Mass Transfer Behaviors and Battery Performance of a Ferrocyanide-Based Organic Redox Flow Battery with Different Electrode Shapes

Abstract

The ferrocyanide-based organic redox flow battery (ferrocyanide-based ORFB), based on electrochemistry, has become a potential energy storage technology due to its low price, eco-friendliness, safety, and convenience. However, its low efficiency and poor mass transfer performance hinder the application of the ORFB. The influence of the electrode shape (trapezoid, sector, and rectangle) on the mass transfer and battery performance are studied based on a numerical model, which is verified by the experiments. The results show that battery performance of the trapezoid electrode is better than that of the sector and rectangle electrode. The discharge voltage of the rectangle battery is the lowest, and the discharge voltage of the trapezoid battery is the highest. The discharge voltage of the rectangle battery is 4.47% lower than that of the trapezoid battery. The uniformity factor value of the trapezoid battery is 26.9% higher than that of the rectangle battery. The trapezoid shape is the best design for the electrode, contributing to the application of the ferrocyanide-based ORFBs.

1. Introduction

As the climate change has become increasingly serious over past decades, wind energy and solar energy have received greater attention. The disadvantages of intermittent discontinuity and instability hinders their large-scale application. Energy storage technology, based on the electrochemistry, is regarded as one of the best methods to solve this problem [1,2,3]. With its safety, high efficiency, and scalability, the redox flow battery stands out among many large-scale energy storage technologies [4]. The ferrocyanide-based ORFB offers the advantages of lower cost and improved environmental protection [5,6], making it increasingly popular [7,8,9]. However, the low solubility of the organics leads to poor mass transfer behaviors in the porous electrodes, which results in the low battery performance of the ferrocyanide-based ORFB. It is significant to investigate the mass transfer performance in the porous electrodes for the commercial application of this technology.

It is well known that the transport and distribution of the reactants are influenced by the flow fields of the electrodes [10]. In the past few decades, researchers have conducted numerous investigations on electrodes with different flow fields. Xu et al. [11] studied the battery without a flow field and with a serpentine flow field, discovering that the energy efficiency of the battery with the serpentine flow field was 5% higher than that of the battery without a flow field. Macdonald et al. [12] investigated the battery performance of a interdigital flow field and a serpentine flow field using a two-dimensional model based on Darcy’s law and reported that the pressure drop of the interdigital flow field was lower than that of the serpentine flow field. Luo et al. [13] combined the serpentine flow field with the interdigital flow field to obtain a new flow field, determining that the system energy efficiency of the new flow field is 2%-20% higher than that of the interdigital or serpentine flow field. Ali et al. [14] investigated the influence of the number of serpentine flow field channels on the all-vanadium flow battery performance and pointed out that the quadruple serpentine design resulted in the highest power-based efficiency. Lu et al. [15] studied a blocked serpentine flow field and reported that 1.4 mm was the most appropriate block height for all vanadium flow batteries. Lee et al. [16] researched the effects of the serpentine channel size on the all-vanadium flow battery performance and reported that a smaller channel size could lead to a high power-based efficiency. Lee et al. [16] also pointed out that the higher electrolyte flow rate could result in better battery performance.

Besides the researches on the flow fields, the influence of the electrode structure on the battery performance is another research hotspot. Chu et al. [17] studied the effect of electrode shape (trapezoid, sector, rectangle) on battery performance, discovering that the battery and mass transfer performance of the sector-shaped electrode were the best. However, the flow fields were not investigated in the study of Chu et al. [17]. Yin et al. [18] designed a ‘leaf’ shaped electrode and compared it with the bipolar plate interdigital flow field and no flow field. The result showed that the pressure drop of the ‘leaf’ shaped electrode was lower than that of either the interdigital flow field or no flow field. Gurieff et al. [19] investigated the performance of the battery with three different electrode structures (trapezoid, sector, and rectangle) and concluded that the different electrode structures could lead to different battery performance and mass transfer behavior. Ali et al. [20] studied the effect of electrode thickness on the battery performance, and the results showed that when the electrode thickness was 1 mm, the maximum efficiency of the battery was 96.8%. Lu et al. [21] designed a battery with an asymmetric electrode structure. Their research found that this electrode can reduce the capacity reduction of the vanadium ions, increasing the overall energy efficiency from 83% to 87.7%.

In reference [14], the influence of electrode shape on the ferrocyanide-based ORFB battery performance and mass transfer process were studied. However, the electrodes in reference [14] are without flow fields. The flow fields play important roles in the mass transfer process. Therefore, it is important to investigate the influence of the electrode shape, with flow fields, on the mass transfer behaviors in the ferrocyanide-based ORFB. In the previous research, most studies concentrated on the flow fields. Few investigations were conducted on the influence of the electrode shape with flow fields. In this paper, the influence of the electrode shape (trapezoid, sector, and rectangle), with the electrode equipped with flow fields, on the mass transfer and battery performance is investigated,. The charge-discharge voltage, over voltage, polarization curve, uniformity factor, average concentration, power, and efficiency of the three kinds of batteries are presented.

2. Model

The conventional ferrocyanide-based ORFB, consisting of an ion exchange membrane, a positive electrode, negative electrodes, two collectors, two pumps for circulating electrolyte liquid, and two electrolyte flow tanks, is shown in Figure 1,. The type of ion exchange membrane in this work is a Nafion^®212 membrane, and the supporting electrolyte is the KOH. The electrolyte was transferred to the porous electrode by the pumps, and the porous electrode can provide more reaction areas for the electrolyte charging and discharging of the electrode. Then, the electrolyte is pumped from the electrode to the tank. Figure 1 shows the three electrode shapes (trapezoid, sector, and rectangle).

The electrolytes used in this organic redox flow battery are presented as follows. The negative electrolyte is 2,6-DHAQ/2,6-reDHAQ, and the positive electrolyte is Fe(CN)₆⁴⁻/Fe(CN)₆³⁻. The electrochemistry reactions occurring in the charging and discharging process are presented as follows:

$$\mathrm{positive}: \mathrm{F}\mathrm{e}{\left(\mathrm{C}\mathrm{N}\right)}_6^{4-}-e^{-}{\underset{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}}{\overset{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}}{\rightleftarrows }}}^{ }\mathrm{F}\mathrm{e}{\left(\mathrm{C}\mathrm{N}\right)}_6^{3-}$$

(1)

$$\mathrm{negative}: 2,6-\mathrm{DHAQ} +2{\mathrm{e}}^{-}\underset{\mathrm{discharge}}{\overset{\mathrm{charge}}{\rightleftarrows }}2,6-\mathrm{reDHAQ}$$

(2)

In this paper, a three-dimensional steady-state model of the organic redox flow battery is built, and the influences of the different electrode shapes (trapezoid, sector, rectangle) on the battery performance and mass transfer behaviors are investigated. However, the mass transfer process and the coupling of hydrodynamics are very complex. To make the model simple and accurate, the following reasonable assumptions are made:

(1)

It is assumed that only K⁺ can pass through the ion exchange membrane.

(2)

It is assumed that the temperature of all regions of the ferrocyanide-based ORFBs is equal and distributed uniformly along all directions of the electrode and the film.

(3)

Side effects are ignored.

(4)

It is assumed that the electrolyte in the water tank is completely mixed.

(5)

The electrolytes are incompressible fluids and diluted solutions.

2.1. Transportation in the Electrode

During battery charging and discharging, the ions present on the positive side electrode are Fe(CN)₆⁴⁻/Fe(CN)₆³⁻ and K⁺/OH⁻ ions. The ions in the negative electrode side electrode are 2,6-DHAQ/2,6-reDHAQ and K⁺/OH⁻ ions. Each species i in the positive and negative electrode follows the following formula:

$$\nabla ·{\overrightarrow{N}}_{\mathrm{i}}=-S_{\mathrm{i}}$$

(3)

where S_i is the source term of specie I, which can be seen in

Table 1. Kinetic and electrochemistry parameters used in the simulation.

Parameter	Symbol	Value	Source
Specific surface area (m−1)	a	2.5 × 105	measured
Diffusion coefficient of 2,6-reDHAQ (m2/s)	D2,6-reDHAQ	4.8 × 10−10	Ref. [13]
Diffusion coefficient of 2,6-DHAQ (m2/s)	D2,6-DHAQ	4.8 × 10−10	Ref. [13]
Diffusion coefficient of Fe(CN)64− (m2/s)	DFe(CN)64−	6.89 × 10−10	Ref. [22]
Diffusion coefficient of Fe(CN)63− (m2/s)	DFe(CN)63−	7.20 × 10−10	Ref. [22]
Diffusion coefficient of K+ (m2/s)	DK+	1.96 × 10−9	Ref. [13]
Diffusion coefficient of OH− (m2/s)	DOH−	5.27 × 10−9	Ref. [13]
Operating temperature (K)	T	297.3	-
Electrode conductivity (S/m)	σs	1000	Ref. [23]
Membrane conductivity (S/m)	σm	10	Ref. [23]
Anodic transfer coefficients	α+	0.5	Ref. [24]
Cathodic transfer coefficient	α−	0.5	Ref. [24]
Rate constant, negative reaction (m/s)	kneg	7 × 10−6	Ref. [24]
Rate constant, positive reaction (m/s)	kpos	6 × 10−5	Ref. [24]
Standard potential of negative reaction (V)	E0,neg	−0.71	Ref. [24]
Standard potential of positive reaction (V)	E0,pos	0.33	Ref. [13]
Density of active species of the negative electrode	ρneg	1080	measured
Density of active species of the positive electrode	ρpos	1200	measured
Dynamic viscosity of the negative electrode (Pa·s)	μneg	1.17 × 10−3	Ref. [23]
Dynamic viscosity of the positive electrode (Pa·s)	μpos	1.09 × 10−3	Ref. [23]
Pump efficiency	φ	0.9	Ref. [23]
Outlet pressure (kPa)	pout	101.3	-

. The molar flux N_i of each species i can be obtained by the Nernst–Planck equation:

$$\overrightarrow{N_{\mathrm{i}}}=-D_{\mathrm{i}}^{\mathrm{eff} }\nabla c_{\mathrm{i}}-z_{\mathrm{i}}{w}_{\mathrm{i}}{c}_{\mathrm{i}}F\nabla {\varphi }_1+\overrightarrow{u}{c}_{\mathrm{i}}$$

(4)

$D_{\mathrm{i}}^{\mathrm{eff}}$ is the effective diffusion coefficient of specie i, c_i is the concentration of specie i, and z_i is the valence of specie i. The w_i represents the ion mobility. F is a constant of the Faraday exponent, and ϕ₁ is the potential of the electrolyte, respectively. The effective diffusion coefficient of specie i can be obtained by the Brugman equation.

$$D_{\mathrm{i}}^{\mathrm{eff}}={\epsilon }^{3/2}{D}_{\mathrm{i}}$$

(5)

where ε is the porosity of the porous electrode, and D_i is the diffusion coefficient of species i in electrode. K is the permeability of the electrode, given by Kozeny–Carman equation:

$$K=\frac{d_f^2{\epsilon }^3}{K_f{(1-\epsilon )}^2}$$

(6)

where d_f is the pore diameter of the carbon fiber. K_f is the Kozeny–Carman constant, which was set to 4.89.

The formula of the ionic current density in the porous electrode caused by the movement of the point ions in the electrolyte solution is as follows:

$${\overrightarrow{I}}_{\mathrm{l}}=F{\displaystyle \sum _{\mathrm{i}}{z}_{\mathrm{i}}}{\overrightarrow{N}}_{\mathrm{i}}$$

(7)

2.2. Transport in the Membrane

The type of ion exchange membrane was a Nafion^®212 membrane, and the supporting electrolyte was the KOH. Only K⁺ ion was allowed to pass through the ion exchange membrane, and the other ions are ignored. Thus, the molar flux of K⁺ can be obtained by the following formula:

$${\overrightarrow{N}}_{{\mathrm{K}}^{+}}=-\frac{{\sigma }_{\mathrm{m}}}{F}\nabla {\varphi }_{\mathrm{m}}$$

(8)

where σ_m and ϕ_m are the conductivity and potential of the film.

2.3. Electrochemical Kinetics

The overvoltage generated by the electrochemical reaction inside the positive electrode and the negative electrode is expressed by η_n and η_p and can be obtained by the following formula:

$${\eta }_{\mathrm{n}}={\varphi }_{\mathrm{s},\mathrm{n}}-{\varphi }_{\mathrm{l},\mathrm{n}}-E_{\mathrm{eq},\mathrm{n}}$$

(9)

$${\eta }_{\mathrm{p}}={\varphi }_{\mathrm{s},p}-{\varphi }_{l,p}-E_{\mathrm{eq},p}$$

(10)

E_eq,p and E_eq,n in the formula represent the open-circuit voltage of positive and negative reactions. E_eq,n and E_eq,p can be solved by the Nernst equation.

$$E_{\mathrm{eq},\mathrm{n}}=E_{0,\mathrm{n}}+\frac{RT}{nF}\ln (\frac{C_{2,6-\mathrm{DHAQ}}}{C_{2,6-\mathrm{reDHAQ}}})$$

(11)

$$E_{\mathrm{eq},\mathrm{p}}=E_{0,\mathrm{p}}+\frac{RT}{nF}\ln (\frac{C_{\mathrm{Fe}{(\mathrm{CN})}_6^{3-}}}{C_{\mathrm{Fe}{(\mathrm{CN})}_6^{4-}}})$$

(12)

where E_0,p and E_0,n are the typical equilibrium potentials of positive and negative electrochemical reactions.

2.4. Boundary Conditions

The flow rate of electrolyte at the inlet of the positive electrode and negative electrodes is set to $\overrightarrow{u}$, and the outlet pressure is specified as standard atmospheric pressure. The outlet pressure is set to p = p_out. The Neumann condition is set on all the boundary pressures, except for at the inlet and outlet.

$$p\cdot \overrightarrow{n}=0$$

(13)

On the negative side of the electrode, the potential value at the connection between the electrode and the current collector is set to zero.

At the inlet of the flow field, the initial concentration of each species condition is set as follows:

$$c_{\mathrm{i}}=c_{\mathrm{i},0}$$

(14)

2.5. Battery Performance Parameters

The overall electrolyte concentration on the negative side of the battery is set to c₀, and the electrolyte concentration on the positive side is set to c₁. In order to observe the electrolyte concentration in the battery, during charging and discharging, the parameter SOC is defined as follows:

$$C_{2,6-\mathrm{reDHAQ}}=c_0\mathrm{SOC}$$

(15)

$$C_{\mathrm{DHAQ}}=c_0·(1-\mathrm{SOC})$$

(16)

$$C_{\mathrm{Fe}{(\mathrm{CN})}_6^{4-}}=c_1·(1-\mathrm{SOC})$$

(17)

$$C_{\mathrm{Fe}{(\mathrm{CN})}_6^{3-}}=c_1\mathrm{SOC}$$

(18)

In order to study the performance of three kinds of batteries with different shapes, the charging and discharging voltage of the battery is solved by the following formula:

$$E_{\mathrm{ch}}=\left(E_{\mathrm{eq},\mathrm{pos} }-E_{\mathrm{eq},\mathrm{neg} }\right)+\left(\left|{\eta }_{\mathrm{pos} }\right|+\left|{\eta }_{\mathrm{neg} }\right|\right)+I_{\mathrm{avg} }A{R}_{ba}$$

(19)

$$E_{\mathrm{dis} }=\left(E_{\mathrm{eq},\mathrm{pos} }-E_{\mathrm{eq},\mathrm{neg} }\right)-\left(\left|{\eta }_{\mathrm{pos} }\right|+\left|{\eta }_{\mathrm{neg} }\right|\right)-I_{\mathrm{avg} }A{R}_{ba}$$

(20)

In this paper, the method of quantitative analysis is adopted to calculate the uniformity of the electrolyte concentration distribution in the electrode. The uniformity coefficient of distribution of each species i is set as follows:

$$U=1-\frac{1}{c_{\mathrm{i},ave}}\sqrt{\frac{1}{V}{\displaystyle \iiint {(c_{\mathrm{i}}-c_{\mathrm{i},ave})}^{2}dV}}$$

(21)

where c_i,ave is the average concentration of active materials in the solid electrode, and V is the volume of the solid electrode. Owing to the irreversible nature of the electrode reaction, there is a transfer limitation of the active species in the electrode and a loss of battery power caused by the resistance in the battery, namely P_loss, which can be found in our previous work [17].

2.6. Digital Details

In this paper, the secondary current distribution, the transfer of diluted species, and the physical field of the Brinkman equation equipped with steady-state numerical solution are set in COMSOL 5.5. The positive field, negative field, and membrane domain are defined with corresponding variables in the COMSOL Multiphysics 5.5 software. The relative tolerance is specified as 1.5 × 10⁻⁶, and the maximum number of iterations is 150, using the PARDISO solver.

2.7. Model Validation

The experiments are conducted to verify the reliability of the numerical model. In the experiment, the flow field of the ferrocyanide-based ORFB is serpentine shaped, and the geometric size of the experimental electrode is 20 mm × 20 mm × 4 mm. The material used for the experimental electrode is graphite felt. The current density is 100 mA/cm². The inlet flow of the electrolyte is set at 60 mL/min. The active ion concentration of the catholyte is set at 100 mol/m³, and the active ion concentration of the anolyte is set at 200 mol/m³. Other detailed experimental parameters can be seen in Table 1 and

Table 2. Geometric and operation parameters used in the experiments.

Parameter	Symbol	Value
Length of the electrode (mm)	Le	20
Width of the electrode (mm)	We	20
Height of the electrode (mm)	He	4
Length of the membrane (mm)	Lm	20
Width of the membrane (mm)	Wm	20
Height of the membrane (mm)	Hm	0.05
Width of the flow field (mm)	Wc	0.8
Height of the flow field (mm)	Hc	0.8
Initial concentration of 2,6-reDHAQ (mol/m3)	c2,6-reDHAQ	10
Initial concentration of 2,6-DHAQ (mol/m3)	c2,6-DHAQ	90
Initial concentration of Fe(CN)64− (mol/m3)	cFe(CN)64−	20
Initial concentration of Fe(CN)63− (mol/m3)	cFe(CN)63−	180
K+ concentration in half-cell (mol/m3)	cK+	1000
OH− concentration in half-cell (mol/m3)	cOH−	1000
Porosity of the electrode	ε	0.9
Applied current density (mA/cm2)	I	100
Inlet electrolyte flow rate (mL/min)	Q0	60

. The comparison between the numerical results and experimental data is shown in the Figure 2. Figure 2 shows that the simulated and experimental charge-discharge voltage matches the entire charge-state cycle (SOC from 0.1 to 0.9), with a maximum error of 3.9%, which occurs during the discharge process, and the overall average error is less than 2%, indicating that the three-dimensional numerical model established in this paper is accurate and reliable.

3. Results and Discussion

It is very important for the ferrocyanide-based ORFB to clarify the influence of different flow fields and different electrode configurations on the battery performance and the transport process. Three different electrode shapes (trapezoid, sector, and rectangle), with a surface area of 6400 mm², are studied in the subsequent experiments, and the three electrodes are equipped with three flow fields. The geometry data for the different electrodes and the flow fields are listed in

Table 3. Geometric and operation parameters used in the simulation.

Parameter	Symbol	Value
Length of the electrode (mm)	Le	80
Width of the electrode (mm)	We	80
Height of the electrode (mm)	He	3.5
Length of the membrane (mm)	Lm	80
Width of the membrane (mm)	Wm	80
Height of the membrane (mm)	Hm	0.2
Width of the flow field (mm)	Wc	3.5
Height of the flow field (mm)	Hc	3.5
Length of spindle electrode (mm)	Lc	642.91
Initial concentration of 2,6-reDHAQ (mol/m3)	c2,6-reDHAQ	20
Initial concentration of 2,6-DHAQ (mol/m3)	c2,6-DHAQ	180
Initial concentration of Fe(CN)64− (mol/m3)	cFe(CN)64−	40
Initial concentration of Fe(CN)63− (mol/m3)	cFe(CN)63−	360
K+ concentration in half-cell (mol/m3)	cK+	1500
OH− concentration in half-cell (mol/m3)	cOH−	1500
Porosity of the electrode	ε	0.7
Applied current density (mA/cm2)	I	40
Inlet electrolyte flow rate (mL/min)	Q0	600

. The flow fields and the electrodes are presented in Figure 1. The contact areas between the flow field with the electrode for the three ferrocyanide-based ORFBs remain the same. The battery performance and mass transfer behaviors of the three the ferrocyanide-based ORFBs are obtained, which is improtant for the ferrocyanide-based ORFB application.

3.1. Battery Performance of Batteries with Different Electrode Shapes

The battery performance (such as the charging-discharging voltage, overvoltage, and polarization) is the most important parameter to evaluate the ferrocyanide-based ORFBs. It is of benefit to point out the influences of the electrode shapes on the charging-discharging voltage. The variations in the charge and discharge voltage for the three kinds of batteries is shown in Figure 3, which shows that the trapezoid battery possesses the highest discharge voltage during the discharge process. In the charging process, the trapezoid battery requires the lowest charge voltage. The discharge voltage of trapezoid battery is slightly higher than that of the sector battery, while the charging voltage of the sector battery is slightly higher than that of trapezoid battery. The charging voltage of rectangle battery is the highest, reaching the highest value at SOC = 0.9, which is 3.46% higher than that of the trapezoid battery. The discharge voltage of the rectangle battery is the lowest, and its value reaches the lowest value at SOC = 0.9, which is 4.47% lower than the discharge voltage of the trapezoid battery.

Presented in Figure 4, the overvoltage is also a key parameter, which increased in the numerical model. The overvoltage variation in the three different batteries during the discharge process can be seen in Figure 4. Figure 4 shows that the absolute value of overvoltage decreases first and then increases with the increase in SOC, which is due to the lack of reactants and products. In addition, Figure 4 shows that the trapezoid battery has the lowest absolute value of overvoltage during the discharge process, both for the negative and positive electrodes. Therefore, the trapezoid battery can achieve the highest discharge voltage for the ferrocyanide-based ORFB.

In addition, the polarization is also an important factor influenced by the transport process. Figure 5 shows the variations in the polarization under the SOC = 0.8 condition, illustrating that the discharge voltage decreases as the current density increases for the three kinds of batteries. When the current density is set to 40 mA/cm², the discharge voltage of the trapezoid battery is 33.76% higher than the discharge voltage of the rectangle battery. The lower the polarization loss, the higher the discharge voltage, which illustrates that the battery performance of the trapezoid battery is the best.

3.2. Mass Transfer Behaviors of Batteries with Different Electrode Shapes

Mass transfer behavior is one of the important factors affecting the performance of the battery. In this paper, the mass transfer behaviors of three batteries are quantitatively investigated using the uniformity factor. The variations in the uniformity factor during the discharge process are shown in Figure 6, which shows that the concentration distribution of the electrolyte in the trapezoid battery is more uniform than that of the other batteries. Particularly, when SOC = 0.1, the uniformity factor value of the trapezoid battery is 26.9% higher than that of the rectangle battery. The electrolyte contribution distributes more uniformly, and better battery performance can be achieved, which is consistent with the previous conclusion. Therefore, the trapezoid battery exhibits better mass transfer behavior. In addition, it is not difficult to see that the uniformity factors of the three batteries show a downward trend during the discharge process (SOC from 0.9 to 0.1), according to Figure 6. When the SOC is low, there is a poor mass transfer performance and nonuniformity concentration distribution in the battery, which results in low battery performance.

It is well known that the active ion concentration is one of the important conditions for changing and discharging the overvoltage. In order to accurately and clearly evaluate the distribution of the active ion concentration, the negative electrode is divided into four equal parts along the electrode thickness direction, as shown in Figure 7. The concentration of 2,6-reDHAQ is taken as the representative of the active ions concentration in this section. Figure 8 (SOC = 0.2) and Figure 9 (SOC = 0.6) show the distribution of the 2,6-reDHAQ concentration in the 3L/4 plate of different batteries. It is not difficult to see that the 2,6-reDHAQ concentration of the trapezoid battery is the highest, and the 2,6-reDHAQ distribution of the trapezoid battery is the most uniform.

For quantitatively evaluating the concentration distribution in different batteries, Figure 10 and Figure 11 show the average concentration variations of 2,6-reDHAQ for the three batteries in different plates under the conditions of SOC = 0.2 and SOC = 0.6. It can be seen from Figure 10 and Figure 11 that the concentration of 2,6-reDHAQ increases when moving from 1L/4 to 3L/4, which is the result of flow resistance. Whether in SOC = 0.2 or SOC = 0.6, the average concentration of the trapezoid section is the highest, under any condition. Therefore, the trapezoid battery can enhance the transport of active ions, so that the trapezoid battery can obtain better performance.

3.3. Power Performance of Batteries with Different Electrode Shapes

The hydrodynamic characteristics of different electrodes will result in different pressure drops. Figure 12 shows the variations in pressure drop for the negative electrode from a 600 mL/min flow rate to a 1200 mL/min flow rate. The pressure drop of the trapezoid battery is the lowest, while the pressure drop of the rectangle battery is the highest. At the flow rate of 1200 mL/min, the pressure drop of the rectangle battery is 3520 Pa higher than that of trapezoid battery. Taking the loss of pump work into account, Figure 13 shows the variations in the net power with the flow rate for the three batteries, which illustrates that the net power of the trapezoid battery remains the highest as the flow rate increases from 600 mL/min to 1200 mL/min. Under the flow rate condition of 600 mL/min, the power of the trapezoid battery is 7.58% higher than that of the rectangle battery. Figure 14 shows the variations in the power-based efficiency with the increase in the flow rate for the three batteries, from which it can be seen that the efficiency based on the power of the trapezoid electrode is the highest. Therefore, the trapezoid battery can achieve the best net output power. At the flow rate of 10 mL/s, the efficiency of the trapezoid battery is 4.96% higher than that of the rectangle battery. Therefore, the performance of the trapezoid battery is better than that of the rectangle battery.

4. Conclusions

In this paper, the mass transfer and battery performance of three electrodes (trapezoid, sector, and rectangle), with flow fields, are studied based on a numerical model, which is verified by the experiments. The results show that the discharge voltage of the ferrocyanide-based ORFB with a trapezoid electrode is highest, and that with the rectangle electrode is lowest, and the ferrocyanide-based ORFB with a trapezoid electrode requires the lowest charge voltage. The charging voltage of the rectangle battery is the highest, and the charging voltage of the trapezoid battery is the lowest. The charging voltage of the rectangle battery is 3.46% higher than that of the trapezoid battery at SOC = 0.9. The discharge voltage of rectangle battery is the lowest, and the discharge voltage of the trapezoid battery is the highest. The discharge voltage of the rectangle battery is 4.47% lower than that of the trapezoid battery. The uniformity factor value of the trapezoid battery is 26.9% higher than that of the rectangle battery, and the power of the trapezoid battery is 7.58% higher than that of the rectangle battery. Therefore, the trapezoid shape is the best design for the electrodes of the ferrocyanide-based ORFBs.