ ion concentration c0

VðIIÞ 1200 mol m�3

Initial V3+ ion concentration c0VðIIIÞ 300 mol m�3

Initial VO2+ ion concentration c0VðIVÞ 300 mol m�3

Initial VOþ2 ion concentration c0VðVÞ 1200 mol m�3

HSO�4 ion concentration c0HSO�4

3000 mol m�3

Initial H+ ion concentration c0Hþ

3000 mol m�3

Current density I 400 A m�2

Electrolyte volumetric flow rate Q 5 ml s�1

Pump efficiency wpump 0.9 1

Fig. 3. Comparison between the numerical solutions and experimental data.

5 10 15 200.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Uni

form

ity

Flow rate (ml/s)

SerpentinePorousParallel

Fig. 4. Variations in the VOþ2 concentration uniformity at the outer surface ofelectrode with electrolyte flow rates.



the reduction in concentration overpotential is achieved at the costof the increased pumping power in the VRFB system.

3.3. Overpotential distribution

The average values of overpotentials in both positive and nega-tive electrodes of the VRFBs with no flow field and with two typesof flow field are listed in Table 4. As can be seen, the parallel flowfield shows the largest overpotential, while the serpentine flowfield and the porous electrode with no flow field exhibit almostthe same overpotentials. It is worth noting that the overpotentialdepends on the distribution uniformity of electrolyte in both thein-plane and through-plane directions. For the porous electrodewith no flow field, the in-plane distribution of electrolyte is moreuneven than that in the through-plane direction. With the serpen-tine and parallel flow fields, the electrolyte is distributed onto theouter surface of the porous electrode (the interface of the flowchannel and the electrode), and transported through the electrodetowards the membrane by diffusion and convection. For a givenflow rate, as the flow velocity in the parallel channel is smallerthan that in the serpentine channel, the weaker convective effectin the parallel flow field leads to a more uneven distribution ofthe electrolyte in the through-plane direction and low concentra-tions adjacent to the membrane, which result in the high overpo-tential. On the other hand, the stronger convective effect withthe serpentine flow field enables a more even distribution of elec-trolyte in the through-plane direction, lowering the overpotential.When the flow rate increases from 5 ml/s to 20 ml/s, the distribu-tion of electrolyte becomes more uniform and the overpotential(absolute value) of the positive electrode significantly reducesfrom 0.249 V to 0.205 V for the VRFB with the serpentine flow field,as shown in Tables 4 and 5. However, the reduction in overpoten-tial is achieved at the cost of the increased pumping power in theVRFB system, as can also be seen in Table 5. The variations in over-potential in the positive electrode with electrolyte flow rate for theVRFBs with no flow field and with two types of flow field areshown in Fig. 5. The change trends are similar: The overpotentialdecreases with increasing the flow rate, and the decrease rate ofoverpotential slows down at higher flow rates. At the same timethe differences are also distinct: The decrease rates of overpoten-tial for the serpentine and parallel flow fields are more rapid thanthat for the porous type; and the parallel flow field exhibits thehighest overpotential at all flow rates. These phenomena can be ex-plained as follows. As discussed in preceding sections, for the VRFBwith the serpentine and parallel flow fields, the vanadium ions inthe electrolyte are supplied to the active sites of electrode not onlyby diffusion but also by convection. On one hand, the increasedflow rate can increase the average electrolyte concentration alongthe flow direction, which enlarges the concentration difference be-

tween flow channel and the active sites such that enhances the dif-fusion effect. On the other hand, the convection effect inside theporous electrode is improved with the increased flow rate and evendistribution of electrolyte in the through-plane direction can beachieved. That is the reason why the decrease rate of overpotentialwith the flow rate for the VRFBs with serpentine and parallel typesis faster than that with the porous type. In addition, Eq. (19) indi-cates that an exponential relationship exists between the inverseof ion concentration and the overpotential for a given current,explaining why the decrease rate of overpotential becomes smallerat higher flow rates.

3.4. Pressure drop and corresponding pumping power

The pressure drop is an important parameter for selectingpumps in the VRFB system. Table 4 shows the pressure drops forporous electrode and two types of flow field at the flow rate of5 ml/s. As can be seen, the porous electrode has the highest pres-sure drop (957 Pa), whereas the parallel flow field shows the low-est (83 Pa). This result indicates that at small flow rates, including aflow field can significantly reduce the pressure drop, especiallywith the parallel type. Table 4 also shows the corresponding pump-ing powers for the porous electrode with no flow field and withtwo types of flow field. When the flow rate becomes higher, e.g.20 ml/s, the pressure drops for porous electrode and two types offlow field increase, as listed in Table 5. Furthermore, the increasepercentage of the pressure drop for the parallel and serpentineflow fields is much larger than that for the porous electrode withno flow field. For the first two flow fields, the pressure drops in-crease from 83 to 1170 Pa and from 330 to 4768 Pa, respectively,while for the porous electrode, from 957 to 4210 Pa as the flow rateincreases from 5 to 20 ml/s. These phenomena can be explained asfollows. For the flow in porous media, Eq. (8) indicates that thepressure drop increases linearly with the velocity, while for thefully developed flow in the square channel without a porous side-wall, the frictional factor is proportional to the square of velocity[34,35]. In the practical operation of VRFBs with the parallel andserpentine flow fields, a porous sidewall (the interface betweenthe flow channel and the electrode) exists in the flow channel. Un-der this condition, part of the electrolyte will be pushed into theporous electrode; the shear stress at the surface of the porous side-wall becomes smaller than the shear stress at the impermeablewalls. The development of fRe is similar to that without a poroussidewall, and the ratio of fRe/fRe0 (fRe0 is the fully developed valuefor the channel without porous sidewall) depends on the ratio ofthe thickness of porous layer to the width of flow channel [36].Therefore for the parallel and serpentine flow fields, the pressuredrop increases faster than that for the porous electrode as thevelocity increases.

Table 4Calculated pressure drop, pumping power, overpotential and efficiency at the positive and negative electrodes at the flow rate of 5 ml/s.

Dp (Pa) Ppump (W) Np (V) Nn (V) Qact (W) Qohm (W) Ptotal (W) wpower

Parallel 83 4.15e�4 �0.279 0.046 1.30 0.014 6.054 0.783Serpentine 330 1.65e�3 �0.249 0.039 1.152 0.014 6.056 0.807Porous 957 4.8e�3 �0.244 0.039 1.132 0.014 6.065 0.810

Table 5Calculated pressure drop, pumping power, overpotential and efficiency at the positive and negative electrodes at the flow rate of 20 ml/s.

Dp (Pa) Ppump (W) Np (V) Nn (V) Qact (W) Qohm (W) Ptotal (W) wpower

Parallel 1170 2.34e�2 �0.229 0.038 1.068 0.014 6.101 0.815Serpentine 4768 9.54e�2 �0.205 0.032 0.936 0.014 6.244 0.827Porous 4210 8.42e�2 �0.212 0.033 0.972 0.014 6.222 0.819



3.5. Power and energy-based system efficiencies

In this work, power and energy-based system efficiencies aredefined by taking account of the consumed pumping power/en-ergy. At a specific operating point (here we use SOC of 0.8 and dis-charging current density of 400 A/m2, which is a common currentdensity in practical operations), the power-based system efficiencycan be defined as [37]:

wpower ¼Pnet

Ptotal¼ 1� Ppump þ Ploss

Ptotalð60Þ

where Pnet is the system net output power, Ppump is the pumpingpower and Ploss is the lost power. The pumping power depends onthe flow rate Q, the pump efficiency wpump and the pressure dropacross the hydraulic circuit. The pumping power is expressed as:

Ppump ¼ Q � Dp=wpump ð61Þ

where Dp represents the total pressure drop through the piping andthe battery. However, as the pressure drop through piping is typi-cally much smaller than that through the battery, the total pressuredrop can be approximated by that through the battery. It should benoted that the Ppump includes the total pumping power of twopumps at both the positive and negative sides.

The lost power can be expressed as:

Ploss ¼ IAðg1 þ g2Þ þ ðDUmemÞ2 �Armem

lmemð62Þ

where the first term in the right hand represents the polarizationloss, while the second term represents the Ohmic loss, with I isthe current density, A is the electrode surface area, DUmem is thevoltage drop through the membrane and lmem is the thickness ofthe membrane.

The variations in the power-based system efficiency with flowrate for the VRFBs with no flow field and with two types of flowfield are shown in Fig. 6. With the serpentine and porous flowfields, the power-based system efficiency initially increases withthe flow rate, followed by a rapid decrease. The maximum effi-ciency with the serpentine flow field is 0.833 at 15 ml/s, whilethe maximum efficiency with the porous electrode is 0.824 at20 ml/s. However, with the parallel flow field, the power-basedefficiency increases monotonically with the flow rate. The variationin the power-based efficiency with increasing the flow rate is theconsequence of the competition between the reduced lost powerand the increased pumping power. When the flow rate is low,the reduction of lost power with the flow rate is fast (see Fig. 5),

which overcomes the increase of the pumping power. However,when the flow rate becomes larger, the reduction of the lost powerslows down, whereas the increase of the pumping power becomesprominent. Once the increase in the pumping power surpasses thereduction in the lost power, the efficiency begins to decline. Itturns out that there exists an optimal flow rate at which themaximum power-based efficiency is achieved. As the increase inthe pumping power with the flow rate with the serpentine flowfield is faster than that with the porous electrode, the efficiencydecreases much faster with the serpentine flow field once theflow rate exceeds the optimal value. For the VRFB with the parallelflow field, as the pumping power is far smaller than that with theother two, the optimal flow rate is beyond the region discussedhere.

Consider the energy balance during a charge–discharge cycling,the energy-based system efficiency can be defined as:

wenergy ¼Wdisch �Wpump;disch

Wchar þWpump;charð63Þ

where Wdisch is the output work during discharging and Wchar is theinput work during the charging process. To quantify Wdisch andWchar, the cell voltages at various SOCs are needed. Fig. 7a showsthe variation in the positive and negative overpotentials for theVRFBs with no flow field and with two flow fields operating at theirrespective optimal flow rates. Since the local electrochemical reac-tion rate is highly influenced by the concentrations of two ions (SeeEqs. (18) and (19)), the minimum value of overpotential appearsaround the SOC of 0.5, while the maximum value is at the lowestor highest SOC. The corresponding cell voltages are shown inFig. 7b. It can be found that the VRFB with the serpentine flow fieldexhibits the lowest charging voltage and the highest dischargingvoltage for all SOCs. Moreover, Fig. 8 shows the energy-based effi-ciencies for the VRFBs with no flow field and with two types of flowfield: 0.831 with the serpentine flow field, 0.801 with no flow fieldand 0.789 with the parallel type.

For comparison, the widely-used round trip efficiency (RTE) isdefined as:

RTE ¼R t0;disch

0 IdischðtÞEdischðtÞdtR t0;char0 IcharðtÞEcharðtÞdt

ð64Þ

where t0 represents the total charge (or discharge) time.As shown in Fig. 8, the value of the energy-based efficiency is

lower than that of the RTE, as the energy-based efficiency eluci-dates the energy production and consumption in a VRFB system

5 10 15 20 25 30

0.20

0.22

0.24

0.26

0.28

|Pos

itive

ele

ctro

de o

verp

oten

tial (

V)|

Flow rate (ml/s)

Serpentine Porous Parallel

Fig. 5. Variations in the positive electrode overpotential with the electrolyte flowrates.

5 10 15 20 25 300.78

0.79

0.80

0.81

0.82

0.83

0.84

0.85

Serpentine Porous Parallel

Pow

er-b

ased

effi

cien

cy

Flow rate (ml/s)

Fig. 6. Power-based efficiency for the VRFB with different flow fields operating atvarious flow rates.



more comprehensively. In addition, the differences between thetwo efficiencies for the VRFB with no flow field, with serpentineand parallel flow fields are 0.015, 0.015 and 0.002, respectively,owing to the fact that the pumping work during a charge–discharge cycling for the VRFB with parallel flow field is far lessthan that with the other two.

4. Conclusions

In this work a three-dimensional numerical model is proposedand applied to the study of flow field designs for a vanadium redoxflow battery (VRFB). VRFBs with no flow field and with serpentineand parallel flow fields are numerically tested. The salient findingsare summarized as follows:

1. Compared with the conventional battery configuration with theporous flow-through electrode, including a flow field canimprove the distribution uniformity of electrolytes throughthe electrode, particularly at smaller flow rates.

2. The overpotential in the VRFB depends on the distribution uni-formity of electrolytes in both the in-plane and through-planedirections. The VRFB with the serpentine flow field shows thelowest overpotential due to the more even electrolyte distribu-tion over the electrode surface and the enhanced convectivemass transport towards the membrane.

3. The power-based efficiency varies with the electrolyte flowrate. With an increase in the flow rate, the increased pumpingpower is offset by the lowered overpotential, indicating thatthere is an optimal flow rate for each of the flow fields at whichthe maximum efficiency can be achieved. Operating at the opti-mal flow rate, both the energy-based efficiency and RTE for theVRFB with the serpentine flow field are found to be the highest.Therefore, the serpentine type appears to be a more suitableflow field design for VRFBs than the parallel flow field.

Acknowledgement

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

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0.0 0.2 0.4 0.6 0.8 1.0

-0.27

-0.24

-0.21

-0.180.02

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0.70

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987.0 197.0

0.8160.801

0.846

ParallelSerpentine

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gy b

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effi

cien

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Porous

System efficiency Round-trip efficiency

0.831

Fig. 8. Energy-based efficiencies and RTEs for the VRFBs with different flow fields.



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Numerical Investigations of Flow Field Designs for VRFB 2013

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