# A One-Dimensional Stack Model for Redox Flow Battery Analysis and Operation

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## Abstract

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## 1. Introduction

^{−1}) and sulfuric acid (ca. 0.09 USD kg

^{−1}) and the use of a common redox-active species on both sides of the cell, enabling recovery from crossover-driven capacity fade [4,9,10]. Related RFB research areas that seek to increase energy density and/or employ inexpensive reactants, such as nonaqueous systems [11,12,13], and bromine- [14,15,16], sulfur- [17,18,19], and iron-based chemistries [20,21,22,23], generally focus on development of materials that overcome the undesirable VRFB performance attributes, namely the cell voltage (E

^{θ}= 1.26 V) and the cost of vanadium (ca. 20 USD kg

^{−1}) [1,4,24,25]. Many of these emerging storage concepts and novel redox chemistries have demonstrated the potential to surpass the VRFB in the future; however, to date, none has shown sufficient progress to displace the VRFB as the premier RFB for large-scale energy storage [6,17,22,23,25].

^{−1}) [8], and poor stability at elevated temperatures due to the precipitation of vanadium as V

_{2}O

_{5}[32]. In addition to these RAE-focused activities, several reports have examined membrane properties and their improvement [41,42,43,44,45,46,47,48], the effects of electrodes and their modification [49,50,51,52,53,54,55,56], flow field design [57,58,59,60,61], flow rate [62,63,64], temperature [65,66], and electrolyte rebalancing [10,67]. Although these studies are typically systematic and conclusive for the individual system of interest, the varied experimental configurations between research groups can lead to challenges when trying to combine these individual studies and broadly assess system performance and sensitivity. The field as a whole lacks an accepted control experiment comparing baseline performance of the non-focal components. Thus, attempts to compare, validate, or build upon component advances (e.g., electrode design) reported by different research groups can often be complicated by the differences in operating conditions (e.g., flow rate or electrolyte velocity, potential limitations), in cell design (e.g., flow field type and geometry, reservoir volume, electrode thickness, membrane selection), or in electrolyte conditions (e.g., temperature, composition, sparging conditions). As such, a model with which researchers can readily input their numerous system properties (e.g., electrolyte transport, kinetics, flow field geometry) would aid in the assessment of constituent component contributions and their impact on full system performance. Arguably the most accredited VRFB models have been developed through the complementary work of Shah et al. and You et al. [68,69,70,71], which incorporate numerous physical parameters into a 2D model of the vanadium flow cell. These cell-level models include kinetic (Butler-Volmer), conductivity, and mass-transfer losses in a similar manner to the one used here [72,73]. Each of these models leverage some prior developments from Newman and Tiedemann to describe the porous electrode [74,75]. In addition, modeling efforts have been extended beyond the individual component or cell level to stack systems several times [24,25,76,77,78,79,80,81,82], but each report employs different empirical, inaccessible descriptors of performance-driving parameters (e.g., mass-transfer correlations). Thus, although multiple models exist, they are generally difficult to use, incorporating an overwhelming number of parameters, the selection of which is always subjective even when well-informed.

## 2. System Models

#### 2.1. Electrode Polarization

_{e}) is shown in Figure 1, where L

_{e}is the electrode thickness.

^{+}vs. 1.5 M V), and the proton mobility is higher than any of the vanadium ions (${D}_{{H}^{+}}$ (93) >> D

_{V}(3.9) ≈ D

_{IV}(3.9) > D

_{III}(2.4) > D

_{II}(2.4) × 10

^{−6}cm

^{2}s

^{−1}[87,88]), where D

_{i}with i as a roman numerical indicates the diffusion coefficient of the vanadium ion in oxidation state i. Proton diffusivity is denoted as ${D}_{{H}^{+}}$. Additionally, omission of the protons from the kinetic expression enables a simple translation from the VRFB to a generic redox pair. For the high potential (positive) electrode, the primary electrode reaction is described by Equation (1), in which E

^{θ}refers to the standard state reduction potential [1].

_{n}) at the surface of the porous electrode as a function of cathodic (k

_{c}) and anodic (k

_{a}) rate constants, surface concentrations of the oxidized (${c}_{ox}^{s}$) and reduced (${c}_{red}^{s}$) active species, and the potential of the solid (${\varphi}_{1}$) and liquid (${\varphi}_{2}$) phases [72,89].

_{q}are the universal gas constant, the absolute temperature, the Faraday constant, the transfer coefficient, and the number of electrons transferred, respectively. Note that oxidation corresponds to positive current (i

_{k}> 0) by this convention. Considering a pseudo-steady state condition within the porous electrode, this faradaic current can be related to the oxidized species mass-transfer coefficient (${k}_{m,ox}$) and bulk concentration (${c}_{ox}^{b}$) according to Equation (4).

_{n}in the subsequent analysis. Relating the local reaction current density to the ionic current density (i

_{2}) through the liquid-phase RAE requires consideration of the volume-specific electrode surface area, or electrode area per volume, (a

_{e}) as described by Equation (7), which is derived from a charge balance on a differential planar control volume within the porous electrode.

#### Mass-Transfer Coefficients

_{i}are dimensionless, empirical, user-defined constants, and Sh, Re, and Sc are the Sherwood ($Sh={k}_{m}{d}_{f}/D$), Reynolds ($Re=\rho {v}_{e}{d}_{f}/\mu $), and Schmidt numbers ($Sc=\mu /\rho D$). The default fiber diameter (d

_{f}) is 7 μm [94], and the viscosity (µ) and density (ρ) values for both electrolytes are taken to be 5 mPa s and 1.5 g mL

^{−1}, respectively [8]. For simplicity, we assume that electrolyte properties are constant and do not vary as a function of SoC. In our base case for all of the included analysis, we use the correlation in Equation (11) from Barton et al., as it has specifically been demonstrated with the interdigitated flow fields [73].

_{e}) is calculated based on the number of channels (n

_{ch}), the flow rate through the electrode (Q

_{cell}), the channel length (L

_{ch}), electrode thickness, and the electrode porosity using Equation (12).

#### 2.2. Membrane Effects

_{mem}) and thickness (L

_{mem}), and the membrane voltage drop (ΔV

_{mem}) is estimated according to Equation (13).

^{2+}crosses over to the high potential (positive) side, or VO

^{2+}crosses to the low potential (negative) side, the ions will interact with those already present to achieve the intermediate oxidation state, V

^{3+}, as described by Equation (14). Similar self-discharge reactions are also listed in Equations (15) and (16), and the combination of these three reactions describe the self-discharge events that are assumed to occur quickly and allow each side of the battery system to maintain mass and charge balances. Additionally, inclusion of these reactions simplifies the system by allowing for the consideration of only one electrochemical reaction for each side of the cell as no more than two oxidation states of vanadium will exist for any significant length of time.

_{X,i}) to Faraday’s law of electrolysis for estimation of active species concentration at each new time step during the simulation according to the Equation set (17) [83]. These equations are derived by simple consideration of the series of events occurring following crossover. For example, we describe the case in which a V

^{2+}ion crosses over from the low potential (negative) side, with a reservoir volume V

_{low}, to the high potential (positive) side with a reservoir volume V

_{high}. Within the high potential (positive) reservoir the ion interacts with a VO

_{2}

^{+}ion according to Equation (15) producing a V

^{3+}and a VO

^{2+}ion. From this point, the V

^{3+}ion interacts with another VO

_{2}

^{+}ion to form two VO

^{2+}ions according to Equation (16). In total, this crossover of a single V

^{2+}ion produced three VO

^{2+}ions and consumed two VO

_{2}

^{+}ions, and this is reflected by the coefficients on the crossover flux of V

^{2+}in Equation set (17). Note that the relative impact of crossover on the composition profile depends on the ratio of the cell area (A) to the reservoir volume (V

_{i}). Also note that if one were to introduce the effect of a side reaction with a rate specified on a volumetric basis, such as vanadium precipitation [32], then that tank reaction could be incorporated within the expressions in Equation set (17).

^{2+}, V

^{3+}, VO

^{2+}, and VO

_{2}

^{+}permeabilities (Κ

_{i}) through Nafion 117 [41]. The flux through the membrane may be modeled according to Equation (18) whether an active separator/membrane (active sites and surface charge play a role) or passive separator (porous separator without additional chemical functionality or interactions) [83]. Arguably, this treatment is incomplete as we neglect effects from supporting species (e.g., H

^{+}, H

_{2}O, etc.), but it is practical and provides a qualitative approximation of the crossover flux for each species i (Ν

_{X,i}) as a function of the primary system properties, such as ion valence (z

_{i}), membrane thickness, ion concentration, and the potential gradient across the membrane, which scales with the current density.

_{i,}

_{0}is the concentration of species i at the membrane surface [83]. For open-circuit conditions, Equation (19) is singular, and the simple diffusion problem is solved instead, resulting in Equation (20). These expressions are evaluated for each individual cell and assumed to be uniform over the cell area. Additionally, the flux is taken to be positive for each species as it leaves its native side of the cell, consistent with Equation set (17). Estimation of c

_{i,}

_{0}requires a priori knowledge of the membrane behavior. In the case of an active separator/membrane (e.g., Nafion), some condition of active site saturation may be assumed based on the solution conditions, 1.5 M V here [83,84,85]. In the case of a passive separator with a non-specific selectivity (e.g., Daramic), the concentration may be taken as a planar average of the active species concentration using the separator porosity (${c}_{i,0}={c}_{i}{\epsilon}_{mem}$) as its actual liquid-phase concentration is effectively depressed by the inclusion of solid volume.

#### 2.3. Shunt Resistance Network

_{p}) and 2(N − 1) manifold currents (I

_{m}), which total 5N − 2 currents. Given the system geometry and material properties, we can estimate manifold (R

_{m}) and port (R

_{p}) resistances for each side of the cell. The application of Kirchhoff’s current and voltage laws to this resistor network leads to a set of five repeating equations, Equation set (21) corresponding to the resistor network near cell n based on the cell voltage (E

_{cell}).

_{tot}is the total current applied through the stack.

_{cell}) to the incremental change in cell current due to the shunt currents. This approximation is shown in Equation (23), and the value of r

_{cell}is evaluated numerically using two function evaluations according to Equation (24), which come from the definition of the derivative and require a model decision in the selection of the current step (ΔI) for the evaluation of this derivative.

_{port}). For context, in the base case, where we cycle the cell at 100 A, this value is on the order of 3 mA.

#### 2.4. Hydraulic Losses

_{CK}) value of 4, which corresponds to randomly aligned fibers [95]. We note here that in a real system, the permeability and mass-transfer resistance are inherently related; however, the mass-transfer model incorporated here does not fully capture this behavior, ignoring the effect of k

_{CK}on the mass-transfer losses.

_{cell}) [58,96]. Intermediate steps include the calculation of the channel hydraulic diameter (d

_{h}) in Equation (28) based on the channel width (w

_{ch}) and height (h

_{ch}), as well as a dimensionless geometry-informed permeability factor (ξ) described in Equation (29) that captures the combined effects of permeability and design parameters, such as the channel length (L

_{ch}) and rib width (w

_{rib}).

#### 2.5. Model Framework

_{cell}within the shunt current matrix, then solving for the cell, port, and manifold currents at each time step. Additionally, when a half-cycle (charge or discharge) finishes, the element capturing the applied current within the shunt current matrix descriptor must also be updated (as the sign changes). After this solution, the active species concentrations are updated according to Equation set (17). Finally, at each time step, the currents and cell voltages relative to ground are all stored as data accessible by opening the exported Microsoft Excel file or MATLAB workspace. The names of these files are taken from user inputs on the “Simulation” tab of this function. In the case that a single cell is analyzed, this employs a slightly shortened algorithm as shunt currents are not present. After completion of this forward walk through time, a few variables are plotted within the application, specifically, the species concentrations and the system total and cell average voltages are plotted as a function of time. All of the employed code as well as the free-standing application are included in the supporting information (SI) for complete documentation and future implementation. The selected default values of all system parameters are listed in Appendix A in Table A1 along with clarifying illustrations of system geometry in Figure A1.

## 3. Results and Discussion

#### 3.1. Single Cell Performance

^{−1}) are the same, there is a large variation in the predicted mass transfer coefficients from these empirical correlations. In recognition of the large variation and its impact on estimates of cell performance, we incorporated several empirical correlations [58,73,91,92,93] directly within the application as well as the option to enter a user-specified correlation in the format of Equation (10). The more recent correlations predict smaller mass-transfer coefficients, and this should be kept in mind throughout the subsequent analysis in which the correlation from Barton et al. [73] is selected as it has been demonstrated specifically with interdigitated flow fields. The frequently cited correlation from Schmal et al. was developed specifically for mass-transfer to an individual fiber, and the experimental setup used in 1985 may not be an accurate representation of the conditions achievable within current-generation VRFBs [92]. In this prior work, the commonly cited correlation was developed for carefully aligned fibers in a flow-through geometry, but comparison of their reactor bed with a modern RFB reveals a few quantifiable differences: typical fiber electrodes are not fully aligned with the direction of flow, interdigitated flow fields have significant flow orthogonal to the electrode plane, carbon paper porosity is typically lower than in the felt employed (ca. 80% vs. 96%), and the fibers are typically smaller than those found in the felt (ca. 7 µm vs. 15 µm) [92]. Perhaps the most interesting aspect of this particular figure is the indication that there is room for improvement in the characterization of mass transfer for RFBs. All three of these prior publications considered d

_{f}to be the relevant length scale, but it may be important to consider separately the electrolyte path length, electrode thickness, and possibly an independent descriptor for the pore diameter or other details of the electrode microstructure. In particular, the path length within the porous electrode varies significantly with flow field design and experimental setup [73,91,92]. All three of the polarization curves in Figure 5a use literature values for vanadium reaction kinetics [88], which happen to use the same value for the anodic and cathodic reaction directions although different values are used for the high potential and the low potential reactions. As a result, these curves are symmetric about the open-circuit voltage (V

_{OC}) as the difference in individual mass transfer coefficients is nearly imperceptible at the cell level. In Figure 5b, we manipulate these rate constants to illustrate the impact that a 10-fold increase or decrease of the kinetic rate constants would have on the cell polarization. This is meant to emphasize that if the kinetic rates previously demonstrated can be achieved within a scaled cell, there is little incentive to catalyze the reactions further as an order-of-magnitude increase produces only marginal improvement in the polarization. Performance-driven research would likely be better served furthering membranes, flow fields, or electrode architecture. Conversely, if those rate constants cannot be achieved, their increase could significantly improve cell performance. Figure 5c shows a single (second cycle) cell discharge for three different current densities at a fixed flow rate. Due to the moderate value of the mass-transfer coefficient estimated, the cell reaches 81% of the theoretical capacity (20.1 Ah L

^{−1}) at low current densities, and as this increased to a more moderate value of 300 mA cm

^{−2}, this drops sharply to 30% of the theoretical. For comparison, Jiang et al. report accessing 72% of the theoretical capacity at 40 mA cm

^{−2}, and decreasing to 25% at 320 mA cm

^{−2}[97]. Figure 5d illustrates the influence of current density and flow rate on the cell voltaic efficiency and accessed capacity. The voltaic efficiency (black, left) decreases from ca. 95% to 88% as the current density is increased with a scaled flow rate from 100 mA cm

^{−2}to 300 mA cm

^{−2}at 2.12 L min

^{−1}and 6.36 L min

^{−1}, respectively. This particular area-specific flow rate (2 mL min

^{−1}cm

^{−2}) is comparable to reported and optimized values within the field [58,62]. If scaled in this manner for this 1000 cm

^{2}cell, the accessed capacity falls from 88% to 67% of the theoretical in contrast to Figure 5c, where flow rate was held constant. The increase in flow rate increases the accessed capacity as well as the voltaic efficiency due to the increases in the mass-transfer coefficients. Practically, the pumping losses also increase linearly with flow rate from 16 kPa to 48 kPa for 2.12 L min

^{−1}to 6.36 L min

^{−1}. Since the pressure drop scales linearly with flow rate, the pump work scales with its square, and in terms of pump power, this corresponds to an increase from 1.6 W to 15 W (3

^{2}×) with a 70% pump efficiency. Although this may seem a subtle point, it illustrates that the hydraulic losses within a flow battery grow quickly as flow rate is increased in an attempt to overcome mass-transfer resistance. This relationship poses a dynamic problem for the efficient operation of VRFBs as the SoC and system load change. High-power discharge requires some minimum flow rate, which negatively impacts the efficiency.

#### 3.2. Membrane Crossover

^{2}cell operating at 100 mA cm

^{−2}with 2.5 L tanks and flow rate of 2.12 L min

^{−1}. The relatively small tank size was selected to keep the simulation time short and approximate common experimental energy-to-power ratios. Permeabilities are listed in Table A1 [41]. Crossover is easily removed by setting the membrane permeability to each active species to 0. The theoretical half-cycle for this configuration is 1 h, and over the 50 cycles shown, the active membranes (Nafion 212 and 117) show low capacity fade. In the case of these cation-exchange membranes, the varied solubilities (listed in Table A1 [84]), or saturation concentrations, of vanadium within the membrane bias crossover in one direction adding to the imbalance of total vanadium on either side. For the conditions shown, the Nafion coulombic efficiency remains high, and the capacity fade is less than many reports of experimental results [97,98,99], likely due to the omission of supporting species crossover. Here, the capacity fade over 50 cycles was 10.7% and 1.9% of the initial accessed capacity for the Nafion 212 and Nafion 117, respectively. For comparison, Jiang et al. show a capacity loss of over 50% and 20% for Nafion 212 and Nafion 117 over 50 cycles [97]. Figure 6b shows the self-discharge from 50% SoC at open-circuit conditions with the same flow rate and tank volume. Practically, this long self-discharge time for a system with an energy-to-power ratio of 1 h is promising for the field. If this low permeability to active species is genuinely achievable, losses due to crossover and self-discharge over the course of a cycle should remain small, and the selectivity of these membranes may be sufficient to satisfy performance metrics for coulombic efficiency. If other losses were included, such as side-reactions, these may present convoluting effects on the system performance, for example, shifting of the average vanadium oxidation state, and may further reduce the system capacity. As it stands, omission of supporting species crossover from the model likely causes the underestimation of capacity fade observed here but the trends are qualitatively correct. We also attempted to include a comparison of a passive separator with a MacMullin number of 4.5 and thickness of 25 μm, similar to Celgard 2500 [100]. The results are not shown here as the capacity fade was rapid enough to prevent meaningful discussion. We expect that the error in the model is greater for the high fluxes experienced with a passive separator due to the omission of supporting species transport.

#### 3.3. System Performance

^{−1}per cell) to maintain a constant energy-to-power ratio (4 h at 100 mA cm

^{−2}) and electrolyte velocity. Figure 7a shows the shunt currents within the ports and manifold as a function of position within the stack. These current magnitudes and trends are similar to past work [78,82]. The edge conditions force the manifold current to match the port current, and this is directly observable at cell N for each stack size. At cell 1, the port and manifold currents are equivalent in magnitude and direction, but opposite in sign merely due to the convention chosen in Figure 2. The applied current, 100 A, is much larger than the port and manifold currents, and as a result, the impact of these shunt currents on the cell voltage is minimal as an individual cell voltage is relatively insensitive to a 0.1% change (100 mA vs. 100 A) in the current passing through it. Practically, the port currents may be considered leakage currents from individual cells, and this leakage is summed within the manifold to produce a maximum manifold current near the center of the stack. The port currents are the greatest near the end of the stack and they are minimal near the middle of the stack. Physically, this phenomenon arises due to the fact that each successive cell drives current through the port in the opposite direction of the previous cell. However, at the ends the ports only see one cell thus the leakage (port) currents can grow to a greater value. Figure 7b shows the increase in mean discharge power (left) as a function of stack size alongside the power lost to shunt currents and pump work (right). Across the cases explored, pump work and shunt losses grow significantly with the stack size. However, the relative magnitude of these losses remains small. The shunt current magnitude is consistent with prior reports, both in terms of the relative scale of these currents and the dependence on position within the stack [78,81]. Practically, quantitative comparison of these values is difficult as simple variations in system geometry can significantly affect the results.

^{−1}per cm

^{2}of active area is near the point of maximum efficiency. In this case, the system operates at slightly higher efficiency at 1.5 mL min

^{−2}per cm

^{2}of active area, so if we consider roundtrip energy efficiency to be our primary variable to maximize and constrain the system to an invariant flow rate, then this would be the optimal condition. However, both logic and prior art suggest that to maximize efficiency, a variable flow rate should be employed [101,102]. The system is generally insensitive to mass-transfer coefficient while the reactant is being consumed from an enriched state (i.e., discharging at high SoC or charging at low SoC), thus operation at low or moderate flow rates is optimal here. When the reactant becomes depleted, it then becomes worthwhile to increase the flow rate, enhancing the mass-transfer coefficient and enabling greater accessible capacity. We also note the slope of the efficiency curve to either side of the flow rate corresponding to maximum efficiency. To the left of the maximum, decreases in flow rate produce a sharp decrease in efficiency, whereas to the right, the decrease is slower, suggesting that in the case of uncertainty, erring towards higher flow rates may be favorable. Increasing current density increases the flow rate of maximum efficiency due to the increase in mass-transfer losses (data not shown). Additionally, higher current densities reduce the maximum efficiency due to an increase in all sources of losses. Figure 9 examines the degree to which current density influences the system performance.

^{−1}) as the current density increases from 50 mA cm

^{−2}to 300 mA cm

^{−2}. The voltage efficiency also decreases from 96% to 81% with this increase in current density. A thorough comparison with the literature reveals that these efficiencies are definitively overestimated at low current densities, but within the range of reported values at high current densities, as results are widely varied in this regime [103,104,105,106,107]. For comparison, Zhou et al. report a voltaic efficiency of 90% and 67% at 80 mA cm

^{−2}and 320 mA cm

^{−2}, respectively, but the authors use Nafion 115, which should have a resistance of 2.5× the value used here, representing Nafion 212 [103]. Additionally, this same work reports similar accessed capacities at both low and high current densities (82% and 20% of the theoretical at 80 mA cm

^{−2}and 320 mA cm

^{−2}, respectively) [103]. Wang et al. report voltaic efficiencies ranging from 92% to 67% at 40 mA cm

^{−2}and 200 mA cm

^{−2}, respectively [104]. Recent demonstration and analysis of controlled-power cycling of a system with multiple vanadium flow battery stacks at 50 kW to 200 kW showed voltage efficiencies of 91% to 80% [105]. In this case, the membrane was not reported, and the estimation of the average current density is unclear. The overestimation of the voltage efficiency at low current density suggests that the kinetic losses may be underestimated and, although these kinetic parameters have been used in past work, their true values likely vary greatly between systems, especially as a function of electrode selection and pretreatment. At high current densities, differences within the literature are more likely explained by variations in the mass-transfer coefficient between systems. For the default conditions employed, this model estimates a limiting current density of a little over 400 mA cm

^{−2}as shown in Figure 5a, but within the literature, one can find limiting current densities for similar electrolyte compositions as low as 60 mA cm

^{−2}or greater than 2600 mA cm

^{−2}[106,107]. These differences arise as a result of the large variations in experimental setup, and serve as motivation for our incorporation of multiple mass transfer correlations as detailed in Section 2.1. As the model stands, these results are generally bound by previous reports of system performance and emphasize the sharp decrease in efficiency and accessed capacity as the limiting current density is approached.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Symbol | Description | Dimensions ^{1} | Default Value ^{2} |
---|---|---|---|

L_{e} | electrode thickness | L | 260 μm |

k_{c} | cathodic rate constant | L t^{−1} | 8.5 × 10^{−4} cm s^{−1} [88] |

k_{a} | anodic rate constant | L t^{−1} | 5.3 × 10^{−4} cm s^{−1} [88] |

${c}_{i}^{b}$ | bulk concentration of species i | N L^{−3} | 0.75 mol L^{−1} |

d_{f} | fiber diameter | L | 7 μm [94] |

k_{m,i} | mass-transfer coefficient of species i | L t^{−1} | m s^{−1} |

n_{q} | number of electrons transferred | 1 | |

n | cell index | - | |

N | total number of cells | 35 | |

N_{X,i} | crossover flux of species i | N L^{−2} t^{−1} | mol m^{−2} s^{−1} |

x | position within the porous electrode | L | m |

i | current density | I L^{−2} | 100 mA cm^{−2} |

I | current | I | A |

a_{e} | volume-specific electrode surface area ^{3} | L^{−1} | 85700 m^{−1} |

R_{i} | resistance of element i | M L^{2} I^{−2} t^{−3} | Ω |

Re | Reynolds number | - | |

Sh | Sherwood number | - | |

K_{i} | membrane permeability of species i | L^{2} t^{−1} | m^{2} s^{−1} |

i = V^{2+} | 3.39 × 10^{−12} m^{2} s^{−1} [41] | ||

i = V^{3+} | 1.87 × 10^{−12} m^{2} s^{−1} [41] | ||

i = VO^{2+} | 2.84 × 10^{−12} m^{2} s^{−1} [41] | ||

i = VO_{2}^{+} | 2.32 × 10^{−12} m^{2} s^{−1} [41] | ||

c_{i,sat} | membrane solubility of species i | N L^{−3} | mol m^{−3} |

i = V^{2+} | 113 mM [83] | ||

i = V^{3+} | 52 mM [83] | ||

i = VO^{2+} | 28 mM [83] | ||

i = VO_{2}^{+} | 18 mM [83] | ||

ε_{mem} | membrane porosity | 0.39 [108] | |

L_{mem} | membrane thickness | L | 50 μm |

K_{CK} | Carman-Kozeny constant | 4 [95] | |

w_{rib} | flow field rib width | L | 0.89 mm [58] |

w_{ch} | flow field channel width | L | 1.17 mm [58] |

L_{ch} | flow field channel length | L | 28 cm |

d_{ch} | flow field channel depth | L | 0.76 mm [58] |

n_{ch} | number of flow field channels | 175 | |

d_{p} | port geometric diameter | L | 8 mm [82] |

d_{m} | manifold geometric diameter | L | 10 mm [82] |

L_{p} | port length | L | 100 mm [82] |

L_{m} | manifold interport distance | L | 6 mm [82] |

V_{res} | reservoir volume | L^{3} | 350 L |

V_{OC} | open-circuit voltage | M L^{2} I^{−1} t^{−3} | 1.4 V [109] |

E_{low} | lower voltage limit for cycling | M L^{2} I^{−1} t^{−3} | 5.5 |

E_{high} | upper voltage limit for cycling | M L^{2} I^{−1} t^{−3} | 8.5 |

T | temperature | T | 22 °C |

Sc | Schmidt number | - | |

Pe | Péclet number | - | |

D_{i} | diffusivity of species i | L^{2} t^{−1} | m^{2} s^{−1} |

i = II; species is V^{2+} | 2.4 × 10^{−6} cm^{2} s^{−1} [88] | ||

i = III; species is V^{3+} | 2.4 × 10^{−6} cm^{2} s^{−1} [88] | ||

i = IV; species is VO^{2+} | 3.9 × 10^{−6} cm^{2} s^{−1} [88] | ||

i = V; species is VO_{2}^{+} | 3.9 × 10^{−6} cm^{2} s^{−1} [88] | ||

v_{e} | electrolyte average velocity | L t^{−1} | m s^{−1} |

Q | volumetric flow rate of RAE ^{4} | L^{3} t^{−1} | 74.2 L min^{−1} |

Subscripts | |||

ox | oxized redox-active species | ||

red | reduced redox-active species | ||

1 | solid (electrode) phase | ||

2 | liquid (electrolyte) phase | ||

high | high-potential side | ||

low | low-potential side | ||

Greek | |||

α | transfer coefficient | 0.5 | |

ν | kinematic viscosity | L^{2} t^{−1} | m^{2} s^{−1} |

π_{i} | mass transfer correlation parameter | - | |

κ | RAE conductivity | I^{2} t^{3} M^{−1} L^{−3} | 270 mS cm^{−1} [8] |

κ_{eff} | effective RAE conductivity ^{5} | I^{2} t^{3} M^{−1} L^{−3} | S m^{−1} |

κ_{mem} | membrane conductivity | I^{2} t^{3} M^{−1} L^{−3} | 67 mS cm^{−1} [110] |

μ | dynamic viscosity | M t^{2} L^{−1} | 5 mPa s [8] |

η_{pump} | pump efficiency | 70% | |

Δt | time step for cycling | t | 20 s |

ε | electrode porosity | 85% [73] | |

Δϕ_{m} | overpotential at the membrane | M L^{2} I^{−1} t^{−3} | V |

ΔP | pressure drop | M t^{−2} L^{−1} | Pa |

ρ | RAE density | M L^{−3} | 1.5 g mL^{−1} [8] |

σ | electrode conductivity | I^{2} t^{3} M^{−1} L^{−3} | S m^{−1} |

ϕ_{i} | potential in phase i | M L^{2} I^{−1} t^{−3} | V |

^{1}Dimension key: mass (M), length (L), time (t), temperature (T), current (I), matter (N).

^{2}In the case of a time-variant variable, the initial value is given. In the case of multiple values being assumed or estimated based on other system parameters, only the default SI units are given.

^{3}Estimated based on long thin-fiber characteristics based on a

_{e}= 4 × (1 − ε)/d

_{f}[73,91].

^{4}This flow rate was selected by scaling the flow rate from a few references [58,62], as well as our own experimental practices. Heuristically, it appears that a flow rate of ~2 milliliters per minute per square centimeter of cell area is fairly standard practice.

^{5}Estimated using a Bruggeman relation κ

_{eff}= κε

^{1.5}[72,73,90].

**Figure A1.**Clarifying illustration of system geometry parameters that enable extension beyond single-cell analysis.

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**Figure 1.**Porous electrode model domain and boundary conditions. The electrode reaction described by Equation (1) for the high potential electrode or Equation (2) for the low potential electrode occurs throughout this domain.

**Figure 2.**System resistor network near a generic cell n as used to estimate shunt currents. Arrow directionality indicates positive current flow.

**Figure 3.**Model graphical user interface (GUI). (

**a**) Material properties, such as electrode properties, bulk redox-active electrolyte (RAE) properties, and membrane characteristics. (

**b**) System properties, such as flow field geometry and operating conditions. (

**c**) Simulation options, such as time step and output file names, as well as simulation outputs, such as pressure drop and voltage history. (

**d**) Assorted interface components, such as manifold geometry, membrane behavior, a cell polarization tool, and a place for user-input notes.

**Figure 5.**Single cell performance. Discharge (○) and charge (△) polarization at 50% state-of-charge (SoC) and 1.5 M total active material concentration and a fixed flow rate of 2.12 L min

^{−1}using (

**a**) three different mass-transfer correlations [73,91,92] and (

**b**) three different sets of kinetic parameters, where literature (black) kinetic values are compared to a 10× increase (red) or decrease (blue). (

**c**) Galvanostatic discharge curves for three different current densities at 2.12 L min

^{−1}. (

**d**) Single cell voltaic efficiencies (black, left) and discharge capacities (blue, right) as a function of current density and flow rate.

**Figure 6.**Single cell crossover impact. (

**a**) Charge (△) and discharge (▽) capacity as a function of cycle number at 100 mA cm

^{−2}. Inclusion (black, red) and omission (blue) of crossover clearly indicate its impact on capacity fade with stable current efficiency. (

**b**) Change in active species concentrations as a function of time at open-circuit conditions due to the rate of crossover. Electrolytes contained 1.5 M V initially.

**Figure 7.**Stack losses during discharge at as a function of the number of cells when cycled at 100 mA cm

^{−2}with a flow rate of 2.12 L min

^{−1}cell

^{−1}and 1.5 M V. (

**a**) Mean port (△) and manifold (○) currents as a function of position within the stack. (

**b**) Mean stack discharge power (○, left) alongside energy losses due to shunt currents (△, right) and pump work (□, right).

**Figure 8.**Influence of flow rate on 35-cell stack performance at 100 mA cm

^{−2}. (

**a**) Impact of flow rate on accessible system capacity (black, left) and discharge energy density (blue, right). (

**b**) Impact of flow rate on the average discharge electrochemical power (○, black, left) and the pump power (△, black, left) alongside the roundtrip efficiency with (△, blue, right) and without (○, blue, right) accounting for pumping losses.

**Figure 9.**Performance of the 35-cell stack over a range of current densities at 74.2 L min

^{−1}. (

**a**) Discharge curves and (

**b**) accessed capacity (○, black, left) and voltage efficiency (△, blue, right) as a function of applied current density.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Barton, J.L.; Brushett, F.R.
A One-Dimensional Stack Model for Redox Flow Battery Analysis and Operation. *Batteries* **2019**, *5*, 25.
https://doi.org/10.3390/batteries5010025

**AMA Style**

Barton JL, Brushett FR.
A One-Dimensional Stack Model for Redox Flow Battery Analysis and Operation. *Batteries*. 2019; 5(1):25.
https://doi.org/10.3390/batteries5010025

**Chicago/Turabian Style**

Barton, John L., and Fikile R. Brushett.
2019. "A One-Dimensional Stack Model for Redox Flow Battery Analysis and Operation" *Batteries* 5, no. 1: 25.
https://doi.org/10.3390/batteries5010025