Current Density Scaling in Electrochemical Flow Capacitors

The principal quantity that is desired to be predicted by modeling is the steady-state current, I, that can be achieved by a particular EFC for a given cell voltage, flow rate, etc. Three different scales (i.e. characteristic current values associated with EFC performance) exist to nondimensionalize this current; depending on the context, one particular scale may be more useful than the others.

The analogous appearance of Equation 3 to advection-diffusion heat transfer leads to the first such scale, I_diff (i.e. the diffusion-controlled current). Analogously to the conduction heat transfer rate into a channel driven by a temperature difference, I_diff is a current that is driven by the total potential difference across the half-cell (ΔV₀) divided by the diffusion-controlled resistance, R_diff.

Here η_i is the uniform overpotential at the inlet; it is included in order to take into account the attenuation to the achievable current that results from partially charged particles being reintroduced into the cell when a two tank approach is used. R_diff is equal to a length-over-area ratio multiplied by a resistivity value. Equation 2 shows the appropriate resistivity to be equal to the quantity (1/σ + 1/κ). R_diff is hence,

I_diff is a measure of the current that results from the applied cell voltage as mediated through restrictions imposed by growth of the overpotential boundary layers. For a velocity profile that is fully developed by the time the flow reaches the leading edge of the electrode/membrane, dimensional analysis of the governing equations requires the quantity I/I_diff to be a function of at most three nondimensional variables: the capacitive Péclet number (Pe_c = Uδ/α_e), the aspect ratio of the cell (AR = L/δ), and a ratio of the ionic and electronic conductivities (γ). The ratio of the conductivities could take various forms; κ/σ is the most obvious one. However, a more advantageous way to express this ratio is as γ = (κ + σ)(1/σ + 1/κ) – the reason for the selection of this particular expression will be discussed shortly. In general,

In analogy to the Graetz problem in heat transfer though,⁷ scaling in the longitudinal direction allows for Pe_c and AR to be combined into a single composite parameter, the capacitive Graetz number

Gz_c is a newly defined nondimensional number similar to the Graetz number in heat transfer. The capacitive Graetz number characterizes the growth of the laminar overpotential boundary layers within the channel. Small capacitive Graetz numbers imply that the overpotential boundary layers will extend across the entire channel gap upon reaching the end of the current collector (hence fully utilizing the slurry). I/I_diff is therefore only a function of two independent nondimensional parameters. The functional relationship appears as follows:

The second current scale arises when considering the maximum charge that can be accepted by the double layers of the slurry particles as they flow through the cell. At the fully utilized limit (where the double layers at the exit of the cell are equal to the half-cell voltage), this current takes the form

Here is the total volumetric flow rate of the slurry. A similar term has been called a "hydraulic current".⁴ After rearrangement, this capacitive current scale can be related to the diffusional current scale by the following relationship:

The capacitive current scale I_cap is particularly useful when considering EFCs operating in a fully-utilized, single-pass mode (as depicted in Figure 1). An EFC operating as such must maintain a sufficiently high I/I_cap ratio (as near to one as possible) in order to perform as intended, and must therefore be operated at a small capacitive Graetz number.

I_diff and I_cap both have direct analogies in heat transfer and arise from consideration of overpotential advection-diffusion behavior. The third current scale, I_ohmic, however, does not have a direct analogue. Unlike in heat transfer where increasing flow rates inexorably lead to higher heat transfer rates, I_ohmic is an upper limit of the achievable current that is imposed by IR-losses alone (regardless of increases in flow). This limit comes into play at high capacitive Graetz numbers where the overpotential boundary layers become very thin and the utilization of the slurry is low. Under these circumstances, the parallel ionic and electronically conductive paths are both fully accessible and are not impeded by the presence of overpotential boundary layers. Similarly to I_diff, this current can be expressed as a total potential difference across the cell divided by a resistance:

The resistance in the Ohmically-controlled limit, however, takes the form

This form is equivalent to the high-frequency limit of electrochemical impedance spectroscopy (R_hf).²⁰ I_ohmic can be expressed in terms of I_diff as

The simplicity of this expression is the reason for selecting the specific form of the conductivity ratio γ that was adopted earlier. γ has a minimum value of 4 when κ/σ = 1, which shows that the Ohmic current scale is always larger than the diffusional scale. One additional advantage of this definition of the conductivity ratio is that γ is unchanged if the numerical values of the ionic and electronic conductivities are swapped. Swapping the respective conductivities changes whether the boundary layer principally grows outward from the current collector or from the membrane,⁶ but has no effect on the total current that is achieved; thus, having identical γ values for these two respective scenarios eliminates the need to run simulations for cases where, for example, σ > κ (as an equivalent case with κ > σ can be found).

It should be noted that if, in analogy to the convective heat transfer coefficient, an overall "capacitive current transfer coefficient" based on the inlet value of (ΔV₀-η_i) were introduced such that:

(where A_s is the geometric surface area of the current collector) then the quantity I/I_diff can be seen to be equivalent to a capacitive Nusselt number, , defined as

Equation 17 is thus an analogous form of Newton's law of cooling as applied to capacitive slurry electrodes. A similar development based off of the log-mean difference between the inlet and outlet values of ΔV₀-η_m (where η_m is the bulk value of the overpotential) is also possible.

Through the use of this scaling approach, an approximate solution for the variation of the steady-state current, I, with respect to changes in flow rate, cell geometry, and slurry properties can be found. This approximate solution is based off of the combination of individual solutions for three distinct operational regimes of an EFC.

The first regime is that of the fully-utilized slurry where the overpotential distribution across the entire channel at the exit of the cell is nearly equal to the voltage applied across the half-cell. As mentioned, this occurs at low capacitive Graetz numbers where the overpotential boundary layers from the current collector and membrane have sufficient space to grow and join together. In this regime, asymptotically . The nondimensional currents can thus be expressed in the following forms, where the subscript '1' indicates that this relationship is for the first regime (i.e. the fully utilized regime)

The second regime is for moderate Graetz numbers where the overpotential boundary layers are able to freely grow but do not have sufficient time/space within the cell in order to extend across the entire channel gap. Due to the direct analogy to heat transfer implied by Equation 3, the scaling relationship governing the total EFC current is identical to the scaling relationship controlling the heat transfer rate in the Graetz problem.⁷ The scaling of the Lévêque solution to the Graetz problem²¹ therefore applies at sufficiently large Gz_c. In general though, the boundary conditions on the overpotential differ from those of the canonical Graetz problem (i.e. Dirichlet or Neumann specifications) as the boundary conditions for the EFC model are specified in terms of Φ₁ and Φ₂, and not in terms of η directly. However, when IR losses become negligible (i.e. I/I_ohmic is less than roughly 0.1), the sum of the overpotentials at the current collector and membrane nearly equals ΔV₀ (i.e. a uniform value). Under these circumstances then, the EFC model acts identically to the standard uniform-temperature Graetz problem in a channel. As I/I_diff is directly analogous to the average Nusselt number from heat transfer, the standard Lévêque result for for thermally developing flow in a channel can be used,²² resulting in the expression

Here c₀ is equal to (3/2)^2/3/Γ(4/3) = 1.467. Extensions to this Lévêque result are possible (i.e. the inclusion of higher-order terms of Gz_c),²³ but this has not been pursued. This specific value of c₀ is valid for the fully-developed parabolic velocity profile of the Newtonian fluid; an alternative rheological model (such as the Ostwald–de Waele relationship) can be expected to alter the value of c₀ somewhat (as has been shown for heat transfer²⁴), but an examination of this effect has been left to a future paper.

Through their definitions, I/I_cap and I/I_ohm can be similarly expressed in this regime as

The third and final regime is the Ohmically limited regime. In this regime IR-losses dominate and, as discussed, the current becomes invariant of flow. Asymptotically, . Therefore

Each of these regimes therefore possesses a unique limitation that controls the magnitude of the current. For the fully utilized regime (regime 1), it is the inability to charge the slurry beyond the applied potential. At intermediate values of Gz_c (in regime 2), it is the rate of development of the overpotential boundary layers. At high Gz_c (in regime 3 — the Ohmically limited regime) it is the prevailing magnitude of the IR-losses. The three separate solutions for these respective operational regimes can be combined into a single composite curve to predict the steady-state current if it is assumed that smallest of these three unique limitations is the sole controlling factor. Hence, the quantity I/I_diff (as a function of the capacitive Graetz number and γ) can be expressed as by an approximate scaling solution as

where, again, the numerals 1, 2, and 3 refer to the expressions for the fully utilized, diffusivity limited, and Ohmically limited regimes respectively. Similar constructions can be performed for I/I_ohmic and I/I_cap. A qualitative plot of I/I_diff (equivalently, γ(I/I_ohmic)) resulting from this assumption is shown in Figure 3. Plotted versus log scales, the results for each regime are represented as a line.

Zoom In Zoom Out Reset image size

The plot of I/I_cap is shown in Figure 4. This plot is useful for consideration of single-pass EFCs where the slurry is desired to be as fully utilized as possible.

Zoom In Zoom Out Reset image size

In both of these plots, the transition from the fully utilized regime to the diffusivity limited regime can be calculated to occur at Gz_c = c₀^3/2. The transition from the diffusivity limited regime to the Ohmically limited regime can be calculated to occur at (γ/c₀).³ In reality the transition between regimes does not occur abruptly at distinct capacitive Graetz numbers, but is spread across intermediate transitional zones. Comparison with numerical results from computational fluids dynamics simulations will be presented in a subsequent section to delimit this more fully. In general, the family of curves that result from this composite construction can serve as good upper estimates of the expected performance from a given EFC.

In order to determine the validity of the above scaling relationships, computational simulations of the governing equations were run for a large number of combinations of flow rate, ionic conductivity, electronic conductivity, and specific capacitance of the slurry. The simulations were done in terms of dimensional quantities, and the results were subsequently nondimensionalized. As mentioned, in order to simplify the numerical/scaling comparison, in this section the slurry properties are considered to be uniform constants.

A plot of the numerically determined curves for I/I_diff versus the capacitive Graetz number for various values of γ are shown in Figure 5. As suggested by the scaling approach, the results were found to be dependent only on the capacitive Graetz number and the conductivity ratio γ. The composite scaling solution for each curve is also shown in the figure. The agreement between the scaling and numerical results is very good.

Zoom In Zoom Out Reset image size

In the fully utilized regime (away from the transitional zone), all of the numerically predicted curves match the scaling result in Equation 19 extremely well. For the diffusivity limited regime, the γ = 200 curve follows the scaling results of Equation 21 for a considerable range of Gz_c, while the γ = 20 briefly does so. The γ = 4 case (where κ/σ = 1), however, does not. The reason this occurs is because the low Ohmic limit (relative to I_diff) in this case never allows the overpotential boundary layers to grow freely; thus, the two intermediate transitional zones that exist between the three respective regimes overlap, and the boundary layer growth behavior and scaling is never able to be expressed. As γ is increased (i.e. as the ratio of the Ohmic to the diffusional current scale is made larger) to values beyond those shown in the figure, the numerical solutions follow the c₀Gz_c^1/3 scaling result for greater and greater extents of Gz_c. Once Gz_c is further increased all of the curves do begin to asymptotically approach the Ohmic limit from Equation 24 (although the γ = 200 case must be pushed to larger Gz_c than those that were simulated in order to more fully achieve its limiting value).

A plot of the numerically determined curves for I/I_cap versus the Graetz number for various values of γ are shown in Figure 6. This is the same data as in Figure 5 except related through Equation 13. The composite scaling solutions for each curve are again also shown in the figure. Not surprisingly, the agreement between the scaling and numerical results is again quite good. As mentioned, plotting the steady-state current scaled to I_cap is useful when seeking to determine performance in the context of the four tank, single-pass EFC approach. For example, from these plots it can be ascertained that in order to utilize the slurry to a sufficient degree, the capacitive Graetz number must be maintained to be sufficiently low. For a desired 95% utilization requirement (i.e. if it is wished that the bulk overpotential of the fluid leaving the cell is at least 95% of the total half-cell potential; equivalently, I/I_cap = 0.95), the numerical plots show that Gz_c must be less than roughly 0.8 regardless of the conductivity ratio. For given slurry characteristics and a specified cell aspect ratio the only way to achieve a Graetz number this low is through the use of a low flow rate (which, in turn, could imply a very small steady state current). The scaling solution predicts a somewhat higher criterion for 95% utilization (Gz_c = 1.91) because it does not account for the transitional zone between the two regimes that leads to earlier attenuation of the performance.

Zoom In Zoom Out Reset image size

In general, the results in Figure 5 and 6 suggest that the scaling relationships can be used as reasonable upper estimates of the numerically predicted steady-state current from an EFC; a family of numerical curves for a variety of values of γ can be used when better predictions are required. In either case, for certain slurries possessing micropores the results must be applied cautiously when the residence time within the cell is small as sufficient time may not be available for ions to diffuse into the interior of the particles (thus making the assumption of a constant specific capacitance regardless of flow rate questionable).

Contour plots of the overpotential as calculated by the numerical model are shown in Figure 7 in order to demonstrate the characteristics of the three regimes qualitatively. For the fully utilized regime in Figure 7a, the overpotential of the slurry has uniformly reached the half-cell potential across the entirety of the channel as it proceeds downstream. At higher capacitive Graetz numbers, growth of the boundary layer of overpotential becomes far more visible, as in Figure 7b. As the Graetz number is made larger still, eventually the boundary layer becomes very thin as in Figure 7c, and the current becomes Ohmically controlled.

Zoom In Zoom Out Reset image size

As the scaling relationships have been shown to be consistent with the simplified numerical results (where the slurry properties were considered to be spatially uniform), experimental data has been taken in order to examine the applicability of these relationships to real-world EFCs. The experimental data consisted of the steady state currents that resulted from operation of the Case 1 and Case 2 single-tank EFCs at specified flow rates. Typical chronoamperometric data for Case 2 is shown in Figure 8 for a single flow rate (100 ml/min). As can be seen, an initial spike in the current upon the application of the cell voltage at t = 0 s quickly decays away until the current assumes a steady-state trend; the sinusoidal appearance of the data even during steady-state occurs due to the pulsatility of the flow as provided by the peristaltic pump.

Zoom In Zoom Out Reset image size

A plot of the steady-state current that resulted as such for Cases 1 and 2 across a range of flow rates is shown in Figure 9. This data is plotted in its raw, dimensional form. The somewhat limited range of flow rates selected (25 ml/min to 250 ml/min) are those that were permissible by the use of the peristaltic pump. It should be noted that Case 2 has current densities much higher than those reported in literature for EFCs (e.g. 100 times the 2 mA/cm² at 13 ml/min reported in Porada, et al.⁵). This is because a far larger amount of conductive additive (in this case an additive possessing a high aspect ratio) was included, thereby ensuring a sufficiently large electronic conductivity, and, hence, a large electronic diffusivity. Admittedly, the specific capacitance of the Case 2 slurry is lower than that of the slurry used by Porada, et al., but as it is a flow capacitor with arbitrarily large tank size, using a slurry composition that favors high current densities in such a manner may be permissible.

Zoom In Zoom Out Reset image size

These results can be compared to the scaling predictions through use of nondimensionalization of the experimental data. This limited flow rate range from the peristaltic pump leads to an equally narrow range of the resulting Gz_c for any particular case. As it was desired to show the experimental trend across as great a range of Gz_c as possible, slurries of considerably different properties were selected for Cases 1 and 2. Case 1 is comprised entirely of the Nano27 high aspect-ratio conductive additive (and is hence, highly conductive but does not possess a large specific capacitance) while Case 2 is comprised of both Nano27 and the highly porous YP-50 activated carbon (and is hence modestly conductive with a far larger specific capacitance).

As is the case in the Graetz problem in heat transfer, specification of the appropriate fluid/slurry properties to be used in the nondimensionalization is challenging. In heat transfer, the fluid properties can change across the channel gap due to the dependence of viscosity, etc. on the local temperature. This variation leads to concepts such as that of the film temperature.⁷ For the nondimensionalization of the EFC results, similar complications arise.

The principal issue is that the applicable flowing conductivity value is difficult to specify. The ionic conductivity can be assumed to be the literature value of 1 M H₂SO₄ and can also assumed to be unattenuated by any Bruggeman-type effect as the volume fraction of particles in each case is less than 10%. The specific capacitance can also be treated straightforwardly assuming that flow rates are sufficiently low to prevent residence time effects from occurring — in those cases, the steady value of the specific capacitance can be used for each respective slurry.

The flowing electronic conductivity number though is more problematic. Generally the average flowing electronic conductivity value of a slurry as measured across the entire cell is a factor of two or so lower than its static value.²⁵ However, this integrated average value is not necessarily suitable to describe the growth of the overpotential boundary layers as, for the flow rates used, the boundary layers are small and confined to a small region near the current collector. The growth is thus more accurately described by the local conductivity adjacent to the current collector. This near-wall region is where the largest shear rates are present, suggesting that the local value of the electronic conductivity would be lower there than the factor of two reduction alone. As uncertainty in the determination of the numerical value of this local conductivity exists, for the nondimensionalization it was assumed that the electronic conductivity for both experimental cases could be described by the static electronic conductivity value divided by an unknown factor (to be determined from a fit to the data). The single factor that was found to apply to both cases was 5.5.

Figure 9 includes the results from this scaling approach in dimensional form, while Figure 10 shows the experimental data that has been nondimensionalized accordingly along with the scaling solution. From Figure 10 it can be discerned that both of the sets of experimental results were in the range of Gz_c such that the Lévêque scaling was applicable. As can be seen, the 1/3-power Gz_c dependence was matched well by both sets of data. For Case 1, the exponent was determined from a least-squares fit to be 0.32 ± 0.01. For Case 2, the exponent was found to be 0.28±0.01. The slightly smaller exponent for Case 2 may be the result of residence time effects lowering the effective specific capacitance of the slurry at higher flow rates (as much of the surface area of the YP-50 slurry constituent comes from micropores).

Zoom In Zoom Out Reset image size

Alternative explanations also exist to account for the fact that the experimental current is much lower than predicted. For example, the presence of a contact resistance at the slurry/current-collector interface could also act to reduce the achievable current density. Earlier papers have noted that this contact resistance can be somewhat large; Dennison, et al. found it to be up to 40% of the high-frequency resistance of their EFC cell.³ Compared to the low-frequency resistance of the cell though (which is what is pertinent to actual EFC charge/discharge operations), the contact resistance they reported was negligible. Furthermore, reference 3 also reported that the contact resistance is significantly reduced with higher flow rates. As the flow rates used in the Case 1 and Case 2 experimental studies are much larger than those in reference 3, these combined facts suggest that the contact resistance would be negligible for our EFC system. Another resistance that could also act to reduce the charging current is that from the separator. Both the contact and membrane resistances, however, act in series with the effective resistance of the EFC that is predicted by the scaling approach. The exponent of the scaling solution for the experimental current with respect to capacitive Graetz number would thus be altered from the expected 1/3-power if either of these resistances were appreciable.

Thus, the close agreement of the scaling exponents from experiment and modeling (i.e. the fact that they are so close to 1/3) suggests that the EFC performance in our experiments is indeed governed primarily by overpotential boundary layer growth within a region near the wall where a linear velocity gradient prevails. It is unclear, however, why the electronic conductivity had to be reduced by a larger than expected factor from its static value in order to make the experimental results overlap with the scaling law result. More investigation must be performed in order to confirm that it is indeed localized attenuation of the electronic conductivity that is occurring due to the presence of the higher shear rates in the boundary layer region.