Patent Application: US-201414514954-A

Abstract:
flow battery . the battery includes high energy density fluid electrodes having a selected non - newtonian rheology and structure for providing intermittent flow pulses of controlled volume and duration of the fluid electrodes , the structure adapted to promote interfacial slip to improve flow uniformity . the battery disclosed herein provides a potential solution to large - scale electrical energy storage needs .

Description:
in this patent application , these loss mechanisms are simulated , and provide four strategies by which the energetic efficiency of suspension - based flow batteries can be maximized ( fig1 ). the components of this strategy are : ( 1 ) the control flow volume per pumping stroke , ( 2 ) promotion of slip at interfaces , ( 3 ) tailoring of suspension rheology , and ( 4 ) selection of active - material thermodynamics . a computational model of flowing electrochemistry is developed , generalized from previous work in which the model was tested against experimental results for one aqueous semi - solid electrochemical couple . 9 we show that , while non - uniform flow velocities diminish capacity and efficiency , losses can be minimized through practical measures . incorporation of all such optimizations suggest flow battery operation modes with up to 98 % energetic efficiency exceeding that of conventional flow batteries , 16 using readily achievable experimental parameters . a schematic of the simulated flowing half - cell appears in fig2 ( a ). the electroactive region of the half - cell is the region between current collector ( orange ) and separator , 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 ( blue ) 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 [ fig2 ( b )] 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 . two of these active materials are solid - state li - ion intercalation compounds lifepo 4 and licoo 2 , and the other is a redox solution ( vo 2 + / vo 2 + ). each equilibrium potential versus state - of - charge ( soc ) curve is illustrated in fig1 ( d ). 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 , lifepo 4 exhibits two - phase equilibrium between 9 % and 97 % soc [ fig1 ( d )] which represents the majority of a practical electrochemical cycle : 24 lifepo 4 → li x fepo 4 +( 1 − x ) li + +( 1 − x ) e − . a consequence of this two - phase equilibrium is that spatial soc gradients can exist at thermodynamic equilibrium because lifepo 4 particulates can have a particular phase fraction with any average soc between 9 % and 97 % and maintain at the same equilibrium potential however , for licoo 2 , 25 , 26 licoo 2 → li 1 - y coo 2 + y li + + ye − 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 , licoo 2 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 lipf 6 in mixed carbonate solvent . 27 we also analyze the cathodic couple of the vanadium - redox aqueous solution ( vo 2 + / vo 2 + ). the following simplified reaction is modeled in which vo 2 + is oxidized to vo 2 + during charging : 25 vo 2 + + h 2 o → vo 2 + 2h + + e − . 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 h 2 so 4 ( 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 . fig3 depicts the evolution of soc 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 . an alphanumeric rubric is used to distinguish the cycle steps for all of the subsequent results , as shown in fig3 . 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 reaches ( 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 &# 39 ; 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 &# 39 ; 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 below , 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 &# 39 ; s bulk . the electrochemical operation of the cell is specified by the upper and lower voltage cutoff limits applicable to the electrochemical couple ( table 1 ) 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 stimulated 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 . the computational model includes simultaneous electrochemical processes ( electronic conduction , reaction kinetics , and redox - species diffusion ) and fluid flow . the symmetry of the simulated cell allows it to be modeled as two - dimensional [ fig2 ]. solution - phase polarization resulting from ion conduction in the electrolyte is neglected . for suspensions containing representative electrolytes this assumption is valid , because ( 1 ) electronic conductivity (˜ 1 ms / cm ) dominates polarization , being at least ten - fold less than ionic conductivity and ( 2 ) the time - scale for salt depletion through the electrode thickness exceeds the cycling time by more than ten - fold ( see conductivities and diffusivities in refs . 27 , 30 ). fluid pulses in the intermittent operation mode are assumed to occur infinitely fast , enabling the isolation of electrochemical cycles into two distinct states governed by either ( 1 ) electrochemistry without flow or ( 2 ) flow without electrochemistry . these processes are described in detail below , as well as the boundary conditions imposed and the numerical discretization techniques implemented . in the absence of flow , electronic conduction occurs via the conductive - particle percolating - network suspended in the electrolyte , and the solid - phase potential φ x is governed by charge conservation : where φ 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 cell 23 ( 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 - aqueous 31 and aqueous 9 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 times 33 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 ., lifepo 4 and licoo 2 ) 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 , v s is the volume fraction of active material , and f is faraday &# 39 ; s constant . the values of c s , max for lifepo 4 and licoo 2 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 1 . 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 ., h 2 so 4 ( ref . 28 ) or chloride solutions 29 ) minimizes the electric field that drives migration . therefore , mass conservation of vo 2 + and vo 2 + in the absence of migration sufficiently describes the electrochemical processes that occur in the v - redox system : where c j and d effj are the concentration and effective diffusivity of redox species j in the electrolyte . the soc of the electrode is equivalent to the concentration of vo 2 + . 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 &# 39 ; s porosity ε is approximately 100 %. the diffusivities for both v - redox species are assigned bulk values of 3 . 9 × 10 − 01 m 2 / 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 fig1 ( d ) with respect to soc . the ionic potential φ 0 is assumed uniform at the value of the chosen counter - electrode ( table 1 ). 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 i 0 is the exchange current density and rt has its usual meaning . in general , the exchange current density i 0 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 i 0 depends on the type of reaction . table 2 summarizes the kinetic parameters and volumetric surface area , a ( m 2 / m 3 ), 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 i 0 = fkc s , max ( c e ) 0 . 5 ( 1 − x li ) 0 . 5 ( x li ) 0 . 5 , ( 6 ) 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 lifepo 4 43 and 4 μm diameter licoo 2 37 particles . the exchange current density i 0 for the cathodic v - redox reaction depends on the concentration of redox species j at the reaction surface , c j s : 28 i 0 = fk ( c vo 2 + s ) 0 . 5 ( c vo 2 + s ) 0 . 5 . ( 7 ) pore - scale mass - transfer resistance causes surface concentrations , c j s , 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 m 2 / 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 : where { right arrow over ( v )} 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 fig2 ( b )]: where î is the unit vector along the channel &# 39 ; s axis ( taken as the x - coordinate ). results for a variety of velocity profiles are presented below . to simulate galvanostatic charge / discharge conditions a time - invariant total current i applied is imposed at current - collector / suspension boundaries ( denoted , γ cc ): where d { right arrow over ( γ )} 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 takes 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 fig2 ( a ). 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 [ fig2 ( b )]. the governing equations for electrochemistry without flow were discretized with the finite volume method 46 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 &# 39 ; 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 &# 39 ; 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 . first , the temporal variation of voltage and soc during the cycling of two aliquots of suspension is shown for three flow scenarios to elucidate efficiency - loss mechanisms : ( 1 ) plug flow of a unit aliquot ( m = 1 ), ( 2 ) newtonian flow of a unit aliquot ( m = 1 ) in the absence of slip , and ( 3 ) newtonian flow of a half - aliquot ( m = 0 . 5 ) in the absence of slip . effects of increasing total flow - volume are also simulated . subsequently , optimization with respect to aliquot factor is addressed for fixed flow profiles . finally , performance for flows having various degrees of slip and bulk shear is assessed for optimized aliquot factors . charge capacity (%)— the ratio of stored charge to the theoretical maximum , coulombic efficiency (%)— the ratio of discharge capacity to charge capacity , average polarization ( mv )— half the difference between the time - averaged cell voltage during charge and discharge respectively , discharge energy (%)— the ratio of delivered energy to the theoretical maximum , and energetic efficiency (%)— the ratio of discharge energy to charge energy . 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 . 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 fig4 ( a ) 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 3 , each of the suspensions exhibits near - theoretical charge capacity , but the coulombic efficiency does increase in the order of v - redox to licoo 2 to lifepo 4 . this ordering corresponds to increasing soc - range of two - phase stability [ see fig1 ( d )]. 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 the soc field [ fig4 ( b )] and potential field of the electron - conducting phase [ fig4 ( c )] 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 [ fig4 ( b )], because continuity of the electron - conducting phase drives its potential beyond the electroactive region [ fig4 ( c )]. this potential induces reactions outside the cell &# 39 ; 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 lifepo 4 , diffuse soc bands are not visible in fig4 ( b ), but the potential front propagates well beyond the immediate electroactive region [ fig4 ( c ) and video s1 ]. coulombic loss is suppressed for lifepo 4 , 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 licoo 2 , 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 [ fig5 ( a )] is dramatic . charge capacity and coulombic efficiency are reduced well below the values seen for the corresponding plug - flow cycle ( table 3 ). 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 . fig5 ( a ) and table 3 show that under newtonian flow without slip coulombic efficiency is sensitive to the voltage - capacity relationship for the active material . lifepo 4 with its wider two - phase coexistence ( flat voltage - capacity curve ) is much more efficient ( 96 %) than the two other suspensions ( 80 %) which have small ( licoo 2 ) 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 licoo 2 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 fig5 ( b )], 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 lifepo 4 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 & lt ; 1 ), in a pseudo - continuous mode . the effect of such m & lt ; 1 aliquot cycles is illustrated in fig6 , 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 3 ). 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 fig5 ( b ) and 6 ( b )]. however , the coulombic efficiency for m = 0 . 5 newtonian flow without slip remains less than for plug flow at m = 1 ( table 3 ), due to persisting charge dispersion outside of the electroactive region , manifested as residual soc after the last discharge step [ discharge e3 , fig6 ( b )]. 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 coulumbic efficiency improves . this is seen in the last two rows of table 3 , 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 [ fig7 ( b ), charge s7 ], limiting charge capacity , even as coulumbic 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 coulumbic 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 . 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 fig8 . the calculated suspension displacement profiles are shown in fig8 ( e ). a 4 - aliquot system was modeled . in addition , half of the system &# 39 ; 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 . a critical aliquot factor { tilde over ( m )} 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 ū and u max are the mean and maximum axial velocities of the flow . for the no - slip newtonian case , { tilde over ( m )}= 2 / 3 . for newtonian flow without slip , fig8 ( a ) shows that charge capacity drops sharply above this critical aliquot factor , because discharged suspension is pumped past the electroactive region if m & gt ;{ tilde over ( m )} [ as illustrated in fig5 ( b )]. below this critical aliquot factor , the charge capacity [ fig8 ( a )] is weakly dependent on the aliquot factor . fig8 ( b ) shows that the cell polarization decreases monotonically with increasing aliquot factor . this scaling is primarily due to the constriction of current when the electroactive region is not completely replenished . residue left behind from prior cycle steps results in a heterogeneous distribution of soc within the electroactive region . consequently , current becomes localized on region of the current collector nearest fresh suspension . due to this localization of current , heightened ohmic drop occurs across the section of fresh suspension , manifesting as polarization at the cell level . this interpretation is supported by the good agreement between the calculated polarization and that predicted by a simplified model of current localization [ red - dotted line , fig8 ( b )]. in this simplified model current is distributed uniformly over a fraction of the current collector &# 39 ; s length , ml cc , with a time - averaged ohmic drop of w 0 /( 2mσ s ), where 0 is the uniform current density associated with a unit aliquot . for the continuous - flow limit ( extrapolated to m → 0 ), because the mean cell voltages on charge and discharge approach their respective cut - off voltages , the average polarization scales roughly with the magnitude of the voltage cut - off window [ fig8 ( b )]. the extrapolated polarization for continuous flow is a lower bound , because flow - induced impedance 14 , 32 will further increase polarization in the continuous - flow limit . the average polarization computed here for small aliquots agrees well with those predicted for licoo 2 and lifepo 4 under continuous flow at lower c - rates in ref . 23 . for larger aliquot factors , additional effects influence the average polarization , including the specific kinetic and thermodynamic properties of the active material . the reasons why the discharge energy [ fig8 ( c )] and the energetic efficiency [ fig8 ( d )] should have a maximum near a critical aliquot factor can be explained . discharge energy is a compromise between reduced charge capacity for larger aliquot factors ( m & gt ;{ tilde over ( m )}) and increased polarization at smaller aliquot factors ( m & lt ;{ tilde over ( m )}). the former process naturally reduces the available discharge capacity for m & gt ;{ tilde over ( m )}, while the latter process for m & lt ;{ tilde over ( m )} reduces the mean voltage at which discharge capacity is delivered to the external circuit . in contrast , energetic efficiency has a maximum because the coulombic efficiency is decreased for larger m ( due to the transverse dispersion of protruded charged suspension for m & gt ;{ tilde over ( m )}) and polarization is increased for m & lt ;{ tilde over ( m )}. these results also show that the intermittent flow mode can reach higher energetic efficiency and discharge energy than the ( conventional ) continuous flow mode . the trends in fig8 show why intermittent flow is preferred . note that even though the smallest aliquot factor simulated explicitly ( m = 0 . 125 , pseudo - continuous ) produces highly uniform soc distributions within the electroactive region , the detailed analysis we have presented here shows that is has several percent lower energetic efficiency than does operation at the critical aliquot factor [ fig8 ( d )]. in the continuous - flow limit ( extrapolated to m → 0 ) energetic efficiency losses for all chemistries are double those achieved by operating at critical aliquot factors (˜ 10 % versus ˜ 5 %, respectively ). if flow - induced impedance arises under continuous flow ( as reported in refs . 14 , 32 ), efficiency losses under continuous flow will be even larger . 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 know to slip at the walls along which they flow , 55 and this process increases flow uniformity even when the suspension undergoes bulk shear . 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 . 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 fluid , for which viscosity μ varies with shear rate { dot over ( γ )} as μ = μ p + τ 0 /∥{ dot over ( γ )}∥, and the flow is rigid ( i . e ., ∥{ dot over ( γ )}∥= 0 ) for shear stresses less than the yield stress τ 0 . this rheology exhibits shear - thinning behavior ( i . e ., viscosity μ decreases monotonically with increasing shear - rate magnitude ∥{ dot over ( γ )}∥), with viscosity converging to the material - dependent plastic viscosity μ p at high shear rates ( i . e ., μ (∥{ dot over ( γ )}∥→∞)= μ p ). 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 [ bn = τ 0 w /( 2μ p ū )], and the slip number ( sl = 2μ p β / w ). bn is a characteristic scale of elastic shear stresses ( given by yield stress τ 0 ) relative to the characteristic contribution from viscoplastic stress ( given by 2μ p ū / w ). sl is a measure of the flow &# 39 ; s slipperiness and is the ratio of the slip extrapolation length ( see ref . 59 ) to the channel &# 39 ; s half - width in the high - velocity limit ( bn → 0 ). fig9 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 &# 39 ; s yield radius r y ( half the width of the flow &# 39 ; s rigid core ) and the slip ratio s ( ratio of the slip velocity u w to the mean velocity ū ). 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 fig9 ). 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 fig9 ). 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 ( fig9 ) 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 ( fig9 , black - dashed lines ). fig9 shows such curves for several slip numbers ( 0 , 10 − 2 , 10 − 1 , and 10 0 ). points are marked along each constant - sl curve by triangular symbols that indicate the corresponding bingham number bn ( see fig9 , 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 ū 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 . the set of possible velocity profiles for bingham - plastic flow with slip comprise a two - dimensional space ( fig9 ). superimposed on this map are red - dotted curves along which critical aliquot factor { tilde over ( m )} [ defined for each point on the map by eq . ( 14 )] is constant ; the particular curves shown in fig9 are for { tilde over ( m )} 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 cycling the total system &# 39 ; s suspension was simulated . the simulated pumping sequence can be viewed in video s5 for the lifepo 4 - based suspension with sl = 0 and bn = 0 . fig1 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 [ fig1 ( c )] exceeds 95 % of the theoretical value for all three chemistries , whereas without slip [ fig1 ( c ), 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 →∞[ fig1 ( d )], compared to 91 - 95 % without slip [ fig1 ( d ), 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 & gt ; 2 energetic efficiency can be realized within 1 % of that for a perfecting slipping suspension . expressed in terms of material properties and channel width , this condition is β & gt ; w / μ p . this result shows that the slip coefficient to achieve a sufficient level of energetic efficiency depends on the channel &# 39 ; s size and rheology . in like manner , fig1 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 & gt ; 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 τ 0 & gt ; 100μ p ū / w . this result shows that the yield stress to achieve a sufficient level of energetic efficiency depends on the suspension &# 39 ; 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 . maximizing efficiency is essential to the practical utilization of energy - dense flow batteries for large - scale energy storage . a model of electrochemical kinetics and flow was developed to identify operating conditions and rheological behavior that maximize electrochemical performance . the results suggest that electrochemical efficiency can be maximized through ( 1 ) flow volume control , ( 2 ) tailoring of suspension rheology , ( 3 ) promotion of interfacial slip , and ( 4 ) selection of active - material thermodynamics . precisely tuned flow volumes , large yield stresses , large navier slip coefficients , and two - phase - like active - materials produce the greatest electrochemical efficiencies . these considerations provide a critical aliquot size for intermittent flow mode operation . three active - material systems were modeled ( lifepo 4 , licoo 2 , and v - redox ). in the worst case ( unit aliquots of newtonian flow in the absence of slip ), coulombic and energetic efficiencies can be as low as 80 %. however , by flowing critically - sized aliquots in a plug - like manner , discharge energy as a percentage of the theoretical value , and energetic efficiency , can both exceed 95 %. understanding the present results in the wider context of design and operational constraints of suspension - based flow batteries is essential to their useful integration in scaled devices : the tradeoff between losses due to electrochemical processes and due to mechanical processes must be accounted in the practical design of flow cells and materials - engineering of suspensions . the incorporation of slippery surfaces is expected to have auxiliary benefits for flow cell operation , including ( 1 ) reduction of flow resistance that will reduce mechanical energy losses and ( 2 ) minimization of microstructural rearrangement 31 that can have deleterious effects in suspension - based flow cells . though high yield stress may be beneficial to electrochemical performance by inducing plug flow ( which maintains microstructure ), such a strategy will increase the mechanical energy required to pump suspensions . slip promotion strategies ( e . g ., with surface roughness 51 , 57 ) are beneficial but electrical continuity between the suspension and current collector must be considered . we have shown that an intermittent flow mode maximizes electrochemical efficiency relative to the continuous flow mode employed in conventional flow batteries based on redox solutions . this flow mode has been demonstrated at the lab - scale , 5 , 9 but specialized pumps , flow - 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