Engineering Electrode Rinse Solution Fluidics for Carbon-Based Reverse Electrodialysis Devices

Natural salinity gradients are a promising source of so-called “blue energy”, a renewable energy source that utilizes the free energy of mixing for power generation. One promising blue energy technology that converts these salinity gradients directly into electricity is reverse electrodialysis (RED). Used at its full potential, it could provide a substantial portion of the world’s electricity consumption. Previous theoretical and experimental works have been done on optimizing RED devices, with the latter often focusing on precious and expensive metal electrodes. However, in order to rationally design and apply RED devices, we need to investigate all related transport phenomena—especially the fluidics of salinity gradient mixing and the redox electrolyte at various concentrations, which can have complex intertwined effects—in a fully functioning and scalable system. Here, guided by fundamental electrochemical and fluid dynamics theories, we work with an iron-based redox electrolyte with carbon electrodes in a RED device with tunable microfluidic environments and study the fundamental effects of electrolyte concentration and flow rate on the potential-driven redox activity and power output. We focus on optimizing the net power output, which is the difference between the gross power output generated by the RED device and the pumping power input, needed for salinity gradient mixing and redox electrolyte reactions. We find through this holistic approach that the electrolyte concentration in the electrode rinse solution is crucial for increasing the electrical current, while the pumping power input depends nonlinearly on the membrane separation distance. Finally, from this understanding, we designed a five cell-pair (CP) RED device that achieved a net power density of 224 mW m–2 CP–1, a 60% improvement compared to the nonoptimized case. This study highlights the importance of the electrode rinse solution fluidics and composition when rationally designing RED devices based on scalable carbon-based electrodes.


Resistance calculations
The internal resistance of the RED stack (R i ) consists of several resistances, divided into ohmic resistances (R ohmic ) and non-ohmic resistances (R non-ohmic ): .
(Eq.S1)  i =  ohmic +  non -ohmic Ohmic resistance refers to the resistances of all RED components: HC and LC saline solution compartments, AEM, CEM (all of which stands for cell pair resistance), AEM shielding membrane, ERS, electrodes and electrode connections (all of which stands for electrode compartment resistance).Therefore, the cell pair resistance (r) equals: , (Eq.S2)  =  AEM +  CEM +  HC +  LC where membrane resistances (R AEM and R CEM ) are material properties, and HC and LC compartments resistances (R HC and R LC , respectively) can be calculated using the formula: , (Eq.S3)  HC/LC = f obs ℎ σ HC/LC  where σ HC/LC (S m -1 ) is the ionic conductivity of HC and LC compartments, respectively, h (m) is the channel height, A (m 2 ) is the area of direct one-side contact between membrane and HC and LC, and f obs is an obstruction factor (for channels without spacer f obs equals to 1).
For calculating the resistance of HC and LC compartment the average conductivity of the solution within the cell is considered.For this reason, the solutions' conductivity is measured before being pumped into the RED device and after exiting the RED stack, enabling average calculations.The obtained values for HC compartment are: before σ HC = 87 mS cm -1 and after σ HC = 44 mS cm -1 leading to an average value of 66 mS cm -1 .For LC compartment are: before σ LC = 1 mS cm -1 and after σ LC = 53 mS cm -1 leading to an average value of 27 mS cm -1 .Interface area equals membrane area in direct contact with HC or LC, A, equals 110 cm 2 .Given that, R HC is calculated to be 4.2 mΩ and R LC = 10.5 mΩ for n= 1 with h= 300 µm.Specific membrane resistance (Ω m 2 ) is given by the membrane producer, 0.3 mΩ m 2 and needs to be multiplied with the membrane area, A, to obtain the membrane resistance (Ω).Thus, R AEM = R CEM = 27.3 mΩ.
.  = 27.3 + 27.3 + 4.2 + 10.5 = 69.3(mΩ) The resistance of the whole electrode compartment can be calculated by extrapolating the resistance values from various numbers of cell pairs to a RED stack with n= 0. Experiments are conducted for n= 1 -6.1 Ω, and n= 5 -17.1 Ω, leading to a n= 0 value of 3.4 Ω for h= 300 µm channel height.Given that, R ohmic can be calculated: . (Eq.S4) Deviations between calculated and measured values are probably caused by deviations from theoretical assumptions, i.e., constant ionic conductivity within a channel, uniform distribution of HC and LC solutions within all fluid channels, etc. Calculated ohmic resistances of the RED stack composed of different gaskets (channel heights, h) are summarized in Table S1.

Electromotive force and voltage calculations
The electromotive force created at the membrane/saline solution interface can be calculated using the following formula: , (Eq.S5) where α CEM, AEM is the apparent permselectivity of the membrane (-), R-universal gas constant The total EMF of the RED device is equal to: U max =(EMF CAM +EMF AEM )n, and gives

Gross, Pumping and Net Power Density calculations.
Power, P, can be predicted using Ohm's law, combining voltage, U, current, I, and resistance, R: The power P (W) needs to be later recalculated considering the number of cell pairs, n, and one-side direct area between membrane and saline solution, A, to obtain gross power density, Pd gross, in mW CP -1 m -2 to enable comparison with experimental data and other literature reported values.For the voltage, one can consider EMF multiplied by number of cell pairs, n, 0.224 V • 5= 1.12 V. Maximum power is obtained when external load matches R i , thus, R i is taken into calculations.Computation of current, I, is difficult as Faraday's law cannot be applied straightforwardly, as no deposition or dissolution processes are ongoing at the electrode material.The redox system should run effectively, without deposition/dissolution reactions and it varies depending on the parameters, such as redox species concentration, activity, electrode material, porosity, flow conditions and many others.Therefore, for the purpose of computing the real RED stack power metrics maximum voltage and current have been considered for calculations using a semiempirical approach, leading to the feasible maximum power output of this device with various channel heights.Taking it into account that the theoretical Pd gross equals 420 mW CP -1 m -2 for h= 300 µm, 347 mW CP -1 m -2 for h= 125 µm, and 222 mW CP -1 m -2 for h= 100 µm.Thus, gross power density can be plotted against channel height.
Theoretical pumping power (P in ) predictions use straightforward computation, as pressure drop, Δp, and flow rate, V, need to be considered for each solution flowing through RED device, i.e., LC, HC and ERS. (Eq.S8) This equation (Eq.S8) takes into account hydrodynamic resistance of rectangular channel and recalculate it to cylindrical pipe cross sectional diameter [1], η-fluid dynamic viscosity (Pa s), L-channel length (m), w-channel width (m), h-channel height (m), V-flow rate (m 3 s -1 ).Maximum power input is calculated for a pump setting with VḢ C = VL C = 0.21 cm 3 s -1 flow rate, that is maximal for HC and LC solutions, and VĖ RS = 0.05 cm 3 s -1 .P in divided by n and A gives Pd in .
The net power density is obtained when pumping power density is subtracted from gross power density, Pd net = Pd gross -Pd in .It directly shows the available power that can be used by external devices.

Fluidics
The Reynolds number (Re) of a fluid can be calculated using the equation: where: A-cross-sectional area of the channel (m 2 ), P wet -cross-sectional perimeter wetted by the shear stress (m), w and h-channel width and height (m).For calculations density and viscosity of HC and LC solutions were assumed to be the same as for pure water (298 K), namely: ρ= 999.8 kg m -3 and η= 1.79 mPa s.Width of channel w= 1.3 cm, height, h=100, 125 or 300 µm, respectively.Given that, the Reynolds number is calculated for each system with n= 5 and equals: 0.7 for h= 300 µm; 0.4 for h= 125 µm and 0.7 for h= 100 µm.Thus, the Re number for all experiments of RED with n= 5 independently on a h is lower than 1.

Passive mixers design
Passive micromixers increase Pd gross , without substantial cost increase-as no external energy input to run them is required.Passive micromixers disturb the fluid flow by physical manner [2].For this purpose, one or more walls of the fluid channel are designed with notches or protrusions, often in the form of grooves and ridges.One of these structure designs is the staggered herringbone mixer (SHM), which generates a chaotic flow even at low Reynolds numbers, Re < 100 [3].We focus on this mixing structure as Re <1 for our RED device with n= 5. SHM structures consist of V-shaped grooves that are installed at regular intervals in the fluid channel.The V is asymmetrical and has a long and a short arm.The term SHM-cycle is used when several grooves are installed one after the other and the orientation of the asymmetrical grooves changes halfway through one cycle (two half-cycles).The geometrical parameters that fully define the SHM structures are: • groove intersection angle θ (°) • width of the short arm w s (m) • width of the long arm w l (m) • the asymmetric factor P = w l /w (-) • the number of grooves per half-cycle N (-) • the fraction of the membrane area that is covered by the ridges z (-).

S9
Different studies shows that the mixing effect of the SHM is strongly dependent on the ratio of channel height to channel width h/w and on the ratio of groove width to ridge width b/a.The specific model is proposed to calculate the optimum groove width to ridge with ratio b/a [3], while the channel height to width ratio h/w seems to be optimal in the range of 0.46-0.48[4,5].The asymmetric factor P should be 2/3 as at this value most of the cross-sectional area is involved in the chaotic flow [6].The influence of the groove intersection angle θ has been addressed in several papers, and suggested optimum value lies between 36.8• [5] and 51.4• [4].A smaller angle leads to a greater pressure drop in the channel [5].Most of the papers state that the number of grooves is found to only have a minor [4] or no effect at [7,8] all onto the mixing performance.The fraction of the wall that is covered with the mixing promoter does not play a role in most of the applications (e.g., heat/mass transfer [9]) but when used in RED it has an influence on the performance of the system.If part of the membrane is covered, this impedes contact and ion exchange between the feed water and the membrane interface (socalled shadow effect) [10].The shadow effect is more prominent the higher the concealed fraction [11], so z should be kept as low as possible.
Passive micromixers could decrease resistance of the feed solutions, especially LC .Thus, their could find their potential application in RED device.
where: ρ-fluid density (kg m -3 ), υ-fluid velocity (m s -1 ), L-characteristic length of the fluid channel (m), η-fluid dynamic viscosity (Pa s).The characteristic length of a fluid channel with rectangular cross section equals the hydraulic diameter d h (m), defined as:

•
height of the fluid channel h (m) • width of the fluid channel w (m) • dept of the grooves d (m) • width of the grooves a (m) • width of the ridges b (m)

Figure S2 .
Figure S2.Engineering RED device n= 5 with different channel height, h: a) Pressure drop computation for average pressure drop in LC and HC compartment (calculated from equation Δp= (12ηLVL C, HC )/h 3 , with dynamic viscosity, η= 1.79 mPa s, channel length L= 0.865 cm, and channel height, h) for various flow rates, VL C, HC , and membrane separation distance, h, together with experimental data obtained with h= 100 µm for average of LC and HC solutions (Δp LC, HC = (Δp LC + Δp HC )/2); b) pressure drop of LC and HC for various channel height, h.

Figure S3 .
Figure S3.Enhanced mixing at the membrane/saline solution interface.Design of passive mixer structure for fixed membrane separation distance (h= 300 µm) and various saline solution channel configurations in a RED device with n=5: a) passive mixer design with herringbone structure designed for a channel width, w= 1.3 cm; b) schematic of the studied configurations within LC and HC saline solution compartment with maintained h= 300 µm (orange line represents AEM and/or CEM with a thickness of 75 µm) by applying one NBR gasket (300 µm) -NBR, three Folex gaskets (3x 100 µm) -F-F-F and Folex passive mixer sandwiched between two Folex gaskets (3x 100 µm) -F-S-F; c) pressure drop recorded for HC and LC compartments for three configurations studied (NBR, F-F-F and F-S-F); d) U vs. I; e) Pd gross vs. I; f) Pd net vs.I with Pd in values.

Figure S5 .
Figure S5.Engineering of the ERS composition in the RED device n= 5 with h=300 µm.LC= 0.01 M NaCl, and HC= 1 M NaCl both with VL C, HC = 0.21 cm 3 s -1 , ERS with VĖ RS = 0.05 cm 3 s -1 and various composition represented in: a) U-I curve; b) Pd gross vs. I.

Table S1 .
Ohmic resistance of various RED device with n= 1 and n= 5 and various h