Achieving stable and reliable assembly of flow battery stacks through equivalent mechanical models

Abstract

The transition to a low-carbon society demands energy conversion and storage devices with high efficiency. Redox flow batteries are promising candidates; however, their stacks' energy efficiency (EE) remains constrained, and one of the main reasons is the sub-optimal assembly force. Inadequate assembly force can elevate contact resistance among components, and heighten ohmic losses. Conversely, excessive assembly force can overly compress porous electrodes, reduce effective permeability and notably elevate pumping loss. Both scenarios adversely impact the stack's energy conversion efficiency. Moreover, insufficient assembly force may lead to seal failure and electrolyte leakage, while excessive force can jeopardize the battery's structural integrity and potentially harm internal components. Furthermore, any modifications to the stack size or assembly materials necessitate a complete repetition of this process, further increasing the complexity and resource demands. Despite its critical significance, the systematic investigation of the impacts of assembly force optimization in flow batteries remains largely neglected within existing research. To overcome these challenges, this study develops an equivalent mechanical model for RFB stacks, facilitating the determination of the optimal assembly force during stack assembly. This optimization accounts for electrochemical performance, sealing integrity, and the structural strength of individual components. Results indicated that maintaining the tightening torque within the calculated range ensured the sealing integrity of the RFB stack. Furthermore, the stack achieved a coulombic efficiency of 95% and an EE of 84% during cycling operations at a current density of 100 mA cm⁻². These findings confirm the effectiveness and practicality of the proposed method for achieving precise and reliable assembly of RFB stacks, ensuring that the battery operates stably while maintaining high EE.

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Lei Wei

Dr Lei Wei is an Associate Professor in the Department of Mechanical and Energy Engineering at Southern University of Science and Technology. He obtained a doctorate from Hong Kong University of Science and Technology after earning his bachelor's and master's degrees from Xi'an Jiaotong University. Currently, he serves as Associate Director of the Shenzhen Key Laboratory of Advanced Energy Storage and Associate Editor of the Journal of Energy Storage. His research focuses on high-performance flow batteries, thermal runaway mitigation strategies for electrochemical storage systems, and AI-driven materials design, with fundamental investigations into enhanced energy-mass transfer mechanisms in electrochemical systems. He has published over 100 SCI papers, with an H-factor over 40, and more than 30 Chinese patents have been authorized.

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Introduction

In recent years, the escalating demand for transitioning from fossil fuels to renewable energy sources, such as solar and wind power, has provided a strong impetus for the development of advanced energy conversion and storage devices with high efficiency. The redox flow battery (RFB) is a promising candidate developed in the 1970s and has garnered substantial academic and industrial attention.^1,2 Over the decades, the RFB has remained a subject of significant interest due to its potential in addressing large-scale and long-duration energy storage challenges.^3–5 As illustrated in Fig. 1a, the RFB is a rechargeable energy storage system in which energy conversion occurs through reversible electrochemical reactions between two redox pairs.⁶ These redox pairs are generally dissolved in liquid flowable electrolytes, which serve as the energy carrier for charge transfer. The RFB system comprises two primary components: a stack where electrochemical reactions take place and external tanks that store the electrolytes. The flow of electrolytes between the stack and the tanks is facilitated by pumps that ensure continuous circulation.^7,8 The total energy storage capacity of an RFB is determined by the concentration and volume of the electrolytes in the tanks, while the power output is controlled by the operating current density and the total electrode area within the stack. The flexible and customizable design of RFBs enables them to meet varying energy-to-power ratios, making them particularly suitable for grid-scale energy storage applications.^9,10

Fig. 1 Schematic diagram of a RFB (a) system, (b) stack, (c) partial unfolded view.

Despite these advantages, the practical application of RFB still requires higher energy efficiency (EE), which has become a research focus. At present, various strategies have been proposed to improve the EE of lab-scale RFB, including flow field design, electrode and membrane material development, etc.^11–15 Although the EE of lab-scale RFB has reached over 80% at a current density above 400 mA cm⁻², most stack-scale RFBs' EEs are still relatively low (less than 80% at a current density of 100 mA cm⁻²).^16,17 The huge gap in EE between stack-scale and lab-scale RFB is attributed to the complexity of the stack's structure and assembly.¹⁸ As shown in Fig. 1b and c, a typical RFB stack is a composite mechanical structure that encompasses sealing elements, fixed electrodes, electrode frames, inlet plates, current collectors, end plates, and various fasteners.^19,20 Within this framework, multiple single cells, each comprising bipolar plates, carbon felts, and ion exchange membranes, collectively dictate the output power. Bipolar plates play a pivotal role in connecting the positive and negative electrodes of adjacent single cells, facilitating current collection during charge/discharge cycles, polarity isolation, and provision of electrolyte flow channels. Each single cell is constructed with two porous electrodes separated by an ion-exchange membrane and operates using aqueous electrolytes containing active materials. Detailed elucidation of the single-cell structure is presented in Fig. S1.†

It is worth noting that the optimal assembly force plays a critical role in the EE of the RFB stack. Since the RFB stack has such a complex structure, it becomes difficult to determine the appropriate assembly force, and excessive or insufficient assembly force will result in low EE (Fig. 2). When excessive assembly force is applied, the porous electrode becomes overly compressed, drastically reducing its porosity and resulting in decreased effective permeability and increased concentration polarization losses. Moreover, it elevates the pressure drop between the battery's inlet and outlet, along with pumping losses, ultimately diminishing the battery's system efficiency.²¹ Additionally, excessive assembly force may cause electrode fibers to puncture the polymer membrane, weakening its ability to block active ions and reducing coulombic efficiency (CE).^22–25 On the other hand, insufficient assembly force results in poor contact between the components of current collectors, bipolar plates, and membranes. This poor contact increases contact resistance and ohmic losses, leading to a decline in EE. In addition to affecting the battery's EE, the assembly force also plays a role in its stable and reliable operation. For example, excessive assembly force can also create localized stress concentration, potentially causing plastic deformation or component cracking. In contrast, inadequate assembly force may result in liquid electrolyte leakage. Hence, precise control of the assembly force is essential for optimizing battery performance.^19,26 Tang et al. systematically investigated the electrochemical–mechanical coupling behavior during RFB stack assembly, revealing critical structure–property relationships: electrode compressive strain exhibits linear growth with assembly pressure, resulting in a decrease in electrode porosity and a corresponding decrease in electrode permeability, triggering active species concentration gradients and localized current density fluctuations. Obviously, excessive assembly force will significantly reduce the EE of the battery, and it may even lead to the exacerbation of side reactions. Contrastingly, the contact resistance exhibits nonlinear characteristics with the decrease of assembly pressure, but when the pressure exceeds a certain value, the decrease in resistance becomes very slow and it can be regarded as almost no longer decreasing. This requires that the assembly force meet the minimum threshold of ohmic loss.²⁷ These findings highlight the multi-objective optimization dilemma: balancing permeability maintenance with contact resistance suppression requires precise identification of the critical assembly force window. Thus, determining an optimal assembly force range is essential for ensuring the RFB stack's EE. Despite its importance, limited research has been conducted on the role of assembly force in RFB performance.

Fig. 2 The impacts of assembly force on the RFB's energy efficiency.

During the initial development phase of RFB stacks, the determination of assembly torque predominantly relies on empirical trial-and-error approaches by engineers. This process typically involves preliminary torque settings based on historical data or analogous designs, followed by iterative adjustments to refine parameters. A single debugging cycle consumes several days (including leakage tests and performance validation), prolonging the new stack development timeline. Subjective judgments by engineers may introduce unexpected torque deviations, triggering seal failure or excessive electrode compression and even causing component failure. The assembly force magnitude is intricately linked to the materials, dimensions, and structural parameters of the stack. Even minor variations in these factors require a restart of the assembly process, leading to time and labor intensity. This challenge significantly hinders the optimization of the RFB stack structure and materials. Experience-dependent methods hinder standardized process documentation, elevating manpower costs by 2–3 times during production line commissioning and impeding assembly consistency in 100 MW-scale stacks. Developing a finite element model (FEM) with multiscale structural characteristics can roughly guide the clamping process, though it presents significant computational challenges. Achieving accurate results requires cross-scale fine mesh sizes: on the order of micrometers for membrane elements, approximately 10 micrometers for electrode-bipolar plate contact elements, and 100 micrometers for bipolar plates, while larger endplates and bolts require meshing up to one centimeter. Consequently, FEM analysis for large RFB stacks involves complex nonlinear contact mechanics, with models frequently reaching millions of elements and necessitating extensive computational resources. Even for a single cell, computation steps can take several hours. Simplifying the geometry for FEM analysis is thus essential for efficiently assessing assembly forces. For example, Xiong et al.^28,29 conducted a force analysis on a 20-cell RFB stack, demonstrating the relationship between assembly force, electrochemical performance, and mechanical failure risk. The study's simplification of internal components as flat plates reduced its applicability to practical conditions. Peng Lin et al.^30,31 introduced an equivalent stiffness model, calculating the stiffness of individual components to determine total stack stiffness, significantly reducing computational demands. However, the structural and component differences between RFB and fuel cells limit the model's applicability.

Accurate and efficient determination of assembly force in the stacks is essential for the effective engineering of RFBs. This work proposes a mechanical model that substantially reduces the development costs and accelerates the design cycle of RFB stacks, effectively bridging a critical technological gap in this research domain, and conceptualizing the RFB stack system as a series or parallel arrangement of elastic elements (springs). The stiffness coefficients for each spring were calculated through material mechanics analysis of periodic structures. Based on the calculated stiffness values considering RFB stack sealing requirements, contact resistance, and electrode permeability, optimal assembly force ranges were determined. The established model enabled a comprehensive analysis of the factors influencing the system, evaluating their interactions and impacts on the overall structure. Finally, we validated this method on the RFB stack containing 5 cells. The stack is well sealed and maintains a CE of over 95% in cycle tests of charge and discharge at 100 mA cm⁻². Experimental results indicate that the RFB stack assembled under this model's guidance can maintain high CE and EE while ensuring the stability of the energy conversion process. This method facilitates the determination of ideal assembly forces, eliminates mechanical failures of the stack and achieves ideal EE. This method can provide valuable guidance for practical RFB stack construction. In addition, digital twin technology is a virtual replica of physical systems; Feng et al. proposed its enormous potential for application in energy storage systems.^32,33 It can enable real-time monitoring and simulation of RFB systems. The application of equivalent mechanical models in RFB digital twins shows broad prospects, and we will consider future research in this area.

In the following part of the work, Basic model descriptions presents the idea, the establishment process, and validation of the equivalent mechanical model. Application of the model to stack assembling presents the guiding process of the model for the assembly of the RFB stack, comprehensively considering the sealing requirements, structural strength, contact resistance and electrode permeability. The calculation results are presented in Calculation results and the stack performance verification is discussed in Experimental validation. Finally, in the Conclusion, the work contributions are summarized and the technical significance of this work for industrial RFB is presented.

Basic model descriptions

Stiffness in solids, defined as the resistance of an elastic body to deformation, quantifies the force required to achieve a unit displacement. Its value is intrinsically linked to the structural dimensions, geometry, and material properties of the object under consideration.³⁴ As illustrated in Fig. S2,† axial stiffness is modeled using a spring analogy, with its magnitude determined by eqn (1). Considering the symmetry of components within a RFB stack, the three-dimensional configuration can be effectively reduced to a two-dimensional cross-sectional representation, which is further simplified into a one-dimensional system of spring combinations. In such a representation, parallel components are modeled as parallel springs, while vertically aligned components are represented as springs in series.

The equivalent stiffness (k_eq) is described by:

k_eq = EA/L (1)

where E denotes Young's modulus, A is the cross-sectional area, and L is the axial length of the component. The present study focuses primarily on the mechanical behavior of RFB stacks under assembly forces. The comprehensive mechanical model is constructed by initially analyzing individual cells, which are then cyclically assembled to represent the entire stack. Employing the equivalent stiffness model allows the determination of the deformation-assembly force relationship. The equivalent stiffness model is based on fundamental physical principles: the material constitutive equation (stress–strain relationship) and geometric compatibility conditions (deformation coordination equation), ensuring its inherent scalability to accommodate variable cell counts. Replacing components with different materials or sizes only requires adjustment of material properties (e.g., Young's modulus) and dimensional parameters to meet operational requirements. Additionally, the equivalent mechanical model applies to various flow field configurations. To illustrate, the interdigitated flow field was processed by counting rib quantity as independent units in parallel before calculation. Similarly, serpentine and parallel flow fields can be mathematically modeled this way. Furthermore, unconventional flow fields may have their stiffness calculated directly through surface area and thickness parameters.

Basic assumptions

To establish an accurate equivalent stiffness model, specific assumptions are introduced to balance theoretical rigor with practical applicability:

(1) Rigid end plate approximation: the end plate is modeled as a rigid body with adequate mechanical strength. While achieving ideal rigidity in practice is challenging, proper design can approximate this condition.³⁵

(2) Uniform clamping load: during assembly, each clamping screw is assumed to experience an identical clamping load and undergo uniform elongation.

(3) Flow channel configuration: bipolar plates are engraved with flow channels, incorporating N flow channels and N − 1 ribs.

(4) Bipolar plate and flow frame regions: the bipolar plate is segmented into three regions—BPP-insertion, BPP-rib, and BPP-basis—while the flow frame is divided into frame-seal and frame-insertion regions, as depicted in Fig. 3.

Fig. 3 Schematic diagram of component structure division: (a) bipolar plate, (b) flow frame.

(5) Uniform interface thickness: before assembly, all interface components, including electrodes, membranes, and sealing gaskets, are assumed to have a uniform thickness, with no gaps present at the connection points of the RFB stack.

By integrating these assumptions, the modeling framework achieves both simplification and fidelity, enabling an accurate representation of the mechanical response of RFB stacks to assembly forces.

Equivalent mechanical model of a single RFB

Fig. 4a illustrates the cross-sectional view of a single cell. To facilitate subsequent calculations, the single cell is partitioned into three regions: the core area, the insertion area, and the sealing area. The RFB stack is correspondingly divided based on this segmentation. Based on the single-cell structure divided in Fig. 4a, we established an equivalent mechanical model of the single-cell as shown in Fig. 4b, where δ is the elongation after tightening the screw, C is the number of screws, and the preset upward direction is the positive displacement direction.

Fig. 4 (a) Single cell area partition. (b) Schematic of the equivalent mechanical model of the single cell.

Subsequently, a simplified model, depicted in Fig. 5a, was established by integrating the electrodes, membrane, and ribs of a single cell into a unified computational unit referred to as the “cell-kernel.” The springs associated with the gasket-seal and frame-seal were subsequently combined in series to constitute the “cell-seal.” Similarly, the BPP-basis and cell-kernel springs were integrated in series to form the “cell-center”, while the BPP-insertion, gasket-insertion, and frame-insertion springs were also combined in series to generate the “cell-insertion”. These components were ultimately arranged in parallel, thereby streamlining the equivalent mechanical model. As a result of these simplifications, the mechanical system of a single cell could be reduced to a more concise form, as illustrated in Fig. 5b.

Fig. 5 Schematic of the equivalent mechanical model of the single cell: (a) simplified, (b) final.

Based on the model shown in Fig. 5b, the equivalent stiffness of an RFB single cell is given:

k_single-cell = k_cell-center + k_{cell-insertion} + k_cell-seal (2)

where k is the equivalent stiffness of the corresponding component, and subscripts distinguish the components. The cell-center's equivalent stiffness can be calculated as:where the equivalent stiffness of the cell-kernel can be calculated as:where i is the serial number of the ribs. Similarly, the equivalent stiffness of the cell-insertion is:

Due to the static mechanical equilibrium conditions, the governing equation can be written as:

Ck_screw(δ + Z^F_single-cell) + k_single-cellZ^F_single-cell = 0 (7)

So the deformation displacement of the single cell under assembly force is:

From eqn (8), we can calculate the assembly force of the three parts of a single cell:

F_cell-center = k_cell-centerZ^F_single-cell (9)

F_cell-insert = k_cell-insertZ^F_single-cell (10)

F_cell-seal = k_cell-sealZ^F_single-cell (11)

Equivalent mechanical model of a RFB stack

As shown in Fig. 1b and c, the RFB stack connects multiple internal batteries in series through bipolar plates. The equivalent mechanical model of a stack can be formed by arranging single cells in sequence, as shown in Fig. 6.

Fig. 6 Schematic of the equivalent mechanical model of the RFB stack.

Then the stiffness of the RFB stack can be calculated as:

k_stack = k_stack-center + k_{stack-insertion} + k_stack-seal (12)

where the stiffness of the stack-center, stack-insertion, and stack-seal can be described as:where i, j, and x refer to the serial numbers of the components. According to the static mechanical equilibrium conditions, the governing equation of the model of the stack can be described as:

Similarly, the assembly forces applied to the stack are:

F_stack-center = k_stack-centerZ^F_stack (17)

F_{stack-insertion} = k_{stack-insertion}Z^F_stack (18)

F_stack-seal = k_stack-sealZ^F_stack (19)

F_assembly = F_stack-center + F_{stack-insertion} + F_stack-seal (20)

Due to specific assumptions and simplifications inherent in the proposed equivalent mechanical models, discrepancies inevitably exist between the analytical results and the actual physical behavior. To evaluate the accuracy of the equivalent mechanical model, finite element analysis (FEA) was employed. The calculations were based on the three-dimensional (3D) model of the RFB stack, as illustrated in Fig. 1b. The structural parameters of the RFB stack utilized in the analysis are detailed in Table 1.

Table 1 Primary geometric parameters of the RFB stack

Components	Length (mm)	Width (mm)	Thickness (mm)
Electrode	626	166	0.5
Gasket	756	311	0.5
Bipolar plate	642	182	3.2
Flow frame	756	311	6
Membrane	626	166	0.05

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Calculations were performed on individual components and the single-cell assembly, respectively. Computational simulations for individual components require only tens of seconds per step, which is deemed acceptable. However, to ensure accuracy, the mesh for the single-cell assembly was refined. Following rigorous mesh sensitivity verification, the final computational grid comprises 927 144 nodes and 414 092 elements (see Fig. 7). Each simulation case executed on an Intel Core i7-10700F CPU with 32 GB RAM requires approximately 40 minutes. Table 2 lists the mechanical properties of each component, assuming that all components exhibit linear elastic behavior.

Fig. 7 FEA contact model: (a) schematic diagram of component deployment, (b) calculation results of displacement generated by the clamping force.

Table 2 Material properties of the RFB stack

Components	Material	Young's modulus (GPa)	Poisson's ratio	Yield strength (MPa)	Source
Electrode	Carbon felt	0.0075	0.1		27
Gasket	EPDM	0.0078	0.47	20	28
Bipolar plate	Graphite plate	3.44	0.25	130	27
Flow frame	PVC	2.41	0.38	40	27
Membrane	Nafion	0.2	0.25	32	27

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Stiffness values were derived by independently computing the compressive deformation displacement of each component along the force direction under varying pressures via FEA. The displacement exhibited proportionality to the applied force (Fig. S3†), where the slope corresponds to the derived stiffness. This identical methodology was applied to the single-cell assembly, with FEA results detailed in Fig. S3.† Concurrently, an equivalent mechanical model was constructed in Simulink to calculate component-level and assembly-level stiffness using identical stack parameters. Comparative validation between FEA-derived and Simulink-modeled stiffness values (Table 3) demonstrated high consistency, as evidenced by relative deviations below 1% for individual components and under 3% for the single-cell assembly. When compared with the FEA results, the stiffness obtained from the equivalent mechanical model has a relative error of approximately 3%. Although minor discrepancies are observed, they are considered acceptable within the context of engineering assembly. Additionally, considering the computational time of several tens of minutes required for FEA and the challenges associated with achieving convergence, the equivalent mechanical model in Simulink completes the calculations within a few seconds. Therefore, the equivalent mechanical model provides significant practical advantages for the analysis and optimization of RFB stacks.

Table 3 Result comparisons of the equivalent mechanical model and the FEA

Components	Equivalent stiffness (N m^−1) of FEA	Equivalent stiffness (N m^−1) of the equivalent mechanical model	Relative error (%)
BPP-base	4.4282 × 10^10	4.4684 × 10^10	<1
BPP-insertion	1.3995 × 10^10	1.3898 × 10^10	<1
BPP-rib	1.4376 × 10^9	1.4276 × 10^9	<1
Electrode	1.5448 × 10^9	1.5587 × 10^9	<1
Membrane	4.6551 × 10^11	4.6738 × 10^11	<1
Frame-seal	4.7886 × 10^10	4.7506 × 10^10	<1
Frame-insertion	7.0456 × 10^9	7.0810 × 10^9	<1
Gasket-seal	1.8283 × 10^9	1.8450 × 10^9	<1
Gasket-insertion	2.0027 × 10^8	2.0168 × 10^8	<1
Single-cell	1.6889 × 10^9	1.7374 × 10^9	<3

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Application of the model to stack assembling

The assembly force for large RFB stacks is a critical parameter influenced by a multitude of interrelated factors, each playing a pivotal role in the stack's operational stability and structural integrity. This force must be meticulously determined and maintained within a well-defined range to balance several essential aspects, including sealing integrity, adequate permeability, minimal contact resistance, and structural safety. To address this, an equivalent mechanical model is employed to ascertain the optimal range for the assembly force, providing a systematic and robust methodology.

The RFB stack typically comprises three primary components: (A) the power generation core, located at the center of the stack, (B) the bipolar plate assembly with flow frames, designed to prevent internal leakage, and (C) the external seal area, located at the stack's periphery. In the equivalent mechanical model, these components are referred to as the stack kernel, stack insertion, and stack seal, respectively.

For the stack kernel, variations in the assembly force during construction exert a significant influence on parameters critical to power generation, such as electrode permeability and interfacial contact resistance. These adjustments in assembly forces directly affect the efficiency of charge transfer and overall energy conversion, underscoring the necessity for precise control. Specifically, in the context of stack insertion, variations in assembly force primarily impact internal sealing efficiency, particularly in mitigating electrolyte cross-contamination between cells. This factor is crucial for preserving electrochemical stability and ensuring consistent performance. Furthermore, alterations to the clamping load on the stack seal predominantly govern the external sealing integrity of the stack, thereby ensuring the containment of electrolytes and gases under operational conditions.

The subsequent calculations are primarily grounded in structural strength analysis, which assesses the impact of these force variations on the operational performance and sealing efficacy of the stack.

Tightening torque of nuts

Under the assumption that the end plate is an undeformable rigid body, the stack-kernel, stack-insertion, and stack-seal undergo the same deformation after the assembly force is applied, which can be calculated as:where:

F_whole-stack = F_assembly (22)

The left-hand side of eqn (22) represents the total compressive force applied to the stack, whereas the right-hand side quantifies the total tensile force exerted on all the screws. Although these forces are opposed in direction, they are equal in magnitude. By substituting eqn (22) into eqn (21), we can derive:

Within the domain of mechanical design, a linear relationship exists between the tensile force of a screw and its tightening torque, succinctly captured by:³⁴

T = 0.2Fd/C (24)

where T denotes the tightening torque of the screw, F represents the tensile force of the screw, C is the number of screws and d is the nominal diameter of the screw.

By combining eqn (23) and (24), the relationship between tightening torque and assembly force can be obtained:

Assembly force design based on the strength of the stack-seal

In the RFB stacks, to completely eradicate electrolyte leakage, gaskets are commonly interposed between flow frames, utilizing materials such as fluoroelastomers and ethylene propylene diene monomer (EPDM) rubbers. A pivotal factor influencing seal integrity is the mean contact pressure exerted on the sealing interface. Insufficient mean contact pressure may precipitate leakage, whereas excessive pressure risks damaging the sealing components or adjacent components. Consequently, to ensure the integrity of the sealant, the mean contact pressure (P_seal) must adhere to stringent engineering standards and design guidelines. For P_seal to be optimal, it must be calibrated within a predefined range that balances the prevention of leakage against the potential for material failure, thereby safeguarding the operational efficiency and safety of the RFB stack. This can be expressed as:

P^min_seal ≤ P_seal ≤ P^max_seal (26)

where P^min_seal denotes the minimum threshold of mean contact pressure essential for preventing any leakage, whereas P^max_seal signifies the maximum mean contact pressure that the sealing material can withstand without failure. Based on the empirical formula,³⁶ the range of sealing force is:where P_electrolyte is the electrolyte pressure of the RFB stack, b is the thickness of the gasket, ξ is the transient coefficient of the gasket from elastic to plastic deformation (roughly having a value of 3 for most polymeric materials³¹), and σ^yield_gasket is the yield stress of the gasket material.

Therefore, the allowed range of the tightening torque applied to a single screw based on the strength requirement of the stack-seal can be written as:

Assembly force design based on the strength of the stack-insertion

Step surfaces are commonly designed on the bipolar plates and flow frames to ensure correct alignment during the assembly of RFB stacks (Fig. S5†). O-rings are used to seal between cells to prevent internal leakage of the stack.

The compression ratio of the O-ring is controlled by changing the depth of the sealing ring groove, and its calculation process is as follows:

When compressed, the O-ring produces a significant restoring force (F_O-ring) according to the compression ratio [see eqn (1)], as shown in Fig. 8a. In this study, the compression ratio is 20%, as the total length of the H 80 O-ring for the present RFB stack is 1.75 m and the section diameter is 3 mm. The calculated restoring force of the O-ring is about 45 N cm⁻¹, and the total restoring force is calculated to be 8000 N as shown in Fig. 8b.

Fig. 8 (a) Restoring force of O-ring type gaskets according to compression ratio.³⁷ (b) Schematic showing the equivalent restoring force of the O-ring.

In addition, the application of force in the stack-insertion phase necessitates meticulous regulation to avert the stress surpassing the compressive strength threshold (σ_frame) of the flow frame material in this region.

Combining the above two constraints, the force range of stack-insertion can be expressed as:

P^min_seal ≤ P_seal ≤ P^max_seal (30)

where s_frame is the safety factor of the bipolar plate and here we set it to 2. Thus, the viable interval for the tightening torque applied to each screw, in compliance with the stack-insertion's strength specifications, is delineated as:

F_O-ring(k_stack-kernel + k_{stack-insertion} + k_stack-seal)0.2d/Ck_{stack-insertion} ≤ T ≤ σ_frameA_{BPP-insertion}(k_stack-kernel + k_{stack-insertion} + k_stack-seal)0.2d/Ck_{stack-insertion}s_frame (31)

Assembly force design based on the strength of the stack-kernel

In the assembly process of RFB stacks, upon contact between the BPP and the electrode, a circuit is established. If the pressure applied to the contact surface is insufficient, there arises an unreasonably high contact resistance, leading to unexpected ohmic polarization within the circuit. Consequently, the minimum clamping load applied to the stack-kernel should exceed the load induced by the electrolyte pressure. However, this minimum clamping load may share the same lower limit as the stack-seal, as specified by eqn (28). Conversely, the assembly force exerted on the internal stack components should not be so great as to push the stress in the components to their strength limits, ensuring the integrity of the components and the proper operation of the stack. In this segment of our investigation, we've pinpointed the proton exchange membrane as the most susceptible component to damage. Drawing from our prior research, we've uncovered that excessive assembly force can trigger a cascade of issues. Specifically, it causes the carbon fibers within the electrodes to perforate the membrane, leading to a phenomenon known as the “crossover” of vanadium ions. This crossover event significantly diminishes the CE of the battery.³⁸ The stress required to cause these micro-pinhole defects is much lower than the yield stress of the membrane material. Therefore, the assembly force for the stack-kernel should meet the following criteria:

F_stack-kernel ≤ γσ^yield_membraneA_rib/S_membrane (32)

where γ is the number of ribs on one side of BPP, S_membrane is the safety factor of the membrane and here we set it to 5 to minimize the occurrence of micro pinhole defects. From this, it can be inferred that, considering the strength of the stack-kernel, the maximum allowable tightening torque applied to a single screw is:

T ≤ (k_stack-kernel + k_{stack-insertion} + k_stack-seal)γσ^yield_membraneA_rib0.2d/Ck_stack-kernelS_membrane (33)

Strength design of the screws

In the structure of RFB stacks, the screws serve a pivotal role by primarily withstanding axial tensile stress. It's noteworthy that an increase in torque results in a corresponding increase in stress. Grounded in the fundamentals of mechanical design,³⁹ the axial tensile stress experienced by each bolt must align with the design standards of material strength. The formula is as follows:where d₁ is the minor diameter of the screw, and S is the safety factor (usually taken as 1.3 when checking tight screws³⁹).

Relationship between electrode BPP interface resistance and tightening torque

Through rigorous strength design analysis, the optimal assembly force within prescribed safety margins has been identified. The endeavor to determine the ideal assembly force for RFB stacks through precise control of tightening torque represents a complex and academically significant challenge. The pursuit of an optimal clamping load constitutes a multi-objective optimization problem, widely regarded as a critical issue in computational mechanics. This complexity stems from the intricate interactions between assembly forces and stack performance parameters. Nevertheless, leveraging the theoretical framework of equivalent mechanical models, this study adopts a systematic methodology to determine the optimal assembly force for a given RFB stack, thereby offering a practical solution to this multifaceted problem. The contact resistance between two solid surfaces is predominantly governed by the multiscale topographical characteristics of the surfaces and the intrinsic material properties. When two solid surfaces are brought into contact under minimal applied pressure, only the asperities at the surface roughness contribute to the interaction, resulting in an exceptionally small real area of contact. Under such conditions, interfacial resistance is notably high.^40–42 As the compressive force increases incrementally, the real contact area expands, leading to a corresponding reduction in ohmic resistance. For RFB stacks, the assembly force must be adequately high to ensure that the overall ohmic resistance within the circuitry remains within acceptable limits.

Research on interface contact resistance in RFBs has been extensively documented in the literature. For instance, Xiong et al.²⁷ employed a custom-designed fixture to systematically measure the contact resistance of RFB interfaces under varying contact pressures. Their investigation meticulously plotted the characteristic curve illustrating the relationship between resistance and applied force. Based on this, the relationship between contact pressure and contact resistance is derived as follows:

where R_contact is the interface contact resistance and P_contact is the contact pressure at the electrode, which can be calculated as:

Therefore, the interfacial contact resistance can be determined by combining eqn (25), (35) and (36):

R_contact ≈ 0.098004/(1 + ((CTk_stack-kernel − 0.2dP_electrolyteA_electrode(k_stack-kernel + k_{stack-insertion} + k_stack-seal))/(0.0004472d(k_stack-kernel + k_{stack-insertion} + k_stack-seal)γA_rib)^1.197713)γA_rib) (37)

Relationship between electrode permeability and tightening torque

In the process of the compression effects on porous electrodes within battery assemblies, the alteration of the porosity can be succinctly described by the pivotal equation (eqn (38)):⁴³where V signifies the geometric volume of the electrode and θ₀ denotes the initial porosity of the electrode in the absence of external forces. The parameter ε_V is defined as the volumetric strain of the porous electrode, taking on negative values when the electrode is subjected to compressive forces. As a pivotal parameter reflecting the fluid transport capabilities within electrodes, permeability exhibits a close and intricate relationship with porosity. The mathematical representation of this relationship is delineated in eqn (39):⁴⁴

Calculation results

Based on the established model, a computational program was developed to analyze the design parameters of RFB stacks. A demonstration of an RFB stack design comprising multiple cells is provided in the subsequent sections. Table 1 summarizes the primary dimensional specifications of the stack, which includes design quantities of 5, 10, 20, 30, 40, and 50 cells. The material parameters required for the demonstration are listed in Table 2. Initially, all fasteners were selected from the standard M12 series, including washers, nuts, and screws.

The stiffness in the compression direction of the stack kernel, stack insertion, and stack seal was calculated for designs with varying numbers of cells. As shown in Fig. 9, the stiffness of the stack seal was consistently the highest among the three components. However, all stiffness values decreased nonlinearly as the number of cells increased. In addition, this model can easily adapt to different materials by modifying parameters. For example, Fig. S6† shows the comparison of stack stiffness calculated using flow frames made of PVC and PP. The equivalent stiffness of different flow fields was also compared (Fig. S7†).

Fig. 9 Variation in the equivalent stiffness with the number of cells.

To ensure the structural strength and sealing requirements of RFB stacks under all possible conditions, the maximum and minimum torques required for stacks containing different numbers of cells were evaluated. As illustrated in Fig. S8,† the torque range remained relatively stable across varying cell numbers, with values spanning from 30.04 N m to 219.03 N m. Notably, the upper limit of this range was typically not utilized due to its impracticality. For subsequent calculations, a 20-cell stack was selected as the reference design.

Further calculations were performed to determine the torque demands for the stack insertion section, yielding a range of 18.92 N m to 615.40 N m. This range aligns with empirical observations from assembly processes. The force required for O-ring sealing was lower than that for gasket sealing pressure, making its lower limit relevant for reference, whereas the upper limit was seldom reached.

The maximum torque for the stack kernel section was calculated based on the membrane's strength, resulting in a value of 103.36 N m. Although a safety factor was incorporated to minimize the occurrence of micro-pinhole defects, damage to the membrane could still occur due to puncturing by carbon fibers in the electrode, even if the upper limit torque was not exceeded. Unfortunately, no established reports currently exist on the strength threshold of the membrane in this context, necessitating further research in this area.

The upper limit of tightening torque, determined by the strength of the bolts, was calculated to be 159.93 N m using 12.9-grade screws.

To maximize the performance of the RFB stack, minimizing interface resistance is essential, which necessitates increasing the tightening torque. However, ensuring high electrode permeability requires limiting the tightening torque. These two factors are inherently contradictory and must be carefully balanced. Calculations of interface resistance and electrode permeability under varying torques are presented in Fig. 10a. The shaded region in the figure represents the optimal allowable range, where interface resistance and permeability remain within acceptable limits, spanning from 20 N m to 45 N m. Considering all relevant factors, the optimal tightening torque range for the studied RFB stack was determined to be between 30.04 N m and 45.00 N m, as depicted in Fig. 10b.

Fig. 10 (a) Interface resistivity and permeability with the tightening torque. (b) The optimal tightening torque range.

Experimental validation

To ensure the accuracy of the computational results, an experimental verification of the RFB stack was conducted. The assembled stack, comprising five cells, is illustrated in Fig. 11, with the relevant dimensional data provided in Table 1. High-pressure gas testing was employed to evaluate the sealing performance of the stack. During the test, if the stack is not adequately sealed, the significant pressure difference between the interior and exterior will cause gas leakage. The results of the pressure test and pressure retention are presented in Table 4.

Fig. 11 RFB stack assembled for testing purposes.

Table 4 Gas tightness test of the RFB stack

Tightening torque	Initial pressure	Hold time	Final pressure
25 N m	150 kPa	3 h	65 kPa
32 N m	150 kPa	3 h	141 kPa
40 N m	150 kPa	3 h	149 kPa

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The results indicate that when the tightening torque was set to 25 N m, the internal high-pressure gas began to leak, with the pressure dropping from 150 kPa to 65 kPa within 3 hours. Upon increasing the tightening torque to 32 N m, the pressure after 3 hours stabilized at 141 kPa, demonstrating a notable improvement in the pressure retention capability. Furthermore, when the tightening torque was raised to 40 N m, the pressure remained almost unchanged at 149 kPa after 3 hours, which is in close agreement with the calculated permeability values derived from the equivalent mechanical model.

After assembling the RFB stack under the tightening torque of 32 N m and 40 N m, electrochemical performance tests of the stack were conducted. The charge and discharge cutoff voltages of the stack were set at 8.25 V and 4.5 V, respectively, corresponding to approximately 1.65 V and 0.9 V for a single cell, which aligns with the values reported in previous studies.⁴⁵ The stack was cycled at a current density of 100 mA cm⁻², and the charge and discharge profiles are depicted in Fig. 12c. As observed, the voltage increased during charging and decreased during discharging, a trend attributed to the concentration variation of active species, which induces changes in both the open-circuit voltage and the polarization voltage. Our attention was first focused on evaluating the stack's CE. This parameter is a key indicator of the battery's internal sealing effectiveness. If there is an issue with the internal O-ring sealing, the positive and negative electrolytes within the battery can intermingle, leading to a substantial decrease in CE. The test results reveal that the CE of the stack consistently remained above 95%, which is approximately 1% lower than the value observed for a single cell under the same current density.³⁸ This minor reduction is attributed to the shunt current loss that occurs in multi-cell stacks, which leads to a partial loss in CE. Furthermore, the EE, defined as the ratio of discharged energy to charged energy, was maintained at approximately 84%, as shown in Fig. 12b. The stack assembled at 32 N m showed marginally lower CE and EE compared to the 40 N m assembly. This discrepancy may arise from superior sealing and lower contact resistance at 40 N m. Both CEs and EEs remained stable throughout the whole testing period, with no degradation observed. These results provide strong evidence that the battery is operating in a stable and efficient manner under the designed tightening torque.

Fig. 12 RFB stack test results: (a) voltage variation curve, (b) battery efficiencies, (c) charge and discharge curve.

This completes our entire workflow of using equivalent mechanical models to guide the assembly of large-scale RFB stacks. In summary, the parameters of the RFB stack, including geometric dimensions of components and material properties, are imported into the equivalent mechanical model. Limit conditions such as sealing constraints and material strength limits are then input, after which the equivalent mechanical model derives the optimal assembly torque range, as illustrated in Fig. S9.† And then the stack is assembled using the torque calculated using the equivalent mechanical model.

Conclusion

RFBs contain numerous internal mechanical structures, including fastening screws, metal endplates, plastic plate frames, and conductive bipolar plates. While previous research has predominantly focused on internal electrochemical reactions and material properties, the design and assembly of these mechanical components also play a critical role in determining the stability and efficiency of the system. Unlike prior studies, this work emphasizes the impact of mechanical assembly forces on the performance of the electrochemical stack. To achieve this, a large RFB stack is simplified into an equivalent mechanical model comprising multiple elastic elements connected in parallel or series. Beginning with a single RFB unit, an equivalent stiffness model for the entire stack is constructed. The effects of structural parameters on internal stress, contact resistance, and sealing pressure are analysed, allowing for the determination of the optimal clamping torque range for the stack. Numerical analysis employing a three-dimensional nonlinear contact model demonstrates that the prediction error for the stiffness of individual components within a single cell is less than 1%, while the error for the stiffness of the overall single-cell structure remains below 3%. These results validate the accuracy of the proposed method. Based on this model, the optimal clamping torque for RFB stacks is derived by considering contact resistance and permeability values. In practical applications, the optimal clamping torque is determined by balancing these factors to accommodate various potential influences. Guided by this approach, the assembled RFB stack successfully passed high-pressure gas tightness and battery performance tests, exhibiting stable CE above 95% and EE exceeding 84% during cycle tests at the current density of 100 mA cm⁻². These findings confirm the efficacy of the proposed method. The design methodology outlined in this study not only provides a robust framework for the development of mainstream RFB stacks but also offers insights applicable to other types of RFB systems. This approach ensures system stability and reliability, contributing to advancements in the design and optimization of RFB technologies.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Nomenclature

k	Equivalent stiffness of the component (N mm^−1)
F	The load applied on the component (N)
E	Young's modulus (N mm^−2)
A	Cross-section area (mm^2)
L	Length of the component along the clamping load direction (mm)
C	Number of the clamping screws
N	Number of flow channels on one side of the bipolar plate
M	Number of cells of a large stack
Z ^F	Clamping deformation (mm)
S	Safety factor

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 12426307, 524B2078, 52206089), Guangdong Major Project of Basic and Applied Basic Research (2023B0303000002), Guangdong Basic and Applied Basic Research Foundation (2023B1515120005), Natural Science Foundation of Shenzhen (JCYJ20241202125327036, JCYJ20240813100103005), Research Center on Energy Storage Technology in Salt Caverns Program (TO2203001), and high level of special funds (G03034K001). The computation in this work is supported by the Center for Computational Science and Engineering at the Southern University of Science and Technology.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ta03767k

‡ Honghao Qi and Lyuming Pan contributed equally to this work.

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