Different from the static battery setup, in RFBs, the reactants are continuously pumped to the electrochemical cells while the products are removed from the cells, and the battery performance is significantly influenced by the mass transport process [22]. Specifically, mass transport is a complicated process that is coupled with electrolyte flow and electrochemical reactions. In addition, mass transport is affected by flow configurations, electrode/electrolyte properties, and assembling/operating conditions. Herein, we mainly discuss the influence of flow fields on mass transport. The flow-through and flow-by structures are the two main cell configurations to distribute electrolytes. In the flow-through configuration, the electrolyte directly flows into the electrode domain, while in the flow-by configuration, the electrolyte is transported in the flow channels and then convects into electrodes. Different flow patterns and geometric parameters lead to distinct velocity distributions, and a higher velocity magnitude contributes to lower concentration loss and higher ion transport flux. The ion flux of active species can be expressed according to the Nernst-Planck equation:

which constitutes the diffusion, migration and convection terms, where i indicates the species, ${D}_{i}^{eff}$ is the effective diffusivity, ${c}_{i}$, is the molar concentration, ${\mathrm{z}}_{i}$ is the charge number, ${u}_{i}$ is the mobility, F is the Faradaic constant, ${\phi }_{l}$ is the ionic potential in the electrolyte, ${u}$ is the electrolyte velocity. In a flow battery system, the diffusivity is relatively small while the convective item dominates the total ion flux. Therefore, higher concentration and velocity lead to a higher flux of ions [42]. In addition, the concentration overpotential can be expressed as [43]:

where ${\eta }_{c}$ is the concentration overpotential and ${c}_{b}$ is the reactant concentration in the bulk. Generally, the mass transfer coefficient can be evaluated by considering the velocity magnitude:

where b and n are empirical constants, v represents the velocity magnitude of electrolyte in the porous electrode.

To determine the values of the constants, researchers calculate ${k}_{m}$ values at different electrolyte velocities and describe the correlation according to the fitting curve.

Specifically, ${k}_{m}$ values can first be calculated on the basis of the measured limiting current and basic parameters:

where a is specific area, $L$, ${t}_{e}$ and $W$ are electrode length, electrode thickness and electrode width, respectively. ${I}_{lim}$ is measured limiting current. ${c}_{i}^{in}$ is the concentration of species in the electrolyte stored in the reservoir.

By adjusting the electrolyte flow rate, ${k}_{m}$ is calculated under a wide range of electrolyte velocities. A fitting curve is then generated to describe the dependence of mass transfer coefficient on electrolyte velocity, and the constants b and n in Eq. 3 are accordingly obtained [44].

It can be found that local concentration loss is directly influenced by the electrolyte velocity and concentration. In addition, the concentration of active species, as we can see from Eq. 1, is influenced by the velocity field. More sophisticated mass transport at the interface between electrolytes and electrodes is not discussed here. From the aspect of flow fields, the flow field pattern should be well-designed to lower the overall concentration overpotential. More significantly, the problem of non-uniform distribution of active species is exacerbated as the flow field is scaled-up, resulting in elevated concentration overpotential and performance degradation, which necessitates the implementation of effective and efficient scale-up strategies to mitigate the issue.

In addition to mass transport, different cell configurations or flow patterns lead to different pressure drops and hence affect the pumping work. In fact, the pumping work constitutes a non-negligible part of the system power loss, which is dependent on the flow rate, cell architecture and dimensions. The flow field structure not only affects the flow resistance of channels, but also influences the electrolyte velocity and convection path within electrodes. The dependence of pumping work and VE_pump on the pressure drop are given in Table 1. The pumping loss scales linearly as the pressure drop increases, and it affects both the charge and discharge processes.

Hereinto, we first use the conventional flow-through configuration with the simplest flow process to demonstrate the effect of cell architecture and geometric dimensions. The pressure drop of the flow-through structure can be given by [18]:

By substituting the expression of the pressure drop into the relationship between pumping work and pressure drop (listed in Table 1), the pumping work of the flow-through structure can be given by [18]:

where $\mu$ is the electrolyte viscosity, $L$, ${t}_{e}$ and $K$ are the electrode length, electrode thickness and electrode permeability, respectively.

As illustrated in Eq. 6, the pumping work scales dramatically as the flow rate and convection distance increase. Therefore, there must be a critical flow rate over which the increased pumping work cannot be counteracted by the reduced electrochemical polarizations. In addition, compressing the electrode results in lower electrode thickness and permeability, thereby increasing the pressure drop of the battery system.

When the flow-by structures are applied to the RFBs, the convection distance within the electrode is greatly decreased since the electrolyte can be transported across the whole active area via the channels. Serpentine and interdigitated flow fields are two most widely used flow fields for distributing the electrolytes, while they have completely different characteristics and represent two types of flow field structures: one has connected inlet and outlet, and the other has disconnected inlet and outlet.

For the serpentine flow fields, the flow resistances of channels and electrodes between adjacent channels are in parallel relationship, hence only a portion of electrolyte penetrates into the electrode driven by the pressure difference of neighboring flow channels. Increasing the cross-sectional area of channels can lead to a lower pressure drop but increases the mass transport polarization due to the reduced penetration rate.

As for the interdigitated flow fields, the electrolyte flow enters the manifold from the inlet, disperses into inflow branch channels, flows into the electrode and across the under-rib region, then flows out through the outflow channels. Therefore, the interdigitated flow field is intrinsically a combination of several flow-through configurations in which all the electrolyte penetrates into the porous electrode [45]. Decreasing the channel width not only increases the flow resistance within channels, but also elongates the convection distance between inflow and outflow branch channels. In addition, the pressure drop in the main channels may increase considerably when scaling up the flow battery. It was found that a 400-cm² cell with interdigitated flow field has a pressure drop of 0.05 MPa at a low flow rate of 2 mL min⁻¹ cm⁻², over 40% of which is induced by the flow resistance between the inlet pipe and branch channel end [46]. As a result, designing flow channels that can transport the electrolyte in a low pressure drop is an applicable approach to further improve the battery performance.

In addition to the conventional aqueous redox flow batteries, novel flow battery systems have emerged, including hypersaline slurry flow batteries and aqueous organic flow batteries. These systems exhibit unique kinetics and hydraulic properties compared to conventional electrolytes, necessitating the implementation of distinct flow fields.

For slurry flow batteries, different from conventional aqueous electrolytes which can be treated as Newtonian electrolyte with a constant viscosity, the slurry exhibits non-Newtonian behavior which can be described by the power law viscosity $\mu =K{(\dot{\gamma })}^{n-1}$,

Where $\dot{\gamma }$ is the shear rate, K and n are power law constants, and are fitted to be 5.72 and 0.18, according to the literature [47]. In addition, the electrolyte viscosity mainly ranges from 100-1000 cP [48,49,50,51], which is much higher than that of the conventional electrolyte (below 10 cP [27]). It is found that due to the increased flow resistance, the slurry flow battery is firstly demonstrated with only one straight flow channel [48,49]. Recently, zero-gap flow cells with serpentine flow fields were applied in the slurry flow cells [50,51]. Due to the high viscosity of the slurry electrolyte, the channel width needs to be increased to reduce the pressure drop. For conventional flow field, the channel width is relatively small (e.g., 1 mm [27]). In contrast, the channel width adopted in simulation of slurry flow batteries is 4.8 mm [47]. However, the investigation of flow fields is still in its initial stage since for slurry flow batteries, in addition to dramatically increased electrolyte viscosity, the construction of continuous electrolyte conductivity network is also critical for the steady operation. Currently, the cell is operated in continuous or intermittent mode. The flow field configuration, electrode properties, electrolyte properties under different operating conditions need to be further optimized and studied. Furthermore, due to the unique flow properties of slurry flow batteries, a numerical model that can accurately describe their behavior still needs to be established.

For aqueous organic RFBs, the viscosity and diffusion coefficient of aqueous organic electrolytes are compared with conventional electrolytes and found that both parameters fall within the same range [52,53,54,55,56,57,58]. Therefore, it is reasonable to assume that flow field design for aqueous organic RFBs can be borrowed from conventional RFBs, resulting in the use of flow-field-structured configurations such as serpentine and interdigitated setups. However, the specific geometric parameters of the flow field for a particular system may vary due to different hydraulic properties and kinetics of the electrolytes. Indeed, the properties of the electrolyte and the design of the flow field can affect mass transport and pumping loss from different aspects. In particular, viscosity directly impacts the flow of electrolyte through channels and electrodes, and higher viscosity can lead to greater pressure drop and pumping loss. Consequently, the degree of pumping loss in different systems can vary depending on the electrolyte’s viscosity. When it comes to mass transport, the ion flux is determined by a combination of convection, diffusion, and migration as shown in Eq. 1. The design of the flow field can directly impact the convection term in mass transport, while the ion diffusivity plays a significant role. In summary, while the flow field design for aqueous organic RFBs can be adapted from conventional RFBs, it is important to consider the different kinetics and hydraulic properties. The specific flow field design should take into account both electrochemical performance and pumping loss, which altogether are reflected in the system voltage efficiency.

In this section, we will first compare the battery performance of flow-through and flow-by structures. The flow-through configurations are the earliest adopted cell architectures in RFBs, and there are mainly two types of flow-through configurations. In the first configuration, the electrolyte enters the electrode and exits from diagonal corners [69] and the second type employs inlet and outlet distribution channels on two sides [30], as shown in Fig. 7a1. Typically, the electrolyte distribution in the second type flow-through configuration exhibits higher distribution uniformity than the first type. Zheng et al. numerically optimized the flow-through structures by introducing guidance and distribution channels at inlets and collection channels at the outlets (corresponding to the two types of flow-through structure). As shown in Fig. 7a2, such a design enables the plug flow of electrolytes, which leads to a more uniform distribution in the flow-through structure [70]. The flow-by-structured configurations engrave flow channels on the graphite plate to distribute electrolytes, and then the electrolytes convect into the porous electrode. The electrodes are tightly attached to the flow fields without gaps, therefore, such configuration is also called “zero-gap” architecture.

We will compare the battery performance with and without flow fields from the aspect of hydraulic and electrochemical performances. As shown in Fig. 7b2, since all the electrolyte transports inside the porous electrodes, the electrolyte velocity is relatively high and convection distance is rather long, which results in a high pressure drop in the flow-through structures [69]. As for the influences on the electrochemical performance, the structures with and without flow field affect the cell ohmic loss and the electrolyte distribution inside the electrodes. As discussed in the introduction part, the flow-by configuration enables the adoption of thin electrodes, which significantly reduces the ohmic loss. Xu et al. compared the RFB performance with and without flow fields. The flow-through structure can lead to a more uniform distribution in the through-plane direction, while the flow-by configuration leads to a more uniform in-plane distribution, The through-plane electrolyte supply is mainly from the convection and diffusion, and such transport can be enhanced through increasing the flow rates. Therefore, the battery with serpentine flow fields shows a higher discharge voltage and, thereby, a higher VE at high flow rates [72]. Xu et al. also applied a numerical model to compare different flow configurations, and results show that the flow-field-structured batteries exhibit a higher electrolyte distribution uniformity than the flow-through structure [26]. Since the electrolytes are directly pumped into the electrodes without the guidance of flow channels over the convection path, the electrolyte may be maldistributed inside the electrode in the flow-through structure, which leads to poor battery performance, especially when the electrode active area is enlarged, as reported in Ref. [69]. Overall, the flow field configurations can reduce the pressure drop and lead to more uniform distribution of electrolyte. More importantly, there leaves great flexibility and space to design the flow patterns, adjust the geometric parameters in the flow field structures.

Characteristics of different flow fields In this section, we will compare the most widely studied flow fields, which include parallel, serpentine, interdigitated, and spiral flow fields. These flow fields have been extensively studied in fuel cells [73,74,75]. However, there are fundamental differences in transport properties (e.g., the diffusivity of the fluids) between liquid electrolytes in RFBs and the gas in fuel cells. The transport mechanisms and optimal flow field design for RFBs need to be thoroughly investigated. As depicted in Fig. 7b, the parallel flow field consists of parallel channels, all of which are connected to both the inlet and outlet manifold. Typically, this flow field shows the lowest pressure drop but also weak under-rib convection, leading to nonuniform distributions of electrolytes [67]. The interdigitated flow fields also exhibit parallel flow channels, but the branch channels are alternately connected with the inlet or outlet manifold, forming two disconnected parts. The electrolyte enters the inflow channel and is forced into the electrode, then flows out from the outflow channels, leading to increased uniformity of electrolyte distribution. The inlet and outlet of the serpentine and spiral flow field are connected by one channel but arranged in different path configurations.

Both the hydraulic and electrochemical performance of different flow fields have been studied. MacDonald et al. modeled the flow distribution and pressure drop within the flow-through, parallel and interdigitated flow fields. The hydrodynamic models show that the flow-through structure suffers from excessive pressure drop while the parallel flow field shows the lowest pressure drop and poor penetration of the electrolyte into the electrodes. In addition, optimizing the dimensions and geometric parameters of the interdigitated flow field can yield both high performance and lower pressure drop [76]. In addition to the hydrodynamic models, Darling et al. experimentally compared flow-through, parallel, and interdigitated designs assembled together with graphite felt or carbon paper electrodes. The cell with parallel flow fields shows lower performance even at relatively high flow rates due to the low penetration rate of electrolytes into the electrodes [71]. Huang et al. compared the parallel, serpentine, spiral flow fields using a multiphysics model. The cell with spiral flow fields shows the smallest voltage loss and, therefore, higher VE than the other two types of flow fields. However, the spiral flow fields have longer flow channels, which leads to increased pumping loss and hence a lower VE_pump [29] (Fig. 7c). Due to the weak convection of the parallel flow field and the large flow resistance of the spiral flow field, these two types of flow fields receive the least research interest, while the serpentine and interdigitated flow fields are the most widely researched through both the numerical simulations and the experimental investigations.

Comparison between serpentine and interdigitated flow fields under different working conditions The serpentine and interdigitated flow fields are the most widely used configurations in RFBs. However, the characteristics of electrolyte flow in these two types of flow fields are quite different. Since the inlet and outlet are connected in the serpentine flow field, the electrolyte can preferentially transport in the channels or in the electrodes, which are decided by the pressure difference between adjacent channels. In contrast, the channels in the interdigitated flow fields are not connected. The electrolyte is distributed from the manifold to the inflow branch channels, then transported through electrodes via under-rib convection, and finally collected through the outflow branch channels and the manifold. Therefore, due to the distinct features of the two types of flow fields, both the pressure drop and electrolyte distribution in the two flow patterns are quite different. From the pressure drop aspect, a large proportion of the pressure drop in the interdigitated flow fields comes from the manifold [46]. In many cases, the pressure drop of interdigitated flow fields is smaller than the serpentine pattern due to the split electrolyte and shorter flow distance in branch channels. However, the pressure drop in the interdigitated flow field sharply increases when the electrode with lower permeability is used. From the electrolyte distribution aspect, the non-uniformity of reactant distribution in the serpentine flow field mainly comes from two parts. Firstly, the reactants are continuously consumed along the direction from inlet to outlet. Therefore, even though the velocity distribution under different ribs is similar, only the electrolyte near the inlet is rich in reactants and regions near the outlet usually demonstrates higher mass transport polarization. Secondly, the velocity under one rib is not uniform due to the weak convective regions at the turns of the pattern, since the electrolyte penetration into electrodes is driven by the pressure gradient in the two neighboring channels. In addition, the non-uniformity of electrolyte distribution in the interdigitated flow field also comes from two parts. Firstly, the flux of electrolytes in different branch channels is uneven. Secondly, the electrolyte velocity varies along a single branch channel. Messaggi et al. adopted a macro-segmented flow battery setup to perform locally resolved electrochemical characterizations. The local polarization curves corresponding to the different segments in serpentine and interdigitated flow fields reveal that the single serpentine channel shows decreased electrochemical performance from the inlet to the outlet, while the interdigitated configuration is limited in the central region of the active area, as shown in Fig. 7d [59]. It is difficult to determine which type of flow field, either serpentine or interdigitated, leads to better battery performance due to their distinct features in pressure drop and electrolyte distribution. The comparative conclusion may vary depending on the operating and assembling conditions.

Zhang et al. built a two-dimensional model and compared the serpentine and interdigitated flow fields with an active area of 9 cm². The simulation results reveal that the interdigitated flow field shows a lower pressure drop due to the shorter channel length and a more uniform distribution of electrolytes than the serpentine configuration [23], as shown in Fig. 7e. Actually, the battery performance with serpentine and interdigitated flow fields may vary under different scales. Kumar et al. found that the comparative results get reversed for the serpentine and interdigitated structures with different active areas. On a 100 cm² scale, their experimental results demonstrate that the serpentine flow fields show the highest electrochemical performance and the lowest pressure drop. However, on a smaller scale of 80 mm × 51 mm, the interdigitated flow fields show higher power density and less pressure drop. Such phenomenon is explained by the flow maldistribution that occurred when the interdigitated flow field is scaled-up [69]. In addition to the reversed battery performance induced by the scales, a more complex interplay between cell architecture and electrode properties and operating conditions is investigated by Houser et al., as shown in Fig. 7f. The interdigitated flow field outperforms the serpentine structure with a low flow rate of 20 mL min⁻¹ and a thin electrode, but the battery performance with the serpentine structure can match that with the interdigitated at increased flow rates (90 mL min⁻¹) and thicker electrodes. The interdigitated flow fields exhibit enhanced convection of electrolytes compared with the serpentine configurations since the electrolytes are forced into electrodes for the interdigitated architecture. Therefore, enhanced convection leads to better performance of the cell with interdigitated flow fields at low flow rates, however, the difference between flow fields is minimized at increased flow rates [27]. A similar conclusion was also reported in the work that Houser et al. compared serpentine, interdigitated, and flow-through structures with guidance channels [30].

It is noteworthy that cell size plays a critical role in influencing the performance comparison between cells with and without flow fields, as well as different flow patterns. During scale-up, pumping loss constitutes a significant portion when evaluating cell performance, and VE_pump serves as a more comprehensive indicator in this regard. Owing to the distinct features of various cell configurations and their impact on mass transport and pressure drop, different flow configurations may necessitate distinct scale-up strategies, which will be discussed in greater detail in subsequent sections.

Scaling up the flow cell with serpentine flow field usually suffers from the large pressure drop due to the excessively long flow distance in channels. Applying the split design to serpentine flow fields is regarded as an effective method to lower the pressure drop and maintain the even electrolyte distribution by decreasing the flow distance within each flow channel [23,77,78,79]. Gundlapalli et al. proposed several novel patterns including enhanced cross-flow serpentine and flip-flop serpentine flow fields, as shown in Fig. 9a. The adoption of flip-flop serpentine flow field greatly shortens the flow distance within the active area and reduce the system pressure drop compared to the conventional serpentine flow field. Additionally, the uniform electrolyte distribution leads to the timely replenishment of reactants, thereby improving the SE by over 2% [80].

Other than the application of split channels, some novel configurations were also developed to improve the mass transport of the serpentine flow field. As previously mentioned, the mass transport polarization of serpentine channels is partially caused by the non-uniform velocity distribution under one rib. For the adjacent flow channels, the pressure difference at the connected end is nearly zero, while that at the open end is large due to the longer flow distance. Based on this, Sun et al. proposed three distinct manners (by introducing sloping channels, partially sloping channels and stepwise channels) for making the pressure gradient on the two sides of the rib uniform, as shown in Fig. 9b. By using the modified serpentine flow fields, the SE with pumping consumption considered is improved by about 4% compared to that with the conventional serpentine flow field at 100 mA cm⁻² and 30 mL min⁻¹ [81]. In some studies, researchers still use straight channels but rearrange the flow path to improve the uniformity of electrolyte distribution. Xu et al. were the first to propose a convection-enhanced flow field by tailoring the flow paths based on the conventional serpentine flow field in fuel cells. The pressure difference between adjacent channels is significantly increased due to elongated flow distance via channels [83]. Such a design was then introduced to RFBs and expected to reduce the electrochemical polarization since the mass transport of liquid electrolyte is more dependent on convection (Fig. 9c). Experimental results show that the EE with new flow field adopted is enhanced by 10% as opposed to that with conventional serpentine flow field, which can be attributed to larger convection speed and more uniform reactants distribution in electrodes [36]. Lu et al. numerically compared the convection-enhanced flow field with conventional serpentine and interdigitated flow fields based on the mass transport polarization and pressure drop. Although the cell with convection-enhanced flow field exhibits higher pumping consumption than that with interdigitated flow field, the new flow field still outperforms the other two structures in overall SE due to its significant advantage in reducing the mass transport polarization [60]. Generally, both the serpentine and spiral flow fields have connected inlet and outlet. In the study of Yang et al., the conventional spiral flow field shows inferior electrochemical performance due to the weak convection near the center of electrodes. To circumvent the issue, they developed two double spiral flow fields with outlet and inlet at the center, respectively, as shown in Fig. 9d. With the same pumping power, the application of the two novel designs improves the EE by 7.40% and 6.21% as opposed to the serpentine flow field [82].

It is worth noting that although some patterns are proposed with lab-scale active areas, the design principle still works for stack-scale batteries. Specifically, the pumping consumption accounts for a more significant proportion of power loss during the scale-up, which can be effectively mitigated by splitting channels. Moreover, the non-uniform electrolyte distribution becomes even more severe due to the elongation of flow distance. Enhancing the under-rib convection is always promising to improve the electrochemical performance of batteries at any scales.

Novel patterns based on interdigitated and parallel flow field Different from the serpentine flow field in which the electrolyte flow across the whole length of the flow channel, interdigitated flow field usually has a shorter flow distance since the electrolyte is distributed into branch channels from the inflow main channel and flows out directly via the outflow main channel after the convection process. However, it is still found that the pumping consumption may increase considerably when the cell scales up, and the pressure drop between the inlet pipe and distribution channel end accounts for 42% of the total pressure drop in a 400-cm² cell [46]. Therefore, a hierarchical interdigitated flow field, which has independently regulated channels to lower the pumping loss and further enhance the mass transport, was proposed. Specifically, the main channel and primary branch channels with large sectional areas are designed to transport the electrolyte across the entire length of the active area, and the secondary branch channels with a small sectional area serve to inject the electrolyte into the adjacent electrode with a relatively high velocity to ensure good mass transport, as shown in Fig. 10a. Accordingly, experimental results show that the pumping loss is reduced by 65.9% and the VE_pump is increased from 73.8% to 79.1% at 240 mA cm⁻² and 3.0 mL min⁻¹ cm⁻² when the traditional interdigitated flow field is replaced by hierarchical interdigitated flow field [18]. Differently, Guo et al. proposed a bifurcate interdigitated flow field to alleviate the pumping consumption. As shown in Fig. 10b, the electrolyte is divided equally through hierarchical bifurcate channels outside the active area before entering the branch channels, which reduces the pressure drop during the electrolyte distribution/collection processes and maintains the uniform distribution of reactants. As a result, the pumping consumption is reduced by 45% with an active area of 105 cm², hence the VE_pump increases from 69.95% to 73.10% at 250 mA cm⁻² and 2.4 mL min⁻¹ cm⁻² [84].

Compared to the interdigitated flow field, the inflow and outflow main channels of parallel flow field are directly connected by branch channels, thereby leading to a low pressure difference between adjacent channels and weak under-rib convection. Therefore, Lisboa et al. replaced the straight parallel channel with corrugated channel systems, which employ periodic throttling of the flow to optimally deflect the electrolytes into the porous electrode, as shown in Fig. 10c. An improvement of up to 102% in net power density is obtained even with the adverse effect of increased pumping work [85].

Except for the optimization of flow field configurations, conventional flow fields are also applied to non-rectangular active area for improving the battery performance. For example, the trapezoid flow field shows great superiority in enhancing mass transport and improving the VE compared to the conventional rectangular flow field [87]. However, the relatively uneven distribution of electrolyte flow can still occur in the four corners of the trapezoid region. Hence the radial oriented quasi-parallel and quasi-interdigitated channels were applied to trapezoid flow fields, as shown in Fig. 10d. The introduction of channels improves the spatial distribution uniformity of electrolyte and accelerates the fluid velocity in electrodes, and thus reduces the polarization and increases the rate capacity of RFBs [86]. The comparison of flow batteries with novel flow field patterns and classic low fields is summarized in Table 2.

As for the scale-up of interdigitated-based flow fields, the novel patterns are expected to demonstrate more significant advantages compared to the conventional interdigitated flow field. This is because the scale-up of interdigitated structure suffers from two aspects: large pressure drop in manifolds and uneven electrolyte distribution among branch channels. Both the issues become more serious with larger active areas. The hierarchical and bifurcate patterns can continuously alleviate the pumping loss as well as non-uniform electrolyte distribution, and the superiority can be more extraordinary after the scale-up.

Aside from the intuition-based design strategies, optimization algorithms and machine learning are used to aid in designing the unconventional flow field design. Specifically, some researchers describe the flow fields with two- or three-dimensional computational domains and evolve the optimal configurations iteratively using topology optimization method. For example, Yaji et al. numerically evaluated the fluid flow and mass transport behaviors for a fixed design domain with a two-dimensional model. Hereinto, the species generation rate, which is typically estimated by the Butler-Volmer equations, is simplified to a linear function of mass transfer coefficient and local ions concentration. The topology optimization was then applied to maximize the electrochemical reaction rate of corresponding flow fields. Schematics of the simplified computational domain and optimization history are shown in Fig. 11a1 and a2. The authors then investigated the dependencies of the optimized configurations with respect to the porosity of the electrode and the pressure loss. It was found that the electrode region, where electrochemical reactions take place, becomes thicker with larger porosity and pressure settings, as shown in Fig. 11a3 and a4. Such results are reasonable since the electrolyte permeates the electrode more easily under a larger permeability and pressure difference [37]. Chen et al. constructed a three-dimensional computational domain, which includes the flow channels caved on the graphite plate and electrode adjacent to the graphite plate, to simulate the flow cell more accurately. The topology process was treated as a maximization problem of the electrode surface concentration in the negative electrode. The computational domain and iteration history are provided in Fig. 11b, in which only the inlet and outlet are set for the initial design. However, the optimized design does not show superiority in reactants distribution as opposed to the conventional interdigitated flow field configuration under a relatively small active area [88]. Lin et al. demonstrated that the interdigitated flow field design exhibits a better performance through tuning the channel and land dimensions, and such a process can be realized automatically. Therefore, the topology optimization was performed with the conventional interdigitated flow field as the initial state. The optimization target was to minimize the sum of the electrochemical power loss and pumping loss. The convergence was achieved after about 30 iterations and the total power loss was significantly reduced, as depicted in Fig. 11c [89].

Based on the aforementioned studies, topology optimization provides a potential design pathway for generating irregular configurations by combining them with numerical models. The selection of initial patterns and maximized/minimized indicators significantly affects the final optimized configuration. It might be reasonable to use the total power loss or VE_pump as the objective indicator for striking the balance between the electrochemical and hydraulic performance. However, it is worth noting that the irregular structure is hard to process with conventional fabrication approaches, and 3D printing technology may facilitate the application of topology optimization to the flow field design in RFBs. In addition, the topology optimization is currently applied to lab-scale flow fields. When applying topology optimization to stack-scale cells, there will be a significant increase in computational cost, which can be mitigated by using simplified numerical modeling and considering geometry similarity. Besides, the effect of optimization method still needs to be validated experimentally in the scale-up of RFBs.

Different from the iterative optimization of flow field configuration, Wan et al. constructed a flow field library, predicted the performance with machine learning and selected structures with high performance. Specifically, the first procedure was to generate a large amount of flow field designs. The active area as well as channel to rib ratio were fixed, and the inlet and outlet were connected to avoid excessive variety of configurations. As shown in Fig. 11d1, the active area is divided into several blocks. The initial flow channel region is represented in yellow. The random movement of the flow channel was then set until the channel to rib ratio attains 0.5 and a total of 11,564 configurations were generated. Next, 1164 flow fields were imported into the three-dimensional numerical model to obtain the reactants distribution and pressure drop. The relationship between the performance and 1164 flow field patterns were then extracted to train the convolutional neural network (CNN), based on which the performance of all the generated configurations were predicted with a low resource cost. In the end, eight configurations with a uniform reactant distribution and low pressure drop were determined and validated with both experiments and numerical simulation, as depicted in Fig. 11d2-d4. It should be noted that the flow channel arrangement of the eight configurations is similar to the reported convection-enhanced flow field in Fig. 9c [38]. The rotary channels induce a larger pressure difference between the adjacent flow channels, thereby enhancing the under-rib convection and battery performance. To sum up, the machine learning-assisted design pathway performs well in high-throughput screening of configurations due to the high accuracy and low resource cost. The advantages also make it promising in optimizing the geometric parameters and designing the configurations with unconnected inlet/outlet (such as interdigitated flow field). The design principles also apply to stack-scale batteries. With an enlarged active area, there will be more patterns in the flow field library, which increases the cost of dataset construction and neural network training. Moreover, as the channel length increases and the patterns become more complex, there might exist more promising flow fields that cannot be generated at small-scale sizes.

Modification in serpentine flow field As introduced in Section 3.1, the serpentine and interdigitated flow fields are the most widely researched configurations. However, dead zones still exist with weak convection of electrolytes in the two patterns, which leads to nonuniform distribution of electrolytes. In this section, we will summarize strategies by modifying the channels to enhance the distribution uniformity in serpentine and interdigitated flow fields. Typically, blocks, obstructions, or pillars are introduced to enhance the electrolyte penetration into the electrodes. The designs of such blocks differ for the serpentine and interdigitated patterns due to the different characteristics of the two structures.

For the serpentine flow field, the pattern is repeated by U-shaped segment, and the connected end leads to a lower pressure difference between the neighboring channels while the other end exhibits higher, such variation leads to the nonuniform electrolyte distribution in the electrodes, as shown in Fig. 9b. Accordingly, Lu et al. placed blocks with different heights at the turning short channels to intensify the under-rib convection, which inevitably leads to increased pressure drop. As optimized by the author with a three-dimensional model, the block with a height of 1.4 mm leads to the highest SE [95] (Fig. 14a). Differently, Pan et al. modified the channel depth along the whole flow path to improve electrochemical performance. This is because although all the ribs have a similar velocity distribution, the downstream electrolyte lacks reactants due to the continuous consumption of electrochemical reactions, thereby inducing an uneven reactants distribution over the entire electrode. Therefore, the channel depth is linearly reduced from the inlet and outlet for speeding up the electrolyte velocity and enhancing the convection near the outlet, which leads to an increase in EE by 5% at 12 mL min⁻¹ cm⁻² and 400 mA cm⁻² [96].

Modification in interdigitated flow field Akuzum et al. investigated the effects of varying channel depths using ramps or obstructions on both the electrochemical performance and pressure drop in the interdigitated flow field, as shown in Fig. 14b. Four different channel configurations were investigated, including a ramped channel and channels with varying numbers of obstructions. It was found that ramped channel leads to an improved electrochemical performance by increasing the electrolyte penetration depth in the electrode, especially on the electrode-membrane interface. However, adding obstructions in the channel leads to even worse electrochemical performance, which is possible due to the “by-pass” of electrolyte that the electrolyte travels to the outlet after encountering the obstructions. In addition, it is interesting to find that the tapered channel and channels with obstructions lead to an even smaller pressure drop compared with the flow field without any modifications. These conclusions of adding obstructions in interdigitated flow channels differ from that in the serpentine flow field, which may come from the different flow paths in the two distinct geometries [97].

A systematic study of the location of the obstructions and their effectiveness in improving the cell performance in interdigitated flow fields is carried out by Messaggi et al. Specifically, two types of obstructions located on the channel wall or the electrode interface were investigated, as shown in Fig. 14c. As revealed from the simulation, obstructions on the channel side contribute to the electrolyte penetration into the porous electrode, while obstruction on the electrode interface allows part of the electrolyte to reach the channel end. Therefore, the channel side obstruction is more effective to be placed close to the electrolyte outlet to compensate for the consumption of the reactants. Obstructions at the electrode interface are more effective to be placed close to the electrolyte inlet and induce higher under-rib convection, which leads to higher overall performance [39].

Typically, pillars, obstructions, or blocks are placed in the channels to enhance the electrolyte convection into the electrodes. In addition to such functions, the pillars can work as supports for electrodes under compression, reducing the electrode intrusion into channels. Actually, the issues of electrode intrusion into channels become even more severe when the flow field scales up, which may lead to dramatically increased flow resistance and uneven electrolyte distribution. As reported by Gundlapalli et al., pillars were placed in the manifold of the interdigitated flow field to mitigate the electrode intrusion in the 900 cm² and 1500 cm² cells [98] (Fig. 14d).

In addition to placing obstacles in the channels to increase the electrolyte velocity, adjusting the geometry of the channel cross-section can also enhance the electrolyte velocity. However, the enhancement in reactant transport versus the increase in the pumping loss should be discussed. García-Salaberri et al. modeled the effect of channel tapering on the pressure drop and flow distribution, as shown in Fig. 14e. Since the pressure drop in the interdigitated flow fields comes from both the channel and the electrodes, the increase in pressure drop in tapered channels is tightly related to the electrode properties. Such effect was also discussed in García-Salaberri et al.’s findings. For electrodes with high permeability, pressure drop in channels accounts for the total pressure drop. Therefore, the pressure drop in the cells with tapered flow channels would be much higher than that in the constant cross-section flow channels, and the enhancement in the electrochemical performance may not compensate for the increased pumping consumption. However, for electrodes with low permeability, tapered flow fields can lead to a higher SE with a similar pumping loss [99].

Other modifications in the channels include changing the shapes of the cross-section, e.g., triangular, semi-circle or trapezoidal channels [100]. Either adding obstructions or changing the shapes of the channel cross-section is aimed to enhance the electrolyte penetration into the electrode. However, the parasitic influence on the pressure drop should be considered together with the electrochemical performance. In addition, the results may vary for different flow patterns, operating conditions, and cell scales. Currently, it is difficult to reach a consensus for optimizing channel modification, and more systematic research work involving both experimental and numerical strategies is expected to give more fundamental insights. Furthermore, in addition to the tradeoff between possibly increased pressure drop and electrochemical performance gains, manufacturing and assembly issues must be taken into account when modifying channels during scale-up. These considerations may be combined with parametric optimization for flow field scale-up.

Even though the flow-by cell architecture leads to both improved electrochemical performance and a lower pressure drop, the high manufacturing cost and sealing pressure hinder the scale-up of flow fields. In this regard, some researchers proposed engraving channels in electrodes to distribute electrolytes instead of flow channels. There are mainly three different types of engraving channels on electrodes. The first type is engraving channels through the whole thickness of the electrode. Yin et al. studied electrodes engraved with interdigitated flow fields using a 3D numerical model and compared it with flow fields in graphite plate and flow-through configurations, as shown in Fig. 15a. It is simulated that the electrodes cut with flow field exhibit the lowest pressure drop. Even though the cell with engraved electrodes shows slightly lower SE than the flow-through configurations at a low flow rate (1 mL min⁻¹), it performs much better at increased flow rates. Actually, there exists an optimal flow rate to reach the highest SE for a particular cell configuration, and the optimal flow rate differs for different flow field structures [101].

Cutting channels over the whole thickness of the electrode may lead to difficulties in assembling the electrodes in the cells since the electrodes are composed of separating pieces. Therefore, the second type of engraving channels on electrodes is to cut a certain depth in the channel while the electrode can keep its integral structure. Bhattarai et al. experimentally investigated four types of electrodes with channels, which were rectangular open channel, interdigitated open cut channel, interdigitated circular poked channel and cross poked circular channels in cells with an active area of 100 cm², as shown in Fig. 15b. For the first two types, channels were made on the surface of electrodes, while the latter two constructed channels inside the electrode. Results show that engraving interdigitated type channels, including on the electrode surface and bedding inside electrode, can lead to improved cell performance, while the former mainly reduces the pressure drop and the latter mainly improves the electrolyte distribution uniformity [102]. The same group later scaled up the electrodes with interdigitated flow channels to a 20-cell 1 kW VRFB stack, which demonstrates stable performance in achieving an EE of 80.5% at 80 mA cm⁻², showing the potential for electrodes with channels to be scaled-up [103] (Fig. 15c). Hao et al. constructed some parallel channels with different lengths and one open end on the electrode, which demonstrates higher performance than electrodes with interdigitated channels (Fig. 15d) [104].

The third type of engraving channels on the electrode is to cut part of the channels on the graphite plate and part of the channels on the electrodes. As reported by Sun et al., staggered serpentine channels were constructed on the plate and electrode (near membrane side), which enhance the mass transport in both the in-plane and through-plane directions, as shown in Fig. 15e [105]. Compared with the conventional interdigitated flow field, the detached serpentine configuration leads to improved mass transport, though slightly increased pressure drop. Therefore, a 3.2% increase in SE (v.s. interdigitated flow field) is achieved at a current density of 100 mA cm⁻², an electrolyte flow rate of 20 mL min⁻¹, and a compression ratio of 40%. Compared with the serpentine flow field, the detached serpentine configuration leads to lower pressure drop and enhanced electrolyte mass transport.

As summarized above, all three types of constructing channels in the electrodes are effective in reducing the pressure drop and improving the electrolyte distribution uniformity. Considering the construction feasibility and effectiveness, the second type of engraving open channels on electrodes seems more applicable in practical applications, especially for scaled-up stacks.