As shown in Figure 1, the VRFB single cell consists of a peristaltic pump, graphite carbon felt, proton exchange membrane, current collector, and electrolyte storage tank. Both graphite carbon felts will undergo oxidation and reduction reactions during the charge and discharge processes. The positive electrode side is composed of $$ \mathrm{VO}_{2}^{+} $$ and VO²⁺, and the negative electrode side is composed of V³⁺ andV²⁺. The peristaltic pump transports the electrolyte into the battery for circulation, and the activity of the proton exchange membrane is to transfer hydrogen ions and avoid cross-mixing contamination of the electrolyte. These reactions are:

Positive electrode:

(1) $$ \mathrm{VO}^{2+}+\mathrm{H}_{2} \mathrm{O} \leftrightarrow \mathrm{VO}_{2}^{+}+2 \mathrm{H}^{+}+e^{-} \\ $$

Negative electrode:

(2) $$ \mathrm{V}^{3+}+e^{-} \leftrightarrow \mathrm{V}^{2+} \\ $$

During the charging process, the negative electrode V³⁺ will be reduced to V²⁺, and the positive electrode VO²⁺ will be oxidized to $$ \mathrm{VO}_{2}^{+} $$, while the discharge process is the opposite. The state of charge (SOC) is defined as:

Positive electrode:

(3) $$ S O C=\frac{c_{V}}{c_{V}+c_{I V}} \\ $$

Negative electrode:

(4) $$ \text { SOC }=\frac{\mathrm{c}_{\text {II }}}{\mathrm{c}_{\text {II }}+\mathrm{c}_{\text {III }}} \\ $$

To conduct numerical simulation of VRFBs, this study established 3D single-cell models of four flow channels. The 3D model mainly consists of graphite carbon felt, current collector, and proton exchange membrane. Figure 2 shows the structure diagram of the 3D VRFB model, and the dimensions of the 3D VRFB are provided in Table 1.

• Battery temperature is isothermal during the simulation and is not affected by time.

• Porous electrodes and membranes are isotropic in all directions (Isotropic).

• The side reactions of hydrogen and oxygen evolution and absorption are ignored.

• The flow of electrolyte is incompressible and laminar, and the electrolyte is regarded as a dilute solution.

The volume flow rate Q_pump driven by the peristaltic pump controls the electrolyte flow rate v_in into the cell inlet:

(27) $$ v_{\text {in }}=\frac{Q_{\text {pump }}}{A} \\ $$

where A is the cross-sectional area. It is assumed that the inlet flow is a fully developed flow:

(28) $$ \frac{\partial u}{\partial x}=0 \\ $$

The electrolyte flows out of the battery, flows into the electrolyzer, and flows back to the battery from the inlet side of the battery through a peristaltic pump. The inlet concentrations $$ c_{\mathrm{V}^{2+}}^{i n} $$, $$ c_{\mathrm{V}^{3+}}^{i n} $$, $$ c_{\mathrm{VO}^{2+}}^{i n} $$, and $$ c_{\mathrm{VO}_{2}^{+}}^{i n} $$ are given by the following ordinary differential equation:

(29) $$\frac{d}{d t}\left(c_{i}^{\text {in }}\right)=c_{i, \text { int }} V-\left(\int_{\text {outlet }}\left(c_{i}^{b} u\right) d L-\int_{\text {inlet }}\left(c_{i}^{b} u\right) d L\right) \\ $$

where c_i,int is the initial concentration of vanadium substance, V is the volume of electrolyte, $$ c_{i}^{b} $$ is the concentration of ions in the electrolyte, and u is the flow rate of the electrolyte. When doing pressure calculations, the outlet pressure is set to 0 Pa.

(30) $$ P_{\text {out }}=0 \\ $$

Since VRFBs usually operate in constant current mode, constant current conditions are applied to the sidewalls of the current collector^[42]:

(31) $$ -\sigma_{s}^{\text {eff }} \nabla \Phi_{s} \cdot \vec{n}=\left\{\begin{array}{l} -\frac{I}{A_{c c}}(x=0) \\ \frac{I}{A_{c c}}(x=L) \end{array}\right. $$

where I is the current, and the opposite is true for the discharge state. The other outer surfaces of the current collector are insulated and the surface potential outside the negative electrode is zero.

(32) $$ \Phi_{c c}=0 $$

In order to benchmark the accuracy of the 3D VRFB model established in this study, a 3D VRFB model with SFC was used and compared with the research results of Ali et al.^[43]. Figure 3 shows the numerical results of this study and the trend diagram of battery voltage during the charging and discharging states studied by Ali et al.^[43]. For the discharging state through numerical simulation, the battery voltage decreases as time increases. From the SOC state, the voltage at SOC = 0.9 is 1.4093 V, which drops to 1.225 V at SOC 0.11. The experimental results are consistent with the trend of the numerical results. Therefore, the numerical simulation of the 3D model of VRFB in this study has high accuracy for studying the effects of other factors on the performance of VRFB. To quantitatively support the model accuracy, the maximum relative deviation between the numerical simulation and experimental data was calculated. The results show that the maximum relative deviation is less than 1.7% (at the end of the discharge cycle), which confirms the high fidelity of the proposed model.

The grid structure of VRFB is composed of a tetrahedral grid. In the grid independence study, the 3D VRFB model with the SFC was used as the benchmark, and the current density of I = 60 mA cm^-2 and electrolyte flow rate of Q = 60 mL min^-1. The grid numbers are varied from 55,396 to 1,661,709. Among them, the grid number of 1,661,709 is used as a benchmark to calculate the relative errors of other grid numbers and to analyze the time it takes for the VRFB to discharge from SOC when the SOC is 0.9 and the voltage is 1.15 V. The results of this study are presented in Table 4. It can be seen that the error of the grid number of 783,097 is only 1.8258% compared with the grid number of 1,661,709, but the calculation time is greatly reduced, so the grid number of 783,097 is the most cost-effective.

This study simulated the pressure distribution of VRFBs with four flow channel structures at the same electrolyte flow rate of 60 mL min^-1. The result is shown in Figure 4. Since the outlet pressure is zero, the pressure drop in the flow channel can be calculated from the average pressure at the inlet. The pressure drops of the four types of flow channels in descending order are 5,023.4 Pa, 332.75 Pa, 203.07 Pa, and 135.79 Pa for SFC, IFC, IFC with pins, and PFC, respectively. Larger flow channel pressure drops affect the pump and create more burdens. Among them, the flows of the PFC are straight lines without any obstructions, so they have the smoothest and most uniform pressure distribution. In order to add a convex part to the IFC, taking structural design and other factors into consideration, the number of IFCs with pins was reduced, and the inlet area of the flow channel was increased. The convex part provides greater fluid resistance compared to the IFC. This design helps to improve the mixing effect of the electrolyte and fluid distribution in the flow channel. Although overcoming the fluid resistance will cause the pressure drop to increase, due to the interdigitated inlet, the cross-sectional area is smaller than that of the IFC with pins, so the pressure at the entrance of the finger-shaped flow channel is greater than that of the IFC with pins. Finally, because the SFC has only two inlets, the flow channel has many turns, and the length of the flow channel is much longer than the other three flow channels. As a result, the SFC has the largest pressure range distribution.

This study simulated the influence of flow channel geometry on battery charge and discharge processes when the current density of four flow channels was 60, 80, and 100 mA cm^-2, respectively, and for a charging cut-off voltage of 1.55 V and a discharge cut-off voltage of 1.15 V. The results are shown in Figures 5 and 6. In Figure 5, the charge and discharge time of all flow channels shortens as the current density increases, and increases as the current density decreases. In addition, the figure shows that the charge and discharge curves of the IFC, IFC with pins, and SFCs have hyperbolic characteristics. The charge-discharge curve of the PFC is a curve, and the slope of the flow channel at the beginning of charging and discharging is small. Because the design of the PFC will make the electrolyte in the flow channel evenly distributed, the initial concentration gradient of vanadium ions is established slowly, so the initial concentration polarization effect is not obvious. However, after entering the middle stage of charging and discharging, the electrolyte still flows along the flow channel and lacks a mechanism to force it to flow into the electrode surface. The electrolyte cannot fully participate in the reaction and cannot effectively increase the electrochemical reaction rate, resulting in the electrolyte in the PFC having low utilization efficiency. In Figure 6, the charge and discharge capacities of all flow channels decrease as the current density increases and increase as the current density decreases. The distance between the different current density curves of the IFC and SFC is small, while the distance between the different current density curves of the PFC and the IFC with pins is larger, which means that when the current density increases, the charge and discharge capacity loss rates for IFC and SFC are smaller than those of the PFC and the IFC with pins.

To better analyze the effect brought by the flow channel geometry, the data are obtained at i = 80 mA cm^-2, and these data are shown in Figure 7. At i = 80 mA cm^-2, the SFC has the longest discharge time of 1,820.23 s, followed by 1,721.56 s for the IFC, 1,705.75 s for the IFC with pins, and 1,414.2 s for the PFC. Among them, it can be found from Figure 7A that the PFC completes the discharge reaction before reaching the cut-off voltage of 1.15 V. Because the PFC does not have the forced diffusion characteristics of the IFC and the IFC with pins, the channel length is also much shorter than that of the SFC, which causes the electrolyte in the PFC to flow through the electrode quickly and cannot fully carry out a sufficient reaction to convert chemical energy into electrical energy, resulting in the ions in the electrolyte not being fully utilized and immediately flowing out of the battery. For the next cycle, this design will result in poor electrolyte utilization and faster voltage drop.

The charge and discharge capacity is obtained by multiplying the battery charge and discharge time, current density, and battery reaction area. At i = 80 mA cm^-2, the SFC has the largest discharge capacity of 1,016.79 mAh, followed by 956.42 mAh for IFC, 947.64 mAh for the IFC with pins, and 671.22 mAh for the PFC. Figure 7B and C show that although the voltage efficiency of the SFC is 90.65%, slightly lower than the 90.68% of the IFC, its single charge and discharge capacities are the largest, demonstrating the SFC’s superior overall performance. The IFC and the IFC with pins have similar values in terms of capacity, but the IFC has an advantage in terms of voltage efficiency. The performance of the PFC is the worst among the four geometries.

Among the four flow channel geometries, the VRFB with SFC has the best performance, so it is selected for further discussion at i = 60, 70, 80, 90, and 100 mA cm^-2. The effects of current density on the starting voltage of the battery during charging and discharging are investigated in this section. At i = 60, 70, 80, 90 and 100 mA cm^-2, the corresponding discharge starting voltages are 1.409 V, 1.4 V, 1.392 V, 1.383 V, and 1.374 V respectively; the charging starting voltages are also 1.295 V, 1.303 V, 1.312 V, 1.321 V, and 1.329 V, respectively. Figure 8A shows that during the charging and discharging processes, a higher current density has a lower discharge starting voltage and a higher charging starting voltage, and a lower current density has a higher discharge starting voltage and a lower charging starting voltage. The starting voltage will affect the time from the battery charging and discharging voltage to the cut-off voltage. The phenomenon of starting voltage is mainly affected by ohmic polarization and electrochemical active polarization. As the current density increases, the resistance of the internal materials of the battery, such as electrodes, electrolyte, and proton exchange membrane, leads to greater voltage loss. In addition, electrochemical reactions all need to overcome the activation overpotential before they can start to react. A higher current density will intensify the active polarization effect of electrochemistry, causing the battery to overcome a larger activation overpotential, resulting in lower discharge starting voltage and higher charging starting voltage.

The detailed charge and discharge capacity data for SFC at different current densities are provided in Table 5. In addition, Figure 8B shows that as the current density increases, the charging capacity of the battery decreases from 1,056.2 to 999.43 mAh, and the discharge capacity decreases from 1,036.98 to 989.45 mAh. Due to the lower initial discharge voltage and higher initial charge voltage caused by ohmic polarization and electrochemical active polarization, the charge and discharge processes with high current density will reach the cut-off voltage faster. This reduces the charge and discharge time, and as a result, the charge and discharge capacities are also decreased. Concentration polarization can also affect the capacity. Under high current density, the reactants near the electrode are rapidly consumed in large quantities, causing the concentration of the reactants near the electrode to decrease, and a concentration gradient appears, causing the electrochemical reaction rate to decrease. This provides the additional voltage need for battery to overcome the concentration polarization overpotential, causing the battery to reach the cut-off voltage faster, reducing the charge and discharge time, and decreasing the charge and discharge capacity.

Coulombic efficiency is related to the charge loss caused by battery side reactions and electrolyte cross-contamination. Energy efficiency reflects the losses in the battery energy conversion process, including voltage loss, Coulombic loss, etc. Voltage efficiency is mainly affected by ohmic polarization, concentration polarization, and electrochemically activated polarization. Coulombic efficiency (CE), energy efficiency (EE), and voltage efficiency (VE) are expressed as follows:

(33) $$ C E=\frac{\operatorname{Discharge} \operatorname{energy}(\mathrm{mAh})}{\operatorname{Charge} \operatorname{energy}(\mathrm{mAh})} \times 100 \% \\ $$

(34) $$ E E=\frac{\operatorname{Discharge} \operatorname{energy}(\mathrm{Wh})}{\operatorname{Charge} \operatorname{energy}(\mathrm{Wh})} \times 100 \% \\ $$

(35) $$ V E=\frac{E E}{C E} \times 100 \% $$

As shown in Figure 8C, the CE of the SFC increases with increasing current density, from 98.2% to 99%. As the current density increases, the reaction rate of the VRFB accelerates, causing the vanadium ions in the electrolyte to be consumed more quickly, thus shortening its time on both sides of the proton exchange membrane, which reduces the chance of cross-contamination of the electrolyte. VE decreases with increasing current density, from 92.4% to 89%. When the current density increases, the ohmic polarization, concentration polarization, and electrochemical active polarization inside the battery gradually increase, resulting in greater voltage loss and reduced VE. EE also decreases with increasing current density, from 90.7% to 88.1%. EE is calculated by multiplying CE and VE, showing the impact on overall cell performance. Although high current density can improve CE and reduce electrolyte cross-contamination and side reactions, the significant drop in VE shows that increasing current density brings greater voltage loss, ultimately leading to a downward trend in overall EE.

From Figure 9A, the effects of different electrolyte flow rates on the starting voltage of the battery in charge and discharge processes can be observed. The electrolyte flow rate was 20, 40, 60, 80, and 100 mL min^-1 under charge and discharge processes. When the electrolyte flow is 20, 40, 60, 80, and 100 mL min^-1, the corresponding discharge starting voltages are 1.379 V, 1.388 V, 1.392 V, 1.393 V, and 1.394 V respectively; the charging starting voltages are 1.325 V, 1.316 V, 1.312 V, 1.31 V, and 1.309 V. It can be seen from Figure 9A that at the beginning of the discharge process, the initial discharge voltage increases slightly as the electrolyte flow rate increases. When the flow rate increases from 20 mL min^-1 to 100 mL min^-1, the initial discharge voltage is increased from 1.379 V to nearly 1.394 V. At the beginning of the charging process, the initial charging voltage decreases as the electrolyte flow rate increases. When the flow rate increases from 20 to 100 mL min^-1, the initial discharge voltage is decreased from about 1.325 V to 1.309 V. The reasons for the trend of the initial voltage caused by adjusting the flow rate are different from those caused by adjusting the current density. The trend caused by adjusting the current density is mainly affected by ohmic polarization and electrochemical active polarization, while the trend caused by adjusting the flow rate is mainly affected by concentration polarization. As the electrolyte flow rate increases, the mass transfer effect of the electrolyte near the electrode increases, which can effectively reduce the concentration gradient caused by insufficient replenishment of reactants, showing that a higher electrolyte flow rate helps to reduce concentration polarization.

The detailed charge and discharge capacity data for the case of SFC at different electrolyte flow rates are provided in Table 6. It can be seen from Figure 9B that when the electrolyte flow rate increases from 20 to 100 mL min^-1, the charging capacity of the battery increases from 979.2 to 1,049.78 mAh, and the discharge capacity increases from 961.23 to 1,036.39 mAh. When the electrolyte flow rate decreases from 100 to 20 mL min^-1, the charge and discharge capacities of the battery decrease. Supplying electrolyte with a high flow rate can quickly replenish the vanadium ions consumed near the electrode, reduce the concentration overpotential, allow more vanadium ions to be utilized within the same voltage difference range, and lengthen the reaction time of the flow battery, so it has more charge and discharge capacities. When the electrolyte flow rate increases from 20 to 40 mL min^-1, the battery discharge capacity increases by 4.08%. When the electrolyte flow rate increases from 40 to 60 mL min^-1, the battery discharge capacity increases by 1.63%. When the electrolyte flow rate increases from 60 to 80 mL min^-1, the battery discharge capacity increases by 1.07%. When the electrolyte flow rate increases from 80 to 100 mL min^-1, the battery discharge capacity increases by 0.85%. When the electrolyte flow rate becomes larger and larger, although the battery discharge capacity will increase, the increase will become smaller and smaller. The electrolyte flow rate will also greatly affect the power of the pump, so it cannot continue to increase without any limitation.

As shown in Figure 9C and Table 6, the CE of the SFC increases with increasing electrolyte flow rate, from 98.2% to 98.7%. When the electrolyte flow rate increases, the time for side reaction products to stay in the electrode and flow channel is reduced, reducing the chance of electrolyte cross-contamination. VE also increases with increasing electrolyte flow rate, from 90.1% to 90.8%. When the electrolyte flow rate increases, the electrolyte can quickly replenish the ions that complete the reaction near the electrode, reducing the possibility of concentration gradients, improving the efficiency of the electrochemical reaction, and reducing the voltage loss of the battery owing to concentration polarization. Finally, EE is a comprehensive index of the overall efficiency of the battery, which is obtained by multiplying CE and VE. Therefore, the EE increases from 88.4% to 89.6% by increasing these two parameters. As with discharge capacity, when the electrolyte flow rate increases, the efficiency of the three batteries tends to increase, but the increase rate becomes increasingly gentle. The electrolyte flow rate cannot continue to increase indefinitely; otherwise, it will impose a significant burden on the pump.

Electrolyte velocity vector plots at different electrolyte flow rates are shown in Figure 10. The results indicate that increasing the electrolyte flow rate enhances electrolyte permeation and convective transport within the porous electrode, allowing the reactants to be delivered more effectively to the electrode interior. Consequently, the non-uniform distribution of active species is alleviated, thereby reducing concentration polarization.