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Three dimensional multi-physical modeling study of interdigitated flow field in porous electrode for vanadium redox flow battery
Journal of Power Sources 438 (2019) 227023
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Three dimensional multi-physical modeling study of interdigitated flow field in porous electrode for vanadium redox flow battery Cong Yin a, b, *, Yan Gao a, b, Guangyou Xie c, Ting Li c, Hao Tang a, b, ** a
School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, 610023, China Hydrogen and Fuel Cell Institute, University of Electronic Science and Technology of China, Chengdu, 610023, China c Energy Conversion R&D Center, Central Academy of Dongfang Electric Corporation, Chengdu, 611731, China b
H I G H L I G H T S
� A 3D multi-physical model of VRB with 57.5 cm2 active area is developed. � Models with different flow fields are simulated, analyzed and compared. � Interdigitated flow field in electrode shows better performance than in channel. � Leaf shaped distribution patterns are observed for mass transport and voltage. A R T I C L E I N F O
A B S T R A C T
Keywords: Vanadium redox flow battery 3D multi-physical model Interdigitated flow field Flow channel in electrode
The flow field design of vanadium redox flow battery is a critical issue for performance optimization of the battery. In this work, an interdigitated flow field is designed in carbon felt porous electrode with active area of 57.5 cm2 and simulated by a three-dimensional multi-physical model. The “leaf” shaped ionic concentration and voltage distribution patterns are observed in the numerical results and analyzed to understand the cell perfor mance. Compared with other two conventional designs, the interdigitated flow field in bipolar plate and no flow field designs, fluid pressure drop of flow field designed in electrode is the least and the system efficiency shows the best over the major operation range of electrolyte flow rates. The flow field designed in electrode greatly lowers sealing pressure demand for vanadium redox flow battery with better reliability and shows potential benefit for large-scale cell stack design and energy storage system application.
1. Introduction Energy storage technologies are critical for delivery of efficient renewable energy sources such as wind and solar. Among various existing energy storage technologies, vanadium redox flow battery (VRB) shows great potential for large scale energy storage due to its advantages of high energy efficiency, long cycle life and independence of capacity and power ratings. Much progress has been made during the commercialization process of VRB in these years, including key mate rials, cell stack design and system integration technologies and a number of demonstrations of VRB power plants have been installed [1–10]. However, VRB system performance and lifetime still need to be improved which attracts more and more attention of researchers all over the world [11–32].
As a flow battery, VRB stores its electrochemical energy in the electrolyte solutions with vanadium ions of different valences. For supplying ionic reactants during charge and discharge processes, flow field design of the cell stack plays a significant role to enhance the battery performance and lower the pump consumption for recirculation the electrolyte fluid. Much effort has been devoted to various flow field studies, including serpentine, parallel and interdigitated flow channels [33–53]. Among the mentioned designs, the interdigitated flow field is beneficial for both distributing even ionic reactants and lowering fluid pressure drop, which attracts much research interest recently [33–47]. M. Messaggi et al. developed a three-dimensional (3D) VRB model coupling fluid dynamics and electrochemistry to study 25 cm2 inter digitated and serpentine flow fields [33]. S. Maurya et al. evaluated serpentine, interdigitated and conventional (without flow pattern) flow
* Corresponding author. School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, 610023, China. ** Corresponding author. Hydrogen and Fuel Cell Institute, University of Electronic Science and Technology of China, Chengdu, 610023, China. E-mail addresses: [email protected] (C. Yin), [email protected] (H. Tang). https://doi.org/10.1016/j.jpowsour.2019.227023 Received 23 October 2018; Received in revised form 28 June 2019; Accepted 14 August 2019 Available online 5 September 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.
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Table 3 Parameter values for the calculation. Symbol D2
Symbol
Parameter
Value
a
Channel width
1 mm
b
Electrode land width
5 mm
c
Channel height
4 mm
Lch
Channel Length
45 mm
d1
Current collector thickness
2 mm
d2
Membrane thickness
0.1 mm
Lx
Width of reactive area
115 mm
Ly
Length of reactive area
50 mm
am
Width of manifold
5 mm
ion diffusivity in electrolyte
2.4 � 10
10
m2/s [54]
ion diffusivity in electrolyte
2.4 � 10
10
m2/s [54]
3.9 � 10
10
m2/s [54]
3.9 � 10
10
m2/s [54]
V
D3
V
3þ
D4
VO2þ ion diffusivity in electrolyte
D5
VOþ 2 ion diffusivity in electrolyte
DHSO4
HSO4 ion diffusivity in electrolyte
k1
Standard rate constant: positive
9.31 � 10 m2/s [54] 1.33 � 10 9 m2/s [47] 6.8 � 10 7 m/s [54]
k2
Standard rate constant: negative
1.7 � 10
Proton diffusivity in electrolyte
7
m/s [54]
1.004 V
Equilibrium potential: VO
ε
Porosity of the electrode
0.9
df
Carbon fiber diameter
1.76 � 10
KCK
Carman-Kozeny constant
5.55 [54]
As
Active specific surface area
10000 m
ρ
Density of the electrolyte
1300 kg/m3
μ
Viscosity of the electrolyte
4.928 � 10
σs
Conductivity of the solid phase
1000 S/m
σmem
Conductivity of membrane
10 S/m
T
Temperature
298 K
cinitial þ
1.5 mol/L
cinitial þ;Hþ
Initial concentration of vanadium ions (positive) Initial concentration of vanadium ions (negative) Initial concentration of proton (positive)
cinitial ;Hþ
Initial concentration of proton (negative)
4 mol/L
iinward
Operating current density
60 mA/cm2
Pump efficiency
0.9 [33,47]
cinitial
ψ Pump
2þ
9
E01
E02
Table 1 Default parameters of the VRB cell’s geometric structure.
Value
2þ
DHþ
Fig. 1. Geometric structure of VRB modeling: (a) simulated components do mains of a single cell including current collector, porous electrode with inter digitated flow channels, membrane separator, and electrolyte fluid; (b) detailed mesh at the corner; (c) xz cross section.
Parameter
=VOþ 2
Equilibrium potential: V2þ =V3þ
0.255 V 5
m [54]
1
3
Pa s
1.5 mol/L 4 mol/L
Table 2 Source and sink terms for each species. Source term
Positive electrode
Negative electrode
SV2þ
–
j2 =F
SV3þ
SVO2þ
SVOþ2
j1 =F
j1 =F
SHþ
SH2 O
SSO24
j2 =F
–
2j1 =F j1 =F 0
–
Fig. 2. Comparison between the numerical and experimental performance of single VRB cell with flow through design during charge and discharge pro cesses. The electrolyte flow rate is constant at 6 ml/s and the operating current density is 60 mA/cm2.
– – – 0
flow field designs with the 1 kW VRB cell stack experiment [40]. S. Kumar et al. experimentally studied the effect of flow field for flow batteries with large cell sizes [41]. M. D. R. Kok et al. developed a model of a hydrogen-bromine redox flow battery cathode with interdigitated flow channels to investigate the effect of electrode morphology and overall architecture of the cell [42]. E. Knudsen et al. modeled the three-dimensional (3D) fluid dynamics of interdigitated flow fields for VRB, respectively [44]. R. J. Kee et al. developed a general model to evaluate flow uniformity and pressure drop within interdigitated-channel structures of redox flow batteries [45]. Q. Xu et al. studied the performance of VRB with different flow fields numerically [47]. In our previous work, interdigitated flow field in bipolar plate (FFplate design) of VRB was investigated with internal parameters distri butions by a 3D coupled model [46]. The mentioned studies mainly
field types with carbon felt electrodes and compare their performance with a serpentine flow field using carbon paper electrodes [34]. J. L. Barton et al. studied both flow-through and interdigitated flow fields experimentally to quantify the mass-transfer coefficient as a function of electrolyte velocity and viscosity [35]. J. Houser et al. studied flow field architecture influence on mass transport and system performance in redox flow batteries [36,43]. A. Bhattarai et al. investigated different designs of flow channels in porous electrode of VRB and relevant energy efficiency of the system [37]. Y. K. Zeng et al. reported the performance enhancement of iron-chromium redox flow battries with interdigitated flow fields due to the uniform catalyst distribution and high mass transport limitation [38]. X. You et al. developed a 3D model of a half-battery with an active area of 900 cm2 to explore the design rules of flow fields [39]. D. Reed et al. studied performance of interdigitated 2
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Journal of Power Sources 438 (2019) 227023
Fig. 3. Simulated distribution patterns of VRB cell with interdigitated flow channels designed in porous electrode at SOC of 0.5 during discharge under electrolyte flow rate of 1 ml/s: electrolyte fluid velocity (a) and pressure (b) distribution patterns in the xy plane with z value of 8.1 mm inside the positive porous electrode; volume flow rate distributions of inlet channels (c) and outlet channels (d) at various channel walls.
focus on flow field designs in the bipolar plate rather than in the porous electrode. Only several literatures concern flow channels inside porous media [37,40]. Actually, it is possible to construct flow field in VRB carbon felt porous electrode due to its thickness of several millimeters for better reactants distributions and lower fluid pressure drop. In this study, interdigitated flow field is designed in carbon felt porous elec trode (FF-electrode design) and cell performance is analyzed in terms of numerical parameters distributions. Meanwhile, VRB system efficiency of FF-electrode design is compared with that of FF-plate design and no flow field design (No-FF) which includes no flow field in plate or elec trode. Based on the simulation results, the coupling factors including flow field design, reactants mass transport and cell stack performance are better understood for optimization of VRB system efficiency.
defined as inlet channels (chin) and the other ten connected to the outlet manifold are outlet channels (chout). The detailed dimensions are listed in Table 1. 2.2. Model assumptions and boundary conditions The assumptions for the proposed model are listed below: (1) SOC (state of charge) which defines vanadium ion concentrations varies with time. When the electrolyte tank is sufficiently large, the change of SOC is relatively small during a very short period of operation. Thus the dy namic model could be simplified as a stationary one to reduce solving time for the large geometry without losing much accuracy; (2) The presented model is isothermal; (3) The diluted-solution approximation is applied; (4) The fluid flow is treated as incompressible flow; (5) The material properties of electrode, electrolyte and membrane domains are homogeneous; (6) Only protons could cross over through the membrane; (7) Side reactions such as hydrogen and oxygen evolutions are neglec ted; (8) Electrolyte volume change due to water permeation or water drag through membrane is ignored. As the presented model is stationary, voltages during charge and discharge processes are calculated with various inlet boundary condi tions of SOC, which are expressed as:
2. Model development 2.1. Model geometry The 3D numerical model of VRB is based on a single cell structure as shown in Fig. 1, of which the dimensions are comparable to those of the experimental ones [16,46]. The VRB cell is composed of bipolar plates, porous electrodes and a proton exchange membrane. Electrolyte solu tions with vanadium ions of different valences, VO2þ /VOþ 2 for the pos itive and V 2þ /V 3þ for the negative, are indicated with blue and green color respectively. The main half-reactions of the two electrodes for VRB are as follows:
cinlet ¼ cinitial ð1 3
SOCÞ
(4)
Negative electrode: V 3þ þ e ⇌V 2þ
(1)
cinlet ¼ cinitial ð1 4 þ
SOCÞ
(5)
þ Positive electrode: VO2þ þ H2 O⇌VOþ 2 þ 2H þ e
(2)
cinlet ¼ cinitial SOC 5 þ
(6)
initial initial cinlet SOC þ;H þ ¼ cþ;H þ þ cþ
(7)
initial initial cinlet SOC ;H þ ¼ c ;H þ þ c
(8)
(3)
cinlet ¼ cinitial SOC 2
The electrolyte fluid flows into and out of the reaction area through the transverse manifold shown in Fig. 1 (a). There are twenty flow channels constructed in the porous electrode for both positive and negative sides, ten of which are connected to the inlet manifold and 3
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Journal of Power Sources 438 (2019) 227023
where K is the permeability coefficient of the porous electrode. The permeability of porous media is given by the Carman-Kozeny equation as [8]: K¼
d2f ε3
(11)
εÞ2
KCK ð1
where ε is the porosity; df is the fiber diameter; KCK is the CarmanKozeny constant, which characterizes the fibrous material and is used as a fitting parameter. ! The molar fluxes N i of ionic species i (with i indicating V 2þ , V 3þ , þ VO2þ , VOþ 2 , H and HSO4 ) in the fluid flow through porous electrode is given by a modified Nernst-Planck equation [8]: ! Ni ¼
ε3=2 Di rci
zi ci ε3=2 Di FrΦl ! þ u ci RT
(12)
Thus, the conservation for species i in the porous electrodes can be written as: ! r⋅ Ni ¼
(13)
Si
where Si is the source term of species i (defined in Table 2); ci, Di and zi are the bulk concentration, diffusion coefficient and charge of species i, respectively; Φl is the ionic potential; F is the Faraday’s constant and R is the ideal gas constant. With the assumption that the electrolyte is electroneutral, it gives: X zi ci ¼ 0 (14) i
For charge balance, the charge that enters the electrolyte equals to the one which leaves the solid phase: r ⋅ is þ r⋅il ¼ 0 Fig. 4. Two-dimensional distribution patterns of ions concentrations at mid of 2þ positive electrode in the xy plane with z value of 8.1 mm: (a) VOþ and 2 , (b) VO (c) Hþ. The simulated VRB cell is designed with interdigitated flow channels in porous electrode and operated at SOC of 0.5 during discharge under electrolyte flow rate of 1 ml/s.
! ! where i l and i s are the local current densities corresponding to the liquid and the solid phases, respectively. For electrolyte, the flow of charged species results in the current in ! solution i l expressed as: X ! ! il¼ zi F N i (16)
inlet inlet where cinlet and cinlet are the inlet concentrations, corre 5 2 , c3 , c4 initial sponding to V 2þ , V 3þ , VO2þ and VOþ and cinitial 2 ions, respectively; cþ are the initial concentrations of vanadium ions at positive and negative inlet inlet portions with 1.5 mol/L; cinlet þ;Hþ and c ;Hþ are the proton concen
cinitial þ;Hþ
i
For solid phase, the current density is calculated with the Ohm’s law: ! is¼
cinitial ;Hþ
trations at positive and negative inlets, while and are the initial proton concentration values at two reaction sides both set as 4 mol/L. The incipient concentration of SO24 in the positive electrolyte is slightly different from the negative one.
2.3.1. Transport in flow channels Navier-Stokes equations for incompressible fluid are applied to simulate the electrolyte flow in channel and manifold domains as follows: rp þ μr2 ! u
(9)
u¼
rp
j1 ¼ r⋅il ¼
r⋅is
(18)
j2 ¼ r⋅il ¼
r⋅is
(19)
! N Hþ ¼
σmem F
rΦmem
(20)
where σmem and Φmem are the conductivity and electric potential of the membrane, respectively.
2.3.2. Transport in porous electrodes For the fluid flow in porous media, velocity ! u can be expressed with Darcy’s law: K
(17)
2.3.3. Transport in membrane For membrane, transport process of proton is considered and the current conservation equation is expressed as:
where ρ is the fluid density; ! u is the velocity; p is the pressure and μ is the dynamic viscosity of the fluid.
μ!
εÞ3=2 σ s rΦs
ð1
where σ s indicates conductivity and Φs indicates electric potential for solid phase. For positive and negative electrodes, j1 and j2 are transfer current densities for the electrochemical reactions at the surfaces of porous electrodes as current sources,
2.3. Governing equations
ρð! u ⋅ rÞ! u¼
(15)
2.3.4. Reaction kinetics As for the reversible electrochemical reactions, the transfer current densities of the positive electrode j1 and the negative electrode j2 are
(10) 4
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Journal of Power Sources 438 (2019) 227023
Fig. 5. Electric voltage distributions in the xy plane with varied z values: (a) 12.1 mm (at the outer surface of the positive current collector), (b) 8.1 mm (at the midlayer of the positive electrode), and (c) 4 mm (at the mid-layer of the negative electrode). Local current vector distributions at the outer surface of the positive current collector: (d) x direction, (e) y direction and (f) z direction. The simulated VRB cell is designed with interdigitated flow channels in porous electrode and operated at SOC of 0.5 during discharge under electrolyte flow rate of 1 ml/s.
given by the Butler-Volmer laws [54]: � � �� � � pffiffiffiffiffiffiffiffi F η1 F η1 j1 ¼ As Fk1 cs4 cs5 exp exp 2RT 2RT j2 ¼ As Fk2
� � � pffiffiffisffiffiffiffisffi F η2 c2 c3 exp 2RT
� exp
F η2 2RT
The quantities csi indicate vanadium-species concentrations at the liquid-solid interfaces in the porous regions, which are generally different from the bulk values due to additional transport resistance (from the bulk solution to the interfaces). They can be related to the bulk concentrations, ci , by approximately balancing the rate of reaction with the rate of diffusion of reactant to (or from) the electrode surface at steady state. For the positive reaction, the balance is: � � � � �� εk1 s F η1 F η1 c4 cs4 ¼ c4 exp cs5 exp (27) γ4 2RT 2RT
(21)
�� (22)
of which, As is the specific active surface area, cs2 , cs3 , cs4 and cs5 are the surface concentrations of V 2þ , V 3þ , VO2þ and VOþ 2 ions which are generally different from the bulk concentration ci, k1 and k2 are the re action rate constants. η1 and η2 represent the activation overpotentials of positive and negative reactions which are defined as:
η1 ¼ Φ s
Φl
E1
(23)
η2 ¼ Φ s
Φl
E2
(24)
c5
E2 ¼ E02
� � � F η1 cs5 exp γ5 2RT
εk1
� cs4 exp
F η1 2RT
�� (28)
where γ4 ¼ D4 =dp , γ5 ¼ D5 =dp , dp is the average pore diameter of the porous structure, Di is the diffusion coefficient for species i in solution. The quantities γi (in m/s) measures the rate of reactant delivery to or from the surfaces by diffusion from the bulk [55,56]. Combining equations (27) and (28), the concentrations of VO2þ and VOþ 2 ions at the liquid-solid interface are expressed as:
where E1 and E2 are the equilibrium potentials for the positive and negative reactions, calculated by the Nernst equations: � � RT c5 E1 ¼ E01 þ ln (25) F c4 � � RT c2 ln c3 F
cs5 ¼
cs4 ¼
c þ εk1 e FðΦs Φl E1 Þ=ð2RTÞ ðc4 =γ5 þ c5 =γ4 Þ �4 . . � 0 0 1 þ εk1 e FðΦs Φl E1 Þ=ð2RTÞ γ5 þ eFðΦs Φl E1 Þ=ð2RTÞ γ 4
(29)
cs5 ¼
0 c þ εk1 eFðΦs Φl E1 Þ=ð2RTÞ ðc4 =γ5 þ c5 =γ4 Þ �5 . . � 0 0 1 þ εk1 e FðΦs Φl E1 Þ=ð2RTÞ γ5 þ eFðΦs Φl E1 Þ=ð2RTÞ γ 4
(30)
0
(26)
where E01 and E02 are the standard potentials for the positive and negative reactions with temperature of 298.15 K and pressure of 1 bar. 5
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Fig. 6. Schematics of three VRB design geometries: (a) flow field in electrode (FF-electrode); (b) flow field in plate (FF-plate); (c) no flow field (No-FF). Electrolyte pressure distributions of the three designs under flow rate of 10 ml/s: (d) FF-electrode; (e) FF-plate; (f) No-FF. Numerical results for comparison of the three designs: (g) charge and discharge cell voltage curves with 1 ml/s electrolyte flow rate; (h) pressure drop under various electrolyte flow rates.
Similar expressions of cs2 and cs3 could be obtained for the negative electrode. Relevant parameters for the simulation are shown in Table 3.
results of charge and discharge voltages, the linear relationship between t and SOC are applied as follows: SOC ¼ SOC0 þ
2.4. Numerical details
t ðSOCe t0
SOC0 Þ
(31)
where t0 represents the total charge or discharge time; SOC0 and SOCe indicate the beginning and ending SOC status. In the experiment, SOC is determined by open-circuit cell voltage VOC , and the SOC range is 0.2–0.8 for charge or discharge processes. The experiment used carbon felt electrode dimension is 96 mm � 60 mm � 4 mm (reaction area: 57.6 cm2), while for comparison, the calculated No-FF electrode size is 115 mm � 50 mm � 4 mm (reaction area: 57.5 cm2) according to Lx and Ly in Table 1. The starting electrolyte solutions for positive and negative vanadium half-cells are aqueous solution of 1.5 mol/L vanadyl sulphate VOSO4 and 1.5 mol/L tri-sulphate V2(SO4)3, respectively, with 2 mol/L sulfuric acid H2SO4 for both sides. Constant electrolyte flow rate of 6 ml/ s is applied and the operating current density is 60 mA/cm2. As shown in Fig. 2, the numerical results fit the experimental one well with small deviations less than 1% during the charge and discharge processes. It is known that the aspect ratio affects overall pressure drop of the stack significantly with constant reaction area, which eventually im pacts on the VRB system efficiency. The aspect ratio of 115:50 shows a decrement of about 12% fluid pressure drop compared with that of 96:60, while the numerical charge and discharge cell voltage curves for the two aspect ratios show almost identical profiles with voltage devi ation less than 1 mV. Besides, the fluid pressure decreases almost evenly from the inlet to the outlet due to the well distributing function of the
The variables in model equations were solved with the COMSOL Multiphysics® package using finite-element method. The sources terms and physical properties of the governing equations are written in the user’s code. The relative error tolerance was set to 1 � 10 6. 3. Results and discussion In our previous work, interdigitated flow field constructed in bipolar plate of VRB was investigated with various design and operation pa rameters. In the present study, we focus on the interdigitated flow field designed in porous electrode and analyze the cell performance and in ternal parameters distributions. Furthermore, VRB performance and system efficiency of FF-electrode design are compared with those of FFplate and No-FF designs. 3.1. Model validation A single VRB cell with no flow field is assembled for charge and discharge test to validate the proposed modeling. The cell structure, VRB test system and detailed experimental procedures can be found in our previous work [16,46]. To compare the experiment and numerical 6
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Journal of Power Sources 438 (2019) 227023
Fig. 7. The cross section views of VOþ 2 ion concentration distributions at inlet, mid and outlet of porous electrode: (a) flow field in electrode; (b) flow field in plate. The FF-electrode and FF-plate VRB cells are operated at SOC of 0.5 during discharge under electrolyte flow rate of 1 ml/s.
Fig. 8. Efficiency comparison of three different VRB cells of FF-electrode, FF-plate and No-FF designs: (a) stack efficiencies and (b) system efficiencies under various electrolyte flow rates.
transverse channels (width: 5 mm).
channel direction (X axis). Fig. 3 (b) shows the fluid pressure distribu tion in porous electrode, where chin(n)_end and chout(n)_end indicate dead end walls of the nth inlet and outlet channels; chin(n)_left and chin(n)_right indicate the left and right side walls of the nth inlet channel; chout(n)_left and chout(n)_right indicate the left and right side walls of the nth outlet channel; Tm_inlet and Tm_outlet indicate the inlet and outlet transverse manifold, respectively. The electrolyte pressure drop be tween the channel dead end (chin(n)_end/chout(n)_end) and the trans verse manifold (Tm_inlet/Tm_outlet) shows much larger than that between the neighbored inlet and outlet channel walls (eg. chin(n)_right and chout(n)_left). For the volume flow rates, values of liquid flowing through the channel side walls show an order of magnitude larger than those of dead end walls shown in Fig. 3 (c and d). To note, volume flow rate through right side wall of inlet channel is about 20% larger than that of left side wall, due to the asymmetric manifold structure where inlet locates at bottom left corner of the VRB cell and outlet at the top right corner.
3.2. Parameter distributions of VRB with FF-electrode design In this part, parameter distribution patterns of FF-electrode design including fluid flow, reactant ion concentration, electric voltage and current density are presented and analyzed at SOC of 0.5 during discharge with electrolyte flow rate of 1 ml/s. Fig. 3 (a) shows fluid velocity distribution in porous media which is critical to convective mass transport of vanadium ions for electro chemical reactions at interface of porous electrode and electrolyte. At the dead end of the inlet and outlet channels indicated by brown dash circles (a1) and (a2) respectively, the fluid velocities show much larger than elsewhere and the electrolyte flows along the channel direction (Y axis). While for the two sides of the nth inlet channel (chin(n)) indicated as (a3) and (a4), fluid velocities show almost an order of magnitude less than those of (a1) and (a2) and the electrolyte flows perpendicular to the 7
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Fig. 4 shows the two-dimensional distribution patterns of ions con centrations in positive electrode. During discharge, electrochemical re þ 2þ leading to similar action consumes VOþ 2 and H and generates VO distribution patterns for the former two ions and pattern with reversed color for the third ion. Taking the VOþ 2 ion as example, the concentration distribution forms parallel multiple “leaf” shaped patterns (Fig. 4. (a3)) with each outlet channel as the symmetrical line of the “leaf”. The VOþ 2 ion concentrations at the dead end of inlet channels (Fig. 4. (a1)) show larger values than those at the dead end of outlet channels (Fig. 4. (a2)). The VOþ 2 ion concentrations at right sides of the multiple “leaf” regions are lowest all over the pattern, of which the values are even less than those at the outlet portion of transverse manifold. The reactant ions concentration distributions have strong impact on the local electric potential and current collections. Electric voltage at the outer surface of the positive current collector (Fig. 5 (a)) shows the largest value near the inlet manifold, medium around the outlet mani fold and lowest at the mid of the current collector. This distinct electric voltage distribution is due to the lowest reactive ions concentrations in the multiple “leaf” regions. Electric voltage distributions inside the positive and negative porous electrodes (Fig. 5 (b) and (c)) accord well 2þ with the reactive ions concentration distributions of VOþ 2 and V . At right sides of the multiple “leaf” regions (each outlet channel as the symmetrical line), the lowest electric potential for positive electrode and highest value for negative electrode indicate the maximum localized reaction polarizations. As for the current at the outer surface of positive current collector, the current vector at z direction shows much larger than those at x and y directions. The z direction currents at the land areas are about 15–20% higher than those at channel areas shown in Fig. 5 (f), which leads to periodic current flows at x direction for each channel-land cycle in Fig. 5 (d). As for the y direction vector in Fig. 5 (e), current shows around zero at mid of the plate (along y direction in xy plane) and flows toward the mid from both up and bottom of the plate. The maximum values of current vectors at x and y directions reach 30% and 40% that of z di rection, respectively.
compared in Fig. 6 (e). At the same operating flow rate, the three de signs’ pressure drop performs as No-FF > FF-plate > FF-electrode. Pres sure drop of No-FF design increases linearly with flow rate, but the curve shows non-linearity for FF-plate and FF-electrode designs. At flow rate of 1 ml/s, pressure drops of No-FF and FF-plate design are 7 and 3.5 times that of FF-electrode design respectively, while the numbers become 4.9 and 3.0 at flow rate of 5 ml/s and 3.6 and 3.4 at flow rate of 10 ml/s. The overall pressure drop of a cell stack is composed of that of manifolds, flow channels and porous media. For No-FF design, pressure drop of manifold is neglectable compared to that of fluid penetrating the porous electrode, however for FF-plate and FF-electrode designs, pressure drop through the channel is dominant and that of manifold is comparable to that of porous media. To analyze the performance difference between FF-electrode and FFplate designs, the cross section views of ion concentration distributions at inlet, mid and outlet of porous electrode are shown in Fig. 7. Generally speaking, reactive ions distributes more evenly in FF-electrode cell than in FF-plate cell, with comparisons of distribution patterns and ion con centration range at the same location. For the FF-electrode design with channels embedded in the porous electrode, the VOþ 2 concentration decreases from each inlet channel to the neighbored outlet ones. The much lowered reactants concentrations appear at two thin layers near the upside and bottom side of the porous electrode as a result of the noslip wall boundary effect in Fig. 7 (a). In the FF-plate cell with channels on the top porous electrode, the electrolyte flow path from inlet channel to the outlet one is different from that of FF-electrode leading to much lowered reactants concentrations near the membrane (Fig. 7 (b)). More detailed parameter distribution characters of FF-plate design can be found in our previous work [46]. The greater reactant concentration uniformity of FF-electrode than that of FF-plate results in better charge and discharge performance of the former. 3.4. Energy efficiency analysis With the cell voltage performance and pressure drop characteristic during a charge-discharge cycle, the energy efficiencies of VRB stack and system are investigated. The round trip efficiency of the stack is defined as:
3.3. Comparison study of different designs In the former part, the distribution patterns of FF-electrode design are investigated. In this section, cell stack performance and system ef ficiency are analyzed with comparisons of FF-electrode, FF-plate and No-FF designs, of which schematic design structures are shown in Fig. 6 (a, b and c). With identical operating current density (60 mA/cm2) and electrolyte flow rate (1 ml/s), charge and discharge cell voltages of the three designs differ from each other, of which the cell performance sequence is No-FF > FF-electrode > FF-plate. At charge SOC range of 0.1–0.9, voltages of FF-plate and FF-electrode show 10–32 mV and 1–4 mV larger than that of No-FF design, respectively. While at discharge SOC range of 0.9–0.1, voltages of FF-plate and FF-electrode show 10–33 mV and 1–4 mV less than that of No-FF design, respec tively. In the No-FF design of this work, the transverse manifolds for electrolyte inlet and outlet are constructed as is commonly applied in an actual VRB cell stack design. Those No-FF designs without transverse manifolds may lead to significantly uneven electrolyte flow distributions and lowered cell performance [47,50] which is not considered in this study. This is the major reason that No-FF design performs better than those with flow channels. To note, the applied current density is calculated with operating current divided by electrode area (xy plane) which are not the same for identical width (Lx) and length (Ly) of reactive area in the model geometry. For No-FF and FF-plate designs, the electrode area isLx � Ly , but that of FF-electrode design writes as ðLx �Ly Þ ða �Lch �nch Þ where nch is the total number of the channels. As a result, the FF-electrode design loses a part of electrode area (area of the channels: a � Lch � nch ) compared with No-FF and FF-plate designs, although its cell voltage performs as well as the No-FF design. The cell stack pressure drop with various electrolyte flow rates are
ψ stack ¼
Wdis Wch
(32)
Wdis and Wch are the output energy during discharge and input energy during charge of the stack, respectively: Z t0;dis Wdis ¼ Idis ðtÞVdis ðtÞdt (33) 0
Z
t0;ch
Wch ¼
Ich ðtÞVch ðtÞdt 0
(34)
where I and V are the stack current and voltage, respectively. Considering the pump consumption Wpump during the chargedischarge cycle, the system energy efficiency can be defined as:
ψ system ¼
Wdis Wpump;dis Wch þ Wpump;ch
The pump consumptions are calculated by: R t0;dis Qdis ðtÞΔpdis ðtÞdt Wpump;dis ¼ 0
ψ pump
R t0;ch Wpump;ch ¼
0
Qch ðtÞΔpch ðtÞdt
ψ pump
(35)
(36)
(37)
where Q is the electrolyte flow rate and Δp is the total fluid pressure drop, which are time independent in this work. 8
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Journal of Power Sources 438 (2019) 227023
As shown in Fig. 8 (a), the stack efficiency increases with electrolyte flow rate as more efficient mass transfer of reactants enhances the electrochemical reactions. The FF-plate cell performs worst, especially at low flow rates, and stack efficiency of FF-electrode is close to that of No-FF design which is also suggested by the charge-discharge voltage curves. The system efficiency of FF-electrode is a little less than that of No-FF cell at low electrolyte flow rate (1 ml/s), but performs much better than the No-FF one at high flow rate (10 ml/s). In addition, VRB stack with large active area shows 3 to 5 times higher pressure drop for the flow through electrode design (No-FF) than its interdigitated (FFelectrode) counterpart [40], while the value is about 3 times for small active area stack shown in Fig. 6 (h). Therefore, the benefit of FF-electrode interdigitated design concluded by the small stack in the presented work tends to be consistent or even enhanced for the large cell stack design. In short, the pump work plays a significant role in the system efficiency of VRB and the low pressure drop design of the cell stack is helpful for the overall system efficiency optimization.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
4. Conclusion
[22]
In this work, VRB cell with interdigitated flow field constructed in carbon felt porous electrode (FF-electrode) is studied with a threedimensional (3D) coupled modeling. The “leaf” shaped reactants con centrations and electric potential distribution patterns appear in the FFelectrode design, leading to worse local performance in the mid part than that around inlet and outlet portions of the porous electrode. The stack and system efficiencies of FF-electrode design are compared with those of interdigitated flow field in bipolar plate (FF-plate) and no flow field (No-FF) designs. In terms of convective mass transport, FFelectrode stack voltage performs as well as No-FF design with trans verse inlet and outlet manifolds and FF-plate stack shows lowest stack voltage efficiency. Considering pump consumption required for the overall pressure drop, system efficiency of FF-electrode is a little less than that of No-FF cell at low flow rate, but shows much better at high ones and FF-plate design performs lowest system efficiency. On one hand, the FF-electrode design greatly lowers sealing pressure demand for VRB stack and enhances system performance; on the other hand, this design wastes a part of electrode for reaction which is constructed with flow channels. Furthermore, the benefit of FF-electrode interdigitated design concluded by the small stack in the presented work is reliable for the large cell stack development. The robust VRB stack design with FFelectrode and the system featured with optimized operating conditions will be discussed in the forthcoming work.
[23] [24] [25]
Acknowledgement
[44]
This work is sponsored by National Key R&D Program of China (No.2018YFB1502700), Science and Technology Program of Sichuan Province (No.2019YFG0002 and No.2017CC0017) and Initiative Sci entific Research Program of University of Electronic Science and Tech nology of China (No.ZYGX2018KYQD207 and No. ZYGX2018KYQD206).
[45] [46] [47] [48] [49]
[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
[50] [51] [52]
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Contents lists available at ScienceDirect
Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
Three dimensional multi-physical modeling study of interdigitated flow field in porous electrode for vanadium redox flow battery Cong Yin a, b, *, Yan Gao a, b, Guangyou Xie c, Ting Li c, Hao Tang a, b, ** a
School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, 610023, China Hydrogen and Fuel Cell Institute, University of Electronic Science and Technology of China, Chengdu, 610023, China c Energy Conversion R&D Center, Central Academy of Dongfang Electric Corporation, Chengdu, 611731, China b
H I G H L I G H T S
� A 3D multi-physical model of VRB with 57.5 cm2 active area is developed. � Models with different flow fields are simulated, analyzed and compared. � Interdigitated flow field in electrode shows better performance than in channel. � Leaf shaped distribution patterns are observed for mass transport and voltage. A R T I C L E I N F O
A B S T R A C T
Keywords: Vanadium redox flow battery 3D multi-physical model Interdigitated flow field Flow channel in electrode
The flow field design of vanadium redox flow battery is a critical issue for performance optimization of the battery. In this work, an interdigitated flow field is designed in carbon felt porous electrode with active area of 57.5 cm2 and simulated by a three-dimensional multi-physical model. The “leaf” shaped ionic concentration and voltage distribution patterns are observed in the numerical results and analyzed to understand the cell perfor mance. Compared with other two conventional designs, the interdigitated flow field in bipolar plate and no flow field designs, fluid pressure drop of flow field designed in electrode is the least and the system efficiency shows the best over the major operation range of electrolyte flow rates. The flow field designed in electrode greatly lowers sealing pressure demand for vanadium redox flow battery with better reliability and shows potential benefit for large-scale cell stack design and energy storage system application.
1. Introduction Energy storage technologies are critical for delivery of efficient renewable energy sources such as wind and solar. Among various existing energy storage technologies, vanadium redox flow battery (VRB) shows great potential for large scale energy storage due to its advantages of high energy efficiency, long cycle life and independence of capacity and power ratings. Much progress has been made during the commercialization process of VRB in these years, including key mate rials, cell stack design and system integration technologies and a number of demonstrations of VRB power plants have been installed [1–10]. However, VRB system performance and lifetime still need to be improved which attracts more and more attention of researchers all over the world [11–32].
As a flow battery, VRB stores its electrochemical energy in the electrolyte solutions with vanadium ions of different valences. For supplying ionic reactants during charge and discharge processes, flow field design of the cell stack plays a significant role to enhance the battery performance and lower the pump consumption for recirculation the electrolyte fluid. Much effort has been devoted to various flow field studies, including serpentine, parallel and interdigitated flow channels [33–53]. Among the mentioned designs, the interdigitated flow field is beneficial for both distributing even ionic reactants and lowering fluid pressure drop, which attracts much research interest recently [33–47]. M. Messaggi et al. developed a three-dimensional (3D) VRB model coupling fluid dynamics and electrochemistry to study 25 cm2 inter digitated and serpentine flow fields [33]. S. Maurya et al. evaluated serpentine, interdigitated and conventional (without flow pattern) flow
* Corresponding author. School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, 610023, China. ** Corresponding author. Hydrogen and Fuel Cell Institute, University of Electronic Science and Technology of China, Chengdu, 610023, China. E-mail addresses: [email protected] (C. Yin), [email protected] (H. Tang). https://doi.org/10.1016/j.jpowsour.2019.227023 Received 23 October 2018; Received in revised form 28 June 2019; Accepted 14 August 2019 Available online 5 September 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.
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Table 3 Parameter values for the calculation. Symbol D2
Symbol
Parameter
Value
a
Channel width
1 mm
b
Electrode land width
5 mm
c
Channel height
4 mm
Lch
Channel Length
45 mm
d1
Current collector thickness
2 mm
d2
Membrane thickness
0.1 mm
Lx
Width of reactive area
115 mm
Ly
Length of reactive area
50 mm
am
Width of manifold
5 mm
ion diffusivity in electrolyte
2.4 � 10
10
m2/s [54]
ion diffusivity in electrolyte
2.4 � 10
10
m2/s [54]
3.9 � 10
10
m2/s [54]
3.9 � 10
10
m2/s [54]
V
D3
V
3þ
D4
VO2þ ion diffusivity in electrolyte
D5
VOþ 2 ion diffusivity in electrolyte
DHSO4
HSO4 ion diffusivity in electrolyte
k1
Standard rate constant: positive
9.31 � 10 m2/s [54] 1.33 � 10 9 m2/s [47] 6.8 � 10 7 m/s [54]
k2
Standard rate constant: negative
1.7 � 10
Proton diffusivity in electrolyte
7
m/s [54]
1.004 V
Equilibrium potential: VO
ε
Porosity of the electrode
0.9
df
Carbon fiber diameter
1.76 � 10
KCK
Carman-Kozeny constant
5.55 [54]
As
Active specific surface area
10000 m
ρ
Density of the electrolyte
1300 kg/m3
μ
Viscosity of the electrolyte
4.928 � 10
σs
Conductivity of the solid phase
1000 S/m
σmem
Conductivity of membrane
10 S/m
T
Temperature
298 K
cinitial þ
1.5 mol/L
cinitial þ;Hþ
Initial concentration of vanadium ions (positive) Initial concentration of vanadium ions (negative) Initial concentration of proton (positive)
cinitial ;Hþ
Initial concentration of proton (negative)
4 mol/L
iinward
Operating current density
60 mA/cm2
Pump efficiency
0.9 [33,47]
cinitial
ψ Pump
2þ
9
E01
E02
Table 1 Default parameters of the VRB cell’s geometric structure.
Value
2þ
DHþ
Fig. 1. Geometric structure of VRB modeling: (a) simulated components do mains of a single cell including current collector, porous electrode with inter digitated flow channels, membrane separator, and electrolyte fluid; (b) detailed mesh at the corner; (c) xz cross section.
Parameter
=VOþ 2
Equilibrium potential: V2þ =V3þ
0.255 V 5
m [54]
1
3
Pa s
1.5 mol/L 4 mol/L
Table 2 Source and sink terms for each species. Source term
Positive electrode
Negative electrode
SV2þ
–
j2 =F
SV3þ
SVO2þ
SVOþ2
j1 =F
j1 =F
SHþ
SH2 O
SSO24
j2 =F
–
2j1 =F j1 =F 0
–
Fig. 2. Comparison between the numerical and experimental performance of single VRB cell with flow through design during charge and discharge pro cesses. The electrolyte flow rate is constant at 6 ml/s and the operating current density is 60 mA/cm2.
– – – 0
flow field designs with the 1 kW VRB cell stack experiment [40]. S. Kumar et al. experimentally studied the effect of flow field for flow batteries with large cell sizes [41]. M. D. R. Kok et al. developed a model of a hydrogen-bromine redox flow battery cathode with interdigitated flow channels to investigate the effect of electrode morphology and overall architecture of the cell [42]. E. Knudsen et al. modeled the three-dimensional (3D) fluid dynamics of interdigitated flow fields for VRB, respectively [44]. R. J. Kee et al. developed a general model to evaluate flow uniformity and pressure drop within interdigitated-channel structures of redox flow batteries [45]. Q. Xu et al. studied the performance of VRB with different flow fields numerically [47]. In our previous work, interdigitated flow field in bipolar plate (FFplate design) of VRB was investigated with internal parameters distri butions by a 3D coupled model [46]. The mentioned studies mainly
field types with carbon felt electrodes and compare their performance with a serpentine flow field using carbon paper electrodes [34]. J. L. Barton et al. studied both flow-through and interdigitated flow fields experimentally to quantify the mass-transfer coefficient as a function of electrolyte velocity and viscosity [35]. J. Houser et al. studied flow field architecture influence on mass transport and system performance in redox flow batteries [36,43]. A. Bhattarai et al. investigated different designs of flow channels in porous electrode of VRB and relevant energy efficiency of the system [37]. Y. K. Zeng et al. reported the performance enhancement of iron-chromium redox flow battries with interdigitated flow fields due to the uniform catalyst distribution and high mass transport limitation [38]. X. You et al. developed a 3D model of a half-battery with an active area of 900 cm2 to explore the design rules of flow fields [39]. D. Reed et al. studied performance of interdigitated 2
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Journal of Power Sources 438 (2019) 227023
Fig. 3. Simulated distribution patterns of VRB cell with interdigitated flow channels designed in porous electrode at SOC of 0.5 during discharge under electrolyte flow rate of 1 ml/s: electrolyte fluid velocity (a) and pressure (b) distribution patterns in the xy plane with z value of 8.1 mm inside the positive porous electrode; volume flow rate distributions of inlet channels (c) and outlet channels (d) at various channel walls.
focus on flow field designs in the bipolar plate rather than in the porous electrode. Only several literatures concern flow channels inside porous media [37,40]. Actually, it is possible to construct flow field in VRB carbon felt porous electrode due to its thickness of several millimeters for better reactants distributions and lower fluid pressure drop. In this study, interdigitated flow field is designed in carbon felt porous elec trode (FF-electrode design) and cell performance is analyzed in terms of numerical parameters distributions. Meanwhile, VRB system efficiency of FF-electrode design is compared with that of FF-plate design and no flow field design (No-FF) which includes no flow field in plate or elec trode. Based on the simulation results, the coupling factors including flow field design, reactants mass transport and cell stack performance are better understood for optimization of VRB system efficiency.
defined as inlet channels (chin) and the other ten connected to the outlet manifold are outlet channels (chout). The detailed dimensions are listed in Table 1. 2.2. Model assumptions and boundary conditions The assumptions for the proposed model are listed below: (1) SOC (state of charge) which defines vanadium ion concentrations varies with time. When the electrolyte tank is sufficiently large, the change of SOC is relatively small during a very short period of operation. Thus the dy namic model could be simplified as a stationary one to reduce solving time for the large geometry without losing much accuracy; (2) The presented model is isothermal; (3) The diluted-solution approximation is applied; (4) The fluid flow is treated as incompressible flow; (5) The material properties of electrode, electrolyte and membrane domains are homogeneous; (6) Only protons could cross over through the membrane; (7) Side reactions such as hydrogen and oxygen evolutions are neglec ted; (8) Electrolyte volume change due to water permeation or water drag through membrane is ignored. As the presented model is stationary, voltages during charge and discharge processes are calculated with various inlet boundary condi tions of SOC, which are expressed as:
2. Model development 2.1. Model geometry The 3D numerical model of VRB is based on a single cell structure as shown in Fig. 1, of which the dimensions are comparable to those of the experimental ones [16,46]. The VRB cell is composed of bipolar plates, porous electrodes and a proton exchange membrane. Electrolyte solu tions with vanadium ions of different valences, VO2þ /VOþ 2 for the pos itive and V 2þ /V 3þ for the negative, are indicated with blue and green color respectively. The main half-reactions of the two electrodes for VRB are as follows:
cinlet ¼ cinitial ð1 3
SOCÞ
(4)
Negative electrode: V 3þ þ e ⇌V 2þ
(1)
cinlet ¼ cinitial ð1 4 þ
SOCÞ
(5)
þ Positive electrode: VO2þ þ H2 O⇌VOþ 2 þ 2H þ e
(2)
cinlet ¼ cinitial SOC 5 þ
(6)
initial initial cinlet SOC þ;H þ ¼ cþ;H þ þ cþ
(7)
initial initial cinlet SOC ;H þ ¼ c ;H þ þ c
(8)
(3)
cinlet ¼ cinitial SOC 2
The electrolyte fluid flows into and out of the reaction area through the transverse manifold shown in Fig. 1 (a). There are twenty flow channels constructed in the porous electrode for both positive and negative sides, ten of which are connected to the inlet manifold and 3
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Journal of Power Sources 438 (2019) 227023
where K is the permeability coefficient of the porous electrode. The permeability of porous media is given by the Carman-Kozeny equation as [8]: K¼
d2f ε3
(11)
εÞ2
KCK ð1
where ε is the porosity; df is the fiber diameter; KCK is the CarmanKozeny constant, which characterizes the fibrous material and is used as a fitting parameter. ! The molar fluxes N i of ionic species i (with i indicating V 2þ , V 3þ , þ VO2þ , VOþ 2 , H and HSO4 ) in the fluid flow through porous electrode is given by a modified Nernst-Planck equation [8]: ! Ni ¼
ε3=2 Di rci
zi ci ε3=2 Di FrΦl ! þ u ci RT
(12)
Thus, the conservation for species i in the porous electrodes can be written as: ! r⋅ Ni ¼
(13)
Si
where Si is the source term of species i (defined in Table 2); ci, Di and zi are the bulk concentration, diffusion coefficient and charge of species i, respectively; Φl is the ionic potential; F is the Faraday’s constant and R is the ideal gas constant. With the assumption that the electrolyte is electroneutral, it gives: X zi ci ¼ 0 (14) i
For charge balance, the charge that enters the electrolyte equals to the one which leaves the solid phase: r ⋅ is þ r⋅il ¼ 0 Fig. 4. Two-dimensional distribution patterns of ions concentrations at mid of 2þ positive electrode in the xy plane with z value of 8.1 mm: (a) VOþ and 2 , (b) VO (c) Hþ. The simulated VRB cell is designed with interdigitated flow channels in porous electrode and operated at SOC of 0.5 during discharge under electrolyte flow rate of 1 ml/s.
! ! where i l and i s are the local current densities corresponding to the liquid and the solid phases, respectively. For electrolyte, the flow of charged species results in the current in ! solution i l expressed as: X ! ! il¼ zi F N i (16)
inlet inlet where cinlet and cinlet are the inlet concentrations, corre 5 2 , c3 , c4 initial sponding to V 2þ , V 3þ , VO2þ and VOþ and cinitial 2 ions, respectively; cþ are the initial concentrations of vanadium ions at positive and negative inlet inlet portions with 1.5 mol/L; cinlet þ;Hþ and c ;Hþ are the proton concen
cinitial þ;Hþ
i
For solid phase, the current density is calculated with the Ohm’s law: ! is¼
cinitial ;Hþ
trations at positive and negative inlets, while and are the initial proton concentration values at two reaction sides both set as 4 mol/L. The incipient concentration of SO24 in the positive electrolyte is slightly different from the negative one.
2.3.1. Transport in flow channels Navier-Stokes equations for incompressible fluid are applied to simulate the electrolyte flow in channel and manifold domains as follows: rp þ μr2 ! u
(9)
u¼
rp
j1 ¼ r⋅il ¼
r⋅is
(18)
j2 ¼ r⋅il ¼
r⋅is
(19)
! N Hþ ¼
σmem F
rΦmem
(20)
where σmem and Φmem are the conductivity and electric potential of the membrane, respectively.
2.3.2. Transport in porous electrodes For the fluid flow in porous media, velocity ! u can be expressed with Darcy’s law: K
(17)
2.3.3. Transport in membrane For membrane, transport process of proton is considered and the current conservation equation is expressed as:
where ρ is the fluid density; ! u is the velocity; p is the pressure and μ is the dynamic viscosity of the fluid.
μ!
εÞ3=2 σ s rΦs
ð1
where σ s indicates conductivity and Φs indicates electric potential for solid phase. For positive and negative electrodes, j1 and j2 are transfer current densities for the electrochemical reactions at the surfaces of porous electrodes as current sources,
2.3. Governing equations
ρð! u ⋅ rÞ! u¼
(15)
2.3.4. Reaction kinetics As for the reversible electrochemical reactions, the transfer current densities of the positive electrode j1 and the negative electrode j2 are
(10) 4
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Journal of Power Sources 438 (2019) 227023
Fig. 5. Electric voltage distributions in the xy plane with varied z values: (a) 12.1 mm (at the outer surface of the positive current collector), (b) 8.1 mm (at the midlayer of the positive electrode), and (c) 4 mm (at the mid-layer of the negative electrode). Local current vector distributions at the outer surface of the positive current collector: (d) x direction, (e) y direction and (f) z direction. The simulated VRB cell is designed with interdigitated flow channels in porous electrode and operated at SOC of 0.5 during discharge under electrolyte flow rate of 1 ml/s.
given by the Butler-Volmer laws [54]: � � �� � � pffiffiffiffiffiffiffiffi F η1 F η1 j1 ¼ As Fk1 cs4 cs5 exp exp 2RT 2RT j2 ¼ As Fk2
� � � pffiffiffisffiffiffiffisffi F η2 c2 c3 exp 2RT
� exp
F η2 2RT
The quantities csi indicate vanadium-species concentrations at the liquid-solid interfaces in the porous regions, which are generally different from the bulk values due to additional transport resistance (from the bulk solution to the interfaces). They can be related to the bulk concentrations, ci , by approximately balancing the rate of reaction with the rate of diffusion of reactant to (or from) the electrode surface at steady state. For the positive reaction, the balance is: � � � � �� εk1 s F η1 F η1 c4 cs4 ¼ c4 exp cs5 exp (27) γ4 2RT 2RT
(21)
�� (22)
of which, As is the specific active surface area, cs2 , cs3 , cs4 and cs5 are the surface concentrations of V 2þ , V 3þ , VO2þ and VOþ 2 ions which are generally different from the bulk concentration ci, k1 and k2 are the re action rate constants. η1 and η2 represent the activation overpotentials of positive and negative reactions which are defined as:
η1 ¼ Φ s
Φl
E1
(23)
η2 ¼ Φ s
Φl
E2
(24)
c5
E2 ¼ E02
� � � F η1 cs5 exp γ5 2RT
εk1
� cs4 exp
F η1 2RT
�� (28)
where γ4 ¼ D4 =dp , γ5 ¼ D5 =dp , dp is the average pore diameter of the porous structure, Di is the diffusion coefficient for species i in solution. The quantities γi (in m/s) measures the rate of reactant delivery to or from the surfaces by diffusion from the bulk [55,56]. Combining equations (27) and (28), the concentrations of VO2þ and VOþ 2 ions at the liquid-solid interface are expressed as:
where E1 and E2 are the equilibrium potentials for the positive and negative reactions, calculated by the Nernst equations: � � RT c5 E1 ¼ E01 þ ln (25) F c4 � � RT c2 ln c3 F
cs5 ¼
cs4 ¼
c þ εk1 e FðΦs Φl E1 Þ=ð2RTÞ ðc4 =γ5 þ c5 =γ4 Þ �4 . . � 0 0 1 þ εk1 e FðΦs Φl E1 Þ=ð2RTÞ γ5 þ eFðΦs Φl E1 Þ=ð2RTÞ γ 4
(29)
cs5 ¼
0 c þ εk1 eFðΦs Φl E1 Þ=ð2RTÞ ðc4 =γ5 þ c5 =γ4 Þ �5 . . � 0 0 1 þ εk1 e FðΦs Φl E1 Þ=ð2RTÞ γ5 þ eFðΦs Φl E1 Þ=ð2RTÞ γ 4
(30)
0
(26)
where E01 and E02 are the standard potentials for the positive and negative reactions with temperature of 298.15 K and pressure of 1 bar. 5
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Journal of Power Sources 438 (2019) 227023
Fig. 6. Schematics of three VRB design geometries: (a) flow field in electrode (FF-electrode); (b) flow field in plate (FF-plate); (c) no flow field (No-FF). Electrolyte pressure distributions of the three designs under flow rate of 10 ml/s: (d) FF-electrode; (e) FF-plate; (f) No-FF. Numerical results for comparison of the three designs: (g) charge and discharge cell voltage curves with 1 ml/s electrolyte flow rate; (h) pressure drop under various electrolyte flow rates.
Similar expressions of cs2 and cs3 could be obtained for the negative electrode. Relevant parameters for the simulation are shown in Table 3.
results of charge and discharge voltages, the linear relationship between t and SOC are applied as follows: SOC ¼ SOC0 þ
2.4. Numerical details
t ðSOCe t0
SOC0 Þ
(31)
where t0 represents the total charge or discharge time; SOC0 and SOCe indicate the beginning and ending SOC status. In the experiment, SOC is determined by open-circuit cell voltage VOC , and the SOC range is 0.2–0.8 for charge or discharge processes. The experiment used carbon felt electrode dimension is 96 mm � 60 mm � 4 mm (reaction area: 57.6 cm2), while for comparison, the calculated No-FF electrode size is 115 mm � 50 mm � 4 mm (reaction area: 57.5 cm2) according to Lx and Ly in Table 1. The starting electrolyte solutions for positive and negative vanadium half-cells are aqueous solution of 1.5 mol/L vanadyl sulphate VOSO4 and 1.5 mol/L tri-sulphate V2(SO4)3, respectively, with 2 mol/L sulfuric acid H2SO4 for both sides. Constant electrolyte flow rate of 6 ml/ s is applied and the operating current density is 60 mA/cm2. As shown in Fig. 2, the numerical results fit the experimental one well with small deviations less than 1% during the charge and discharge processes. It is known that the aspect ratio affects overall pressure drop of the stack significantly with constant reaction area, which eventually im pacts on the VRB system efficiency. The aspect ratio of 115:50 shows a decrement of about 12% fluid pressure drop compared with that of 96:60, while the numerical charge and discharge cell voltage curves for the two aspect ratios show almost identical profiles with voltage devi ation less than 1 mV. Besides, the fluid pressure decreases almost evenly from the inlet to the outlet due to the well distributing function of the
The variables in model equations were solved with the COMSOL Multiphysics® package using finite-element method. The sources terms and physical properties of the governing equations are written in the user’s code. The relative error tolerance was set to 1 � 10 6. 3. Results and discussion In our previous work, interdigitated flow field constructed in bipolar plate of VRB was investigated with various design and operation pa rameters. In the present study, we focus on the interdigitated flow field designed in porous electrode and analyze the cell performance and in ternal parameters distributions. Furthermore, VRB performance and system efficiency of FF-electrode design are compared with those of FFplate and No-FF designs. 3.1. Model validation A single VRB cell with no flow field is assembled for charge and discharge test to validate the proposed modeling. The cell structure, VRB test system and detailed experimental procedures can be found in our previous work [16,46]. To compare the experiment and numerical 6
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Journal of Power Sources 438 (2019) 227023
Fig. 7. The cross section views of VOþ 2 ion concentration distributions at inlet, mid and outlet of porous electrode: (a) flow field in electrode; (b) flow field in plate. The FF-electrode and FF-plate VRB cells are operated at SOC of 0.5 during discharge under electrolyte flow rate of 1 ml/s.
Fig. 8. Efficiency comparison of three different VRB cells of FF-electrode, FF-plate and No-FF designs: (a) stack efficiencies and (b) system efficiencies under various electrolyte flow rates.
transverse channels (width: 5 mm).
channel direction (X axis). Fig. 3 (b) shows the fluid pressure distribu tion in porous electrode, where chin(n)_end and chout(n)_end indicate dead end walls of the nth inlet and outlet channels; chin(n)_left and chin(n)_right indicate the left and right side walls of the nth inlet channel; chout(n)_left and chout(n)_right indicate the left and right side walls of the nth outlet channel; Tm_inlet and Tm_outlet indicate the inlet and outlet transverse manifold, respectively. The electrolyte pressure drop be tween the channel dead end (chin(n)_end/chout(n)_end) and the trans verse manifold (Tm_inlet/Tm_outlet) shows much larger than that between the neighbored inlet and outlet channel walls (eg. chin(n)_right and chout(n)_left). For the volume flow rates, values of liquid flowing through the channel side walls show an order of magnitude larger than those of dead end walls shown in Fig. 3 (c and d). To note, volume flow rate through right side wall of inlet channel is about 20% larger than that of left side wall, due to the asymmetric manifold structure where inlet locates at bottom left corner of the VRB cell and outlet at the top right corner.
3.2. Parameter distributions of VRB with FF-electrode design In this part, parameter distribution patterns of FF-electrode design including fluid flow, reactant ion concentration, electric voltage and current density are presented and analyzed at SOC of 0.5 during discharge with electrolyte flow rate of 1 ml/s. Fig. 3 (a) shows fluid velocity distribution in porous media which is critical to convective mass transport of vanadium ions for electro chemical reactions at interface of porous electrode and electrolyte. At the dead end of the inlet and outlet channels indicated by brown dash circles (a1) and (a2) respectively, the fluid velocities show much larger than elsewhere and the electrolyte flows along the channel direction (Y axis). While for the two sides of the nth inlet channel (chin(n)) indicated as (a3) and (a4), fluid velocities show almost an order of magnitude less than those of (a1) and (a2) and the electrolyte flows perpendicular to the 7
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Journal of Power Sources 438 (2019) 227023
Fig. 4 shows the two-dimensional distribution patterns of ions con centrations in positive electrode. During discharge, electrochemical re þ 2þ leading to similar action consumes VOþ 2 and H and generates VO distribution patterns for the former two ions and pattern with reversed color for the third ion. Taking the VOþ 2 ion as example, the concentration distribution forms parallel multiple “leaf” shaped patterns (Fig. 4. (a3)) with each outlet channel as the symmetrical line of the “leaf”. The VOþ 2 ion concentrations at the dead end of inlet channels (Fig. 4. (a1)) show larger values than those at the dead end of outlet channels (Fig. 4. (a2)). The VOþ 2 ion concentrations at right sides of the multiple “leaf” regions are lowest all over the pattern, of which the values are even less than those at the outlet portion of transverse manifold. The reactant ions concentration distributions have strong impact on the local electric potential and current collections. Electric voltage at the outer surface of the positive current collector (Fig. 5 (a)) shows the largest value near the inlet manifold, medium around the outlet mani fold and lowest at the mid of the current collector. This distinct electric voltage distribution is due to the lowest reactive ions concentrations in the multiple “leaf” regions. Electric voltage distributions inside the positive and negative porous electrodes (Fig. 5 (b) and (c)) accord well 2þ with the reactive ions concentration distributions of VOþ 2 and V . At right sides of the multiple “leaf” regions (each outlet channel as the symmetrical line), the lowest electric potential for positive electrode and highest value for negative electrode indicate the maximum localized reaction polarizations. As for the current at the outer surface of positive current collector, the current vector at z direction shows much larger than those at x and y directions. The z direction currents at the land areas are about 15–20% higher than those at channel areas shown in Fig. 5 (f), which leads to periodic current flows at x direction for each channel-land cycle in Fig. 5 (d). As for the y direction vector in Fig. 5 (e), current shows around zero at mid of the plate (along y direction in xy plane) and flows toward the mid from both up and bottom of the plate. The maximum values of current vectors at x and y directions reach 30% and 40% that of z di rection, respectively.
compared in Fig. 6 (e). At the same operating flow rate, the three de signs’ pressure drop performs as No-FF > FF-plate > FF-electrode. Pres sure drop of No-FF design increases linearly with flow rate, but the curve shows non-linearity for FF-plate and FF-electrode designs. At flow rate of 1 ml/s, pressure drops of No-FF and FF-plate design are 7 and 3.5 times that of FF-electrode design respectively, while the numbers become 4.9 and 3.0 at flow rate of 5 ml/s and 3.6 and 3.4 at flow rate of 10 ml/s. The overall pressure drop of a cell stack is composed of that of manifolds, flow channels and porous media. For No-FF design, pressure drop of manifold is neglectable compared to that of fluid penetrating the porous electrode, however for FF-plate and FF-electrode designs, pressure drop through the channel is dominant and that of manifold is comparable to that of porous media. To analyze the performance difference between FF-electrode and FFplate designs, the cross section views of ion concentration distributions at inlet, mid and outlet of porous electrode are shown in Fig. 7. Generally speaking, reactive ions distributes more evenly in FF-electrode cell than in FF-plate cell, with comparisons of distribution patterns and ion con centration range at the same location. For the FF-electrode design with channels embedded in the porous electrode, the VOþ 2 concentration decreases from each inlet channel to the neighbored outlet ones. The much lowered reactants concentrations appear at two thin layers near the upside and bottom side of the porous electrode as a result of the noslip wall boundary effect in Fig. 7 (a). In the FF-plate cell with channels on the top porous electrode, the electrolyte flow path from inlet channel to the outlet one is different from that of FF-electrode leading to much lowered reactants concentrations near the membrane (Fig. 7 (b)). More detailed parameter distribution characters of FF-plate design can be found in our previous work [46]. The greater reactant concentration uniformity of FF-electrode than that of FF-plate results in better charge and discharge performance of the former. 3.4. Energy efficiency analysis With the cell voltage performance and pressure drop characteristic during a charge-discharge cycle, the energy efficiencies of VRB stack and system are investigated. The round trip efficiency of the stack is defined as:
3.3. Comparison study of different designs In the former part, the distribution patterns of FF-electrode design are investigated. In this section, cell stack performance and system ef ficiency are analyzed with comparisons of FF-electrode, FF-plate and No-FF designs, of which schematic design structures are shown in Fig. 6 (a, b and c). With identical operating current density (60 mA/cm2) and electrolyte flow rate (1 ml/s), charge and discharge cell voltages of the three designs differ from each other, of which the cell performance sequence is No-FF > FF-electrode > FF-plate. At charge SOC range of 0.1–0.9, voltages of FF-plate and FF-electrode show 10–32 mV and 1–4 mV larger than that of No-FF design, respectively. While at discharge SOC range of 0.9–0.1, voltages of FF-plate and FF-electrode show 10–33 mV and 1–4 mV less than that of No-FF design, respec tively. In the No-FF design of this work, the transverse manifolds for electrolyte inlet and outlet are constructed as is commonly applied in an actual VRB cell stack design. Those No-FF designs without transverse manifolds may lead to significantly uneven electrolyte flow distributions and lowered cell performance [47,50] which is not considered in this study. This is the major reason that No-FF design performs better than those with flow channels. To note, the applied current density is calculated with operating current divided by electrode area (xy plane) which are not the same for identical width (Lx) and length (Ly) of reactive area in the model geometry. For No-FF and FF-plate designs, the electrode area isLx � Ly , but that of FF-electrode design writes as ðLx �Ly Þ ða �Lch �nch Þ where nch is the total number of the channels. As a result, the FF-electrode design loses a part of electrode area (area of the channels: a � Lch � nch ) compared with No-FF and FF-plate designs, although its cell voltage performs as well as the No-FF design. The cell stack pressure drop with various electrolyte flow rates are
ψ stack ¼
Wdis Wch
(32)
Wdis and Wch are the output energy during discharge and input energy during charge of the stack, respectively: Z t0;dis Wdis ¼ Idis ðtÞVdis ðtÞdt (33) 0
Z
t0;ch
Wch ¼
Ich ðtÞVch ðtÞdt 0
(34)
where I and V are the stack current and voltage, respectively. Considering the pump consumption Wpump during the chargedischarge cycle, the system energy efficiency can be defined as:
ψ system ¼
Wdis Wpump;dis Wch þ Wpump;ch
The pump consumptions are calculated by: R t0;dis Qdis ðtÞΔpdis ðtÞdt Wpump;dis ¼ 0
ψ pump
R t0;ch Wpump;ch ¼
0
Qch ðtÞΔpch ðtÞdt
ψ pump
(35)
(36)
(37)
where Q is the electrolyte flow rate and Δp is the total fluid pressure drop, which are time independent in this work. 8
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Journal of Power Sources 438 (2019) 227023
As shown in Fig. 8 (a), the stack efficiency increases with electrolyte flow rate as more efficient mass transfer of reactants enhances the electrochemical reactions. The FF-plate cell performs worst, especially at low flow rates, and stack efficiency of FF-electrode is close to that of No-FF design which is also suggested by the charge-discharge voltage curves. The system efficiency of FF-electrode is a little less than that of No-FF cell at low electrolyte flow rate (1 ml/s), but performs much better than the No-FF one at high flow rate (10 ml/s). In addition, VRB stack with large active area shows 3 to 5 times higher pressure drop for the flow through electrode design (No-FF) than its interdigitated (FFelectrode) counterpart [40], while the value is about 3 times for small active area stack shown in Fig. 6 (h). Therefore, the benefit of FF-electrode interdigitated design concluded by the small stack in the presented work tends to be consistent or even enhanced for the large cell stack design. In short, the pump work plays a significant role in the system efficiency of VRB and the low pressure drop design of the cell stack is helpful for the overall system efficiency optimization.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
4. Conclusion
[22]
In this work, VRB cell with interdigitated flow field constructed in carbon felt porous electrode (FF-electrode) is studied with a threedimensional (3D) coupled modeling. The “leaf” shaped reactants con centrations and electric potential distribution patterns appear in the FFelectrode design, leading to worse local performance in the mid part than that around inlet and outlet portions of the porous electrode. The stack and system efficiencies of FF-electrode design are compared with those of interdigitated flow field in bipolar plate (FF-plate) and no flow field (No-FF) designs. In terms of convective mass transport, FFelectrode stack voltage performs as well as No-FF design with trans verse inlet and outlet manifolds and FF-plate stack shows lowest stack voltage efficiency. Considering pump consumption required for the overall pressure drop, system efficiency of FF-electrode is a little less than that of No-FF cell at low flow rate, but shows much better at high ones and FF-plate design performs lowest system efficiency. On one hand, the FF-electrode design greatly lowers sealing pressure demand for VRB stack and enhances system performance; on the other hand, this design wastes a part of electrode for reaction which is constructed with flow channels. Furthermore, the benefit of FF-electrode interdigitated design concluded by the small stack in the presented work is reliable for the large cell stack development. The robust VRB stack design with FFelectrode and the system featured with optimized operating conditions will be discussed in the forthcoming work.
[23] [24] [25]
Acknowledgement
[44]
This work is sponsored by National Key R&D Program of China (No.2018YFB1502700), Science and Technology Program of Sichuan Province (No.2019YFG0002 and No.2017CC0017) and Initiative Sci entific Research Program of University of Electronic Science and Tech nology of China (No.ZYGX2018KYQD207 and No. ZYGX2018KYQD206).
[45] [46] [47] [48] [49]
[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
[50] [51] [52]
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