Coupling Microkinetics with Continuum Transport Models to Understand Electrochemical CO 2 Reduction in Flow Reactors

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I. INTRODUCTION
Electrochemical CO 2 reduction (eCO 2 R) is a promising approach for the sustainable production of fuels and chemicals using renewable electricity [1].With the development of CO 2 electrolysis gaining momentum in the past decade, significant progress has been made in understanding how factors such as the electrode composition, electrolyte pH, CO 2 availability, temperature, mass transport, and electrolyte composition affect the reaction rate and efficiency [2].Despite these promising advances, we are still faced with several challenges that have to be overcome in order to enable the commercialization and cost-effective operation of CO 2 electrolyzers [3].For instance, the electrochemical conversion of CO 2 towards high-value C 2+ products like ethylene, ethanol, and propanol still suffers from low-energy efficiencies (i.e., high overpotentials) and product selectivity on state-of-the-art Cu-based catalysts [4].Simpler eCO 2 R products like CO can be produced at industrially relevant current densities (> 300 mA/cm 2 ) with high selectivity on Au and Ag-based catalysts.However, mass transport limitations due to low CO 2 availability and loss of CO 2 as carbonate, the competing hydrogen evolution reaction, and factors affecting long-term stability (surface reconstruction and/or poisoning, salt deposition, and bubble formation) at typical operating conditions still remain major challenges [5][6][7].A lot of these challenges are a result of the existing gaps in resolving the reaction environment at the vicinity of the electrode, which was shown to be crucial in obtaining accurate mechanistic insights [8] and in controlling the overall performance of eCO 2 R [9,10].
In parallel with the experimental progress, modeling studies from the atomistic to the continuum levels have also greatly improved the overall understanding of eCO 2 R.For instance, atomistic simulations based on density functional theory (DFT) and microkinetic modeling have provided crucial insights into the complex reaction pathways and mechanisms involved in eCO 2 R [11], identifying the rate-and selectivity-determining steps towards the various eCO 2 R products [12][13][14][15], and understanding mechanisms behind electrolyte effects (e.g., electrolyte pH, cations, and anions) on eCO 2 R [16,17].Similarly, continuum models have also provided important insights into the structure of the electrical double layer, transport phenomena, and the spatial and temporal distribution of key parameters that control reactor performance [18][19][20][21].Although microkinetic simulations provide detailed information on coverages of reaction intermediates, intrinsic activity of sites, and reaction pathways, they typically do not account for (i) diffusion of CO 2 across the boundary layer, (ii) the buffer equilibria involving CO 2 , (iii) the changes in the pH at the vicinity of the electrode during eCO 2 R, and (iv) double layer charging effects, all of which have been shown to drastically affect CO 2 availability and influence reactivity near the electrode surface [8,20].These observations therefore necessitate the coupling of activities and coverages of species predicted using ab initio-based microkinetic simulations with concentrations of species involved in the reaction from continuum transport models that account for (i)-(iv) to be able to fully resolve the reaction environment at the vicinity of the electrode surface and obtain unbiased insights into eCO 2 R.Such a coupling was successfully performed with a one-dimensional (1D) transport model to study eCO 2 R on Au by Ringe et al. [8], and on Ag by Singh et al. [22].
In this work, we present a multiscale modeling approach that couples ab initio microkinetic simulations with twodimensional (2D) continuum transport models that account for electrolyte flow, migration, diffusion, buffer equilibria, and finite ion size effects to study electrochemical CO 2 reduction to CO on Au electrodes.We find the key parameters of eCO 2 R including the CO 2 concentration and pH at the electrode, the current density towards CO (j CO ), and the simulated Tafel slopes to strongly depend on both the applied potential and the position along the electrode.We further discuss two strategies to improve CO 2 utilization: increasing the shear/flow rate of CO 2 and the introduction of a defect in the middle of the electrode.In both cases, we observe an increase in j CO due to increased CO 2 availability in the mass-transport-limited regimes.We obtain good agreement between the results of the 2D simulations and a 1D model.Finally, we develop a phenomenological Butler-Volmer-like model to estimate j CO that can be used instead of the microkinetic simulations in future studies involving more complex geometries.Our work presents an important step toward the development of multidimensional, multiscale modeling approaches needed to capture the full complexity of electrochemical processes including eCO 2 R and beyond.

A. Description of the model system and coupling framework
Figure 1(a) shows a schematic of the reactor configuration with a volumetric flow rate Q between the planar cathode and the anode, representing an aqueous flow reactor.In this study, we focus on the eCO 2 R reaction at the cathode (Au) of length L (L = 1 cm), as shown in Fig. 1(b), and we note that the configuration considered is different from more complex configurations such as the gas diffusion electrode, which would have a qualitatively different fluid flow.The velocity profile within the full reactor is parabolic, i.e., plane Poiseuille flow, as shown in Fig. 1(a).However, since we are primarily interested in mass transfer near the cathode surface, and the mass transfer boundary layer is expected to be thin, i.e., the Schmidt number here is large, we approximate the velocity profile as a linear shear flow with shear rate γ .This mass transfer problem was previously considered in Ref. [23] for simplified reaction kinetics.We consider γ ranging from 0.1 to 100 (×1.91 s −1 ), which corresponds to Péclet numbers ranging from 10 4 -10 7 , where D CO 2 is the diffusivity of CO 2 .The Péclet number here provides a measure of the relative importance of flow to diffusion.To provide some intuition on these shear rates, we estimate that, for a channel of 1-cm width and 1-cm height, the corresponding flow rates Q are approximately 1.91-1910 mL/min.Note that while we consider this full range of Péclet numbers in our analysis, the highest Péclet number likely corresponds to a physically unattainable flow rate.We also estimate the expected boundary layer thickness (d BL ) averaged over the cathode according to [23] where a = 3 2/3 (1/3)/4.The continuum transport equations are solved on a 2D mesh, with the coupling between microkinetics and continuum transport (see the Appendix for details) performed at each point on the reactive (cathode) boundary, i.e., y = 0, in a self-consistent manner [see Fig. 1(c) and the Appendix section for details on the coupling scheme].The key parameters exchanged during the coupling scheme at the boundary ([CO 2 ] y=0 , pH y=0 , potential φ y=0 , and j CO ) are also highlighted.In what follows, we present results obtained from the coupled 2D simulations of eCO 2 R to CO on Au electrodes.

B. Variation in the key parameters with applied potential and electrode position
Figure 2 shows the variations in the key reaction parameters with the applied potential (U) from the coupled microkinetic simulations and the 2D continuum transport model.The results correspond to flow with Pe = 10 5 (Q = 19.1 mL/min).The solid lines and the shaded regions indicate the average value and variation of the parameter along the length of the electrode, respectively.As can be seen in Fig. 2(a), the CO 2 concentration at the electrode   A3)].Note that the pH at the electrode is generally lower than the bulk pH (7.5 for a 0.5-molar KHCO 3 solution) as a result of electrostatic repulsion of anions (OH − ) (and attraction of cations) from the negatively charged electrode.We only observe a higher pH at the electrode compared to the bulk pH at potentials < −1.0 V.This observation further points to the importance of the exact definition of "local" in the context of pH, as was also highlighted in a previous study by Bohra et al. [20].
The CO partial current density (j CO ) obtained from the coupled 2D simulations is shown in Fig. 2(c).There is a variation in the slope of j CO as a result of the change in the rate-determining step (RDS) with U [8], as expected for multistep electrochemical reactions [13] and confirmed by a degree of rate control [25] analysis.(cf.Fig. S2 within the Supplemental Material [26]).A change in the RDS with U also results in changes in the continuous Tafel slopes (∂U/∂log 10 j CO ), as shown in Fig. 2(d).The dashed lines indicate the Tafel slopes obtained from the microkinetic model for the respective RDS in the absence of mass transport effects (i.e., based only on intrinsic kinetics).At potentials higher (less reducing) than about −0.7 V, the *COOH − → *CO step [Eq.(A1c)] limits the overall rate, resulting in an intrinsic Tafel slope of ca.40 mV/dec.For U < −0.8 V, the adsorption of *CO 2 [Eq.(A1a)] is the RDS, resulting in an increase in the intrinsic Tafel slope as it is the first step in the reaction mechanism (approximately 80-100 mV/dec).The reason for a potential-dependent Tafel slope in the latter step is the quadratic surface-charge dependence of *CO 2 adsorption [8].While the deviations from the intrinsic Tafel slopes are minor at the lower potentials with the *COOH → *CO step being the RDS, significant deviations are observed at U ≤ −0.8 V, where the adsorption of CO 2 is the RDS due to the convolution of CO 2 mass transport with the intrinsic kinetics.This deviation is in agreement with the rapid depletion of [CO 2 ] y=0 with U observed in Fig. 2(a), resulting in mass transport limitations at higher potentials.Here it is worth noting that most experiments for eCO 2 R on Au are performed in the potential regions where there is a convolution of intrinsic kinetics with mass transport limitations [27], which could result in incorrect mechanistic interpretations regarding the (potential-dependent) identity of the rate-determining steps.This aspect was particularly highlighted in a recent study by Gregoire et al. [28], where hydrodynamics of the reactor strongly affects the measured Tafel slopes for eCO 2 R.
We further analyze variations in [CO 2 ], j CO , and pH at the electrode as a function of the position (x) along the electrode for two different potentials (U = −0.8 and −1.1 V) for Pe = 10 5 , as shown in Figs.3(a)-3(c).Contour plots showing a more resolved dependence of these parameters on both the applied potential and spatial position along the electrode are included in the Supplemental Material [26] for the studied flow rates.At U = −0.8V, [CO 2 ], pH at the electrode, and j CO remain fairly constant with electrode position.However, for U = −1.1 V, we note a rapid decrease in [CO 2 ] and j CO with electrode position.This can also be seen in Figs.3(d)-3(f), with the electrode position on a log scale.An initial reduction is observed within the first nanometer of the electrode, with a further reduction from about 10 −2 cm until the end of the electrode.The rapid depletion of [CO 2 ] with electrode position at U = −1.1 V also results in a concomitant increase in the simulated Tafel slopes with electrode position (see Fig. S3 within the Supplemental Material [26]).These observations indicate that having electrodes with smaller lengths might be beneficial, particularly for reactor configurations where flow is stagnant (e.g., H-cells) where there is likely a significant depletion in [CO 2 ] with increasing cathodic potentials.

C. Strategies to increase CO 2 availability
We explore two possible strategies to increase CO 2 availability at higher potentials: (i) using the same reactor configuration shown in Fig. 1(b) and changing γ , i.e., the flow rate Q, to the values shown in Table I, and (ii) the introduction of an inert defect at the middle of the electrode.

The effect of flow
First, we consider the effect of flow on the CO current density j CO .In Fig. 4, we show the increase or decrease in j CO as a function of Pe relative to the case where Pe = 10 5 .As expected, we observe that j CO increases with increasing Pe, and that the effect is magnified as we approach more negative U, i.e., less reaction-limited conditions.When U is small, the effect of flow is expected to be diminished, since [CO 2 ] is not sufficiently depleted on the electrode for an increase in flow to be beneficial.We note that the choice of Pe to compare to here is somewhat arbitrary, and that the potential where mass transfer limitations begin to occur is dependent on Pe.Specifically, as Pe increases, a larger potential is required to deplete the [CO 2 ] at the electrode, and as a result, the comparisons here are not at equivalent levels of mass transfer resistance.Nevertheless, the comparison in Fig. 4 demonstrates the effect of flow rates.
It is interesting to note that the relative increases in j CO observed in mass-transfer-limited conditions are of a similar order of magnitude to the classical Graetz-Leveque problem, i.e., shear flow of a single species over a perfectly reactive surface [29,30].In particular, the total reaction is expected to scale as approximately Pe 1/3 [23]; as an example, when Pe is increased from 10 5 to 10 6 , this simple theory would predict the current density to increase by a factor of approximately 2.2.Of course, the problem considered is not expected to match this simple theory exactly, since we have also considered additional physics, such as electromigration and finite-size effects.Nevertheless, it is reassuring that the right order of magnitude is captured.
While we have primarily focused our attention here on the current density j CO , it may be important in many applications to consider the CO 2 single-pass conversion, or in other words, the fraction of CO 2 that is consumed.In certain situations, it may be desirable to have a single-pass conversion near 1.This can be defined as where A is the area of the electrode, F is Faraday's constant, and C bulk CO 2 is the bulk CO 2 concentration.While we have observed j CO to increase with Pe, it certainly increases slower than linearly at large U-the simple theory predicts Pe 1/3 .Since the flow rate Q scales ∝ Pe, the single-pass conversion is expected to approach 0 as the flow rate increases.For the simple theory, we would have λ ∼ Pe −2/3 .Thus, simply trying to achieve ever higher flow rates can have diminishing returns when considering other factors in the overall design and optimization of electrochemical reactors.

The effect of an inert defect
We explore a second strategy to increase CO 2 availability, and thus current density j CO : the introduction of an inert defect at the center of the electrode.As shown in Fig. 5(a), we introduce a defect of length L defect = 0.1 cm in between the electrode, resulting in two electrodes of length L/2 (i.e., 0.5 cm each).We note that the inert defect being considered is effectively a gap in the electrode without fundamental material properties (e.g. the potential of zero charge, PZC), but this analysis could be extended to consider material properties, including surface charging and electrocatalytic activity that might exhibit stronger influence on the overall performance of the electrode.In Fig. 5(b), we compare [CO 2 ] at the electrode at U = −1.1 V both when the defect is and is not present; the data for the case of no defect are plotted with an imaginary gap in the middle to facilitate comparison.We observe that there is increased [CO 2 ] available to the beginning of the second electrode after the defect.The resulting relative increase in j CO is shown in Fig. 5(c).
The primary mechanism behind the increased [CO 2 ] can be explained as follows.As CO 2 flows past the first electrode, the [CO 2 ] boundary layer grows in thickness as it is continuously used up at the electrode.However, over the inert defect, where no reaction occurs, [CO 2 ] is allowed to replenish via diffusion through the boundary layer.At the second electrode, a second boundary layer grows, and [CO 2 ] continues to deplete.This interaction between the two [CO 2 ] depletion wakes is complex (see, e.g., Refs.[31][32][33]), and as a result, does not permit a simple theoretical approximation.Like the previous case of no defect, we continue to observe an increase in CO 2 availability at the electrode with increasing flow even in the presence of an inert defect, but the rich relationship between defect, flow, and microkinetics merits further exploration.While the resulting increase in j CO due to the defect as shown in Fig. 5(c) is relatively modest, it is important to note that increasing j CO in this way also directly results in spatial modifications of the local pH and a corresponding increase in the single-pass conversion λ.In contrast to the previous section where we increased the flow rate Q, the denominator of λ does not increase with the addition of the defect [cf.Eq. ( 3)].Increasing the flow rate and altering the distribution or patterning of electrode materials, i.e., the addition of defects, are two independent effects that should be balanced depending on whether the ultimate goal is to increase j CO or λ.Additionally, the influence on spatial pH profiles may be a strategy for tailoring local conditions along the electrode for specific reaction pathways or products, particularly for more complex electrochemical reactions that strongly depend on pH (e.g., CO 2 /CO reduction on Cu) [12][13][14][15].This warrants a need for codesign that simultaneously considers the flow rate and electrode patterning to achieve target outputs in electrochemical flow reactors.It is important to note that, for this study, we have chosen the location and size of the defect arbitrarily, since our goal in this work was to demonstrate how our multiscale framework can handle complex configurations.Future investigations will involve a full study of how the configuration of defects interacts with the flow and the microkinetic model.

D. Comparison with lower-dimensional surrogate models
The state-of-the-art multiscale models that attempt to resolve mass-transport effects typically adopt 1D descriptions that use an analytical Butler-Volmer expression for the boundary condition [19,34], or combine ab initio-based microkinetic simulations with continuum mass-transport models [8,35,36].While these approaches are more computationally tractable, they fail to capture the spatial inhomogeneities in the key reaction parameters observed in this study.This limitation can thereby limit the reliability of such models when attempting to optimize electrode geometries for targeted performance.Nonetheless, lowerdimensional models that can predict product current density and selectivity as a function of operational conditions are still desirable, particularly for other computationally intensive modeling efforts such as topology optimization, where complex flow paths, mass transfer resistances, and intricate electrode geometries can arise that cannot be predicted a priori [18,[37][38][39].Thus, extracting and validating simpler models from two-(or higher-)dimensional models with results obtained in lower dimensions is useful for gauging the regimes at which surrogate models can sufficiently describe the properties of interest.

One-versus two-dimensional simulations
We compare the results of our 2D simulations with flow explicitly considered to 1D simulations that assume an effective boundary layer thickness that corresponds to the respective Pe for the 2D simulations (cf.Table I and the Supplemental Material Appendix [26]).As shown in Fig. 6, we find good agreement between the average j CO obtained from the 2D simulation and the 1D simulation when the average boundary layer thickness over the electrode is approximated by Eq. ( 2), where d BL scales as Pe −1/3 [23].For example, when Pe = 10 5 , d BL ≈ 300 µm.While this approximation for the boundary layer thickness was derived assuming single species mass transfer, the agreement observed here demonstrates that, even when electromigration, finite-size effects, and more complex kinetics are considered, a 1D simulation can potentially capture the average properties predicted in a 2D simulation as long as the boundary layer thickness is judiciously chosen.Of course, 2D simulations are still required in order to capture the spatial variation of the parameters, which are found to vary strongly over short distance scales (Fig. 3).Furthermore, such a comparison is only straightforward in simple flow configurations like the one considered in this work, and more complex flows, electrode geometries, and electrochemical reaction pathways would merit further investigation.

Phenomenological Butler-Volmer model to estimate j CO
While the coupling between the microkinetic simulations and the 2D continuum transport model is efficient for the simple reactor geometries considered in this work, it might be computationally expensive to perform the multiscale simulations for more complex geometries and/or in three dimensions.Additionally, for computational topology optimization to target specific electrode and reactor designs, the high dimensionality of the parameter space makes the coupling scheme of microkinetic and continuum transport models computationally prohibitive.Crucial for gradient-based optimization algorithms, derivatives are easily obtained for simple expressions such as Eq. ( 4), but not for an external microkinetic simulation package.This motivates the development of surrogate models that can replace the microkinetic simulations to help accelerate the coupling scheme.To do so, we fit the j CO and U data obtained from parametric sweeps of [CO 2 ] and pH in microkinetic simulations (without the coupling to continuum transport) to a phenomenological Butler-Volmer-type model.The model has the form [40] where j 0,1 and α 1 are the exchange current density and charge-transfer coefficient associated with the first RDS (*COOH − → *CO), j 0,2 and α 2 are the exchange current density and charge-transfer coefficient associated with the second RDS (the adsorption of *CO 2 ), and j lim is the limiting current density.We note that j 0,1 , j 0,2 , and j lim depend on [CO 2 ] and pH (see Sec. S2 of the Supplemental Material [26] for further details on the microkinetic sweeps and the fitting procedure).Furthermore, such a simplified model might not be applicable to more complex reactions involving (potential-dependent) adsorbate coverages [41].We further use the following simplifications while performing the microkinetic sweeps and the coupled 1D simulation: (i) a constant capacitance (17 µF/cm 2 ) was used in the simulations and (ii) the Frumkin correction that relates to an additional potential drop within the electrochemical double layer obtained from the coupled simulations [8] was neglected.We stress that these simplifications strongly influence the agreement between the coupled 1D simulations and the phenomenological model, and further motivate the need for validating surrogate model results with resolved higher-dimensional models.
Figure 7 shows the contribution of the different terms in Eq. (4) towards j CO evaluated with the [CO 2 ] and pH obtained from the 1D simulation with an effective boundary layer thickness of 300 µm.At each potential, the largest term in Eq. ( 4) dominates the overall contribution towards j CO .While the first term dominates for U −0.55 V where the *COOH − → *CO is the RDS, the second term dominates at more reducing potentials when the adsorption of *CO 2 is the RDS.For the 1D models used to compare with Eq. ( 4), we find that, even though the Tafel slope appears to rapidly increase at the most reducing potentials in Fig. 7, it is clear that the j CO remains far below the fundamental limiting current j lim .
Overall, the j CO predicted from the phenomenological model is in good qualitative agreement with the coupled microkinetics and 1D continuum transport simulation, with the degree of quantitative agreement subject to the assumptions or simplifications mentioned previously.Therefore, such a model could be used instead of the microkinetic model while extending the coupling scheme to more complex geometries and higher dimensions, provided that it has been validated by more resolved and higher-dimensional models, as we have shown here.Here j 0 and j 1 refer to the currents associated with the first and second terms in Eq. ( 4), respectively.

III. CONCLUSIONS
By using a multiscale modeling approach that couples ab initio-based microkinetic simulations to continuum transport models, we provide a detailed understanding of how the local environment can influence electrochemical reduction reactions in a flow reactor, focusing on CO 2 reduction to CO on Au electrodes.Our approach is able to spatially resolve key reaction parameters, including the CO current density, CO 2 concentration, and pH at the electrode.We further explore and quantify how these key reaction parameters can be impacted by experimental variables such as flow rate and applied potential, confirming that, while higher flow rates do improve the achievable current densities at a given potential, mass transport effects can limit the utilization of the full electrode length for CO production.Our results show that disruptions of the planar electrode with inert sections or gaps can improve the available CO 2 concentration along the electrode length and offer modest gains in the CO current density that depend on flow rate and potential.This suggests that simple spatial patterning of planar electrodes may be an effective (co)design strategy for improving single-pass conversion efficiencies and help mitigate other constraints set by flow rates and operating conditions.While we focus on CO 2 reduction on Au electrodes in this work, these results are general to a variety of different electrochemical reactions and illustrate the severity of mass-transport limitations encountered in flow reactors.The framework presented in this study of coupling continuum transport with microkinetics is an important step towards future investigations of more complex cell configurations, such as gas diffusion electrodes, where flow patterns may not be as simple.Additionally, we show that this approach can allow for the creation of informed (condition-aware) surrogate models of lower dimensionality and computational cost that can accelerate the design and optimization of electrochemical flow reactors.
The raw data to reproduce the figures in the article and the open-source software package to perform the coupled microkinetic and transport simulations are available from Zendodo [53,54].elements are used for spatial discretization (on quadrilateral cells for the 2D simulations).Typically used to tackle the advective term in the advection diffusion equation, streamline upwind Petrov-Galerkin stabilization [48,49] is used here on the combined advection migration flux (see the Supplemental Material [26] for discretization details).

Coupling between continuum and microkinetic modeling
To complete the continuum model defined in Appendix A 2, the boundary conditions (A4) and (A5) require j CO values at each boundary point.In one dimension, this is a single value, but in two dimensions, it is for every mesh node on the cathode boundary (for a piecewise-linear flux along the cathode).Values for j CO are obtained from solving the microkinetic model described in Appendix A 1. For closure, this model requires surface values for CO 2 concentration, pH, and potential φ, which are all obtained from the solution of the continuum model.
We solve the coupled continuum-microkinetic model in an iterative manner as outlined in Algorithm 1.In the algorithm, j CO refers to the piecewise-linear current density of CO along the cathode surface, and Y refers to the solution vector of the continuum model, i.e., concentrations C i of all species i and the potential φ.The operation continuum(j CO , Y) refers to numerically solving the continuum model with CO current density j CO , and using Y as the initial guess for the Newton solver.The operation mkm((Y) k ) refers to solving the microkinetic model with input values C CO 2 , pH, and φ from the solution vector Y at node k.The relaxation parameters θ 0 , θ 1 , where θ 0 + θ 1 < 1, help with the stability of the iterative scheme.
Because of the nonlinearity of the continuum model, Newton's method will not converge at higher cathodic potential when starting from a constant initial guess.Instead, we use a continuation strategy where cathodic potential is increased incrementally, such that the initiation of CO current density and the solution vector Y is done using the values of the previous voltage (replacing lines 2 and 3).

FIG. 1 .
FIG.1.(a) Schematic of the reactor configuration considered, where a volumetric flow rate Q flows between two plates.In this study, we focus on the mass transfer phenomena near the cathode of length L (L = 1 cm), as highlighted in (b).Near the cathode, the velocity profile can be approximated as a shear flow with shear rate γ .A boundary layer of depleted CO 2 concentration is shown schematically.(c) The continuum transport equations are solved on a mesh, where at each point on the reactive (cathode) boundary, coupling with the microkinetic model is performed.
FIG. 2. Variations of the key parameters with the applied potential measured vs the standard hydrogen electrode [U (vs SHE)] for Pe = 10 5 ( γ = 1.91 s −1 , flow rate = 19.1 mL/min).Average values are in solid lines and shaded regions indicate variation along the electrode length.(a) Concentration of CO 2 ([CO 2 ]) at the interface.Dashed line indicates bulk [CO 2 ] (34 millimolar).(b) pH at the interface.Dashed line indicates the bulk pH (7.5).(c) CO current density j CO and (d) the continuous Tafel slopes.Dashed lines indicate the Tafel slopes corresponding to the RDS being *COOH → * CO (40 mV/dec) and CO 2 (g) → * CO 2 (80-100 mV/dec) from microkinetic simulations in the absence of transport effects.

FIG. 3 .
FIG. 3. Variations of the (a) CO 2 concentration at the electrode, (b) CO current density, and (c) pH at the electrode as a function of position (x) along the electrode at U = −0.8 and −1.1 V vs SHE for Pe = 10 5 .The same variations are shown in (d)-(f), showing the electrode position on a logarithmic scale for better resolution.

FIG. 4 .
FIG. 4. Effects of different flow rates on j CO , with the increase or decrease shown for the different Pe relative to Pe = 10 5 .

7 FIG. 5 .
FIG. 5. (a) Schematic of the cathode with an inert defect (L defect = 0.1 cm) in between the two electrodes, each of length 0.5 cm along with the boundary layer of depleted CO 2 .(b) Increase in the [CO 2 ] at the interface at U = −1.1 V with and without the defect, and with increasing Pe (from 10 5 to 10 7 ).(c) Percentage increase in j CO with the defect as compared to without, shown for different respective Pe.

5 FIG. 6 .
FIG.6.Comparison between the spatially averaged j CO obtained from the 2D simulation with Pe = 10 5 and a 1D simulation with an effective boundary layer thickness of 300 µm.

FIG. 7 .
FIG.7.Contribution of individual terms in the phenomenological Butler-Volmer model towards j CO , plotted with the results from the 1D simulation with an effective boundary layer thickness of 300 µm.Here j 0 and j 1 refer to the currents associated with the first and second terms in Eq. (4), respectively.

TABLE I .
Summary of the shear rates ( γ ) used in this study, as well as their corresponding Péclet numbers (Pe), the effective boundary layer thickness (d BL ), and the approximate flow rate, assuming a 1-cm-wide channel of 1-cm height.