Maximizing Energetic Efficiency in Flow Batteries Utilizing Non-Newtonian Fluids

A schematic of the simulated flowing half-cell appears in Fig. 2a. The electroactive region of the half-cell is the region between current collector (orange) and separator (blue), the volume of which is referred to as a unit aliquot. Because semi-solid suspensions are mixed conductors, the electroactive zone,²³ in which electrochemical reactions occur, will extend outside the immediate electroactive region defined here. The separator facilitates the transfer of ions with an electrolyte bath at uniform, time-invariant potential. The suspension-based electrodes are assigned properties based on laboratory measurements of electrodes comprised of liquid electrolyte, carbon black, and active material. The two-dimensional geometry [Fig. 2b] is specified by the lengths of the inlet L_i, current collector L_cc, and outlet L_o, as well as the width of the channel w. L_cc and w are fixed at 50 mm and 0.5 mm, respectively, while various values of L_i and L_o are simulated depending on system size.

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Three active materials (Table I) were simulated. Two of these active materials are solid-state Li-ion intercalation compounds LiFePO₄ and LiCoO₂, and the other is a redox solution (VO²⁺/VO₂⁺). Each equilibrium potential versus state-of-charge (SOC) curve is illustrated in Fig. 1d. For the redox system, the equilibrium potential increases monotonically with SOC. For the intercalation compounds, there is a range of SOC where the equilibrium potential has a plateau. These plateaus indicate equilibrium between two phases with states of charge given by the limiting plateau compositions. As discussed below, the plateau width (range of SOC) is an important feature in electrochemical performance. LiFePO₄ exhibits two-phase equilibrium between 9% and 97% SOC [Fig. 1d] which represents the majority of a practical electrochemical cycle:²⁴

A consequence of this two-phase equilibrium is that spatial SOC gradients can exist at thermodynamic equilibrium because LiFePO₄ particulates can have a particular phase fraction with any average SOC between 9% and 97% and maintain at the same equilibrium potential.However, for LiCoO₂,^{25, 26}

the two-phase plateau is much smaller. As a consequence, such equilibrium SOC gradients can exist over a smaller SOC window (15–50%). Thus, over most of a full SOC swing, LiCoO₂ exhibits single-phase behavior for which equilibrium gradients cannot exist. We model these solid-state active materials with non-aqueous electrolytes, e.g., 1 mol/L LiPF₆ in mixed carbonate solvent.²⁷

We also analyze the cathodic couple of the vanadium-redox aqueous solution (VO²⁺/VO₂⁺). The following simplified reaction is modeled in which VO²⁺ is oxidized to VO₂⁺ during charging:²⁸

This system is hereafter referred to as "V-redox." The solubility and mobility of redox ions in the electrolyte enables mixing of SOC throughout the cell. We modeled a suspension in which this redox solution contains a percolating network of electronically conductive particles. The resulting suspension has finite electronic conductivity as well as ionic conductivity, and charge transfer reactions are assumed to occur at the conductor/solution interface. Representative aqueous electrolytes for this system include H₂SO₄ (Ref. 28) and chloride solutions (Ref. 29).

Previous work^{5, 8} has demonstrated that operating SSFCs in an intermittent flow mode reduces pumping losses relative to the continuous flow mode in which conventional flow cells are typically operated. Accordingly, we emphasize the intermittent flow mode, although continuous flow behavior can be inferred directly by extrapolating to the limit of many short duration intermittent pulses. Figure 3 depicts the evolution of SOC and cell voltage for an intermittent cycle, defined by an alternating sequence of current and flow impulses. The particular process shown involves the cycling of only two aliquots, which is the shortest complete cycle possible. As we will show, this two-aliquot cycle sets an upper bound on efficiency losses. The cycling of more aliquots demonstrates convergence to a fixed capacity and efficiency.

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An alphanumeric rubric is used to distinguish the cycle steps for all of the subsequent results, as shown in Fig. 3. The first charging step starts from a fully discharged state (charge-S1, not shown) and ends when the terminal charging voltage is reached (charge-E1). Subsequently, suspension is flowed through the cell in the absence of current to eject charged material from the electroactive region. The second charging step starts (charge-S2) and ends after the terminal charging voltage is reached (charge-E2). In steps S1-E2, two aliquots of fluid are charged. To discharge, the current and flow direction are reversed and commences with discharge-S1 (not shown). When this discharge step reaches the limiting voltage (discharge-E1), flow is also reversed to replenish the electroactive region with charged suspension, and the second discharge step starts (discharge-S2). The cycle is completed after the terminal discharge voltage is again reached (discharge-E2).

The cycle above represents the simplest in a class of intermittent cycles for which we now describe the possible operational parameters. We define the pumped volume relative to that of a unit aliquot (i.e., the cell's internal volume) and refer to this quantity as the aliquot factor m. (e.g., m = 1 and m = 0.25 correspond to pumping one full aliquot and one-quarter aliquot per pump stroke, respectively.) The system's total suspension volume is given as a multiple of unit aliquots. The flow rate at which pumping occurs dictates the flow profile shape for a particular suspension rheology, suspension/wall interface, and cell channel design. As we show later, the influence of flow rate, material parameters (both rheological and slip), and channel dimensions can be captured in terms of two dimensionless parameters describing the complete space of velocity profiles for a typical non-Newtonian flow electrode exhibiting both wall slip and steady shear in the suspension's bulk.

The electrochemical operation of the cell is specified by the upper and lower voltage cutoff limits applicable to the electrochemical couple (Table I) and the time dependence of the applied current, I_applied. One of the attributes of flow batteries is that the relative capacities of the stack and tanks can be varied arbitrarily. Here we simulated current rates for the cell of C/3 (full charge and discharge of the stack in 3 hours), corresponding to long-duration storage for a complete system that is some multiple of 3 hours.

In the absence of flow, electronic conduction occurs via the conductive-particle percolating-network suspended in the electrolyte, and the solid-phase potential ϕ_s is governed by charge conservation:

where σ_s is the effective electronic conductivity of the suspension. The second term in Eq. 1 is a source term that couples electronic conduction to the electrochemical surface reaction characterized by the local reaction current density i_n and surface area per unit suspension volume a. Because the entire suspension is electronically conductive, electrochemical reactions can occur outside the immediate electroactive region of the cell²³ (this is implicit in Eq. 1). It has been shown that the electronic conductivity varies widely with carbon black content.^{5, 9, 31} Here, we assume a value of 1 mS/cm for σ_s corresponding to experimental results for ∼1 vol.% Ketjen black in non-aqueous³¹ and aqueous⁹ suspensions. Recent work has also shown that the electronic conductivity of semi-solid suspensions depends on shear rate,³² but such variations will have negligible impact on the intermittent flow mode used here (i.e., charging takes place when the semi-solid electrode is static). In addition, conductivity variations due to microstructural relaxation after a flow pulse are expected to be minimal, since oil-based suspensions of carbon black have exhibited gelation times³³ that are at least five orders of magnitude smaller than the present charge/discharge time-scales.

For suspensions using typical Li-ion intercalation compounds (e.g., LiFePO₄ and LiCoO₂) of fine particle size, intercalated lithium concentration at the particle surface differs by less than 1% of the bulk value at C/3 rate (based on room-temperature diffusivities inferred from Refs. 34, 35). Consequently, the intercalated lithium fraction x_Li at a given time t is assumed uniform and is governed by the following conservation equation:

where c_s,max is the volumetric concentration of intercalated lithium at saturation, ν_s is the volume fraction of active material, and F is Faraday's constant. The values of c_s,max for LiFePO₄ and LiCoO₂ are 22.8 mol/L (Ref. 36) and 51.6 mol/L (Ref. 37), respectively. (It is these high molarities, compared to the 1–2 mol/L concentrations typical of aqueous redox flow batteries,^{15, 16} that allow semi-solid electrodes to have high energy densities.) SOC is defined by the intercalated lithium fraction relative to those at the charge and discharge cutoff voltages listed in Table I.

In the case of the V-redox system, the diffusion time-scale for redox molecules through the electrode thickness is much shorter (∼10 s) than the cycling time-scale. The flux of redox species is dominated by diffusion (i.e., not migration), because the high ionic conductivity of concentrated, acidic electrolytes (e.g., H₂SO₄ (Ref. 28) or chloride solutions)²⁹ minimizes the electric field that drives migration. Therefore, mass conservation of VO²⁺ and VO₂⁺ in the absence of migration sufficiently describes the electrochemical processes that occur in the V-redox system:

where c_j and D_eff,j are the concentration and effective diffusivity of redox species j in the electrolyte. The SOC of the electrode is equivalent to the concentration of VO²⁺. Source terms in Eqs. 3 and 4 couple redox-species diffusion to the electrochemical reaction that occurs at conductor-electrolyte interfaces. The exclusion of electrolyte volume from the suspension by the conductive carbon (1 vol.% loading) is negligible, and the suspension's porosity ɛ is approximately 100%. The diffusivities for both V-redox species are assigned bulk values of 3.9 × 10⁻¹⁰ m²/s from literature.³⁸

The surface overpotential η drives electrochemical reactions at solid-electrolyte interfaces and is given as η = ϕ_s − ϕ_e − ϕ_eq, where ϕ_e and ϕ_eq are the ionic potential of the electrolyte and the equilibrium potential of the active material, respectively. The equilibrium potential models of the three active materials (as a function of x_Li and redox species concentrations) were taken from the literature^39–41 and are shown in Fig. 1d with respect to SOC. The ionic potential ϕ_e is assumed uniform at the value of the chosen counter-electrode (Table I). A Butler-Volmer model describes the surface reactions for all three suspensions. Assuming equal anodic and cathodic transfer coefficients, the reaction current density i_n is:³⁰

where i₀ is the exchange current density and RT has its usual meaning. In general, the exchange current density i₀ depends on the concentration of active species. This dependence reflects the competition between forward and reverse reactions at the conductor/electrolyte interface, and, therefore, the functional form of i₀ depends on the type of reaction. Table II summarizes the kinetic parameters and volumetric surface area, a (m²/m³), for each suspension. For the solid-state active materials, a is the active-particle/electrolyte interface area (per unit suspension volume), and the exchange current density can be expressed as:⁴²

where c_e is the ion-conducting species concentration in the electrolyte (taken as 1 mol/L here), k is the reaction rate-constant, and x_Li is the intercalated lithium fraction (determined by solving Eq. 2). The values of a used here assume 100 nm diameter LiFePO₄⁴³ and 4 μm diameter LiCoO₂³⁷ particles.

The exchange current density i₀ for the cathodic V-redox reaction depends on the concentration of redox species j at the reaction surface, c^s_j:²⁸

Pore-scale mass-transfer resistance causes surface concentrations, c^s_j, to differ from the bulk electrolyte concentrations, c_j, of a given species j:²⁸

where d_p is the pore diameter of the material on which the surface reaction takes place. We utilize the procedure described in Ref. 28 to determine the surface concentrations. For the V-redox suspension 1 vol.% loading of Ketjen black is assumed with a specific surface area of 1453 m²/g (Ref. 44) and a pore diameter of 100 nm (in between the size of the carbon black aggregates and the individual particles comprising them).⁴⁵

When intermittent flow pulses occur much faster than electrochemical processes, pure advection governs both intercalated lithium fraction:

and species concentration fields:

where is the suspension velocity field. Fully developed, axial flows are considered here, the velocity fields of which are non-zero only in the x-direction and depend only on the y-coordinate [see Fig. 2b]:

where is the unit vector along the channel's axis (taken as the x-coordinate). Results for a variety of velocity profiles are presented later.

To simulate galvanostatic charge/discharge conditions a time-invariant total current I_applied is imposed at current-collector/suspension boundaries (denoted, Γ_cc):

where is the inward-pointing differential area vector. The present simulations use an applied current concomitant with the complete charging of the electroactive region in 3 hours (i.e., C/3 stack-level rate). Potential drops due to bulk resistance of the metallic current collector are neglected. Contact resistance at the suspension/current-collector interface is also neglected, as its value is highly material-dependent. We note that contact resistance would increase the effective impedance of the cell and is not expected to change the qualitative trends observed here. Though flow-induced contact resistance in electrochemical flow capacitors has been suggested,¹⁴ their effect in the intermittent flow mode will be minimal because all charge transfer take place when the electrode is static. All remaining surfaces in contact with the suspension are modeled as electronically insulating.

For the V-redox system, a proton-conducting membrane impenetrable to redox species is assumed in place of the separator in Fig. 2a. Zero-flux conditions for the redox species are imposed at each of these separator surfaces. At the inlet and outlet of the simulated cell, periodic boundary conditions are imposed on each solved parameter [Fig. 2b].

The governing equations for electrochemistry without flow were discretized with the finite volume method⁴⁶ with implicit discretization in time and central difference discretization in space. The fully coupled set of electrochemical equations was solved with the aggregation-based algebraic multigrid program.^47–50 Iterative convergence of all overpotentials was achieved within 10⁻⁹ V. Because the transfer of charge and species is purely advective during intermittent pumping, a semi-Lagrangian method was implemented to obtain solutions to Eqs. 10 and 11. Specifically, intercalated-lithium fraction and species concentration were determined using backward-time, nearest-neighbor interpolation along streamlines. The resulting numerical scheme conserves species, because the streamlines (along which nearest-neighbor interpolation is performed) are horizontal and parallel to the flow field's streamwise x-coordinate. The scheme lacks the artificial numerical diffusion that plagues upwind differencing schemes.⁴⁶ The lack of numerical diffusion of the present scheme enables accurate solutions even for coarse meshes in the streamwise direction along the cell's axis (i.e., the x-direction). Therefore, the computational domain was discretized with an anisotropic, rectilinear mesh having cells of length 0.500 mm and 0.010 mm in the x (streamwise) and y (transverse) directions, respectively. A time step of 8.6 s was used to march the solution forward in time, adapted as necessary to ensure convergence of the iterative solver.

When an aliquot of charged suspension is pumped out of the electroactive region of the flow cell, ideal plug flow, defined as uniform translation of charged material, is not typically observed. Instead, the shear-thinning rheology of semi-solids^{5, 31} results in bulk shear, which in turn distorts SOC and concentration fields upon advection.

To illustrate this effect, we compare ideal plug flow to ideal Newtonian flow without wall slip. Plug flow is a reasonable lower bound to the extent of flow non-uniformity because it can be induced in attracting colloidal suspensions by the formation of lubricating liquid layers at walls upon shear.⁵¹ And, shear-thinning fluids adopt some degree of plug flow even without wall slip, as we show later. However, the other extreme is pure Newtonian flow without slip, which results in greater non-uniformity (quantified as the ratio of the centerline velocity to the mean velocity) than is seen for shear-thinning fluids (e.g., Bingham plastics, power-law fluids,⁵² and Cassonian fluids).⁵³ Therefore, Newtonian flow without slip is a reasonable upper bound representing maximum non-uniformity of flow.

Consider first the plug flow of two sequential aliquots each having unity aliquot factor (m = 1). Shown in Fig. 4a is the cell voltage variation with charge time (i.e., the discharge process proceeds in decreasing time) for each of the suspensions simulated. As shown in Table III, each of the suspensions exhibits near-theoretical charge capacity, but the coulombic efficiency does increase in the order of V-redox to LiCoO₂ to LiFePO₄. This ordering corresponds to increasing SOC-range of two-phase stability [see Fig. 1d]. Coulombic efficiency loss occurs as the electroactive zone (in which electrochemical reactions take place) extends beyond the immediate electroactive region (over which the current collector extends). This phenomenon is particular to suspension-based flow batteries, where the working fluid is a mixed conductor.²³

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The SOC field [Fig. 4b] and potential field of the electron-conducting phase [Fig. 4c] show evidence of such electroactive zone extension. These fields are shown at the start (S) and end (E) of charge and discharge steps, and in all figures they are elongated in the transverse direction to aid visualization. The edge of the electroactive region has diffuse SOC bands [Fig. 4b], because continuity of the electron-conducting phase drives its potential beyond the electroactive region [Fig. 4c]. This potential induces reactions outside the cell's electroactive region. At the edges between charged and discharged suspensions, the diffuse SOC bands grow with time and are most readily seen for the V-redox system. This dissipated charge is not recovered during subsequent cycling (see discharge, S2) and accounts for the majority of coulombic inefficiency. In the case of LiFePO₄, diffuse SOC bands are not visible in Fig. 4b, but the potential front propagates well beyond the immediate electroactive region [Fig. 4c and Video S1]. Coulombic loss is suppressed for LiFePO₄, because low SOC (9%) suspension outside the electroactive region can coexist at equilibrium with suspension at dissimilar SOC in the electroactive region (9–97%). Such suppression of charge transfer occurs to a lesser extent for LiCoO₂, because it has a smaller two-phase SOC range (15–50%).

Newtonian flow without slip was simulated for the same cycle. The impact of the greater flow non-uniformity on the cell voltage [Fig. 5a] is dramatic. Charge capacity and coulombic efficiency are reduced well below the values seen for the corresponding plug-flow cycle (Table III). The cause of this behavior is illustrated by the SOC after the first intermittent pumping step (charge, S2). The lack of slip at the wall leaves residual charged material that remains behind upon subsequent pumping. As a result, the second charge step has lower capacity than the first for all suspensions.

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Figure 5a and Table III show that under Newtonian flow without slip coulombic efficiency is sensitive to the voltage-capacity relationship for the active material. Videos S2 and S3 show the time evolution of LiFePO₄ and LiCoO₂ cases, respectively. LiFePO₄ with its wider two-phase coexistence (flat voltage-capacity curve) is much more efficient (96%) than the two other suspensions (80%) which have small (LiCoO₂) or no (V-redox) equilibrium voltage plateaus. This inefficiency results from the transfer of charge outside of the electroactive region after flow. SOC snapshots at the start and end of the second charge (charge, S2 and E2) show this effect most clearly. At the start of the second charging step, SOC gradients induced by non-uniform flow are apparent, but with sufficient time, charge transfer outside of the electroactive region induces equilibration with discharged suspension transverse to the flow direction (charge, E2). The effect is most visible for LiCoO₂ and V-redox suspensions, again because their reactions occur primarily as single-phase transformations. In concert, these processes lengthen suspension aliquots and reduce the SOC inside the aliquot. This process wherein chemical diffusion is apparently enhanced by shear is referred to as dispersion.⁵⁴ On the final discharge step [discharge, E2, in Fig. 5b], the dispersive effect is most obvious. The loss of coherency of the displaced aliquot results in an abundance of incompletely charged material outside of the electroactive region of the cell. In contrast, two-phase LiFePO₄ aliquots remain largely intact during cycling, and nearly all charge is extracted from that suspension.

The previous result demonstrates that non-uniform flow leads to inefficient electrochemical cycling. Though plug flow is ideal, in practice it is not realizable for all suspensions. Thus, in many situations this non-ideal behavior may need to be managed so as to minimize inefficiencies. One strategy is to pump suspension aliquots of lesser volume (i.e., m < 1), in a pseudo-continuous mode. The effect of such m < 1 aliquot cycles is illustrated in Fig. 6, where flow is again Newtonian without wall slip, and the cycle is identical to the previous one except for pumping half-aliquots (m = 0.5). Both charge capacity and coulombic efficiency are improved relative to the m = 1 cycle (Table III). In fact, the charge capacity is now the same as for plug flow at m = 1, with the flow profiles showing that this results from preventing suspension from passing through the electroactive region uncharged. This is seen by comparing the SOC distributions at the start of the second charge step for m = 1 and m = 0.5 [cf., charge S2 in Figs. 5b and 6b]. However, the coulombic efficiency for m = 0.5 Newtonian flow without slip remains less than for plug flow at m = 1 (Table III), due to persisting charge dispersion outside of the electroactive region, manifested as residual SOC after the last discharge step [discharge E3, Fig. 6b].

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The three cases considered so far have cycled one-half of the total system volume (2 aliquots out of 4 total). As more cycles are added, we find an interesting result where the capacity is further reduced due to a different mechanism than already described, but the round-trip coulombic efficiency improves. This is seen in the last two rows of Table III, which compare m = 0.5 Newtonian flow results for pumping 2 aliquots versus 4. Suspension near the centerline that was charged on the first step protrudes into the electroactive region during later charge steps [Fig. 7b, charge S7], limiting charge capacity, even as coulombic efficiency improves. When system size is increased further to 7 aliquots and the system is cycled completely, the capacity is within 2% and the coulombic efficiency is within 0.5% of that in the 4-aliquot system. This suggests that the capacity and coulombic efficiency of large systems operated under equivalent stack-level conditions (flow profile, flow volume, and charge/discharge rate) converge to limiting values near those for the 4-aliquot system shown here.

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The preceding results illustrate that flow velocity profiles, displaced aliquot size, and active-material phase equilibria all influence charge capacity and coulombic efficiency. Extending the comparison of m = 0.5 and m = 1.0, we now test the conjecture that an optimum aliquot factor must exist, at which the total discharge energy and energetic efficiency are maximized. The electrochemical performance for aliquot factors from m = 0.125 to m = 1 are shown in Fig. 8. The calculated suspension displacement profiles are shown in Fig. 8e. A 4-aliquot system was modeled, and a sample pumping sequence of the LiFePO₄-based suspension with half aliquots (m = 0.5) can be viewed in Video S4. In addition, half of the system's suspension was cycled twice during both charge and discharge, limiting coulombic efficiency losses of the m = 0.5 case, for example, to less than 1% versus as much as 5% coulombic efficiency loss with normal cycling.

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Existence of a critical aliquot factor

A critical aliquot factor can be defined that corresponds to the geometric condition where the upstream edge of the displaced aliquot is tangent to the downstream edge (i.e., outlet) of the electroactive region. When this condition is met, the critical aliquot factor can be calculated via streamline integration for laminar flows. For steady (i.e., time-invariant) flow that is one-dimensional, fully developed, and incompressible, the critical aliquot factor is:

where and u_max are the mean and maximum axial velocities of the flow. For the no-slip Newtonian case, . For Newtonian flow without slip, Fig. 8a shows that charge capacity drops sharply above this critical aliquot factor, because discharged suspension is pumped past the electroactive region if [as illustrated in Fig. 5b]. Below this critical aliquot factor, the charge capacity [Fig. 8a] is weakly dependent on the aliquot factor.

Factors affecting polarization

Maximum in discharge energy and energetic efficiency vs. aliquot factor

To this point, we have neglected the departure of velocity profiles from the respective limits of plug flow and Newtonian flow without wall slip. Because semi-solid suspensions exhibit a finite yield stress above which shear-thinning behavior is observed,²⁹ their viscoplastic (i.e., rate-dependent, inelastic) rheology is manifested as a variety of velocity profiles under pressure-driven flow conditions. In addition, concentrated suspensions are known to slip at the walls along which they flow,⁵⁵ and this process increases flow uniformity even when the suspension undergoes bulk shear. In this section, we introduce a model for viscoplastic flow with wall slip to simulate the influence of (1) wall slip and (2) bulk shear on electrochemical performance. For each of several velocity profiles the critical aliquot factor was determined. Two dimensionless parameters are introduced that embody the coupling of flow profile to material properties (describing both rheology and slip behavior), mean flow velocity, and channel width. Comparing these velocity profiles, each operated at the critical aliquot factor, the highest efficiency is found to occur for plug flow. This is realizable in either the limit of (1) highly slippery interfaces or (2) suspension with large elastic stress relative to viscoplastic contributions.

Flow of a viscoplastic suspension with slip

The effects of slip and viscoplastic flow do not occur independently – they are fluid-mechanically coupled through rheological constitutive and momentum balance equations. Consideration of this coupling is necessary to quantify the efficiency trade-offs between the rheological and transport properties of semi-solid suspensions. Slip can be modeled by a linear velocity/shear-stress relationship u_w = βτ_w attributed to Navier,⁵⁶ where u_w and τ_w are velocity and shear stress, respectively, at the channel wall and β is the Navier slip coefficient. Various means can be employed to control the degree of wall slip, including surface roughness^{51, 57} and the volume fraction of suspended particles.⁵⁵ We model a simple viscoplastic case, a Bingham plastic, for which viscosity μ varies with shear rate as , and the flow is rigid (i.e., ) for shear stresses less than the yield stress τ₀. This rheology exhibits shear-thinning behavior (i.e., viscosity μ decreases monotonically with increasing shear-rate magnitude ), with viscosity converging to the material-dependent plastic viscosity μ_p at high shear rates [i.e., ]. The pressure-driven (i.e., Poiseuille) velocity profiles of these fluids are governed by momentum balance, and their shape is uniform where rigid, and quadratic in space where flowing (see analysis in Refs. 52, 55, 58). The critical aliquot factor for a given velocity profile depends on two dimensionless numbers: the Bingham number [], and the slip number (Sl = 2μ_Pβ/w). Bn is a characteristic scale of elastic shear stresses (given by yield stress τ₀) relative to the characteristic contribution from viscoplastic stress (given by ). Sl is a measure of the flow's slipperiness and is the ratio of the slip extrapolation length (see Ref. 59) to the channel's half-width in the high-velocity limit (Bn → 0).

Figure 9 shows the space of suspension displacement profiles (i.e., path of suspension parcels during an intermittent flow pulse) for a Bingham plastic with slip, when displaced at a critical aliquot factor corresponding to the particular velocity profile. Each displacement profile is described geometrically by the flow's yield radius R_y (half the width of the flow's rigid core) and the slip ratio s (ratio of the slip velocity u_w to the mean velocity ). For a fixed yield radius R_y the displacement profile becomes more plug-like as the slip ratio s increases (i.e., along a vertically ascending line on Fig. 9). For a fixed slip ratio s the displacement profile becomes plug-like as the yield radius R_y increases (i.e., along a horizontal line moving rightward on Fig. 9).

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The slip ratio s and yield radius R_y depend on the Bingham number Bn and slip number Sl. In other words, for each point defined by (R_y,s) on the displacement profile map (Fig. 9) there corresponds a pair (Bn,Sl). For a particular slip number Sl, the yield radius R_y and slip ratio s evolve as Bingham number Bn is varied (Fig. 9, black-dashed lines). Figure 9 shows such curves for several slip numbers (0, 10⁻², 10⁻¹, and 10⁰). Points are marked along each constant-Sl curve by triangular symbols that indicate the corresponding Bingham number Bn (see Fig. 9, legend). These curves can be thought of as "flow curves" along which volumetric flow-rate is adjusted continuously, because an increase in Bingham number Bn is equivalent to a decrease in mean flow velocity when material properties and channel width are fixed. For a given constant-Sl curve, both yield radius R_y and slip ratio s increase with increasing Bingham number Bn, i.e., flow uniformity increases with increasing Bn.

Electrochemical performance versus Bingham number and slip number

The set of possible velocity profiles for Bingham-plastic flow with slip comprise a two-dimensional space (Fig. 9). Superimposed on this map are red-dotted curves along which critical aliquot factor [defined for each point on the map by Eq. 14] is constant; the particular curves shown in Fig. 9 are for equal to 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, and 1.00. Thus, given a specific velocity profile (determined by Bingham number Bn and slip number Sl) a critical aliquot factor that maximizes discharge capacity and energetic efficiency can be determined. For all subsequent results, the aliquot factor was adjusted to its critical value based on its coupling to Bingham number Bn and slip number Sl. Specifically, the electrochemical performance of cells operated in two limits of flow is explored with (1) various degrees of slipperiness (specified by Sl) in the high-velocity limit (Bn = 0) and (2) various mean velocities (specified by Bn) in the absence of wall slip (Sl = 0). A 7-aliquot flow cell was simulated, cycling the total system's suspension. The simulated pumping sequence can be viewed in Video S5 for the LiFePO₄-based suspension with Sl = 0 and Bn = 0.

Figure 10 shows that as the slip number Sl increases for Bn = 0, all performance metrics are systematically improved. We see that with strong slip (Sl → ∞, ideal plug flow), the discharge energy [Fig. 10c] exceeds 95% of the theoretical value for all three chemistries, whereas without slip [Fig. 10c, Sl = 0], values are 81–87% depending on chemistry. A reduced difference is seen in the energetic efficiency, which ranges 96–97% among the three chemistries as Sl → ∞ [Fig. 10d], compared to 91–95% without slip [Fig. 10d, Sl = 0]. Thus, increased slip number reduces energetic efficiency losses by, at most, half relative to the Newtonian case without slip (Sl = 0). Though the limit of infinite slip number is unachievable in practice, our results demonstrate that for Sl > 2 energetic efficiency can be realized within 1% of that for a perfectly slipping suspension. Expressed in terms of material properties and channel width, this condition is β > w/μ_P. This result shows that the slip coefficient to achieve a sufficient level of energetic efficiency depends on the channel's size and rheology.

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In like manner, Fig. 11 shows that as Bingham number Bn increases for Sl = 0, all performance metrics are systematically improved. In the slow-flow limit (Bn → ∞) plug flow is realized with high discharge energy and energetic efficiency. Infinitesimally slow flow is impractical, because the time-scale of flow pulses should be short relative to the charge/discharge time in order to be truly "intermittent," as is the objective in this work. Our results demonstrate that for Bn > 50 energetic efficiency can be realized within 1% of that for plug flow. Expressed in terms of material properties, mean velocity, and channel width, this condition is . This result shows that the yield stress to achieve a sufficient level of energetic efficiency depends on the suspension's plastic viscosity, mean velocity, and channel size. In practice, a certain amount of yield stress is optimal, because mechanical energy dissipation increases with increasing yield stress.

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