Modelling of redox ﬂ ow battery electrode processes at a range of length scales: a review †

In this article, the di ﬀ erent approaches reported in the literature for modelling electrode processes in redox ﬂ ow batteries (RFBs) are reviewed. RFB models vary widely in terms of computational complexity, research scalability and accuracy of predictions. Development of RFB models have been quite slow in the past


Introduction
Energy storage research is undergoing a steady increase over the years in response to the need for deploying renewable energy sources for mitigating climate change issues. 1,2 Several nations, 3 such as the USA, 4 UK, 5,6 Europe, 7 China, 8 Japan 9 and others, have taken positive steps forward by allocating signicant public funds for this purpose. 10,11 Amongst several energy storage technologies available, redox ow batteries (RFBs) are considered viable for future renewable energy and grid-scale load levelling applications 12,13 due to favorable qualities, such as independent power/energy sizing, high efficiency, ambient or near ambient temperature operation and projected long charge/ discharge cycle life. 14 Despite these benets, cost and irreversible crossover of active species in asymmetric chemistries (e.g., iron/chromium redox ow battery, or ICFB for short) have limited widespread RFB adoption into the storage market necessitating further research activities. 12 Despite having been in development since 1970s, RFBs have had limited penetration into the energy storage market. 15,16 Since the initial RFB chemistry, several different redox systems have been studied, including aqueousand non-aqueous electrolytic systems, [17][18][19] metallic or non-metallic electrodes and/or Javier Rubio Garcia is a Research Associate at the Department of Chemistry at Imperial College London (UK). He did a PhD in Nanotechnology at the Centre National de la Recherche Scientique (CNRS, France) and Université Paul Sabatier (France) before moving in 2010 to a postdoctoral position at the Energy Research Institute of Catalonia (IREC, Spain). In 2014 he joined Imperial College London to work under the supervision of Prof Anthony Kucernak. His research interests focus on materials and catalysis for electrochemical devices with a particular emphasis on redox ow batteries and low temperature fuel cells.
John Low works as an Associate Professor in WMG (University of Warwick). He leads a team applying the discovery and understanding in electrochemical science through to application-driven technology programmes. He collaborates with UK businesses across power devices supply chain, delivering technology into market and exploiting the rich history of electrochemical engineering. He current collectors, membranes as well as means for scalingup. 20,21 The most widely studied aqueous inorganic systems include the all-vanadium redox ow battery (VRFB), the zincbromine redox ow battery (ZFB), ICFB and the polysuldebromine redox ow battery (PSB). The redox reactions and standard potentials for these battery systems are shown in eqn (1)-(7) below. VRFB: ZFB: ICFB: PSB (positive redox reaction is shown in eqn (4)): RFBs suffer from issues of cross-contamination between electrolytes of both half-cells, 22 poor electrochemical kinetics of some redox couples, shunt current losses 23 and poor energy and power densities. In addition, the growth of non-aqueous RFBs is still in development phase and considerable research is required to match the performance of such systems with their aqueous counterparts. 24 Due to such challenges, only the VRFB and ZFB 25 have seen some commercial success. Moreover, the VRFB is the only system that has been developed to commercial scales as large as 5 MW/10 MW h (this system has been installed to combine with a 50 MW wind farm in Liaoning Province in China). 26 Despite signicant research advances in China and other parts of the world [27][28][29][30][31][32][33][34][35][36][37] (including attempts to increase energy density by introducing polyhalide redox couples in the positive half-cell electrolyte), 38 VRFBs still suffer from several issues related to capacity fading, material degradation, precipitation of active species and others as detailed elsewhere. 6,[39][40][41][42][43][44][45][46][47] Hence, the technology requires further optimisation.
The need to address the optimisation and commercialisation of both VRFBs and other RFB systems raises several important issues for developers of this technology, particularly improvements in ow distribution and operating conditions, enhancement in electrolyte stability as well as enhancement of electrode materials' resistivity towards oxidation. 48 For such cases, it is useful to implement modelling and simulation across multiple length scales in order to optimise the number of laboratory tests required for performance evaluation, to scale this performance to larger form factor cells and to conduct simulations that better inform experimental investigations in this regard. [49][50][51][52] Most key mathematical modelling aspects of VRFBs have been reviewed in depth by Zhang et al. 49 In brief, the models were classied into three types: (i) macro, (ii) micro and (iii) molecular/ atomic approaches. Similarly, the applications of such models were also classied into four types: (i) market level, (ii) stack and system level, (iii) cell level and (iv) material level (Fig. 1). In a VRFB, the utilization of electrolyte is strongly related to its distribution inside an electrode. 49 If the electrolyte ow is not homogeneous, then the rate of the electrochemical reaction is relatively slow, which tends to lead to poor concentration and current distribution, 53 low power density, high overpotential and increased temperatures inside the cell. Although increased electrolyte ow rate may smooth out these effects, this comes at the expense of system efficiency due to the increased parasitic pumping load. 54 Therefore, understanding and controlling the distribution of owing electrolyte within a VRFB is essential for maximizing battery performance and longevity.
Hence, this work builds upon an idea proposed by Zhang and co-workers to determine the effects of redox active species in the porous electrodes at different length scales to effectively analyse and optimise the RFB further. 49 Current research work is thus briey presented and a short perspective on the development of the RFB technology-not solely limited to the VRFB-is discussed. Specically, this review article is divided into seven parts. This introduction is followed by Section 2, which summarises the key results from the mathematical modelling of various ow battery chemistries (excluding VRFBs, which are covered in Section 4). Section 3 highlights major issues related with electrolyte ow and its interaction with porous electrodes inside the RFB, while Section 4 discusses the progress made on the macro-scale modelling of VRFBs. Section 5 considers the progress of research on micro-scale modelling in VRFBs, while Section 6 tackles the development made on the modelling of reaction and transport processes in RFB porous electrodes. Section 7 discusses key aspects involved in the system integration of RFBs from a modelling perspective. Fig. 2 The principle of sodium polysulfide-bromine redox flow battery as portrayed by Zhou and co-workers. Reproduced with permission from Elsevier. 88 Fig. 3 Schematic of (a) closed rectangular flowing Pb cell with electrodes mounted on walls used to carry out experiments, and (b) the computational grid used to carry out simulations. Reproduced with permission from Elsevier. 98 Finally, Section 8 concludes this article and provides a fresh perspective and future directions on modelling of electrodes for RFBs.

Various flow battery chemistries and developments
Since their inception, RFBs have branched from their genesis redox chemistry (iron-chromium 65 ) to a diverse range of redox couples. Thus, to demonstrate the broad applicability of the governing fundamental equations and to present the popular redox couples hitherto, we briey summarise the academic works on these systems. Note, VRFB models are discussed at length in Section 4 and thus are not included in this section.

Brief history
One of the rst models reported on ow cells was by Danckwerts and Hulburt in the 1950s, 55,56 whose work was also adapted to develop boundary conditions, 57 while other models were advanced for ow-through porous electrodes by the 1980s. 58 It was shown that the ow-by porous electrodes provided an improved performance relative to the ow-through counterpart when used in RFBs. 59 Consequently, most subsequent studies employed ow-by electrodes; 60-64 however, Fedkiw and Watts observed that additional enhancements were possible to the model if the effect of geometrical and operational parameters on faradaic efficiency and cell performance was also examined. 65

Zinc-bromine modelling
Mathematical modelling of the zinc/bromine ow battery (ZFB) in a parallel plate conguration took off in the 1980s, 66 with the models by Lee and Selman 67,68 and Evans and White 69,70 providing predictions for many aspects of the ZFB cell of interest to designers. Microscopic models, which focused on zinc dendrite initiation and growth during electrodeposition, were also studied 71 with a macroscopic model of the ZFB ow reactor in combination with a microscopic one describing dendrite growth. 67 These predictions included the current density distributions along the electrode surfaces, the overall battery efficiency and the round trip cell efficiencies. 70 Such work established many of the independent design parameters for an individual cell and showed how to improve cell efficiency via changes in these design criteria. 72-74

Iron-chromium systems
Iron-chromium (ICFB) RFB systems have also attracted industrial attention due to the potential low cost of the electrolytes. However, commercialization has yet to be realised due to issues around electrolyte crossover and hydrogen evolution, which results in accelerated capacity decay. 75 However, many models oen make simplifying assumptions that disregard this process 76 even though the importance of hydrogen evolution on device lifetime is frequently highlighted. Scale-up investigations of ICFBs using shunt current modelling analysis were performed in the early 1990s 77 resulting in an enhanced energy efficiency of ca. 80% for a 0.7 kW battery. This was an improvement over the results reported by Fedkiw and Watts, who showed that membrane resistance contributed signicantly to performance loss 65 albeit not sufficiently enough to warrant further investigations for almost two decades. 39,78 Another investigation described the experimental and numerical modelling applied to a particular Iron Flow Cell prototype. 79 The experimental validation showed numerical errors < 2.25% suggesting this methodological research provided a powerful calibration tool to help engineers in optimisation procedures.

Reversible solid oxide fuel cell
Another class of rechargeable battery composed of a reversible solid oxide fuel cell (RSOFC) and a metal-metal oxide redox cycle unit (RCU), termed solid oxide metal-air redox battery or SOMARB, has been reported. 80 Its high energy-density, high rate-capacity and easy system integration has drawn an increasing interest from the energy community. 81 Since its debut in 2011, signicant progress has been made in the areas of electrical performance, 82,83 new metal-air chemistries 84 and operation optimisation. [85][86][87] 2.5 Polysulde-bromine ow battery Another potentially low cost redox ow cell couple of interest is the polysulde-bromine battery (PSB, as shown in Fig. 4). 88 An investigation of the bromide half-cell found that multiple reaction mechanisms could account for observed behaviour, 89 leading to the mathematical modelling of the PSB system. For the conditions studied, mass transport overpotentials at the bromide electrode were found to limit the performance during discharge. 90 The model showed that signicant dri in conditions could occur due to self-discharge and electro-osmotic effects. For the PSB system using the composite activated carbon electrodes developed by Regenesys, 90 bromide and sulde kinetics rate constants of 4 Â 10 À7 and 3 Â 10 À8 m s À1 were obtained. This model was followed by an investigation on the technical performance of a 15 MW, 120 MW h utility-scale PSB system 91 and was later combined with a simple economic model including the main capital and operating costs to optimise the design and evaluate its commercial viability. Based on 2006 prices, the system was predicted to make a net loss of 0.45 pence per kW h at an optimum current density of 500 A m À2 and an energy efficiency of 64%. The system was predicted to become economically-viable for arbitrage (assuming no further costs were incurred) if the kinetic rate constants of both electrolytes could be increased to 10 À5 m s À1 , for example, by using a suitable (low cost) electro-catalyst. The economic viability was found to be strongly sensitive to the costs of the electrochemical cells and the electrical energy price differential.

Soluble lead redox battery
With regards to the soluble lead RFB (SLRFB), a number of issues have to be addressed before it can be commercialised, 92 such as low charge efficiency 53 and incomplete dissolution of active solids (Pb on the cathode and PbO 2 on the anode), which can accumulate during charge-discharge cycling. Aer a few charge-discharge cycles, the accumulated residue on the electrodes begins to disintegrate and PbO 2 /PbO particles entrain in the electrolyte, resulting in an irreversible loss of the active material that leads to the formation of an insulating layer deep inside the PbO 2 deposits at high acid concentrations.
Consequently, this can cause premature capacity loss thereby yielding a very low cycle life of the SLRFB. 93 Increases in electrolyte ow rates to match the rise in battery efficiency only provide a marginal gain, 93 and attempts to increase battery capacity by enhancing deposit thickness are met with accelerated disintegration of the residue. 94 A resolution of these and other issues requires an understanding of residue build-up on the electrodes. Hence, the previous modelling studies on SLRFB have provided insights into two step charging 95 and effects of temperature, species concentrations 96 and electrode morphology 97 on battery performance. 53 One of these models accounts for ion transport resistance for electron transfer reactions using Butler-Volmer kinetics with tted values of rate constants and non-uniform deposition on electrodes by simulating non-uniform current densities on the electrodes. 53 The model is simulated using COMSOL® for a SLRFB with planar electrodes using fast reaction kinetics, slow reaction kinetics and low transport resistance in the concentration boundary layer. The results suggest that resistance for ion transport in the bulk and concentration boundary layers and resistance for electron transfer at the anode control battery performance. There is a good t with the model and the reported voltage vs. time measurements during charging, relaxation and discharging stages. The model also predicts that increasing the thickness of the concentration boundary layer in the ow direction leads to a signicantly non-uniform Reproduced with permission from Elsevier; 158 (C) the spatial distribution of the velocity (color chart, m s À1 ) and reactant (V 3+ ) concentration (contour line, mol m À3 ) for (a) RFB at 40 mA cm À2 , (b) CFB at 40 mA cm À2 , (c) RFB at 160 mA cm À2 and (d) CFB at 160 mA cm À2 (P in ¼ 4.25 Â 10 4 Pa, P out ¼ 0). Reproduced with permission from Elsevier. 158 distribution of deposit thickness. In the discharge cycle, the deposits near the trailing edge are dissolved completely, which is in agreement with the experimental observations. As time proceeds, the contact line separating the bare portion of the electrode from the active portion recedes into the active region rapidly. This ultimately manifests as a sharp decrease in cell voltage, leading to the cutoff value and termination of the discharge cycle. The incomplete dissolution of the deposited material leaves residues behind, which become thicker with successive charge-discharge cycles.
The model correctly captures the experimentally observed marginal effects of an increase in ow rate on voltage vs. time proles. 53 The simulations show that a change in ow rate and velocity prole have a profound effect on the distribution of the deposits. The simulations carried out with alternating ow directions also show signicantly reduced rate of residue buildup. Replacing the planar electrodes with annular cylindrical electrodes, where the anode is outside, leads to a substantial increase in energy efficiency and also decreases the rate of residue buildup.
Nandanwar and Kumar found that natural convection induced by difference in concentration of Pb 2+ ions near the electrode surface and the bulk of the electrolyte plays a dominant role in the performance of a soluble lead battery with no external circulation. 98 This was later modeled by using a concentration difference driven body force term in the equations of motion. The augmented model, 98 with all the parameter values same as those reported by the same authors elsewhere, 53 very well explains the experimental data obtained in a narrow rectangular cell with no external ow as shown schematically in Fig. 5. 98 The electrode and deposit conductivities in most of the SLRFB models are considered to be innitely large compared to that of the electrolyte. 98 Therefore, these models do not require the potential distribution in the solid phase to be determined. On a microscopic level, the assumption of high conductivity of deposits holds well for the cathode as current passes either through Pb deposits or graphite electrode, both of which are highly conducting. The situation at the anode is different though as the assumption of high conductivity holds for the rst charge cycle as the deposit consists of only PbO 2 . In the rst discharge cycle, some PbO 2 is electro-dissolved and some is converted into insulating PbO solid. Oury and co-workers monitored the change in cycle mass at the anode during discharge using quartz crystal micro-balance and impedance spectra. 99 Their studies conrmed PbO 2 to PbO conversion and formation of an insulating layer that eventually stops the discharge process altogether.
The assumption of highly conducting deposits on the anode irrespective of the fractional content of non-conducting PbO appears unfounded. Oury et al. introduced a revised model that incorporates changes occurring in deposit conductivity on the anode and studied its impact on charge-discharge proles, particularly in the context of charge coup de fouet (the voltage dip shortly aer charging of a fully discharged battery begins), which cannot be explained by previously reported models. 97 Specically, the updated model varied deposit conductivity with compositional changes, from highly conducting to nonconducting when conducting pathways cease to percolate beyond a critical volume fraction of the non-conducting phase. The change in conductivity of a thin layer of deposit on conducting graphite electrodes was used to develop an approximation that permits model predictions to be obtained without determining voltage proles in the composite solid phase. The model effectively explains the observed charge coup de fouet phenomenon and its variation with depth-of-discharge.

Hydrogen-bromine ow battery
For the gas-phase Br 2 -H 2 ow battery, which is again attractive due to the potential low cost, several mathematical models exist. [100][101][102][103] For instance, a mathematical model has been compared to experimental data 101 and predicts the operating conditions of the cell in both fuel-cell (i.e., discharge) and electrolysis (i.e., charge) modes as a function of current, inlet gas composition, ow rate and pressure differential across the membrane. The analysis reveals that gas-phase Br 2 /HBr reactants signicantly enhance mass transfer, which enable higher current densities to be achieved in comparison to a liquid-fed system. A key feature of the model is in accounting for water transport across the membrane, which determines membrane conductivity, reactant concentration and undesired condensation. The model is then used to provide insights into cell operation, including operating conditions needed to avoid water condensation. For example, operating at pressure differentials where condensation is avoided, the model predicts current densities of 1.4 A cm À2 on both charge and discharge states with round-trip efficiencies of approximately 63% and 37% at current densities of 0.5 and 1.4 A cm À2 , respectively. 101

Iron-vanadium chemistry
In the past decade, scientists at the Pacic Northwest National Laboratory (PNNL) introduced a new iron-vanadium (Fe/V) RFB. 104,105 The ow cell performance was demonstrated using a small single cell (active area ca. 10 cm 2 ) operable in a wide range of operating temperatures (0-50 C) with negligible capacity fade over cycles. In addition, this new chemistry ensured reliable and safe operation as a result of negligible hydrogen evolution without any catalysts on the electrode, which was a concern with the previous ICFB chemistry. 106,107 The Fe/V chemistry was validated using a small-scale, single cell at a high ow rate (2 cm min À1 ), although performance validation at the kW-scale at a more reasonable ow rate (<0.3 cm min À1 ) is needed to demonstrate the potential of the technology for large-scale applications. The application of deep eutectic solvents for this chemistry has also been reported. [108][109][110] Further optimisation of both aqueous and non-aqueous Fe/V systems including electrode geometry, electrode design and ow rate are needed to enhance performance. Mathematical models can assist in cell scale-up as well as the optimisation of RFBs by simulating the effects of various cell geometries, varied electrolyte compositions and membranes on cell performance. 111 Furthermore, models aid in cost analysis and optimisation of system control. 106 A zero-dimensional electrochemical model of the aqueous Fe/V RFB has been presented that can model performance at low ow rates (<0.5 cm min À1 ) and varied temperatures. The electrochemical model is appropriate for practical RFBs and shows good agreement with experimental data. 106 In addition, a proposed non-ideal electrode model is introduced that accounts for higher voltage losses at low ow rates. Semiquantitative operational strategies and electrode design guidelines can be obtained from the model. The authors found that ohmic losses associated with the electrolyte were dominating the electrode losses, which meant operating the cell at higher temperatures could reduce electrolyte ohmic losses and viscosity, thus leading to a higher system efficiency. Using thinner electrodes (4.5 mm-thick felt electrodes were used in this study) can reduce ohmic and pumping losses if the same space velocity ‡ is maintained. This electrochemical model could be easily incorporated into system-level and cost models, which could help in system optimisation, system control and pump selection to avoid potential risks that may be involved during large scale RFB system development. 106

Flowable semi-solid lithium-ion battery
In the work of Brunini et al., 112 a porous electrode theory based approach was used and extended from the work of Fuller, Doyle and Newman 113 to incorporate convection of both dissolved ions and active intercalation particles in a system based semi-solid electrodes. They utilised owable mixtures of solid Li-ion storage compound particles suspended in a liquid nonaqueous electrolyte. The semi-solid electrode suspension was made electronically conductive by co-suspending nanoscale conductive particles (e.g., carbon black) along with storage compound particles in the electrolyte. In such semi solid ow cells, conductive particle networks replaced the static current collectors (e.g., carbon felt) used in conventional ow batteries. 114 Additionally, due to the modelling of low ow rate operation added to the fact that the equilibrium voltage functions of Li-ion intercalation compounds vary, the non-uniform current density was modelled by applying potentiostatic boundary conditions while adjusting the voltage with time to match the desired current. 112 As in prior porous electrode models, 113,[115][116][117][118] the transport properties in the solid and electrolyte phases, the equilibrium voltage functions of the intercalation particles and the Butler-Volmer exchange current were employed as model parameters. 112 Of these, the shape of the equilibrium voltage function and the electronic conductivity of the suspension are observed to have the greatest effect on the current density and state of charge distributions in the matched charge and ow rate limit.
An evolved version of this concept has been developed in recent years to mitigate some of the uid dynamics issues. 119 The concept relies on placing the Li-intercalation material in an independent reservoir and employing dissolved species with the ability to mediate the lithiation reaction. 120 Since the mediator can undergo a reversible redox reaction in a ow battery conguration, the reaction overpotential is reduced when compared to the semisolid counterpart. The solid, storing reservoir enables higher volumetric energy density than conventional RFBs. 119 This concept, which has been named redox targeting-RFB or redox solid energy boosters for ow batteries, has been successfully applied for different Liintercalation materials. [121][122][123] Moreover, this redox targeting approach has been demonstrated beyond aprotic Li-based chemistries and aqueous congurations with excellent reported performance, including Prussian blue/ferrocyanide/ ferricyanide mediated system, 124 vanadium chemistries, 125 Naion intercalation materials 126 and polyaniline solid charge storage material applied at both the negative and positive sides of the battery. 127 While no examples of mathematical modelling are available in the literature, charge-diffusion mechanisms within the solid material have been considered in different studies. 128,129 2.10 Quinone-based ow battery A 3D model, including full coupling of the mass balances, momentum balances, ionic current balance, and electronic current balance, was developed for an organic-inorganic aqueous ow cell based on quinone chemistry. 130 Fig. 6(a) is a schematic representation of the ow-through electrode model, which includes ve domains: positive channel, positive electrode, ion exchange membrane, negative electrode and negative channel. From simulations, which were in good agreement with the experimental data it was concluded that six layers of carbon paper was the most appropriate electrode thickness. More than six layers of carbon paper provide more surface area at the expense of ohmic resistance, but less carbon paper cannot provide adequate specic active area. The ow elds in the x and z directions are analysed to account for the electric current density distribution where the high current density at the junction of the land and channel for the owthrough electrode cell is related to the ow eld in the z direction.

Single ow zinc-nickel system
In another paper, the primary characteristics of a single ow zinc-nickel battery is illustrated and based on that, the electrical equivalent circuit model (see Fig. 6(b)) 131 is established for the rst time. 132 The parameters in the battery model are identied by means of a variety of experiments, carried out on a small-capacity battery in the lab. According to the simulation and experimental verications, the model can properly estimate the performance of the batteries under different conditions. The model is also validated for large-scale single-ow, zincnickel batteries.

Summary of ow battery chemistries
Despite the progress reported above, most of the published literature is based on experimental studies while only a few experimental studies on RFBs take into account mathematical ‡ In chemical engineering and reactor engineering, space velocity, refers to the quotient of the entering volumetric ow rate of the reactants divided by the reactor volume or the catalyst bed volume, which indicates how many reactor volumes of feed can be treated in a unit time. modelling to design more efficient ow cells based upon electrolyte interaction with porous electrodes. The next section therefore discusses this important topic and surveys some of the modelling aspects involved.

Introduction
In the last decade, a number of technical advances made in polymer-electrolyte fuel cells (PEFCs) eld have been introduced in the ow battery technology, leading to dramatic performance improvements through enhanced mass transport and reduced area specic resistance (ASR). 133 Indeed, many of the constituent components and designs engineered for PEFCs are directly applicable to RFBs. Consequently, most redox ow cells (especially microuidic designs 134 ) have advanced from lower power densities (ca. 200 mW cm À2 ) to higher, operational values. 11 For example, VRFB power densities greater than 700 mW cm À2 have been reported, despite an operating limiting current density of ca. 1 A cm À2 and requiring relatively high temperatures, 30 which also augment the cost due to the necessary thermal management and/or sequestration. Workers in Dalian achieved doubled power densities for their VRFB system (1.45 W cm À2 ) 135 in comparison to those achieved in Tennessee (0.76 W cm À2 ). 136 Similar high, peak power densities were reported from experiments on quinone-bromide systems conducted in Harvard University (1.00 W cm À2 ). [137][138][139] Despite such improvements, continued advancement of power performance is necessary to support further cost reduction, motivating research and development in the engineering science of RFBs. 133 There are several stimulating features in the engineering of the cell: construction materials, electrode and membrane. 140 VRFBs have been the subject of signicant research efforts, with in-depth analyses being reported by Ma et al. 141 and Secanell and co-workers. 142 The uniformity of the electrolyte ow has a substantial effect upon the effective area of the electrode (ber area of the porous electrode where the actual charge-transfer reaction occurs), 143 the depth, efficacy and lifetime of the battery as well as the electrochemical overpotentials, particularly when the current density varies signicantly.
Miyabayashi et al. showed that a uniform electrolyte ow rate resulted in higher energy efficiencies for RFBs 144 -specically, even if the ow was homogeneous, there were, among other factors, signicant local changes on the electrode surfaces resulting in high pH variations. 145 The signicance of such a velocity study has also been conrmed in works by Bengoa and co-workers 146 and Wragg and Leontaritis. 147 Unfortunately, ow distribution within the battery is a difficult parameter to measure; it is inuenced by several factors, including electrode micro-structure, surface chemistry, compression and electrolyte composition. Most articles to date imply the need of improved ow distribution for enhancing the entire system's efficiency; 148,149 for example, Escudero-González suggested a factor in order to evaluate the ow disorder in a three-dimensional analysis. 150 This involved a fraction of the volume en-route, the foremost ow, and in the reverse course to estimate the uidic dead zones and membrane recirculation, which improved the estimation capability compared to other studies. 79 The factor suggested by Escudero-González and co-workers should allow the modeler to associate geometries by considering ow uniformity through computational uid dynamic (CFD) methods. 150 This validation of the numerical model is of an extra benet, since the real electrolyte is strongly acidic and corrosive that can impair experimental accuracy. The simplied model, however, does not consider these chemical facets and only models the uid hydrodynamic performance. 145 Uniform ow is important for the performance factors in the RFB cell: the electrode effective area; the robustness and efficacy; the useful battery life; and the electrochemical polarization. 63,151 As a result, the effect of various geometries, the ow conditions and the membrane position and conguration 152 on several cells have been studied with the goal of understanding and enhancing RFB operation. 34,146,147,153

Electrolyte ow eld effects
The electrolyte velocity distribution from the ow eld has been an active area of RFB research. 154 A rigorous analysis of ow elds needs two important numerical tools, namely CFD and statistical techniques. As mentioned, CFD aids in the simulation of pressure and velocity distribution providing simple performance indicators and enabling comparisons between different congurations. Statistical techniques can be applied to analyse a large range of individual velocity proles in an accurate manner. The employment of hypothesis tests on data generated by means of CFD enables a quantitative analysis of several features on the velocity elds. 145,155 Even though a broad range of ow elds are possible for fuel cell applications, four patterns are chiey employed in RFB applications. These include the serpentine, ow-through, interdigitated and parallel ow elds. 156 Besides the conventional RFB set-up that includes two, ow-through electrodes (usually carbon or graphite felts) separated by an ion-exchange membrane or separator (non-zero-gap assembly), over the past decade, a PEFC-based design has been increasingly adopted ( Fig. 2 this displays a zero-gap membrane-electrode-assembly that has been extensively tested in the Dalian Institute of Chemical Physics, China). 157,158 In brief, this system usually employs structured ow elds that act as the current collector and compress carbon electrodes (usually felts) against a membrane to yield a zero-gap assembly that dramatically reduces ohmic drop in the cell, 159 thereby yielding high current and power densities. 34,136,160 Flow elds vary with regards to their ability to transport the electrolyte to the reaction zone and the pressure drop necessary to maintain uniform ow. In the ow-through system, the electrolyte is introduced directly through the carbon felt electrode at one end of the cell and is allowed to disperse throughout the electrode before owing out of the opposite end to the outlet channel (ow-through design). This generates a high pressure drop as all the uid is forced through the entire length of porous media and potentially results in hydraulic short-circuiting 77 that hinders cell performance. 15,17,64,161 Thus, uniform distribution of electrolyte over the electrode surface is necessary to minimise pressure losses over the entire surface. 156 Therefore, it is essential to operate at optimal electrolyte ow rates established for balanced system efficiency and capacity. 162 Beyond the simplistic design of the ow-through ow eld, other architectured ow elds have been investigated. The parallel ow eld has been found to exhibit low pressure drops but can present severe ow non-uniformity especially within the channels 163 as well as poor convection under the ribs of the channels that slow mass transport. 164,165 Chen et al. 148 investigated a parallel ow conguration along with CFD analysis for ow allocations in the VRFB and achieved a rather low power density of 15.9 mW cm À2 , which they attributed to non-uniform ow rates throughout the cell. Xu et al. 166 compared the output parameters of different VRFBs with ow-through, serpentine and parallel ow congurations and revealed that the serpentine ow channels elicited the highest round-trip efficiencies. Latha and co-workers 167 also studied the serpentine ow eld of a VRFB by investigating the effect of rib convection and porous electrode compression on ow dissemination and pressure drops. 50 The pressure drop in two, geometrically different serpentine ow elds was also determined over a range of Reynolds numbers, which, combined with CFD simulations, demonstrated the signicance of the compression of the porous medium as an effective parameter for estimating the ow distribution and pressure drop in such ow elds.
It is known from fuel cell studies that the porous electrode can experience uneven compression under the ribs and channels of the ow eld, 168 and this has also been observed in RFBs. In fuel cells, this uneven compression can affect the ow, ohmic resistance and pressure drop. 169 Similarly, uneven compression has also been observed in RFBs. [170][171][172][173][174] Numerical simulations indicate that the uneven compression and intrusion of the porous electrode into the ow channel increased the ow velocity in the channel, enhancing transport processes and electrochemical performance. 172,173 Harper et al. 175 recommended a ow channel design for the bipolar plate in order to reduce pressure drops and to maintain a distributed ow arrangement consistently. The authors also suggested the use of an interdigitated channel design for the ow cell asserting this could improve the cell operation without escalating the pressure drop and promote uniform electrolyte ow through the porous electrode. Tian and co-workers 54 numerically analysed several ow congurations with a single ow inlet and outlet, multiple inlets and outlets, a wide single inlet and outlet as well as a wide single inlet and outlet with branched/staggered interdigitated hollow channels in the porous electrode. They determined that unlike conventional designs, the introduction of ow distribution channels could enable enhanced ow uniformity throughout the electrode, while the introduction of channels inside the porous electrode could decrease pressure drop adversely impacting power output. 156 Tsushima and co-workers 176 studied the inuence of cell geometry and operating parameters on the performance of a RFB with serpentine and interdigitated ow elds. They also observed better performance with an interdigitated ow eld in the VRFB in comparison to the former ow conguration. Several experimental and numerical studies have also been reported for laminar ow fuel cells, which are effectively RFBs with laminar electrolyte ow in single channel ow-through electrodes without an ion-exchange membrane. Kjeang et al. 177 designed a simple microuidic vanadium redox ow cell with a high aspect ratio Y-shaped micro-ow channel over a porous carbon substrate to obtain a peak power density of 70 mW cm À2 , which was lower than membrane-based systems. Zawodzinski and co-workers 34,136,160 adjusted PEFC cell congurations for RFB applications to improve current and power densities. In these designs the electrolyte owed through serpentine channels behind a porous electrode sandwiched to a membrane and counter-electrode (membrane-electrodeassembly).
Further niche ow eld designs have also been explored. Originating in PEFCs, Guo et al. 178 created leaf-like ow elds emulating the venous, biological structure found in plants and blood vessels. Employing a topological optimisation, Behrou et al. 179 proposed a depth-averaging model to maximise power density in PEFCs. The topological optimisation procedure has taken root in RFBs as a numerical study; 180 however, the performance of many of these ow eld designs has yet to be experimentally validated. 181 In summary, numerous studies have shown direct ow eld inuence on cell power performance of the cell, energy requirements and round-trip efficiency. Generally, serpentine and interdigitated ow elds appear superior to the owthrough and parallel ow elds, stemming from the signicant and uniform cross-ow or under-the-rib convection inherent to these designs. 182 While these phenomena have been well-studied in the fuel cell context, 164,183,184 the results are different for RFB applications. Specically, while interdigitated ow elds have not found favor in PEFC applicationspresumably due to the much higher pressure drop compared to the serpentine ow eld 164,183,185 -they appear to be the ow eld of choice for RFBs. To resolve this discrepancy, Latha performed an in-depth, experimental and numerical investigation of the hydrodynamics of serpentine and interdigitated ow elds in typical RFB operating conditions and found that, at the same ow rate, the pressure drop in interdigitated ow eld was lower than its serpentine counterpart. 156 CFD studies further show that strong under-the-rib convection in the reaction zone exists in both ow elds but with a shorter residence time in the case of the interdigitated congurations. These ndings appear to explain the superior performance of interdigitated ow elds in RFB cells; 156 however, there is still considerable research needed to tailor ow eld structure to improve uniformity of electrolyte ow and, consequently, power density.

General mathematical modelling of porous electrodes and their interaction with electrolyte ow
The modelling of RFBs seeks to enhance fundamental understanding and enable high-performing, economical designs of these systems through the development of mathematical equations implemented for numerical simulation of electrochemically reactive ow. One aspect of these models involves predicting the effect of ow distribution and the ow rate schedule on electrochemical performances at porous electrode surfaces. A few dynamic lumped models have been proposed in the literature to address this issue. [186][187][188][189] Nevertheless, given the need to represent the uid dynamics within the ow eld channels and the porous media, higher order models are needed to capture these phenomena. Ke et al. established a twodimensional (2D) macroscopic mathematical model to simulate the dynamic ow patterns in RFBs represented by a single passage of a serpentine ow channel over a porous electrode as shown in Fig. 3(a) 50 with a 2D cross-section of the ow pattern represented in Fig. 3(b) and other general RFB ow congurations shown in Fig. 3(c). The non-dimensional average ow velocity (normalised with respect to the inlet ow velocity) in the ow channel decreased from the entrance to the fully developed ow region while an opposite trend was found in the porous layer. The volumetric ow rate in the porous layer increased as the thickness of this layer rose from 0.041 to 0.287 cm (the number of porous layers ranged from one to seven) for the carbon ber paper electrodes.
Moreover, the authors also developed a description of the maximum current density that estimates the stoichiometric availability of reactant in the porous layer for reaction. 47 Under the inlet condition (Q in ¼ 20 mL min À1 or u in ¼ 33.3 cm s À1 ), the volumetric ow in the electrode layer, Q p increased from 7.81 Â 10 À4 to 1.5 Â 10 À3 cm 3 s À1 when the electrode thickness increased from one to three layers of carbon paper. Under such conditions, the maximum current density increased from 377 to 724 mA cm À2 as the electrode thickness increased. The predicted maximum current densities were found to be in good agreement with experimentally measured limiting current densities, and the same was observed for estimated values of porous layer permeability (377 mA cm À2 predicted vs. ca. 400 mA cm À2 measured for one layer, and 724 mA cm À2 predicted vs. ca. 750 mA cm À2 measured for three layers).
Other work on mathematical modelling and experimental validation shows minimum electrolyte recirculation in the membrane area, which improves the overall interchange efficiency of the cell (ionic interchange). To compare the experimental and computational models for a ow of 50 L h À1 , a methodology is proposed by Escudero-González. 150 Differential pressure and velocity elds are compared in both models with an acceptable convergence that leads to an improved design and key geometric parameters for creating and comparing new cell architectures.
Kee and Zhu performed a study to optimise the ow uniformity and the pressure drop for the interdigitated design via a simple uid dynamics model. 190 The study provides dimensionless charts and analytic representations to estimate the pressure drop and the ux uniformity in a wide range of conditions, providing a rapid assessment of the channel layout as well as design guidelines for its optimisation.
A number of ex situ and in situ studies of ow eld designs have been reported in the literature on RFBs. Hoberecht 191 proposed a conventional ow eld with inlet and outlet manifolds so that the uid can spread out evenly before moving upward through the electrode. This work studied the inuence of cell geometry and design parameters such as cavity thickness, inlet and outlet ow port width, and depth and ow rates on pressure drop. Hoberecht found that the compression of the porous electrode is a major contributing factor and accounts for nearly half of the pressure drop. Miyabayashi et al. 144 have since proposed a cell design with multiple slits in inlet and outlet manifolds to provide a uniform ow distribution pattern and reduce pressure drop while Inoue and Kobayashi 192 proposed a porous electrode with convex, semi-circle, and v-shaped grooves.
In situ studies of ow elds and ow distribution effects in VRFBs have also been reported. Chen et al. performed experimental and numerical investigations of electrolyte ow distribution using a parallel ow eld for a VRFB. 149 Their results showed a highly non-uniform electrolyte distribution over the entire surface with concentrated distribution in the central area of the ow eld and a vortex ow in the inlet and the outlet regions. Simulated results, in agreement with experimental data, suggested that an optimised inner ow eld structure should be designed. Xu and co-workers performed a study on serpentine and parallel ow elds along with a historical conguration (ow through a porous electrode in which the membrane is separate from the electrodes and is not in direct contact with them) using a three-dimensional (3D) numerical model. 84 At optimal ow rates, higher energy and round-trip efficiencies were observed with a serpentine ow eld. Zhu et al. experimentally investigated the effects of ow elds with ow-through and ow-pass (serpentine, interdigitated or parallel ow as shown in Fig. 3) patterns nding that the owthrough pattern increases the electrolyte ow uniformity and the electrode effective area thereby increasing the efficiency. 193 Koeppel et al. 194 redesigned a VRFB for reduced pressure loss using an interdigitated ow eld which Gerhardt et al. 195 further investigated by adjusting the land and channel dimensions to identify an optimal conguration. Other modelling and simulation studies described and accounted for all the relevant phenomena. Escudero-González introduced a methodology comprising three performance indicators (namely the symmetry coefficient, the uniformity coefficient, and the variability range coefficient of velocity front) using CFD simulations to determine the ow distribution in a particular VRFB geometry. 145 Finally, recent numerical and DFT simulations conrmed an experimental study on enhanced VRFB performance due to nitrogen doping of carbon granular electrodes, 196 thereby validating the efficacy of using both experimental and numerical techniques to guide a realizable and cost-effective solution for high-power VRFBs. In this regard, the next section follows up on a survey of macro length-scale modelling of VRFBs.

Macro length-scale modelling of VRFBs
A range of processes may inuence the operational limitations in VRFBs. 197 At the cell level, reaction kinetics, ionic transport through electrodes, electronic contact resistance and masstransfer resistance into and out of the electrodes are expected to impact the power output. Polarization curve analysis is typically implemented to assess the relative magnitude of these resistive contributions over a range of operating currents. 160 Depending on the region of the polarization curve, different losses can dominate cell performance, for example at low currents kinetic limitations are most important (Fig. 7). 197 Extracting all aforementioned parameters from experimental measurements may not be feasible unless coupled with pore-scale resolved kinetic and mass transport models. 111 Here, we survey macro length-scale models for VRFBs due to their widespread investigation in literature. The fundamental equations described are applicable to other singlephase chemistries that follow Butler-Volmer kinetics by changing the parameters and stoichiometric coefficients of reacting species.

Background and literature survey on VRFB models
Macro length-scale models of VRFBs can be generally categorised into two groups: (1) lumped models, which mainly serve as control and monitoring tools, and (2) distributed models, which capture the spatial variations of eld variables within the cell and can be employed for cell design and optimisation. 49,198 Lumped models on the system level address the prediction of stack voltage, state-of-charge (SOC), state-of-power (SOP), ow rate and system efficiency in different SOCs by using global mass and energy balances. 49,52,199,200 These models are  70,168 Similarly, equivalent circuits, which combine various elements, such as capacitors, resistances and voltage sources, have been used to represent the characteristics of the battery. These models are simple and straightforward without considering the inner structure of the battery, but do require experimental data to obtain various operational parameters. 151,[201][202][203] Due to their simplicity, lumped models are widely implemented for practical battery control and for the development of monitoring strategies. Table 1 summarises the main assumptions of typical zerodimensional lumped models. Conversely, distributed models aim to describe the physical and chemical processes occurring within the cells in greater detail. These models are used to enhance the design of the electrodes, membranes, ow elds and to determine the optimal operating conditions such as temperature and current density. The mathematical description of reaction kinetics represents the rst step of the mechanistic modelling of VRFBs. Nonetheless, kinetic models alone are not suitable for describing the complete electrochemical behaviour of the VRFB system due to the coupled transport phenomena in RFBs. Moreover, the appropriateness of different kinetic formalisms are dependent on the reactions in question as well as detailed information of the system unless ab initio calculations, which are seldom applied for VRFBs, are employed to support reaction pathways or species' stability. 43 Additionally, ow models simulate only the momentum transport while the species, charge transport and electrochemical reactions are neglected. [204][205][206] Thus, these ow models are only valid for a limited range of conditions. Higher accuracy can be obtained if the models include detailed physicochemical processes related to the active species concentration, potential drop and current density within the battery. 49 Abu-Sharkh and Doerffel 207 as well as Dees et al. 208 suggest that complete electrochemical models are best suited for optimisation of the physical design aspects 209 of electrodes and electrolytes.
The complete distributed electrochemical models are based on the mathematical description of electrochemical reactions and conservation laws of mass, charge, energy and momentum of electrolyte in the stack and tanks. [210][211][212][213][214][215][216][217][218] These models involve detailed considerations of the dynamic processes of vanadium species, the structure of individual cells and the stack as well as a large number of selected parameters. With the higher order models, the accuracy of the physical domain can be improved but at the cost of computational times. However, depending on the model application, lower dimensional models and other simplications may be acceptable at the discretion of the researcher. The key assumptions for some of these distributed continuum electrochemical models are summarised in Tables  2-4 for 1D, 2D and 3D models, respectively.
Despite the coupled nature of the transport and reaction phenomena, in some cases the solution of the mathematical formulation is performed in three stages (Fig. 8): 198 Distribution of pressure and of ow in the cell; Distribution of chemical species, considering electrochemical interaction; and Distribution of the electric eld and the current density. Following this approach, Bayanov and Vanhaelst 198 solve the uid dynamic and electrochemistry equations by means of the coarse particles method combined with an iteration process. The post-processing of simulation results combined with experimental data enables the determination of the main characteristics of the VRFB-the voltage, the current and the internal resistances. The study of the electrolyte ow in a VRFB is a relatively recent research topic compared to the evaluation of other key parameters of the system. 219 The ow structure is formed by means of a system of inlet and outlet channels as well as porous electrodes. The theoretical models used to simulate the ow behaviour are based on mass and charge conservation. Rudolph et al. 219 analysed the electrolyte ow on the basis of three theoretical models: (1) total ow, depending on the electric current; (2) ow distribution in the inlet/outlet channel system; and (3) ow distribution in a half-cell. For a test RFB, the capacity availability is calculated and measured for a wide range of ow rates and electric currents. The ow distribution in the porous electrode is calculated and the efficiency (affected by the different ow congurations in the half-cells) is measured. The parameters and the conguration of the electrolyte inlet/outlet channels are optimised to obtain a uniform distribution of the electrolyte ow to each half-cell. The results, comprising the SOC of electrolytes, electric current, maximum voltage by charge, power, charge/discharge duration and capacity, correlate with the measured data within an accuracy of AE2%. It is shown that the simple ow design in the electrodes with single inlet and outlet channels allows for an effective use of more than 89% of the electrode surface. A major issue with the existing VRFB models is the inaccurate prediction of the open circuit voltage (OCV), which results in a discrepancy of 131 to 140 mV in predicted cell voltages when compared to experimental data. 220 This deviation is shown to be caused by the incomplete description of the electrochemical double layers within the cell when calculating the OCV using the Nernst equation adopted from fuel cell literature. A more complete description of the Nernst equation has been provided by a couple of authors 220,221 which accounts for two additional electrochemical mechanisms that exist in a VRFB, namely: (1) the proton activity at the positive electrode (V 4+/5+ ) due to the involvement of the protons in the redox reaction, and (2) the Donnan potential due to the proton concentration differences across the membrane (Fig. 9). 220 The more complete form of the Nernst equation closely matches reported experimental data with an average error of 1.2%, and is a signicant improvement over the incomplete Nernst equation (8.1% average error).
Various models take into account capacity losses due to crossover, 51,222,223 side reactions 215,216 and consider different thermal effects 52,162,224,225 as a function of time, for different operating conditions and cell designs. These models assist in understanding the change in electrolyte concentration over longterm operation. Thus, the models can be used for the development of battery control systems that can automatically restore VRFB capacity by periodic electrolyte rebalancing. 171,226,227 Several groups have attempted to investigate mass transport effects through porous materials using experimental techniques to complement model predictions. For example, thermal visualization methods have extensively been exploited for PEFCs and were also extended to investigate the electrolyte convective velocity and distribution in the electrode. 233,234 The detailed data on ow velocity and distribution enabled accurate simulation of mass transport effects using computational models for both serpentine and interdigitated ow eld designs.
While this imaging method provides valuable information regarding liquid reactant distribution, it does not provide any information concerning electrochemical reactions. Aligned with previous experience in PEFCs, Gandomi et al. successfully implemented reference electrodes as probes within the layered porous carbon electrode to map reaction locations associated with potential distribution at the positive electrode of the VRFB. 235 This approach allowed for the identication of regions operating under mass transport limitation (i.e., reactant V(IV) or V(V) starvation) when the system was operated at high current densities. Following a similar approach, Clement et al. implemented a segmented cell with reference electrodes to study the 2D current distribution for a VRFB. 236 The authors correlated the inuence of local electrolyte velocity in the electrode and electrolyte concentration with mass transport limiting conditions. In addition, the authors identied electrode microstructure and wettability as major factors to be considered to successfully understand RFB performance.

Fundamental equations for VRFB macro length-scale electrode modelling
This section summarises the modelling approach for VRFB electrodes, for which the governing equations are provided in Table 5. VRFBs are considered here, but the fundamental equations can be adapted also to other similar chemistries by changing a few specic parameters and stoichiometric coefficients. Going beyond what is currently reviewed in the literature, 76 alternative approaches used in other elds, such as in battery modelling, are also discussed and incorporated in the Table 5 Governing equations for the continuum modelling of VRFB electrodes. The nomenclature used in this table is reported in the ESI (Table  S1) Type and phase Balance equation Transport equation Mass transfer in stagnant layer k s ðc i À c s i Þ ¼ Activity denition a i ¼ g i c i (17) model in order to improve the mathematical description at the macro-length scale. The core of a macro length-scale model of VRFB electrodes consists of a set of conservation equations of mass, charge and chemical species in both electrode and electrolyte as reported in eqn (8a)-(10a). For simplicity, only the faradaic contributions to molar balances are reported in explicit form in the right-hand side of eqn (8a) and (10a). The source term due to the kinetics of sulphuric acid dissociation (S v d,i ) 228 are not explicitly reported, while the source terms due to electric double-layer contributions (S v C,i ), normally neglected in VRFB models, 76,210,221 are briey discussed later in this section. The conservation equations are coupled with transport equations (eqn (8b)-(10b)), which express the uxes as a function of eld variables. Typically, the transport of electrons in the solid phase is described according to Ohm's law (eqn (8b)). The multicomponent transport of chemical species within the electrolyte is based on the generalised Nernst-Planck equation (eqn (10b)), 237 which considers diffusion, migration and convection. The electrolyte is a concentrated solution, thus (i) the mass average velocity w in eqn (9) differs from the molar average velocity v in eqn (10a) 210,211 (the conversion between w and v has been reported by Bird et al. 238 ); and (ii) the diffusion of species is affected by ionic interactions, which are considered using the thermodynamic correction factor vln g i /vln c i in eqn (10b). Alternatively, the generalised Maxwell-Stefan approach proposed by Krishna and co-workers 239 can be used to model mass transport. The difficulty of modelling mass transport in concentrated solutions stems from the complex dependence of diffusion coefficients on species concentrations, which requires experimental characterization. Moreover, when the mass transport is modelled in 1D or 2D, the diffusion coefficient should be corrected for the dispersion as a function of the Péclet number. 76 The electroneutrality equation (eqn (11) ( Table 5)) is typically enforced instead of the Poisson equation to simplify the mathematical treatment. 237 As such, the formation of electric double-layers is not explicitly modeled. Normally the doublelayer contribution, S v C,i , is neglected because its dynamics are faster than the other transport processes. However, this contribution must be taken into account when analysing impedance spectra, either by introducing a phenomenological capacitance into model equations or by properly incorporating diffuse charge in porous electrode theory and Frumkin correction to the Butler-Volmer kinetic expression 240,241 as presented by Bazant and co-workers. 242,243 The transport of electrons and chemical species is affected by the microstructural properties of the electrode, that is, by the effective conductivity factors, k eff , and the permeability, 3 . However, commonly these tensors are replaced by scalar quantities through the adoption of semi-empirical correlations, such as the Bruggeman correlation, 76,210,244 the Carman-Kozeny relation 239 or more sophisticated relationships. 245 However, these correlations, whose reciprocal consistency should be assessed, 246,247 may be too generic for the specic electrode under consideration, and their evaluation through the micro length-scale modelling of the electrode is recommended (vide infra).
The kinetics of the faradaic reactions is typically described by using global Butler-Volmer expressions ( Table 5, eqn (14)), although elementary reaction mechanisms dependent on the electrode surface functional groups have been proposed. 248 Notably, all the reacting species must be included in the kinetic expression, in particular the proton activity in the local equilibrium potential of the positive electrode, E eq + (eqn (15a)), and in the Nernst potential (eqn (13)), as discussed in the previous section. In addition, the Donnan potential is added to the Nernst potential in eqn (13) to accurately compute the OCV as discussed by Knehr and Kumbur. 220 The mass-transport resistance due to the stagnant layer between the concentration in the electrolyte bulk c i and the concentration at the electrode/electrolyte interface, c s i , is considered in eqn (16). Generally, the mass-transfer resistance of protons is negligible through the assumption that the surface concentration of protons, c s H þ equals the bulk concentration, c H +, 221 which additionally simplies the kinetic expression in eqn (14a). In such a case, by rearranging eqn (14) and (16) the surface concentrations can be eliminated in favor of the bulk values. 210,221 Finally, the mass-transfer coefficient, k s , can be estimated through a variety of correlations involving the Sherwood number as a function of Reynolds and Schmidt numbers, according to the electrode micro-structure and ow conditions. 249 Numerical simulations at the micro length-scale can potentially be used in such regard. 250 Although these simulations are computationally expensive, they provide the governing equation with information that would not be available otherwise.

Quantification and modelling of electrode properties at micro-length scale of VRFBs
Quite oen, the available properties for electrochemical reactions are not based on the geometry of actual materials, especially for the electrode, the membrane and the electrolyte. 49,251 Therefore, the continuum models implementing surface or volume averaging (i.e., eqn (10) ( Table 5)) at the cell level cannot be employed to evaluate the local effects of the materials' nanoand micro-structure on the VRFB performance. In order to develop a more precise comprehension of these structural effects on the performance and operation of VRFBs, it is crucial to employ methods that can probe electrode structure at various length scales. 252 Methods such as 3D X-ray tomography and radiography (Fig. 10) 253 can be used to reveal the internal structure of electrodes over a range of length scales (tens of cm to sub-nm). 254,255 The application of tomography to carbon-based materials for energy storage application has been slowly growing. Traditionally, the relatively low Z number of carbon stymies the rendering of electrode images using X-ray computed tomography (XCT). Similarly, preferential sputtering of carbon materials makes it difficult to image using focused ion beam (FIBSEM) tomography. 42,[256][257][258] In other words, the imaging of carbon bers and particles at sub-micron resolution in 3D has been challenging due to (i) the low X-ray attenuation coefficient of graphite and (ii) the interaction of graphite with focused ion beams that may lead to highly non-uniform nano-scale milling. 259 Thus, the precise nano-structure of graphite-based anodes at high resolution remains poorly understood. 260 However, innovations in 3D imaging techniques has meant that the tomography of challenging carbon-based materials is becoming increasingly common in a wide range of studies and numerous metrics, such as specic surface areas, effective conductivity/ tortuosity factors, permeability tensors, pore/particle size distributions, etc., which can now be quantied at high resolutions of the order of sub-100 nm. 261,262 For example, using phase contrast XCT and high-Z compound epoxy-impregnation techniques for FIBSEM, it is possible to achieve good contrast for carbon materials. 263,264 Another technique being utilized is to study the effects of mechanical compression on carbon-based electrodes. 170 Table 6 summarises the range of materials and compression conditions previously reported in the literature (non XCT-based). 265 It can be seen that the nature of the gas diffusion layer (GDL) and the compression range has a signicant effect on the contact resistance measured. The bipolar plate (BPP as labelled in Table 6) material is also known to affect the contact resistance. 266 Weyland and Midgley 267 report, in a review of electron tomography (sub-10 nm resolution), the use of bright eld and energy ltered electron tomography (by means of the Yamauchi et al. 268 stain) to identify complex structural details of a different three-phase polymer. Similarly, Yoshizawa and co-workers have employed electron tomography (3D TEM) to observe the internal structure and connectivity of carbon nanospheres. 269 Fig. 10 Mechanisms of vanadium permeation following stages from (A) dry carbon paper electrode, (B) rapid node-to-node electrolyte infiltration, (C) then fill in of free surfaces and (D) fill in of final remaining porosity (determined via X-ray computed tomographic reconstructions of SGL 10AA carbon paper electrodes). Radiographs (E) illustrate the nodal transport, (F) fill in of free surfaces bypassed earlier and (G) full fill in of outstanding porosity. Using this mechanism, the vanadium electrolyte permeates across the carbon electrode without requiring filling of the whole porous volume at the first step, leaving air pockets which are filled in later. Reproduced with permission from RSC. 253 Thorat and co-workers 270 have developed a method to determine electrode and separator tortuosity, in contrast to its more mundane treatment as an adjustable parameter. 115 Recently, the modication of carbon paper electrodes by reduced graphene oxide for improved battery performance has been studied by means of XCT and correlated with other methods. 271,272 It was demonstrated for the rst time that XCT can resolve between carbon paper and carbon-based deposited layers. Fig. 10 shows tomographic reconstructions of the carbon paper electrodes (SGL 10AA) using results obtained from a separate investigation. 253 Particularly, Fig. 10(a) shows the solid phase (bers) of the electrode itself that may be modelled via Ohm's law (eqn (8) ( Table 5)), thereby determining the conduction properties of the material; Fig. 10(b) shows the ber-pore interfacial surface that could be used for modelling electrochemical reactions using Butler-Volmer kinetics (eqn (14)); and Fig. 10(c) shows the region of pores lled with electrolyte that may be modelled using equations related to diffusion (Fick's law, Nernst-Planck, etc.) and forced convection (Navier-Stokes). Fig. 10(d) shows tomographic re-constructions of the permeation of 1 M VOSO 4 + 5 M H 2 SO 4 (VRFB electrolyte as reported for zero-gap ow cell) through a carbon paper substrate demonstrating how real-time XCT may be used to evaluate and model ow mechanisms in situ for ow batteries. 253 The 3D imaging work on carbon-based materials can capture structures that may be used as geometric inputs to model the behaviour of electrodes and electrolytes. Interestingly, until extensive investigations were performed by Shearing and coworkers, 273 there have been relatively few reports of 3D reconstructions of carbonaceous RFB electrodes in the literature. 235 Qiu et al. have utilised XCT to simulate the effect of concentration, overpotential and charge density within an RFB with a pore-scale resolved model 231 and later demonstrated the effects of real pore/electrode morphology acquired using XCT on electrochemical performance showing that the cell voltage increases with greater electrolyte ow rate as a result of decreasing concentration gradients. 232 Their simulations also suggest that a detrimental effect of performance may occur in the RFB in the event of fuel starvation/low ow rates/low electrolytic concentrations for fresh carbon felt materials. As such, there has been at least one additional study on the tomography of voltage-cycled cells to better understand this behaviour. 274 Reconstructed 3D images of the VRFB electrodes show ber agglomeration and carbon electrochemical oxidation during continuous battery functioning. Key geometric features of the graphite felt samples can also be obtained from these 3D images allowing the estimation of porosity and volume-specic surface area and the changes from operation for each sample. In one study, a signicant decrease (ca. 37%) in the volume-specic surface area of voltage-cycled graphite felts is seen aer only 65 h of continuous VRFB testing, indicating its structural alteration due to carbon oxidation/ber agglomeration. 274 SEM and XPS are employed to verify the structural and surface changes observed by micro-computed tomography. SEM displays bers with a bundle structure commencing to agglomerate aer VRFB cycling and XPS conrms the electrochemical oxidation of graphite bers, demonstrating the formation of an intermediate carbon oxidation product (COOH) on the electrode surface.
The pore-level mass-transfer coefficient is related to the morphology of pore surfaces, electrolyte properties and the local velocity of the electrolyte. 275 The lattice-Boltzmann method (LBM) may be employed to simulate the ow across the pore space. 231,232 The acquired correlation equation for the effective diffusivity of vanadium ions through the porous electrode includes the effects of both the porous electrode structure and ow dispersion (Fig. 11). 231 It was found that the inuence of ow dispersion becomes more signicant with an increase in ow rates and the pore-level mass-transfer coefficient is independent of current density.
Based on the obtained structural information, 3D pore scale models have been developed to explore the effect of electrolyte ow rate, vanadium ion concentration and electrode morphology on VRFB performance. It has been found that overall cell voltage can be improved by increasing ow rate and by using denser electrode structures. In addition, an electrochemical model using the Butler-Volmer equation is employed to provide species ux boundary conditions at the surface of the carbon bers and to deliver the necessary coupling to the local concentration of these species available in the pore space.
One caveat to using 3D imaging of carbon electrodes for electrochemical transport modelling is the large computational expense. One numerical simplication is afforded through pore network modelling (PNM), which converts 3D tomographic images into a connection of pores and throats that are approximated as spheres and cylinders, respectively (Fig. 12). [276][277][278] Recently, PNMs have been used to model the electrochemical transport in electrodes and determine physical characteristics, such as the pore size distribution, permeability, porosity and electroactive surface area. 279 The robustness of PNM has afforded a platform for rapid parametric studies to determine physical insights into electrode microstructure on RFB performance. 278,280 However, further work is necessary to quantify and understand the effects of micro/nano structure on RFB electrode performance and durability.
6. Progress on the modelling of reaction and transport processes in RFB porous electrodes

Modelling for VRFB and similar aqueous-based systems
To enable numerical models to give physically meaningful results of RFB operational parameters, precise transport properties are needed, in addition to a sound formulation. 281 An important transport property is the effective diffusivity, which is necessary to model the mass transport at the representative elementary volume level of porous electrodes in the context of Darcy's law. The widelyused effective diffusivity 210,214,282 is a simple association equation with the inherent diffusivity and the material porosity using a Bruggemann correction; 211 a concern with this correlation, however, is that the inuence of the pore morphology of various porous structures is missed. In parallel, efforts have also been made to experimentally measure the effective diffusivity. A customary approach is to position a porous sample structure between two reservoirs, one of which contains an electrolyte solution, while the other contains de-ionised water, respectively. UV-visible spectroscopic measurements are performed to quantify the change in ion concentration of DI water over a time period. This will enable the extraction of an effective diffusivity value for the ions through the media as a function of porous structural area, thickness and overall solution volume. [283][284][285] A limitation of this approach is the unaccounted inhomogeneous ow distribution that exists in a functional RFB. 275 In addition, there may be other effects not taken into consideration, such as different ionic strengths of two solutions, that may cause a concentration polarization across the porous media. 275 Based on the theoretical context of mass transport within RFB electrodes, diagnostic experimental arrangements have been developed to establish the two, transport properties: the effective diffusivity and the pore-level mass-transport coefficient. The properties were evaluated at a range of electrolyte ow rates through a graphite felt, 286 and the relationship for the effective diffusivity of vanadium ions through the porous electrode includes the contributions of both the porous electrode structure and ow dispersion with the inuence of ow dispersion increasing with greater ow rates. The pore-level, mass-transport coefficient was found to be independent of the current density. In addition, it is worth mentioning that several modelling efforts were dedicated to addressing the effects of compression and other mechanical conditions on effective transport properties, 287-290 even via the development of coupled electrochemical-mechanical models. 291 Xu et al. developed a 2D mass-transport and electrochemical model for a VRFB that accounted for the effect of SOCdependent electrolyte viscosity. 292 The model was used to explore the distributions of vanadium ions concentration, overpotential and local current density for a single VRFB cell. Compared with the outcomes from a constant-electrolyteviscosity model, the results from this model display higher pressure drop (particularly in the positive half-cell) and sharper distributions of overpotential and local current density in the electrodes. The comparison of modelling results shows that the consideration of the SOC-dependent electrolyte viscosity allows more accurate simulations and estimations of pumping energy and system efficiency of VRFBs. A table providing a detailed summary of the governing equations and key ndings of mathematical models on VRFB porous electrodes is given in the ESI (Table S2). † Yu and Chen point out the importance of mass-transfer effects on the overpotential or crossover effects on charge/ discharge efficiency and cycle life. 293 As the current density increases, mass-transfer effects must be considered as depicted in the full Butler-Volmer equation (eqn (14)). However, for small current densities, mass-transfer effects do not play a signicant role and can be removed from eqn (14) (Table 5), thus reducing the computational cost. The conditions in which the effects of mass transfer may be considered Fig. 12 A rendering of a PNM for a hydrogen-bromine fuel cell. Reproduced with permission from ECS and IOP Science. 278 negligible depending on the SOC and other electrochemical parameters. Fig. 13(a) compares the Butler-Volmer equation including the effects of mass transfer (using a mass-transfer limit [MTL] approximation) to the entire current-overpotential equation (as displayed in eqn (14)) at 50% SOC for two dissimilar ow rates as a function of the current density normalised by the limiting current density (the outcome is balanced for anodic currents so it is removed). 293 The Butler-Volmer equation (without masstransfer effects) is fairly accurate at low current densities but slowly deviates as current density rises and mass-transfer effects become signicant. The point at which the two plots intersect is where the MTL approximation becomes more accurate, and at higher current the full Butler-Volmer equation is employed (eqn (14) in Table 5). 241 The overpotential curve is also displayed in the case where ow rate is increased from 1 mL s À1 to 5 mL s À1 . Note, Fig. 13(b) is at 20% SOC and displays an enhancement in the absolute estimation error as compared to the 50% SOC case. 293 Fig. 14(a) compares the voltage output with and without the inclusion of mass-transport effects in the overpotential computation for a 40 min discharge at 600 A m À2 with 80% original capacity. 293 The divergence of ca. 0.05 V aer 40 min of discharge is signicant and will continue to grow as the SOC decreases. When the ow rate is increased by a factor of ve (to 5 mL s À1 ) as in Fig. 14(b), the overpotential losses reduce because of higher mass-transfer effects. In summary, the simulation results reveal that the mass transfer of active species to the porous electrode surface has a signicant impact on the voltage response. 151,213,222 Another simulation paper by Wang and Cho provides: (1) a 3D model outline of dynamic VRFBs; (2) a meticulous elucidation of pore-level transfer resistance and pumping power; and (3) timescale and dimensionless parameter evaluation. 230 At the pore scale, the diffusion time span, which is the approximate timescale for the concentration variation due to diffusion to reach a steady-state, and Péclet number are functions of pore dimensions and are estimated to be ca. 1 s and 1000, respectively. The evaluation also shows that electrolyte pumping accounts for a small part of the VRFB power output (<0.1% for the conditions studied). The model was successfully employed for 3D simulation and was validated with experimental results of charging, idling and discharging. Local working states, such as temperature contours, ion concentration distribution, ow eld and reaction rate, were also evaluated. It was found that the peak temperature occurs near the separator at rst and is cooled by the electrolyte ow as well as the surface of the current collector for both charging and discharging processes. It took ca. 60 s for a steadystate temperature to be achieved. The V(V) concentration at the outlet displays a rapid alteration in the rst few seconds when switching operation. Most transfer current generation was found to occur near the electrode-separator interface; the transfer current represents the local charge transfer rate and is representative of the local electrochemical reaction activity. The Damköhler (Da) number, which relates the chemical reaction and mass transport rates, shows that the macroscopic mass transport rate is higher in the transverse direction, relative to the reaction kinetics. The transfer current uctuates slightly from upstream to downstream of the reaction zone during the early phases of charging and discharging, demonstrating the usefulness of the multi-dimensional method for fundamental analysis of RFBs. 230 Future investigations to further enhance such RFB models include: (1) more accurately assessing multiphase ows, side reaction, heat-ow in electrodes, especially as they pertain to local degradation mechanisms; (2) attaining rigorous experimental data for both material selection, such as electrochemical kinetics and thorough validation (e.g., local distributions); and (3) advancing numerical methods to effectively simulate practical system operation.
A few modelling studies have evaluated the effect of mechanical compression on vanadium redox ow battery performance. 170,173,290,291 Two dimensional models have been used to investigate the effect of changes in the porosity, permeability and ohmic resistance of the porous electrode with compression. In most cases the models have relied on experimental data for the properties of the porous electrode under compression with a notable exception of Xiong et al., 291 who used a multiphysics approach combining mechanical, mass transport and electrochemical processes. All of these studies show that increasing compression leads to a trade-off between increased electrochemical performance (due to reduced ohmic losses) and higher pressure drop, and, consequently, an optimal compression typically exists. In addition, Wang et al. 173 modelled the effect of non-uniform compression (under the ribs and channels of a ow eld) on the electrochemical performance, based on experimental observations of the electrode intrusion into the channels. Further modelling efforts are needed on the effect of compression, especially the nonuniform compression expected with complex rib/channel ow eld designs, in order to enable improved ow eld design, optimisation and scalability.
Finally, the application of mathematical modelling on vanadium-based microuidic cells provides an interesting discussion on the reaction phenomena at the porous electrode and electrolyte interface. 294 The model is dimensionless and is applied to ow-through porous carbon ber electrodes to explore different designs. 295 The working mechanisms of porous electrodes under different operational and geometric parameters were studied. Increasing velocity from the 'insufficient ow' to the 'overow' regime (based on Péclet number as shown in Fig. 15(b) for laminar ow conditions) shows that the electrochemical reaction zone moves from the reactant inlet to the electrode/channel interface. In the operational region ( Fig. 15(b)), the electrochemical reaction zone mainly distributes in the region close to the electrode/channel interface (Fig. 15). These reaction distribution characteristics are generally applicable for cells with microuidic channels, which is in contrast to the situation in a ow-through VRFB modelled separately, in which the reaction was found to occur at the electrode-membrane interface. 230 Accordingly, an optimised dimensional parameter for the electrode length and partially modied porous electrode can be obtained. 295 Most of the studies discussed above account for membrane transport phenomena in greater detail than the inuence of the electrodes for VRFBs. 26,296,297 It is thus necessary for future investigations to focus on the modelling of electrode phenomena, especially considering cases where electrodes are modied with catalysts or physicochemical treatment (e.g. thermal activation) to understand why VRFB performance improves as a result of such electrode treatments.
Other RFB chemistries have been modelled with respect to mass-transfer effects in porous electrodes, which are briey discussed below. The number of studies of modelling that focus on other RFB chemistries is much less than the number that consider VRFBs.

Modelling for non-aqueous and similar ow cell systems
In 2011, Duduta et al. demonstrated a non-aqueous, semisolid, lithium-ion RFB, a hybrid between a traditional ow battery conguration and a rechargeable Li-ion battery. 114 The charge transfer mechanism occurs via dilute yet percolating networks of nanoscale conductors and suspended solid storage compounds. While this mechanism was originally explored by Kastening et al. in 1985 in uidised-bed carbon-based suspensions in sulfuric acid and potassium hydroxide electrodes, 298 such an approach did not receive much attention until the beginning of the last decade when owable semisolid supercapacitors and ow batteries were reported. 299 One of the main advantages of a semisolid electrode formulation is the high concentration of electroactive species (10-20 M) in suspended solid phase and system energy densities that can be achieved when compared to metal redox species in aqueous/nonaqueous solutions (<2 M concentrations and <35 W h L À1 ). 299 Carter et al. modelled the uidics and the electrochemical performance for non-aqueous and aqueous semisolid RFBs using both 1D and 3D models. Their investigation revealed performance issues associated to the use of highly viscous and non-Newtonian owable electrodes. 112 For example, the semisolid conguration necessitates a low resistance interface between stationary current collector and the owable electrode, which is only achievable via intermittent ow. In contrast, a continuous ow mode leads to battery self-discharge processes associated to charge gradients within the uid. 300 Moreover, other capacity degradation issues known for Li-ion intercalation chemistries such as the formation and growth of the solid electrolyte interphase (SEI), are completely linked to cell voltage variations for different SOC. Indeed, the utilization of electroactive materials presenting at potential proles during charge and discharge contribute to the improvement of the battery efficiency. Similar conclusions were experimentally achieved by Tarascon and co-workers. 301 Other electrochemical energy storage devices, such as zincnickel RFBs, 19,302 owable supercapacitors, and Li-polysulde hybrid chemistries, have been demonstrated in recent years. 299 However, few efforts have been devoted to device analysis using mathematical models. In this context, an all-iron aqueous RFB was operated and modelled by Savinell and co-workers 300 who used slurry electrodes to enable separate optimisation of the energy storage capacity and the power delivery capability. However, the negative electrode reaction would need to occur on the slurry particles at high current densities for this electrode to be effective. Thus, mathematical modelling was performed to understand the current distribution in the slurry electrode as a function of the slurry specic area and electrical conductivity. The aim was to obtain >95% plating in the slurry electrode in comparison to that on the at plate at a high current density in excess of 200 mA cm À2 . The model helped in selecting multi-walled carbon nanotubes (MWCNTs) as the most effective slurry material. With the MWCNTs, experimental studies enabled the performance objectives to be achieved. 300 In 2013, Grätzel proposed an alternative lithium-ion, nonaqueous RFB based on redox mediators targeting reactions at both electrodes. 303 A mathematical model based on the conservation of charge and species was later developed by Sharma and co-workers. 304 The reaction and transport processes at the anodic side were of interest, whilst mass and charge transfer at the porous electrode were assumed to be isotropic and the electrode kinetics and regeneration in the storage tank were also considered. In addition, reactions occurring at the positive electrode were assumed rst-order reversible, as conrmed from cyclic voltammetry in a separate study. 305 The model was able to capture the performance of the experimental setup with reasonable accuracy; however, the model did not include temperature effects, 306 detailed reaction mechanisms, effects of electrode compression 171 or shunt currents.
The effect of electrode thickness on the quinone-bromide aqueous RFB was studied in 2015. 130 Polarization experiments on several layers of carbon paper electrodes showed that cells with six to eight layers of electrode thickness gave best overall performance in terms of minimising losses. It was concluded that the variation of the electrode thickness led to a change of both the active surface area and the electrode resistance, thereby enhancing performance. Similar conclusions were obtained by Mench and co-workers for vanadium RFB with serpentine and interdigitated ow eld congurations. 307 They simulated average electrolyte velocity in the electrode domain for increasing layers of carbon paper taking into account the non-uniformity of the distribution in both the in-plane and through-plane directions. Correlations between computational and experimental results for polarization and discharge curves showed how the combination of electrolyte residence time and available electrochemical surface area contributed to alleviate mass-transport limitations regardless of the ow eld design.
A specic, non-isothermal, transient model has also been reported for the quinone-bromide RFB. 308 In this enhanced geometric model, the authors accounted for the inuence of the graphite plate and channel on the cell performance. The interface between the porous carbon electrode and the ow channels were modeled by means of the Brinkman equation. Energy transport was also considered while the temporal effect on voltage and overpotential changes were discussed. At a low applied current density, the ow rate was found to have little effect on cell performance.
A 2D transient, non-isothermal, simulation model for a nonaqueous, hybrid, lithium-oxygen RFB was developed to study the heat and mass-transfer effects within the battery and validate the proposed design. 309 Experiments employed 1 M lithium-based electrolytes in a solvent mixture of propylene carbonate (PC), ethylene carbonate (EC), 1,2-dimethoxyethane (DME), diethyl carbonate (DEC), dimethyl carbonate (DMC), gbutyrolactone (g-BL), tetrahydrofuran, and tetrahydropyran. 310 During operation, the electrolyte was saturated with oxygen in a tank outside of the electrochemical reactor and then pumped into the positive electrode end plate embedded with interdigitated ow channels. Simulation results showed that the convection effect signicantly enhanced oxygen supply in the positive electrode and hence increased battery capacity. However, the model assumed constant positive electrode activation overpotential, and the potential distribution in the electrolyte was neglected. The mass-transport equations for lithium cations and molecular oxygen were based on dilute solution theory. The pressure, the electrolyte velocities, the oxygen concentration, the electrolyte concentration, the electrolyte potential, the electrode potential, the reaction rate, the volume fraction of the solid product and the porosity change caused by Li 2 O 2 precipitation were solved in the computation domain as shown in Fig. 16. 311 An improved model was proposed based upon the porous electrode method and a concentration solution theory was developed by the Newman group. 59,113,115,117,312 Compared to those for traditional RFBs, the improved model considers the effects of the insoluble discharge product deposition in the electrode. 311 In contrast to conventional Li-O 2 battery models, this model includes the effect of convection in species transport. A parametric study was performed to nd the inuence of modelling parameters on the prediction of positive electrode specic capacity and energy. Based on the analysis of the results, two methods: (1) a dual porosity positive electrode structure, and (2) an alternating electrolytic ow method were proposed to further increase the capacity of the aprotic Li-O 2 ow battery. Efforts were made to keep the property data consistent for the same electrolyte type. The model also accounted for organic electrolyte recirculation through the positive electrode to enhance oxygen supply and also incorporated convection effects. Results showed that contrary to conventional static Li-O 2 cells, the electrolyte with a lower conductivity could increase the specic capacity of the Li-O 2 ow cell. The results also revealed that the dual layer positive electrode led to higher capacity than a single layer positive electrode at a current density of 1.5 mA cm À2 and alternating electrolyte ow increased the cathodic capacity by 3.7% at a current density of 0.2 mA cm À2 .
In the past decade, several regenerative fuel cells using a combination of the hydrogen oxidation reaction or the oxygen reduction reaction coupled with soluble redox couples have been reported. [313][314][315][316] Some modelling studies have been performed that have been limited to vanadium-oxygen, 317 hydrogen-vanadium 318 and hydrogen-bromine 319 systems. Considering the versatility of such regenerative fuel cells, especially when employing economically sourced quinonebased couples, 296 there is a need for increased effort in simulation of these systems.
As discussed above, aqueous RFBs have been more thoroughly investigated than non-aqueous RFBs; 24,108,320,321 therefore, a brief summary on how non-aqueous RFBs may be harnessed for practical and commercial applications 322 follows to conclude this section.
(1) Redox molecules for non-aqueous RFB applications should be very soluble (resulting in enhanced capacity and energy density), possess highly positive or negative redox potentials in the electrolyte (causing an increased voltage, energy density, and power density), have fast kinetics (increasing voltage and power density), have stable oxidation states (increasing cycling lifetime), and be economically sourced. High-throughput density functional theory (DFT) computation, physical organic studies, and molecular engineering are effective strategies for molecular design. 323,324 Similarly, redox-active molecular melts, 325 ionic liquids 41 or deep eutectic solvents 61,326 are an alternative to redox molecules dissolved in aqueous or non-aqueous solvents and should possess similar properties. 322 (2) In-depth understanding of the solution chemistry and electrochemistry of redox-active molecules 24,327 and their electrolytes are necessary 328,329 in conjunction with DFT computation for exploring the physicochemical information on redox-active molecules. 323,324,330 Furthermore, computational chemistry can play a role in predicting and mitigating the effect on the degradation of redox active molecules. 322,331 (3) Finally, the effect of ion-transport across ion-exchange membranes on the performance of non-aqueous RFBs needs further evaluation. 322 The subsequent section discusses practical examples of system integration involving mainly aqueous RFBs, although some limited work has been demonstrated with non-aqueous RFBs.

System integration of redox flow batteries
The high energy storage density, quick response time, modularity and long cycle life of RFBs have sparked research interest in developing these systems for various applications, such as grid-scale load leveling/peak shaving, emergency power and renewable energy integration. 35 RFBs have potential integration with wind farms; for example, Turker et al. 333 developed a model for the integration of VRFBs and a medium sized (10 MW) wind farm. 332 In this study, they used the real wind power data, synthetic wind power forecasting tools, and a VRFB soware. The market structure was taken from the Spanish electricity market as they employ deviation penalties. The model demonstrated that VRFBs can be used for the compensation of deviations resulting from the forecast errors in an electricity market bidding structure. The developed model aimed to respond to the deviations between the actual wind farm output and the forecasted electricity demand by varying the battery size and level of deviation penalty. The study showed that a signicant amount of power deviations could be mitigated by utilizing a 2 MW/6 MW h VRFB for the investigated 10 MW wind farm.
RFBs have also found potential integration with various other renewable energy applications. Li et al. integrated an RFB with a photocatalytic, two-step Z-scheme water splitting system for enhancing the solar energy conversion efficiency of the system. 333 The authors experimentally showed the successful integration of these systems and achieved around 0.13% overall solar-to-fuel conversion efficiency.
Liao et al. proposed to integrate solar rechargeable ow cells (SRFCs) with RFBs. 334 SRFCs are electricity generation devices which capture and store the intermittent solar energy via photoelectrochemical reactions. 335 In the study reported by Liao and co-workers, they demonstrated an SRFC integrating a dualsilicon photoelectrochemical (PEC) cell into a quinone/ bromine RFB for in situ solar energy conversion and storage. 334 Yu et al. demonstrated the combination of PEC-conversion and energy-storage functions into one device (aqueous lithiumiodine [Li-I] solar ow battery [SFB]) for efficient utilization of solar energy. 336 For this integration, they incorporated a dyesensitised TiO 2 photoelectrode in a Li-I redox ow battery for simultaneous conversion and storage of solar energy. The photoelectrode and Li-I are linked via an I 3À /I À based positive electrolyte. In this device, iodide anions are photo electrochemically oxidised to I 3À , thereby collecting solar energy and storing it in the chemical form. During the experiments, the device is charged at an input voltage of 2.90 V under 1 sun illumination (AM1.5 G, 100 mW cm À2 ) and discharged at an output voltage of 3.30 V (current density ca. 0.50 mA cm À2 ). 336 Due to this voltage reduction during charging, this device can save up to 20% energy efficiency as compared to traditional Li-I batteries. This work showed that PEC storage using redox species led to higher levels of photon absorption and more effective charge separation, which has been seen in the development of an all vanadium photoelectrochemical storage cell as reported by Wei et al. 337 In this work, a PEC energy storage was combined with a vanadium ow battery, which took advantage of the good round-trip efficiency of the aqueous vanadium redox couples (VO 2+ /VO 2 + and V 3+ /V 2+ ). The system reported enhanced photocurrent and energy conversion efficiencies along with reduced photo corrosion of the photocatalysts. The authors experimentally demonstrated this device and achieved a faradaic efficiency of 95% and the incident photon-to-current efficiency of 12% under 350 nm light. 337 In another effort, Li et al. integrated a regenerated photoelectrochemical solar cell with an organic redox ow battery. 338 They used regenerative silicon solar cells and 9,10-anthraquinone-2,7-disulfonic acid (AQDS)/1,2-benzoquinone-3,5disulfonic acid (BQDS) RFB for this integration, highlighting how organic redox ow batteries have a relative ease with which their properties may be adjusted via target functionalization. 296 These RFBs can also have relatively high energy density exceeding 50 W h L À1 due to a high aqueous solubility (>1 M) of functionalised quinones. In this study, the authors demonstrated the direct charging of the device using solar light without external bias and the discharging of the device was similar to a typical RFB. 338 They achieved solar-to-electricity efficiency of ca. 1.7% and an energy storage density of 1.15 W h L À1 .
Baumann and Boggasch integrated an alkaline electrolyser, PEM-fuel cell and a VRFB in a building automation system. 339 The built system is constructed as a grid-connected hybrid storage with the sole purpose of self-utilization of power produced by photovoltaics. The main components of the integrated systems are the PV arrays, a wind turbine, a microcombined heat and power system, a fuel cell, an alkaline electrolyser, a VRFB, a lead-acid battery, three programmable electronic AC loads and a charging station for electric vehicles (electric vehicle supply equipment). The demonstration of the hybrid system shows how the transient variation of the hydrogen system (electrolyser and fuel cell) and a VRFB integrated into a building automation system affects the performance of the combined system. The results show that such a type of integrated system has limitations in terms of following a load prole. 339 This limitation arises mainly because of the communication amongst the energy management unit, the energy meters, and the local control units. This communication issue can delay the response to high uctuations within a very short timescale and limits their practical use. More research is required to nd possible pathways to mitigate this issue for any successful implementation of such types of system integration.
In short, a lot has been done to enhance the technical feasibility of RFBs for a range of integrated applications. 340 Fig. 17 shows an advanced RFB developed by Sumitomo and New Energy and Industrial Technology Development Organization (NEDO) being applied for system integration in San Diego. 341 As presented, the high energy efficient RFB can achieve simultaneous energy conversion and storage, which further distributes electricity to different end uses. 340 In addition to controlling and regulating the residential energy consumption at home scales, RFBs may even be used for portable electronics with greatly improved energy density comparable with alkaliion batteries, and the liquid-state electrodes are especially applicable for exible devices. 342 Finally, RFBs can be almost instantly recharged by replacing the discharged electrolytes while simultaneously recovering the exhausted electroactive materials separately, providing benecial exibility for electric vehicle applications. However, further investigations on system delay responses are necessary before such benets are fully realised at practical-scales of applications.

Summary and perspective
RFB models are being developed at the cell-and fundamentallevel to address the issues that are important for battery enhancement at the stack/system level, enabling optimisation, reduced cost and improved commercial viability. To attain a more accurate prediction of the battery performance, the models at the cell-and fundamental-level may be improved by including additional physical phenomena, such as water transfer and the electrode microstructure, along with rational simplications. In addition to the optimisation of battery structure and operating conditions, the uniform distribution (electrolyte concentration, current density, overpotential, etc.) inside a device can be achieved via relevant material modication. Signicant variations in local current at the ow channel length-scale suggest that RFBs could operate at much higher current (and hence power) densities if a more uniform current distribution is achieved. Accordingly, modelling work should focus on optimizing electrode structures (3D microscale) based on the objective function of current density, overpotential or electrolyte concentration. In addition, investigation into the molecular/atomic structure and nature of the vanadium electrolyte, leading on to the analogous understanding of other relevant electrolytes (both aqueous and non-aqueous) for RFBs, is of great signicance for achieving a higher energy density via improving the solubility and stability of the electrolytes.
In order to simulate battery operation, multiscale models of VRFB stack/system are needed. Such combination of modelling approaches will enable the integration of phenomena occurring at the wide range of relevant lengths scales in VRFBs. Multiscale models may be derived at the cell-or fundamental-level by considering multiple cells arranged in a VRFB stack/system. In other words, such models should have the capability of connecting microscale processes to cell/stack performance and optimisation levels.
Another topic to be addressed is the roles of micropores and adsorption on RFB performance (depending on the chemistry). Many experimental studies have shown "activation" of the carbon electrode enhances performance, 272,343,344 and this is normally explained in terms of changes in the functional groups on the electrode surface. However, this activation may also generate micropores and increase the surface area, and there are few modelling studies to investigate if the micropores play a role in the enhancement of battery performance or not.
From the system integration perspective, many new congurations have been proposed and tested for the grid-scale power balancing, peak power and integration with other renewable energy technologies. 333,338 Most of these demonstrations have provided promising, initial results for such types of integrations. However, more work is needed to nd the economic feasibility of these congurations. Another important area that requires more attention is the mitigation of delay in the response of these combined systems against the high uctuations in a short duration. Any such delay in the response time decreases the reliability of these systems, thereby limiting their future implementation.

Conflicts of interest
There are no conicts to declare.