Tapered, interdigitated channels for uniform flow

Abstract

A fluid-flow device for providing uniform flow of a fluid is disclosed. The fluid-flow device includes a porous medium, a plurality of inlet channels, and a plurality of outlet channels interdigitated with the inlet channels and in fluid communication with the outlet channels via the porous medium. At least one of the inlet channels and one of the adjacent outlet channels are defined by an outer-boundary profile having a cross-sectional geometry that varies non-linearly along its length. A shortest flow path distance through the porous medium between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is constant. The fluid-flow device provides uniform flow of fluid through the device with improved efficiency.

Description

TAPERED, INTERDIGITATED CHANNELS FOR UNIFORM FLOWRELATED APPLICATIONS

[0001] The present application claims the benefit of priority of U. S. Provisional Patent Application No. 63 / 743,995, filed January 10, 2025 which is hereby incorporated by reference in its entirety.FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under N00014-22- 1-2577 and N00014-25-1-2554 awarded by the Office of Naval Research. The United States Government has certain rights in the invention.FIELD OF THE INVENTION

[0003] This disclosure generally relates to the design of fluid-flow devices and channels utilized in various electrochemical systems.BACKGROUND

[0004] Fluid-flow devices have been used to manage, distribute, or transport fluids through various media in applications such as energy conversion, energy storage, separation, and thermal devices. For example, in electrochemical energy separation or storage devices, a liquid or gaseous reactant may be distributed through porous electrodes. The reactants may be distributed through the electrodes based on the fluid flow field design. In another example, cooling fluid may be delivered to a heat sink comprising of a porous metal foam that rejects heat from an electronic chip. The fluid flow field design can play a significant role in the performance, energy efficiency, and operational costs of such devices.

[0005] One such fluid flow field design includes interdigitated flow fields (IDFFs). They may provide for lower pressure drop and higher electrochemical activity as compared to other flow field designs, including serpentine, parallel, or spiral flow fields. Such IDFFs typically employ straight channels. However, such straight-channeled IDFFs can result in “dead zones” within the flow field. These dead zones result in fluid stagnation, which can degrade the performance of the fluid-flow device. Previous efforts to reduce or eliminate these dead zones by increasing cross-sectional geometry of the flow channel have required increased pumping power requirements, reducing efficiency.SUMMARY

[0006] The present disclosure describes various fluid-flow devices. In some embodiments, the fluid-flow device includes a porous medium and a plurality of inlet channels. The fluid-flow device further includes a plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium. The plurality of outlet channels are interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel. At least one of the inlet channels and one adjacent outlet channel are each defined by an outer boundary profile having a cross-sectional geometry that varies non-linearly along a length of the channel. A shortest flow path distance through the porous medium between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is constant along the length of the inlet channel.

[0007] In some embodiments, the fluid-flow device includes a porous medium, a plurality of inlet channels, and a plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium. The plurality of outlet channels are interdigitated with the plurality of inlet channels so that each inlet channel is positioned adjacent to at least one outlet channel. A shortest flow path distance through the porous medium between an outer boundary profile of the inlet channel and an outer boundary profile of the outlet channel is constant along a length of the inlet channel. Furthermore, at least one of the inlet channels is defined by an outer boundary profile having two-fold symmetry about an inlet intermediate point, and at least one of the adjacent outlet channels is defined by an outer boundary profile having two-fold symmetry about an outlet intermediate point.

[0008] In some embodiments, the fluid-flow device includes a porous medium and a plurality of inlet channels. The fluid-flow device further includes a plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium. The plurality of outlet channels are interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel. At least one of the inlet channels and at least one adjacent outlet channel are each defined by an outer boundary profile having a cross-sectional geometry that varies non-linearly along a length of the channel. The outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel each comprise a plurality of stepwise segments, with each stepwise segment having a constant cross-sectional area along a segment length. The cross-sectional area varies between adjacent stepwise segments.BRIEF DESCRIPTION OF THE DRAWINGS

[0009] The embodiments of the present disclosure may be better understood with reference to the following drawings and description. The components in the figures are not necessarily to scale, with emphasis instead placed upon illustrating the principles of the present disclosure. Moreover, in the figures, like reference numerals are generally used to designate similar or identical features.

[0010] FIG. 1 is a perspective view of a portion of a first example fluid-flow device in accordance with various aspects of this disclosure.

[0011] FIG. 2 is a cross-sectional view of a portion of a second example fluid-flow device in accordance with various aspects of this disclosure.

[0012] FIG. 3 is a cross-sectional view of a portion of a third example fluid-flow device in accordance with various aspects of this disclosure.

[0013] FIG. 4 is an exploded view of an example system that employs one or more fluidflow devices in accordance with various aspects of this disclosure.

[0014] FIG. 5A is a graphical representation showing exemplary axial velocity distributions for different example channel cross-sections, in accordance with various aspects of this disclosure.

[0015] FIG. 5B is a graphical representation showing cross-sectional dimensions of an example channel for an example fluid flow device, in accordance with various aspects of this disclosure.

[0016] FIG. 5C is a graphical representation of a cross-sectional view of a portion of an example fluid-flow device, in accordance with various aspects of this disclosure.

[0017] FIG. 6 is a graphical representation of the thermodynamic efficiency of an example fluid-flow device used in a desalination system, in accordance with various aspects of this disclosure.

[0018] FIG. 7 is a graphical representation of performance metrics for an example fluid flow device used in a desalination system, in accordance with various aspects of this disclosure.

[0019] FIG. 8 is a graphical representation of mean-normalized velocity contours at various longitudinal positions of an example channel of an example fluid-flow device, in accordance with various aspects of this disclosure.

[0020] FIG. 9A is a graphical representation of width profiles for example channels of an example fluid-flow device with no contrast of the flux-velocity between high- and low-pressure channels for different ratios of maximal width to depth, in accordance with various aspects of this disclosure.

[0021] FIG. 9B is a graphical representation of the outer boundary profile of high- and low-pressure channels for the example channels of FIG. 9A, in accordance with various aspects of this disclosure.

[0022] FIG. 10A is a graphical representation of width profiles for example channels of an example fluid-flow device with two-fold higher flux-viscosity product in the high-pressure channels for different ratios of maximal width to depth, in accordance with various aspects of this disclosure.

[0023] FIG. 10B is a graphical representation of the outer boundary profile of high- and low-pressure channels for the example channels of FIG. 10 A, in accordance with various aspects of this disclosure.

[0024] FIG. 11 A is a graphical representation of width profiles for example channels of an example fluid-flow device with five-fold higher flux-viscosity product in the high-pressure channels for different ratios of maximal width to depth, in accordance with various aspects of this disclosure.

[0025] FIG. 1 IB is a graphical representation of the outer boundary profile of high- and low-pressure channels for the example channels of FIG. 11 A, in accordance with various aspects of this disclosure.

[0026] FIG. 12 is a graphical representation of example channel width profiles that provide uniform flow for different values of the dimensionless quantity E / (w0 / sr), in accordance with various aspects of this disclosure.

[0027] FIG. 13A is a graphical representation of the RMS deviation of reaction rates as a function of channel-wise Peclet number for various cross-sectional dimensions of an example channel, in accordance with various aspects of this disclosure.

[0028] FIG. 13B is a graphical representation of the critical Peclet number as a function of Damkohler number, in accordance with various aspects of this disclosure.DETAILED DESCRIPTION

[0029] Conventional fluid-flow devices (e.g., porous electrodes) generally have low hydraulic permeability, and are incapable of supporting high flow rates at low pressures. To achieve increased flow rates, higher pressures may be needed. However, forcing fluid to flow therethrough at substantial rates and / or pressures results in increased energyconsumption. Doing so also has mechanical implications, potentially requiring more robust mechanical designs, increasing manufacturing costs and / or operational costs.

[0030] The present disclosure provides examples of a fluid-flow device (e.g., a porous electrochemical / chemical electrode) having a pattern or arrangement of flow channels configured therein that provide optimal or improved delivery of fluid through the medium. In some examples, a system may employ two or more of these patterned arrangements assembled with various components (e.g., bipolar plates, current collectors, manifolds, etc.) applicable for the system / cell design or functionality.

[0031] As shown in FIG. 1, the fluid-flow device 100 includes one or more high-pressure channels 102 and one or more low-pressure channels 104. The high-pressure channel 102 may also be referred to as an inlet channel(s), and the low-pressure channel 104 may be referred to as an outlet channel(s). The low-pressure channels 104 may be interdigitated with the high-pressure channels 102 such that each high-pressure channel is positioned adjacent to at least one low-pressure channel. In other words, the high-pressure channels and low-pressure channels may alternate along a direction transverse to the length of the channels (along the y-axis shown in FIG. 1). This arrangement allows the device to produce flow fields across an area.

[0032] FIG. 1 shows the high-pressure channels 102 and low-pressure channels 104 in a cross-section along the x-y axis. The high-pressure channels 102 may have a length 105, L, along an x-axis, a width 107, w, along a y-axis, and a depth 109, h, along a transverse z-axis. The high-pressure channel 102 may be defined by an outer boundary profile 120. The outer boundary profile 120 refers to the outer surface of the high-pressure channel 102. The outer boundary profile 120 may be a three-dimensional surface, with the cross-sectional shape as depicted in the planar surface of FIG. 1. The low-pressure channel 104 may likewise be defined by an outer boundary profile 140.

[0033] The high-pressure and low-pressure channels may be separated by a porous medium 106. The porous medium 106 may be composed of various types of materials and have any suitable dimensions. For example, the porous medium may be composed of active material particles and conductive (e.g., carbon) additive particles fastened together by a binding material, with void spaces being filled with an ion-conducting liquid electrolyte. In some examples, the porous medium 106 may include a base portion 130. The base portion 130 may include an impermeable or impervious substrate, such as an insulated, inactive material (e.g., graphite, metal, or other material), upon which the porous medium isaffixed or supported or to which the porous medium is coupled. The porous medium 106 allows fluid to flow through the medium (e.g., gaseous or liquid reactants), and the fluid may generally flow from the high-pressure channel 102 to the low-pressure channel 104 via the porous medium 106.

[0034] The high-pressure and low-pressure channels 102 and 104 may be formed in various media. In some examples, the channels are formed or embedded within the porous medium 106, such as within a porous electrode body, while in other examples, the porous medium is formed to define the high-pressure and low-pressure channels. In some examples, the porous medium 106 may be a monolithic, self-supporting medium. In some examples, the high-pressure and low-pressure channels 102 and 104 are formed in a surface of an impermeable or impervious layer that abuts the porous medium 106. In other examples, the porous electrode may include an impermeable or impervious substrate and a porous medium 106 that is supported by the substrate, where the pattern or arrangement of channels 102 and 104 may be provided in a surface of the porous medium and additionally, or alternatively, in a surface of the substrate.

[0035] In some examples, the high-pressure channels 102 connect to a header inlet channel (not depicted in FIG. 1) that supplies the fluid to be flowed through the porous medium 106. The low-pressure channels 104 may connect to a header outlet channel that receives the fluid after being flowed through the porous medium 106 and the low-pressure channels 104. In some examples, as would be the case in FIG. 1, the header inlet channel and header outlet channel are on opposite ends of the fluid-flow device 100.

[0036] As depicted in FIG. 1, a fluid may enter the fluid-flow device 100 from the left (e.g., at an inlet(s) 108), flow in a rightward direction through the high-pressure channels 102, then flow in a direction transverse to the high-pressure channels (e.g., up or down, as depicted by flow directions 112 in FIG. 1), and then flow in a rightward direction through the low-pressure channels 104 before exiting the fluid-flow device 100 (e.g., at an outlet(s) 114). In other words, the fluid may generally flow in a z-type manner. However, it should be appreciated and understood that other flow path configurations are possible.

[0037] Although only several inlet channels and outlet channels are shown in FIG. 1, it is to be appreciated and understood that, in various embodiments, the fluid-flow device 100 may include more or fewer high-pressure inlet channels and / or more or fewer low-pressure outlet channels, and thus, there may be a repetition of the interdigitated channel pattern along a transverse y-axis. For example, the flow-field device 100 can include 50 or more channels,comprised of 25 or more high-pressure inlet channels 102 and 25 or more low-pressure outlet channels 104. Furthermore, it is to be appreciated and understood that, while FIG. 1 shows a particular interdigitation pattern of inlet and outlet flow channels 102 and 104, the fluid-flow device 100 may include embedded flow channels arranged in other interdigitated patterns or manners.

[0038] The profile or shape (e.g., cross-sectional geometry) of the high-pressure channels 102 and the low-pressure channels 104 may be defined based on certain physicsbased constraints, as explained further herein. In some examples, the profile of the high-pressure channels 102 and / or the low-pressure channels 104 varies non-linearly along a length thereof. This non-linear variation may be, in some examples, continuous. Such shaping may provide uniform or near-uniform flow of fluid into or across a surface transverse to the longitudinal direction of such channels.

[0039] The fluid-flow devices described herein may be used in various applications to improve efficiency. For example, the fluid-flow devices may be used in energy or power sources, such as reduction-oxidation (redox) flow batteries or fuel cells. Further applications include electrochemical separation processes, for example, desalination, electrolysis cells, metal recovery processes, enzymatic reactions, and other areas in which fluid is made to flow through porous media (e.g., with simultaneous electrochemical or chemical reactions or interactions), including electrochemical desalination systems. In some examples, the fluidflow device is a porous electrode.

[0040] The fluid-flow devices described herein enable more uniform flow of fluid through the device, increasing hydraulic permeability and improving efficiency. This allows the fluid-flow device, or the encompassing system or cell, to be operated at significantly reduced pressures, which conserves energy resources. This also reduces or eliminates a need for system designers or manufacturers to expend additional effort on ensuring mechanical or structural integrity of the systems, which can reduce design and manufacturing complexity, time, and associated costs, and can also aid in overcoming technical barriers that may otherwise hinder advancement of existing technologies and the development of new technologies.

[0041] With further reference to FIG. 1, the high-pressure inlet channels 102 may include an inlet 108 (e.g., an inlet end) that is coincident with, or near coincident with, such as within a threshold distance from, an edge of the porous medium 106, and another end 122 that terminates at an electrode-channel gap distance 123 (e.g., g) from an edge of the porousmedium 106. In some examples, the electrode-channel gap distance g may be similar to, for example, within a threshold difference from, the value of the channel spacing, which may yield a relatively sharply-shaped residence time distribution such that, in operation, a substantial portion of fluid flow applied to the fluid-flow device traverses the flow channels and enters the fluid-flow device 100. The low-pressure outlet channels 104 may include an outlet 114 (e.g., an outlet end) that is coincident with, or near coincident with, such as within a threshold distance from, the edge of the porous medium 106, and an opposite end 124. The end 124 of the low-pressure channel 104 may terminate at the electrode-channel gap distance g from the edge of the porous medium 106 or at a different gap distance.

[0042] FIG. 2 depicts a cross-sectional view of an example fluid-flow device 200. The fluid-flow device may include one or more high-pressure channels 202 (also referred to as inlet channels 202) and one or more low-pressure channels 204 (also referred to as outlet channels 204). The fluid-flow device 200 may further include a porous medium 206 separating the high-pressure channels 202 and low-pressure channels 204. As described, the low-pressure channels 204 may be interdigitated with the high-pressure channels 202 such that each high-pressure channel is positioned adjacent to at least one low-pressure channel. Although only several inlet channels and outlet channels are shown in FIG. 2, it is to be appreciated and understood that, in various embodiments, the fluid-flow device 200 may include more or fewer high-pressure inlet channels and / or more or fewer low-pressure outlet channels, and thus, there may be a repetition of the interdigitated channel pattern along a transverse y-axis. Further, it is to be appreciated and understood that, while FIG. 2 shows a particular interdigitation pattern of inlet and outlet channels 202 and 204, the fluid-flow device 200 may include embedded flow channels arranged in other interdigitated patterns or manners.

[0043] One or more of the channels 202 and 204 may be formed of a plurality of stepwise segments 230a-230f along the length 221 of the channel. Each stepwise segment may have a constant cross-sectional geometry (e.g., a constant width 233 and depth into the page or porous medium) along the length 232 of the segment. In some examples, a first, upstream segment (i.e., closer to the inlet 208), such as segment 230a, has a constant cross-sectional area that is larger than one or more downstream segments, for example, segments 230b-230f. This difference may be based on varying one or more dimensions of the cross-sectional geometry.

[0044] For example, in a channel with a rectangular cross-section, one may vary the width, depth, or both. In some examples, in the case of a high-pressure channel, between adjacent segments, one cross-sectional dimension may increase while the other cross-sectional dimension decreases, but with an overall smaller cross-sectional area in the downstream segment as compared to the upstream segment. As one instance, an upstream segment (e.g., segment 230a) may have a width and depth of 2 millimeters, and the downstream segment (e.g., segment 230b) may have a width of 2.5 millimeters and a depth of 1.5 millimeters, such that while the width of the downstream segment increases, its depth decreases, resulting in an overall area that is smaller (3.75 square millimeters compared to 4 square millimeters). In other examples, one dimension may stay the same between adjacent segments, while the other dimension decreases. In further examples, both cross-sectional dimensions may decrease. It should also be appreciated that the dimensions may vary in other manners along the length of the channel, including a downstream segment with an overall area that is larger than upstream segment. It should also be understood and appreciated that the preceding discussion used an example high-pressure channel, a low-pressure channel may be generally dimensioned in a mirrored manner (e.g., an upstream stepwise segment may have a smaller cross-sectional area than a downstream stepwise segment).

[0045] In some examples, the high-pressure channels 202 and low-pressure channels 204 exhibit anti-symmetry. Anti-symmetry may refer to the outer boundary profiles of the high-pressure channels and low-pressure channels exhibiting opposite dimensions, such as mirrored images of each other. Such anti-symmetry is generally depicted in FIG. 3. While the changing cross-sectional geometry was described with reference to the high-pressure inlet channels 202, the same may be true of the low-pressure outlet channels 204, but in an asymmetric manner. For example, a first, upstream segment (i.e., closer to the inlet 318), such as segment 231a, may have a constant cross-sectional area that is smaller than one or more downstream segments (e.g., segments 231b-231f). Again, this difference may be based on varying one or more dimensions of the cross-sectional geometry. For example, in a channel with a rectangular cross-section, the width, depth, or both may be varied. As one example, an upstream segment (e.g., segment 231a) may have a width and depth of 1 millimeter while the downstream segment (e.g., segment 231b) has a width of 1 millimeter and a larger depth of 1.5 millimeters, such that the cross-sectional area increases along the direction of fluid flow within the low-pressure outlet channel 204. While in some examples, the high-pressure channels 202 and low-pressure channels 204 exhibit anti-symmetry, inother examples, there is no symmetry between the high-pressure channels 202 and low-pressure channels 204. The channels (including segments 230a-230f and 231a-231f) may be sized independently of each other.

[0046] The cross-sectional geometry of the plurality of segments 230a-230f and 231a-231f may be sized or dimensioned to achieve a particular hydraulic conductance within the channel. The cross-sectional geometry may be sized to approximate linearly- varying hydraulic conductance. Approximating linearly- varying hydraulic conductance may provide uniform flow, as described further herein. Uniform flow may also be provided by other variations of hydraulic conductance, as described further herein. As one example, the channels 202 and 204 may be dimensioned to vary the hydraulic conductance G based on a cosine series of profiles:∑G (%) = (l / %) / bn[cos(n7r(% — L / 2) / L) — cos(n?r / 2)]

[0047] where bncan be chosen arbitrarily, % is a given position along the length of the channel, and L is the length of the channel. The value of bncan be any real number, positive or negative, that produces real, positive values for hydraulic conductance, G. Here, the hydraulic conductance G is determined from an appropriate Poisuelle-flow relation and is defined in terms of local values of flow rate V along an axis of the channel, its pressure gradient bp / bx along its axis, and dynamic viscosity of the working fluid p G =— pV / (bp / bx). The cross-sectional geometry of the plurality of segments 230a-230f and 231a-231f may also be sized such that a flow velocity of fluid flowing from the high-pressure channel 202, through the porous medium 206 to the low-pressure channel 204 is substantially uniform along the length 221 of the high-pressure channel.

[0048] While the cross-sectional geometry of the channels 202 and 204 has been described with reference to FIG. 2 as rectangular in shape, it should be appreciated and understood that the channels 202 and 204 may have other cross-sectional shapes. For example, the channels may have a circular or ellipsoidal cross-section, with diameter dimension varied between adjacent segments, or can have a trapezoidal, triangular, or other suitable cross-sectional shape. Further, the fluid-device 200, including channels 202 and 204, may be formed in such shapes by any suitable manufacturing process, including one or more of milling, laser machining, microfabrication, additive manufacturing, engraving, stamping, or embossing. Embossing is a forming process where a moderate, distributed force is applied to induce plastic deformation in a localized surface region to create desired geometry

[0049] FIG. 3 depicts a cross-sectional view of an example fluid-flow device 300. The fluid-flow device may include one or more high-pressure inlet channels 302 and one or more low-pressure outlet channels 304 in fluid communication via a porous medium 306. The high-pressure channels 302 and the low-pressure channels 304 may be defined by an outer boundary profile 320 and 340, respectively.

[0050] As described, the low-pressure channels 304 may be interdigitated with the high-pressure channels 302 such that each high-pressure channel is positioned adjacent to at least one low-pressure channel. The outer boundary profiles 320 and 340 of one or more channels 302 and 304 may vary non-linearly along the length of the respective channel. Although only several inlet channels and outlet channels are shown in FIG. 3, it is to be appreciated and understood that, in various embodiments, the fluid-flow device 300 may include more or fewer high-pressure inlet channels and / or more or fewer low-pressure outlet channels, and thus, there may be a repetition of the interdigitated channel pattern along a transverse y-axis. Further, it is to be appreciated and understood that, while FIG. 3 shows a particular interdigitation pattern of inlet and outlet channels 302 and 304, the fluid-flow device 300 may include embedded flow channels arranged in other interdigitated patterns or manners.

[0051] As described, a fluid may flow through the fluid-flow device 300 along the high-pressure inlet channels 302, then traverse to the high-pressure channels 302 through the porous medium 306, and then along the low-pressure outlet channels 304. As shown in FIG.3, in fluid-flow device 300, the profile or shape (e.g., cross-sectional geometry) of the high-pressure channels 302 and the low-pressure channels 304 are sized such that a flow path distance 317 through the porous medium 306 between the outer boundary profile 320 of the high-pressure channel 302 and the outer boundary profile 340 of the low-pressure channel 304 is constant along the length 321 of the channels. This flow path distance 317 may be the minimum or shortest distance between the outer boundary profiles 320 and 340 of the channels via the porous medium 306.

[0052] The profile of the channels may, additionally or alternatively, be sized or dimensioned to achieve a particular hydraulic conductance within the channel (e.g., linearly-varying hydraulic conductance). The profile of the high-pressure channels 302 and the low-pressure channels 304 may additionally be sized or defined to achieve particular physical constraints. For example, the profile of the channels may be sized or dimensioned to provide uniform flow of a fluid transverse to the length of the channels.

[0053] In some examples, the cross-sectional geometry of the high-pressure channel 302 and the cross-sectional geometry of the low-pressure channel 304 are varied non-linearly such that a flow velocity of a fluid flowing between the outer boundary profiles 320 and 340 of the high-pressure channel 302 and the low-pressure channel 304, respectively, is substantially uniform along the length of the channels. In other embodiments, the cross-sectional geometry may be varied such that a residence time of a fluid flowing between the outer boundary profiles 320 and 340 of the high-pressure channel 302 and the low-pressure channel 304, respectively, is substantially uniform along the length of the channels. In some examples, the cross-sectional geometry is defined such that a pressure gradient within the high-pressure channel 302 is substantially equal to a pressure gradient within the low-pressure channel 304 at any given position along the length of the channels.

[0054] To achieve these various physical constraints, one or more dimensions of the channel 302 and 304 cross-sectional geometry may be varied. For example, for channels with a rectangular cross-sectional geometry, the width of the channels may remain constant (i.e., fixed) while the depth of the channels varies, or the width of the channels may vary while the depth of the channels remains constant. In some examples, both the width and depth dimensions vary. One or more of these physical constraints may also be achieved based on various outer boundary profiles 320 and 340 of the channels.

[0055] In some examples, the outer boundary profile 320 of the high-pressure channel 304 may be two-fold symmetric about a point along the outer boundary profile 320. As one example, the outer boundary profile 320 may be two-fold symmetric about an intermediate point (e.g., a midpoint) along the length of the high-pressure channel, as depicted in FIG. 3. Two-fold symmetry generally refers to the symmetry produced when being rotated 180 degrees around a point or axis. As shown in FIG. 3, the shape of the outer boundary profile 320 is two-fold symmetric about the intermediate point 325. As shown in FIG. 3, the intermediate point 325 is located at a midpoint of the outer boundary profile 320. Rotating the upstream portion of the outer boundary profile 320 180 degrees about the intermediate point 325 reproduces the downstream portion of the outer boundary profile 320. In some examples, the intermediate point 325 is located at some other distance along the length 321, such that there is partial two-fold symmetry. The outer boundary profile 340 of the low-pressure channel 304 may, alternatively or additionally, exhibit two-fold symmetry about an intermediate point 327.

[0056] While depicted as two-fold symmetric in FIG. 3, it should be appreciated and understood that the outer boundary profile may exhibit alternative symmetries, or may not exhibit any symmetry. For example, the outer boundary profiles 320 and 340 may exhibit an offset, shifted hyperbolic variation along the length of the channels. As another example, the outer boundary profiles 320 and 340 exhibit a sigmoidal variation along the length of the channels.

[0057] As previously described, the cross-sectional geometry of the channels 302 and 304 may take a variety of shapes. For example, the channels may have a rectangular, circular, trapezoidal, elliptical, triangular, or other suitable cross-sectional shape. The low-pressure channels 304 may have the same, or a different, cross-sectional shape as the high-pressure channels 302. The channels of the fluid-flow device 300 may include a radius at a location of the maximal cross-section, such as at an inlet 308 of the high-pressure channel 302 or at an outlet 314 of the low-pressure channel 304. Such a radius may help to minimize entrance and exit losses due to flow contraction and expansion.

[0058] FIG. 4 is an exploded view of an example system 400 utilizing two fluid-flow devices 410a and 410b. In some examples, the fluid-flow devices 410a and 410b may correspond to, for example, be the same as or similar to, any of the fluid-flow devices 100, 200, and 300.

[0059] As shown in FIG. 4, the system 400 may include a first fluid-flow device 410a and a second fluid-flow device 410b, and a separator 425 disposed between the fluid-flow devices 410a and 410b. The separator 425 may include an exchange membrane, such as an ion exchange membrane, or the like. The separator 425 may be formed as a porous or permeable interface that is either non- selective or that functions with a selectivity mechanism. As described herein, the one or more fluid-flow devices 410a and 410b may be patterned with interdigitated arrays of high-pressure and low-pressure channels, with channel shapes that are optimized, as described elsewhere herein.

[0060] As described, the system 400 may be, or may be included as part of, a power source such as a redox flow battery or a fuel cell, an electrolysis cell, or another construction configured to facilitate enzymatic reactions, an electrochemical separation process, a metal recovery process, a purification process, or another process in which fluid is made to flow through porous media. Accordingly, the system 400 may include one or more other components 430a and 430b, such as bipolar plates, current collectors, manifolds, substrates, and the like, as applicable or needed for the system design or functionality. In someexamples, such as in an example case where the fluid-flow devices 410a and 410b are employed in a fuel cell, other components 430a and 430b may or may not be needed. In further embodiments, the components 430a and 430b that abut the flow devices 410a and 410b, such as bipolar plates or current collectors, may additionally or alternatively include interdigitated arrays of high-pressure and low-pressure channels having shapes that are similarly varied.

[0061] The following describes certain physics-based constraints that may be used to create interdigitated flow channels that provide uniformized flow into the porous media transverse to the longitudinal direction associated with such channels. Channels using the particular profiles that uniformize flow may be employed in arrays of inlet and outlet channels arranged in an interdigitated fashion, such as those described above.

[0062] OPTIMAL TAPERING TO UNIFORMIZE FLOW

[0063] Quasi- ID theory can be used to analyze the flow through IDFFs using either straight or optimally tapered channels. Optimal tapering may be achieved by using crosssection variations that produce a certain functional variation of the hydraulic conductance G. which is defined for Poiseuille flow as G = —Vp / (dp / dx) with V being the volumetric flow rate, p being the dynamic viscosity, and dp / dx being the longitudinal pressure gradient. For laminar, fully developed flow through a rectangular channel cross-section having depth h that is much greater than its width w, hydraulic conductance follows a local-cubic law: G = hw3 / 12. In general, the conductance obtained for laminar, fully developed flow depends on geometry, consistent with its unit of m4. Also, the conductance produced by arbitrarily shaped channel cross-sections can be predicted numerically. This may be performed for trapezoidal cross-sections of flow channels produced by V-bit engraving.

[0064] The results of a quasi- ID analysis of IDFF flows produced using either straight or optimally tapered channels are shown in Table SI. Straight channels used in IDFFs possess channel conductance that remains constant along each channel, whereas optimally tapered channels possess a linear variation of hydraulic conductance. Linear variation of hydraulic conductance does not necessarily dictate linear variation of any individual cross-sectional parameter. For a rectangular cross-section whose width w is varied while its depth h is fixed and where h » w is satisfied, linear conductance scaling indicates that channel width varies with the cubic -root of longitudinal position x: w = w0(l — % / L)1 / 3, where w0is maximal channel width and L is channel length. Ultimately, the conductance variations that a given channel shape possesses may produce a certain transverse velocity distribution between high-and low-pressure channels through porous electrode material. For IDFFs using straight channels transverse superficial velocity us ±deviates from uniformity to an increasing degree when channel length is increased ceteris paribus. In contrast, for IDFFs that use optimally tapered channels transverse velocity may be generally uniform irrespective of channel length, provided that the spacing between channels is large enough that the span of intervening material (e.g., a porous electrode) varies weakly among all flow paths. Sufficient separation between channels may be used when assuming an invariant flow-path length to deduce that linear conductance scaling produces uniform flow.

[0065] The variation of longitudinal flow rate V||(x) within high- and low-pressure channels determines their respective pressure distributions pH(x) and pL(x), which in turn produce the inter-channel pressure difference Ap(x) = pH(x) — pL(x) that drives transverse velocity us ±through porous electrode material. For IDFFs using straight channels longitudinal volumetric flow rate varies in a non-linear fashion to produce channel pressure distributions with an exponential-like decay. The difference in slope between pH(x) and pL(x) at a given position results in the inter-channel pressure difference being non-uniform. In contrast, for IDFFs using optimally tapered channels longitudinal flow rate varies linearly to produce a linear variation of pressure in each channel. Because pH(x) and pL(x) possess identical slopes inter-channel pressure difference becomes uniform. Though the total supply pressure (i.e., pH(x = 0) — pL(x = L)) needed for IDFFs using either straight or tapered channels is the same, straight channels may produce a higher total flow rate than tapered ones with both types of channels using the same maximal cross-section. However, tapered-channel IDFFs possess a smaller volume than straight-channel IDFFs due to a corresponding straight channel’s profile circumscribing that of a tapered channel. In practice, when channel volume fraction is fixed among straight- and tapered-channel IDFFs, tapered-channel IDFFs produce higher apparent permeability. Next, linear conductance scaling may be shown to produce uniform flow.

[0066] Tapering for IDFFs based on the linear scaling of channel hydraulic conductance may be based on the condition that the transverse superficial velocity us ±between high- and low-pressure flow channels is invariant along such flow channels. By subjecting the quasi-1D equations that model flow distribution through IDFFs to a uniform us ±condition, interdigitated channels that possess linear scaling of hydraulic conductance (Eq. 4) satisfy this constraint with the smallest mean-square slope. Hydraulic conductance variations can beproduced by a variety of means, depending on the specific channel cross-section and variation thereof that is sought.

[0067] Design formulas for the tapering of channel cross-sections to produce uniform transverse velocity between high- and low-pressure channels may be obtained. It is assumed that the inter-channel distance s(x) is a weak function of x such that it is approximated using a constant value srthat is representative over the entire range of x: s(x) « sr. Application of Darcy’s law subject to uniform us ±implies that the pressure difference Ap between highland low-pressure channels is invariant along these channels:Ap = p HH— pLL= -kpe

[0068] where pH(x) and pL(x) respectively capture the pressure variations with longitudinal position x within high- and low-pressure channels. kpeand p respectively are the hydraulic permeability of the porous electrode and the dynamic viscosity of the working fluid. The quasi- ID theory also couples the variations of pressure within each channel to the hydraulic conductance variations that may be optimized: GH(x) and GL(x). In general, these effects may be captured using the following differential equations that enforce the conservation of volumetric flow rate within each channel:d GHdpH+ 2 / i|us ±| = 0dx p dx

[0069] The parameter |us ±| may be assumed to be invariant with x, allowing the above equations to be integrated directly to yield explicit functions of x for the longitudinal flow rates ^(x) and ^[( ):GHdpH= - — 2h\us l\x + KHp dxGLdpL^11 (*) = - = 2h\us l\x + KLp dx

[0070] By neglecting flow through the tip of both channels (i.e., (x = L) = 0 and V| ' (x = 0) = 0), the integration constants KHand KLmay be determined:GHdpH(x) = - = 2hL |us ±| (1 — x / L)p dx. GLdpL..E|f (x) = — = 2hL\uSil](x / L)LL CLdC

[0071] Consistent with assuming uniform transverse velocity, which in turn requires that Ap is invariant along channels, the gradient of Ap must therefore be zero:d(Ap) dpHdpLdx dx dx

[0072] Utilizing the preceding expressions for ^(x) and ^[(x), a relation that governs the conductance distributions GH(x) and GL(x) within the respective channels may be obtained:1 — x / L x / LGH(x) GL(x)

[0073] The high- and low-pressure channels may be tapered in the same manner such that their conductance distributions are anti-symmetric: GL(L — x) = GH(x). Consequently, a functional equation that constrains the high-pressure channel’s conductance distribution may be obtained:GH(x) • x = GH(L — X) • (L — X)(Eq. 1)

[0074] Equation 1 defines a space of profiles that includes infinitely many conductance profiles. A necessary and sufficient condition to satisfy this equation is that (%') = (L / 2 ± %') • GH(L / 2 ± %') be an even function, where x' = x — L / 2. Thus, a conductance profile that uniformizes flow can be generated by using any trial profile g(x.H(A _X’ + (i - %) • d(L - x) _ p(x) - g(L - x) g(L - x)W“ 2x ~ 2+2(x / L) (Eq. 2)

[0075] While the above conductance profile may be singular for vanishing x, the following cosine series forms a subspace of profiles that satisfy Eq. 1 without discontinuities or singularities:∑GH(x) = (1 / x) / bn[cos(n7r(x — L / 2) / L) — cos(n?r / 2)]<n=l(Eq. 3)

[0076] where the coefficients bncan be chosen arbitrarily. Using Eq. 2 shows that a linear conductance profile is the lowest-order polynomial profile that uniformizes flow between channels. Further, when negligible flow occurs through channel tips the following linear conductance profile is obtained:GH(x) = Go• (1 - x / L)(Eq. 4)

[0077] where Gois the high-pressure channel’s conductance at its inlet. This linear profile is readily shown to minimize the mean-square slope of conductance ((dGH / dx)2):(dGH / dx)2) = dGH / dx)2+ dGH / dx) — dGH / dx)2

[0078] where (f) = f f / L)dx is an integral average operator. Since dGH / dx is constant for a linear conductance profile, it has no fluctuation from its mean, resulting in all other profiles with the same mean slope having greater mean-square slope. Other conductance profiles that satisfy the uniform flow condition may generally produce larger pressure drops than the linear profile. This may be demonstrated by perturbing the conductance profile of Eq. 4 from linearity, while constraining channel volume and flow rate to be the same among all taper-channel IDFFs, by adjusting the coefficients of its cosineseries expansion:4(— l)n+1GH(x) = G0(l — x / L) = G0~ - — — — [cos(n7r(x — L / 2) / L) — cos(n?r / 2)] x £— <n=1nz7tz

[0079] The pressure drop produced with linear conductance scaling is within 0.2% of all perturbed profiles, demonstrating the ability of linear conductance scaling to enable low-pressure flow.

[0080] Within the following discussion, the above linear channel conductance profile is used. The tapering profile can be determined using the correlation between the local hydraulic conductance and the desired channel cross-section. For example, one can vary the width of the channel while keeping the depth fixed, which results in a cubic-root tapering profile in the width if w « h or a linear tapering in the width if w » h. Similar results can be achieved if the depth is varied and the width is fixed. However, variation in either the width or the depth could lead to feature sizes that are not resolvable by a given manufacturing method. Maximal channel resolution during manufacturing may be obtained by varying depth and width of the high- and low-pressure channels.

[0081] An experimental library 550 of channel cross-sections determined by optical profilometry was generated to correlate hydraulic conductance G with specific microengraving conditions, as shown in FIG. 5A. The porous, intercalative electrodes in which channels were embedded may be used in desalination by symmetric Faradaic deionization. This experimental library 550 was used to design tapered channels (dimensioned according to the approximation of linear conductance 551 in FIG. 5B) having piecewise-constant cross-sections to approximate a linear conductance variation 552 that is sought to uniformize flow, to form the example high-pressure channel 502 (with outer-boundary profile 520) and low-pressure channel 504 (with outer-boundary profile 540) depicted in FIG. 5C. Laser micromachining was also prototyped using similar approaches, demonstrating the versatility of this approach. Precision engraving with a V-shaped cutting tool was used to create channels with widths ranging from 150 to 450 pm and depths ranging from 75 to 200 pm using a desktop CNC machine. Simulated axial velocity distributions resulting from laminar, fully developed Poiseuille flow show that flow in most instances is two-dimensional, justifying the numerical determination of G rather than estimating it with the local-cubic law. To determine the specific piecewise-constant cross-section variation used to uniformize flow, the set of conductance values obtained by varying width and depth jointly were overlaid as segments 552 onto an ideal theoretical curve 553, as depicted in FIG. 5B. Joint variations of width and depth were used because varying either of the two dimensions separately while fixing the other either produced -10% unresolved channel length or a first segment length approaching 50% of the channel’s entire length. The length for the segment having maximal conductance was first chosen to make that segment’s center intersect the ideal curve 553 (FIG 5B). For all other segments an “anchor point” was chosen based on the position at which each segment’s conductance (shown by approximation 552) coincided with the ideal curve 553. The right end point of the interval for each segment was then assumed to equal half the distance between anchor points to the left and right. This strategy approximates the ideal curve 553 to approach a midpoint rule, which produces second-order accuracy for integration. The resulting variations of nominal width, depth, and length for each channel segment are shown to vary non-monotonically along each channel, as identified in Table S2.

[0082] Tapered channels designed using the aforementioned strategy were used to create IDFFs milled onto 45 mm x 45 mm electrodes. In addition to the tapered profile already described, the interdigitation of channels may use the specification of their center-to-center spacing sccand a gap distance g between channel tips and electrode edges. In one example, IDFFs with sccequal to 900 pm and 3.2 mm were produced to probe the impact of spacing on apparent permeability and electrode capacity. A finite, non-zero gap distance may be used to suppress direct flow between the tip of each channel and its adjacent electrode edge, the effect of which was assumed negligible when deriving the associated conductance scaling formula for uniformizing flow. Conversely, too large of a gap may produce dead zones at channel tips. A formula for the minimum gap distance to suppress tip flow to <1% forelectrodes with isotropic permeability may be derived: g > 50wosr / L, where w0and L are the maximal width and length of the channel and sris the representative inter-channel distance. For the IDFF using sccequal to 900 pm the minimum gap distance was 214 pm. A gap distance equal to 500 pm was targeted to produce 44.5 mm long channels, but ~1 mm was engraved in practice due to the finite resolution the engraving technique used in this example.

[0083] To quantify the impact of sccon the local permeability of unpattemed electrode material, four electrodes were engraved with parallel channels extending along the entire length of each electrode, each using different inter-channel spacing (Table S5). Hydraulic permeability was then measured with the associated channels oriented perpendicular to the flow direction, rather than parallel to it, to systematically vary the contiguous flow path lengths through porous electrode material. The measured hydraulic permeability was shown to increase from practically zero for inter-channel distance s > 1.5 mm toward an upper bound of 0.5 pm2for s < 0.5 mm according to a « l / 5thpower law in the deviation from the threshold flow-path length sc. Such scaling is evidence of liquid-phase percolation within the pores of the electrodes. One contributing factor is the hydrophobic nature of thepoly vinylidene fluoride (PVdF) binder used in the electrodes, which could inhibit water from infiltrating micropores. Predicated on the possibility of such a mechanism, the apparent hydraulic permeability of an electrode engraved with parallel channels was measured using both air and water as working fluids with an orientation parallel to the flow direction rather than perpendicular to it. While water produced a permeability of only 140 pm2, air produced a permeability of 240 pm2which is in much better agreement with the calculated theoretical hydraulic permeability of 270 pm2(Table S6). Hence, the percolative dependence of hydraulic permeability on flow-path length is linked to the presence of trapped air within certain pores. Despite this implication, the wettability of V-bit engraved electrode surfaces is shown to be better, in at least some examples, than laser-micromachined surfaces.

[0084] Knowledge of flow-path length dependent hydraulic permeability was used to design tapered-channel IDFFs that maximize apparent permeability. Such IDFFs were created using inter-channel distances below (s = 550 pm) and above (s = 2.7 mm) the percolation threshold. Despite the electrode with larger inter-channel distance using larger channels (w0= 510 pm) than the other (w0= 350 pm), the electrode having smaller channels produced more than three-fold larger apparent permeability. This increase in permeability is attributable at least in part to a shorter flow-path length resulting in greaterlocal, unpattemed electrode permeability. Additionally, the electrode having smaller tapered channels and shorter inter-channel distance produced more than two-fold larger apparent permeability than an electrode patterned with a straight-channel IDFF having similar material removal and inter-channel distance. This increase in permeability may be attributed to the tapering of IDFF channels that also minimizes the effect of dead zones to which straightchannel IDFFs are susceptible.

[0085] Desalination Application using Symmetric, Intercalative FDI

[0086] Desalination experiments were carried out with tapered-channel IDFFs embedded in nickel hexacyanoferrate (NiHCF) cation intercalation electrodes using the in-house symmetric Faradaic deionization (FDI) apparatus. A heated, neutral-pH coagulation bath was used to eliminate PVdF dehydrofluorination at pH > 11. Constant heating of a neutral-pH bath also produced higher electronic conductivity than with transient heating, while producing 20% higher first-cycle charge efficiency than with heated, alkaline wet-phase inversion. During each desalination half-cycle, one such electrode intercalates Na+ions to produce diluate, while the other electrode deintercalates Na+ions to produce brine. In turn, Cl’ ions transfer through an intervening anion-exchange membrane (AEM) to compensate for charge balance. Diluate and brine were recirculated between reservoirs and electrodes at a constant flow rate of 5 mL / min to minimize the state-of-charge gradients resulting from streamwise concentration polarization. The finite specific capacity for cation (de)intercalation within electrodes dictates the use of valve switching to redirect diluate and brine to opposing electrodes when current is switched. While this approach bears similarity with electrodialysis reversal that is used to mitigate fouling and scale formation, such current switching is necessary in symmetric, intercalative FDI to make desalination pseudo-continuous, unless device architecture is modified to eliminate solution switching. The apparent mixing between diluate and brine streams is inevitable to valve switching, but here its effect was minimized by introducing a calibrated delay of 3.2 s between the switching of inlet and outlet valves.

[0087] Cell potential varies between -0.4 V and 0.4 V among galvanostatic charge and discharge half-cycles with 1 mA / cm2current. Such variations of cell potential V and current I produce the electrical contributions to energy consumption: Eetec= f I • Vdt. Pumping energy consumption Epump= f psupply• Vtotaidt results from the supply pressure psuppiy= VtotaiLe,\\E / kappLe, z needed to force water at a total flow rate Vtotaithrough IDFF-pattemed electrodes having certain apparent permeability kapp. Here, Le| and Le ±are respectively the longitudinal and transverse dimensions of the electrode. After each half-cycle there is a sudden change in concentration that is evidence of inter-stream mixing. This effect becomes more prominent with increasing concentration difference between the diluate and brine reservoirs, ultimately reaching a limiting point where the deleterious concentration change due to mixing matches the beneficial concentration change caused by the applied current. Concentration increase within the brine reservoir is suppressed due to an extent of water crossover through the AEM.

[0088] FIG. 6 shows the thermodynamic energy efficiency 600 for desalination experiments including pumping energy alone Epump, including electrical energy alone Eeiec, and total energy Etotaiincluding electrical and pumping energies. Each experiment was conducted in triplicate with the values shown being the average among all experiments, while error bars indicate their standard deviation. Tapered- and straight-channel IDFFs are shown to produce salinity near that of freshwater (17 mM) with 19 mM and 14 mM, respectively. A 62% reduction in volumetric energy consumption (VEC) due to pumping energy is achieved with tapered-channel IDFF electrodes (2.5 kWh / m3) as compared to straight-channel electrodes (6.6 kWh / m3) at maximal salt removal. The energy saved by using electrodes with tapered-channel IDFFs is caused by their 2-3 times increased apparent permeability.Including both electrical and pumping energy consumption, the VEC at maximal salt removal of tapered-channel IDFFs (7.3 kWh / m3) is 32% lower compared to straight-channel IDFFs (11 kWh / m3). Both types of IDFF channels show superior total VEC relative to small-scale seawater reverse osmosis (RO; 15 kWh / m3), multistage flash distillation (20-27 kWh / m3), multi-effect distillation (14-21 kWh / m3), and thermal vapor compression (16 kWh / m3).

[0089] The thermodynamic energy efficiency of tapered-channel IDFFs is shown to be substantially greater than that of straight-channel IDFFs when desalinating feeds with seawater-level salinity. While only a 1.3-fold improvement in efficiency was produced at maximal salt removal, at «50% salt removal tapered-channel IDFFs double the total thermodynamic energy efficiency including electrical and pumping energy (18%) relative to straight-channel IDFFs. While 50% salt removal may be impractical if standalone FDI is used for desalination, applications using conventional desalination processes could use this FDI process for pre-treatment when the feed concentration of source water exceeds the rated concentration (e.g., littoral waters). While tapered-channel IDFFs present promising thermodynamic energy efficiency, the gap between its value including electrical energy aloneand total energy indicates that further improvements in hydraulic permeability could produce total energy efficiency as high as 38% at «50% salt removal and as high as 13% at maximal salt removal.

[0090] Results of desalination experiments using tapered-channel IDFFs fed with brackish water containing 104 mM NaCl were obtained. Only 0.19 kWh / m3of pumping energy was needed to produce 8 mM salinity below freshwater compared to 2.5 kWh / m3for desalination of seawater salinity. Including pumping losses, the total energy consumption of only 0.69 kWh / m3is still lower than that of brackish water RO (1.5-2.5 kWh / m3) and electrodialysis (2.6-5.5 kWh / m3). Furthermore, these tapered-channel electrodes obtain a maximum thermodynamic energy efficiency of 31% with 82% salt removal, outperforming RO, electrodialysis, and capacitive deionization using electric double layers.

[0091] The specific energy consumption (SEC) resulting from tapered-channel IDFF experiments was also compared with that of earlier FDI work that used symmetric Prussian blue analogue electrodes. Previous work either has desalinated brackish feeds with modest salt removal less than 60% or has desalinated feeds with seawater-level salinity with even lower salt removal of 20%. However, the batch-type experiments performed here produces a range of SEC values resulting from the increasing degree of salt removal with increasing number of electrochemical half-cycles. Using brackish feeds with 50% salt removal produced similar electrical SEC contributions to earlier experiments with less than 30% salt removal. Using seawater-salinity feeds with 10% salt removal produced half the electrical SEC of earlier experiments with 20% salt removal. In addition, the tapered-channel desalination experiments performed using seawater-level salinity with salt removal >95% show a 30% reduction in total SEC compared to straight-channel IDFFs.

[0092] In addition to the low energy consumption needed when using tapered-channel IDFFs, their decreased supply pressure reduces the driving force for water crossover through AEMs, as quantified by process water recovery and charge efficiency. Water recovery is defined as the ratio of the diluate volume VDto the total water volume VtotaiWR = VD / Vtotai- Charge efficiency A is defined as the ratio of the moles of salt removed nsaitto the moles Q / F of charge Q transferred. For seawater- salinity feed, both water recovery and charge efficiency decrease from their initial half-cycle values by more than half. However, tapered-channel IDFFs show only very weak drops in both water recovery and charge efficiency for brackish-salinity feeds. The apparent water crossover rates through the AEM were measured as 0.11 L / m2-h and 0.091 L / m2-h respectively with seawater and brackishsalinity levels. These rates are lower than those reported for commercially available membranes, electrodialysis, and flow batteries, which typically range from 0.24-2.5 L / m2-h.

[0093] While channel design does not directly influence water productivity, it enables a scaled-up system to perform more efficiently. To achieve a nominal water productivity of 1 L / h with the present seawater desalination productivity of 0.077 L / m2-h, the present architecture would need 13 m2of membrane area. Such an area could be achieved by either increasing electrode size or by using a stack of multiple electrode pairs. However, an increase in electrode size inherently demands longer channel lengths. Using tapered-channel IDFFs ensure that the transverse superficial velocity remains uniform irrespective of channel length. Increased current density can be used to increase productivity in conjunction with increased electrode area, albeit with increased electrical energy consumption. Patterning of tapered-channel IDFFs (e.g., high-pressure channels 102, 202, 302 and / or low-pressure channels 104, 204, 304) on large-format electrodes may also be used to scale up productivity. Direct milling may be used to form tapered-channel IDFF electrodes. Additionally, high-throughput patterning techniques such as laser engraving may be used to form tapered-IDFF electrodes. Additionally, such milling and laser engraving techniques may be used to form flow channels in other mediums.

[0094] RESULTS

[0095] The tapering of flow channels may be used in IDFFs to reduce or eliminate dead zones to which straight channels are prone. Such tapered flow channels may be for IDFFs embedded in cation intercalation electrodes for electrochemical desalination. In addition to electrochemical separations, channel tapering of this sort can be applied broadly to facilitate the flow of liquid electrolyte or of gaseous feeds in flow batteries and fuel cells that use IDFFs machined into bipolar plates. In some examples, the hydraulic conductance along the flow channels (i.e., high- and low-pressure) vary linearly. This may minimize pressure drop and mean-square slope. It is possible that for certain applications other objective functions could be of interest for minimization, in which case the subspace of conductance profiles can be explored using an infinite cosine series (Eq. 3) or using trial conductance profiles (Eq. 2). This can be extended more broadly by also including the effect of non-uniform flow-path length between channels, though for purposes of this analysis, it is assumed constant.

[0096] Implementation of these designs can be tailored to a chosen method of manufacturing IDFF channels. As one example, linear conductance scaling to produce uniform flow (Eq. 4) imparts one constraint to the geometry of a given channel’s cross-section. For a micro-engraving process, in one experimental example, varying only width or depth led to either an unresolved channel region of -10% or an initial segment covering 50% of the entire channel length. Thus, additional constraints imposed by manufacturing can be included to produce different channel shapes than those provided herein for a particular manufacturing process (e.g., micro-engraving), while still accounting for the aforementioned conductance scaling. An experimental library of channel cross-sections (e.g., library 551 of FIG. 5B) whose hydraulic conductance is simulated can be created for other manufacturing methods to co-design tapered channels having piecewise-constant or continuously varying cross-sections. Laser micro-machining offers, in some examples, improved manufacturing scalability.

[0097] When implementing tapered channels in IDFFs that are embedded within porous electrodes, spacing between channels can be impacted by the micro-environment occurring within electrode pores. Specifically, the hydrophobicity of binder likely causes a percolative path-length dependence of the hydraulic permeability within example porous electrodes. By choosing a small inter-channel distance for tapered-channel IDFFs, the hydraulic permeability between embedded channels could be maximized while limiting material removal to 30%.

[0098] Desalination experiments using symmetric FDI with tapered-channel IDFFs embedded in cation-intercalation electrodes showed significant reductions in energy use relative to straight-channel IDFFs. While both electrodes are effective in producing freshwater from seawater-salinity feeds, tapered-channel IDFFs did so with 62% lower pumping energy due to >2-fold increased hydraulic permeability: 2.5 kWh / m3compared to 6.6 kWh / m3. Additionally tapered-channel IDFFs required 32% less overall energy than their straight-channel counterparts. This lower energy consumption of tapered-channel electrodes produced 2-fold and 1.3-fold increases in thermodynamic energy efficiency respectively at 50% and >95% salt removal from seawater salinity feeds. The total energy consumption for brackish-salinity feeds can be reduced by 10 times compared to seawater- salinity (0.69 kWh / m3compared to 7.3 kWh / m3) by using optimally tapered IDFF channels. For brackish feeds tapered-channel IDFFs maximized thermodynamic energy efficiency at 82% salt removal to 31%, which exceeds the efficiency of conventional small-scale brackish desalination processes. Water recovery was also shown to be limited by water crossover rates through the intervening anion exchange membrane respectively at 0.11 L / m2-h and 0.091 L / m2-h for seawater- and brackish- salinity feeds. While these crossover rates are lowerthan for commercial ion-exchange membranes in electrodialysis and flow battery stacks (0.24-2.5 L / m2-h), reduction of water crossover could further increase water recovery and efficiency.

[0099] MATERIALS AND METHODS OF AN EXAMPLE IDFF

[0100] Synthesis of NiHCF Nanoparticles

[0101] Nickel hexacyanoferrate (NiHCF) nanoparticles were synthesized by a coprecipitation method. Specifically, equal volume of 0.1 M K3Fe(CN)e and 0.2 M NiCh solutions were added dropwise into a beaker containing deionized water using two separate peristaltic pumps (NE-9000G, NewEra Pump System Inc.) at a flow rate of 1.5 mL / min under vigorous stirring. The resulting brown, yellow dispersion of NiHCF nanoparticles was sonicated for approximately 30 minutes and was allowed to settle down overnight.Afterwards, the nanoparticles were collected by centrifugation at 3900 rpm for 5 minutes. This process was repeated several times until the supernatant became clear before it was transferred to a petri dish to dry in an oven at 70°C overnight.

[0102] Electrode Fabrication

[0103] Porous electrodes supported on graphite foil were fabricated using NiHCF nanoparticles, Ketjen black (KB) conductive additive (EC-600JB), and PVdF binder (Solvay Solef 5130) in an 80:5:15 mass ratio. The PVdF binder was dissolved in N-methyl-2-pyrrolidone (NMP, Sigma Aldrich) using a planetary mixer (Thinky, ARE-310) at 2000 rpm for 30 minutes to obtain a viscous solution of 50 mg / mL concentration. NiHCF nanoparticles and KB were finely ground in a vortex mill (Ultra Turra-X, IKA) using 18-20 steel balls (5.0 mm diameter) at 6000 rpm for 30 minutes. This powder mixture was combined with the PVdF solution in a ratio of 1 g of solid material per 3 mL of NMP using the planetary mixer at 2000 rpm for 30 minutes to obtain a homogeneous slurry. The slurry was then cast onto graphite foil (Ceramaterials) current collectors with 120 pm thickness using a doctor blade and a film applicator (Elcometer 4340) to produce wet electrode films of about 1.3 mm thick. These films were then solidified by wet phase inversion in a coagulation bath of deionized water heated to 85°C for 10-15 minutes. The electrodes were then dried in a fume hood for 2-3 hours before being placed in an electric oven at 70°C for about 12 hours. The uncalendared thickness of the dry electrodes, excluding graphite foil thickness, was measured to be 350 ± 50 pm. These electrodes were then calendared using a roll press (MTLXTL) down to 200 ± 25 pm to reduce porosity and to enhance electrical conductivity before they were engraved with IDFFs over a 45 mm x 45 mm area using a desktop computer numerical control (CNC)machine. Electrode porosity and active material (NiHCF) loading were measured, as shown in Table Al.Table Al: Electrode properties with tapered- and straight-channel IDFFs.Channel NiHCF loading Porosity (%) Permeability (pm2) type (mg / cm2) Uncalendared Calendared kpekavv19.1 60.5 32.6 0.35 70apere20.0 58.7 28.8 0.30 6518.2 62.7 35.6 0.11 17Stolg 117.5 64.8 37.9 0.15 32

[0104] Porosity and Permeability Measurement

[0105] The porosity s of each porous electrode was determined as:£= 1 - ^Pc

[0106] where Pf is the electrode’s film density and pcis the mass-weighted density of the electrode’s constituents:1f NiHCF0JKB ™PVdF\^PNiHCF PKB PpVdF ''

[0107] where a)NiHCF, a)KB, and a)PVdFare the mass fractions of NiHCF (0.8), KB (0.05), and PVdF (0.15), respectively. The densities of NiHCF pNiHCp, KB pKB, and PVdF pPVdFwere respectively estimated as 2.0 g / cc, 2.0 g / cc, and 1.75 g / cc.

[0108] The hydraulic permeability of each electrode was measured using a gravity- driven apparatus. Here, a constant pressure head p was maintained and Darcy’s law was used to determine the permeability of the electrodes:, PLe,\\Vk = -; —PAC

[0109] where p is the dynamic viscosity of water (0.890 mPa-s), Le^ is the longitudinal length of the electrode, Acis the superficial normal area of the electrode through which flow occurs, and V is the volumetric flow rate calculated as the ratio of the volume of water passed through the porous electrode to the elapsed time.

[0110] Micro-Engraving of IDFFs into Porous Electrodes and Library Generation

[0111] IDFF channels were embedded into porous electrodes using a piecewise-constant conductance profile as described already. The toolpath for engraving these channels was determined using AutoCAD Fusion 360, and G-code was generated for execution on adesktop CNC machine. The machine was equipped with precision ball screws operated by closed-looped stepper motors controlled by GRBL Mega v.1.1 motion control fireware. A V-bit engraving tool (Drill Bits Unlimited), featuring a 100 pm tip diameter and a 10° tip angle, was used for micro-engraving. A runout of about 80 pm was considered and compensated for due to the asymmetry of the tool bit. The electrode sample was secured on the CNC machine using a perforated aluminum plate (McMaster-Carr) mounted to a vacuum clamp (NEMI).24,000 rev / min spindle speed was used with accompanying HEPA-filtered dust collection. The resulting channel cross-sections were characterized using optical profilometry (Keyence VK-X1000) showing a deviation from nominal dimensions within ± 5%. Representative cross-sections were determined by post-processing profilometry data after aligning each channel’s axis in Cartesian space. The corresponding channel conductance was simulated numerically based on a finite- volume scheme for two-dimensional variation of axial velocity u (y, z) subject to incompressible, laminar, fully developed Poiseuille flow with no-slip boundaries:d2uxd2uxI dpdy2 +dz2p dx

[0112] Symmetric, Intercalative FDI and Desalination Metrics

[0113] Two 250 pm thick graphite foil sheets were used for current collection in the symmetric, intercalation-based FDI flow cell. IDFF-pattemed 45 mm x 45 mm electrodes were then abutted to them as the cathode and anode, separated by a 150 pm thick anion-exchange membrane (Neosepta AMX). Before flow-cell assembly, one electrode was charged to obtain a 0% state-of-charge (SOC, 0.6 V vs. Ag / AgCl) in a flooded cell using a three-electrode setup, while the other was similarly fully discharged to obtain an SOC of 100% (0.1 V vs. Ag / AgCl). The flow-cell assembly was clamped using two 3D-printed end plates. Two Z-type manifolds embedded within gasket material were used to supply diluate and brine to the respective electrodes. A VMP-3 multichannel potentiostat was used to supply 1 mA / cm2current in all desalination experiments, initially with 8 mF of diluate and 2 mF of brine. Salt concentrations were measured using a 930 Compact Ion Chromatography system (Metrohm) while a Conduino system was used to track the real-time concentrations of diluate and brine using calibrated conductance probes.

[0114] Desalination performance metrics were determined as follows. The thermodynamic energy efficiency (TEE) was calculated as the ratio of the minimum specificenergy consumption (SEC) to the cumulative SEC for each half-cycle. The SEC (kJ / mol) was determined using the electrical energy Eetecrequired to remove nscMmoles of salt per cycle.E '-‘e,lec ^Jt0endI{t)V{t)dtSEC = - = -W-salt Tlsalt

[0115] where / (t) and E(t) are the applied current and cell potential as a function of time t. The minimum SEC was calculated as:A c r* / -'A ^^rev{SEC )min= - -W-salt

[0116] where Wrevis the amount of reversible work required for salt separation, determined using non-ideal activity coefficients:Wrev= 2RT[VDCDln( / fDCD) + VBCBln( / ±CfiCB) - {VD+ VB)CE1U( / ±CFCF)]

[0117] where V and C are volume and concentration with the subscripts D, B, and F being diluate, brine, and feed, respectively. The mean activity coefficientfor any given salt concentration was obtained. The volumetric energy consumption (VEC) is the product of the specific energy consumption and the cumulative concentration of the salt removed.

[0118] The water-crossover rate ( / w) through the anion-exchange membrane was also calculated for each experiment as the ratio of the difference between initial (VDo) and final (ko ) diluate volume to the product of the anion-exchange membrane area {AAEM) and the elapsed time At:WAAEMM

[0119] ADDITIONAL MODELING OF TAPERED, INTERDIGITATED CHANNELS FOR UNIFORM FLOW

[0120] Steady-State, ID Analysis of Limiting Current

[0121] The effect of dead zones on the limiting current may be determined by using a one-dimensional (ID) steady-state advection-reaction equation. This may be indicative of a thin electrode section through which flow occurs along a path length s that is identical to the inter-channel distance between high- and low-pressure channels in the transverse direction within interdigitated flow fields. The reactant concentration C(y = 0) at the inlet may be assumed known as Cin. Consequently, the reactant concentration Coutat the outlet depends on the total applied current I and other parameters related to the electrode’s geometry.

[0122] For such a ID problem shown mass conservation of the reactant species requires the following differential equation to be satisfied, assuming equimolar stoichiometry between electrons and reactant species:—+V. (USC+7) = -7?

[0123] where E is the electrode porosity, us= us lj is the superficial velocity vector along the y direction, and J is reactant flux due to diffusion and migration relative to the bulk flow. us ±is the superficial transverse velocity, a is the volumetric reactive area in units of m2 / m3, and inis the pore-scale current density at electrode / electrolyte interfaces.

[0124] An analytical solution for steady- state operation with negligible diffusion and migration flux may be determined, simplifying the associated differential equation to one that is integrable:d, x citn

[0125] This expression is integrated to obtain the variation of concentration C along the flow path from y = 0 to y = s:a fs_Font ^in F’ Ilndyus,±F Jo

[0126] Hereindy is equal to the product of average pore- scale current density in) with inter-channel distance s:a{in)s'-•out '-•in i-.US,LF

[0127] To avoid reactant depletion Cout> 0 may be assumed, such that the following inequality must be satisfied to avoid depletion:a{in)s< 1Fin^s,±F

[0128] The total applied current can be expressed in terms of {in) as / = a{in)hsNL. Using this result a total limiting current can be expressed as:im CinFus ±hLN

[0129] lUmis an advection-limited current that is dictated by the rate of reactant supply via the bulk flow. Alternatively, a limiting current density can be expressed by normalizing the total limiting current with current collector area:. _ h uimrn _ „ / _h\ iLlim, »,us,±'^inrI / LsN \S /

[0130] The above expression shows that the limiting current density scales directly with superficial transverse velocity himus,±> showing the direct effect of dead zones on the device performance. iumis also affected by the geometry of both the electrode and the channels that are used. Linear scaling between the limiting current and the electrode thickness { im)cc~h may be a direct manifestation of neglecting diffusion altogether in the present ID model.

[0131] Quasi-ID Theory oflDFF Flow

[0132] This quasi- ID model for flow through IDFFs may be employed to explain how variations in channel cross-sections can ensure uniform flow distribution.Til (%) — 1^| (% + dx) = 2dl / ±

[0133] where V|| represents the longitudinal flow rate, which depends on the axial location, and dV±denotes the differential transverse flow rate through one side of the control volume. To solve for the velocity distribution, Fy was expressed using a Poiseuille-flow relation between it and the axial pressure gradient dpH / dx in the high-pressure channel:GHdpHp dx

[0134] where GHis the hydraulic conductance in units of m4for a given straight crosssection in the high-pressure channel and p is the dynamic viscosity of the working fluid. Darcy’s law was implemented to obtained dV±in terms of average transverse superficial velocity us l:i i kveh(pH- pL)dV±= |us.j_|hdx = - dx

[0135] where s is the inter-channel distance that defines the flow-path length through the electrode and p is the absolute pressure at a certain location. Subscripts H and L indicate high- and low-pressure channels, respectively. Substituting the expressions for Vy and dV into the above conservation equation yields an ordinary differential equation (ODE) that governs pressure in the high-pressure channel:d 2kpeh(pH- pL) =o— Gdx dx s

[0136] Since the channels may be tapered in order to uniformize flow, this ODE is normalized by the maximal cross-section and introducing a dimensionless position x* = x I with L being the length:d GHdpH2kvehL2— - (pH- pL) = 0dx* Godx*

[0137] A similar expression can be derived for the low-pressure channel. The solution to these equations thus depends only on the dimensionless constant E = kpehL2 / Gos, the ratio between the longitudinal hydraulic resistance and the transverse hydraulic resistance, and the dimensionless conductance functions GH(x) / G0and GH(x) / G0that are the result of variations of channel cross-section for tapered channels or the lack thereof for straight channels.

[0138] Subtraction of the above ODEs for IDFFs using straight channels with GH= GL= Goproduces a single ODE that governs the pressure difference Ap = pH— pL

[0139] The method of trial solution, along with symmetry boundary conditions at both ends of the electrode ( Ap |x=0= Ap|x=L= (Ap)0), produces an expression for the pressure difference between channels that varies with x in the following fashion:sinh ^2VE"(1 — x*)) + sinh(2VE"x*)(Ap)o sinh(2 E")

[0140] The pressure difference Ap = pH(x) — pL(x) is then substituted into each ODE to obtain integral expressions for the absolute pressures pHand pLpH(x*} = - — - ( ^21 f f (sinh (2 E"(1 — x*) + sinh(2 E"x*) dx*dx*Gosrsinh(2yE J J JK K 7 7pL(x*) = - — - ^21 [ [ (sinh (2 E"(1 — x*) + sinh(2 E'x*) dx*dx* Gosrsinh(2yE J J JK K 7 7

[0141] The above expressions may then be solved using no-flux boundary conditions at channel tips (dpH / dx*\x*=1= dpL / dx*\x*=0= 0) subject to the symmetry boundary conditions pH— pL|x»=0= pH— pL|x*=1= (Ap)0and an outflow boundary condition pL|x*=1= 0 to yield the following absolute pressure distributions for straight-channel IDFFs:PH (X,) =2[sinh(2VT)+ 72 (^(272 ) - 1)] M2^1 +^(2^) — 2Vs"%"(cosh(2Vs') — 1) + sinh(2Vs") + 2 ^(cosh(2V3") — l)jPI(X,) =" 2[sinh(2V^TvT(Pc h(2V=) - 1)] [S‘nh(2V"(1” ’’)+ Sinh(2^3+ 2VE"x"(cosh(2VE") — 1) — sinh(2VE") — 2V^(cosh(2VE") — l)j

[0142] For IDFFs using optimally tapered channels, linear conductance scaling may be used together with expressions for the longitudinal flow rate ^(x) to determine the pressure gradient within each channel:dpH_ 2hL|usj (l - Q p _ 2hL\uS J_\ (1 -£) / z _ 2hL\uSil]pdx GH(x) Go• (L — x) / L Go

[0143] Such a pressure gradient thus corresponds to linear variation of pressure in the high-pressure channel:He \ H 2 / IL|IZSJ_| / ZpH(x) = p" - - - X

[0144] By recognizing that Ap is also invariant along such tapered channels, linear variation of pressure pL(x) in the low-pressure channel is also produced:,u„ 2hL\u<. i Ip / L \ P (x) = pH(x) - Ap = p? - -H - X + —o0\ Z£ /

[0145] By introducing the normalized position x* = x / L:Hr n H 2hL2\uSil]up (x ) = p" - - - (x )GoLr *> H 2hL2\uS:±\p / 1 \pL(x*) = p" - L-LTX’ + -O0\ Z£ /

[0146] To express the above pressure distributions in terms of supply pressure, transverse superficial velocity magnitude |izs ±| may be determined in terms of supply pressure, Pin— Pout ■|_ | > ^o(. Pin ~ Pout)lUs.-L I—7 ]" T2phL2(1 +

[0147] Substituting this relation for |us j_| into the associated pressure equations enables their non-dimensionalization:Pin ~ pH(x*J X*Pin - Pout 1+Put ~pL(x*)=** + 2^Pin - Pout 1+

[0148] The associated variations of longitudinal flow rates in high- and low-pressure channels for both straight and tapered channel electrodes were obtained using the corresponding expressions for pressure. The resulting distributions shown in Table SI. Table SI: Parameters for the determination of tapered- and straight-channel IDFFs distributions shown in Fig. 1 of the main text.Parameters Value Unit Channel length, L 50, 100, 200 mm Channel depth, h 0.2 mm Channel nominal width, w 0.35 mm Inter-channel distance, sr0.55 mm Porous electrode permeability, kpe10 pm2Total pressure difference, Ap = p^n— pflt1000 Pa Dynamic viscosity of the working fluid, p 0.00089 Pa — s

[0149] Cosine-Series Conductance Profiles and Perturbation from Linear Scaling

[0150] A cosine-series solution of Eq. 1 may be used to provide uniform flow between tapered-channel IDFFs:GH(x) ■ x = GHL — x) • (L — x)

[0151] From this equation, / (x) = GH(x) • x is an even function about the channel’s midpoint (x = L / 2). Accordingly, (x) may be chosen to be a cosine series that is an even function about x = L / 2:∑ / (x) = XGH(X) = > bnCOS(HTT(X — L / 2) / L)C— <n=0

[0152] A no-flux boundary condition may then be applied at the channel’s tip such that GHx = L) = 0:∑ / (x = L) = L • GH(x = L) = L > bncos(nn / 2) = 0■4-^71= 0

[0153] The term for n = 0 may be considered to obtain:V 100^0 = — y bncos(nn / 2)

[0154] Substitution of this result produces the following expression for (x):100(x) = xGH(x) = > bn[cos(n7r(x — L / 2) / L) — cos(n?r / 2)]<n=l

[0155] To ensure that the resulting conductance distribution is physically realizable, this function may be set to produce a finite conductance Goat x = 0: GH(x = 0) = Go. Thus, the following limit may be set to be finite:G ro= r hm> - x->0 X

[0156] Since the denominator of the above limit approaches zero for vanishing x, each term in the numerator may be set to approach zero to produce an indeterminate limit:cos(— n?r / 2) — cos(n?r / 2) = 0

[0157] Since cosine is an even function, this condition is automatically satisfied for every n > 1. L'Hopital's rule may be applied to determine an expression for Goin terms of the cosine series coefficients bnf xGo= lim - = lim f'(x) = — >nn sinx->0 X x->0n=l ' '

[0158] Since terms associated with even n are found to zero in the above series for Go, the magnitude of Gois shown to result only from odd terms:Go = -vY b2m+1(-l)m(2rn + 1)L*—‘m=0

[0159] Thus, for Goto be non-zero requires the inclusion of at least one mode having odd n in the cosine series. Also, arbitrary choices for bndo not assure that GH(x) is positive over the whole interval nor that GH(x) is non-zero for x E [0, L). Thus, inspection of each profile is needed to ensure such conditions.

[0160] Channels use conductance profiles that are perturbed from linearity according to the cosine series for linear conductance scaling:4(— l)n+1GH(x) = G0(l — x / L = Go— - -: — [cos(nπ(x — L / 2) / L) — cos(nπ / 2)] xZ_in=1nz7Tz

[0161] Perturbations were produced by fractionally varying coefficient bmof the particular mthmode within the associated cosine series. The associated distribution of pressure within each channel was then determined numerically by integrating the pressure gradient produced by each conductance profile, subject to a certain inlet flow rate Vj|j0:dpH= / z2(l - x / L)V||,o= / z2(% / L)V||,0 =Vdx GH(x) GL(x) dx

[0162] Among all perturbations, channel volume was constrained to be equal and pressure drop was normalized by the corresponding value (, Np)Unearfor linear conductance scaling. Consequently, a perturbation bm / bmof zero produces (.^p^perturbed / ^P^Unear = 1. To fix the volume among all flow channels (e.g., high- and low-pressure) relationship between conductance and channel width w, in this example, was assumed. Irrespective of the conductance / width scaling that was assumed, the pressure drop of channels having linear scaling of conductance with longitudinal position are shown to produce pressure drop within 0.2% of the lowest pressure drop observed among all perturbed conductance profiles. When a local-linear law is assumed, linear scaling of conductance with longitudinal position produces lower pressure drop than any other perturbed profile.

[0163] Nominal Cross-Section Dimensions of Piecewise Constant Channels

[0164] In some examples, piecewise-constant tapered high- and low-pressure channels may be used (e.g., channels 302 and 304 of FIG. 3 and channels 502 and 504 of FIG. 5C). Example cross-sectional dimensions of piecewise-constant tapered channels are shown for joint variations of width and depth in Table S2, variations of width alone in Table S3, and variations of depth alone in Table S4. The same is shown in Table S5 for straight parallel channels that were used to characterize the scale dependence of hydraulic permeability.Table S2: Geometric parameters of segments used to design tapered-channel IDFFs on the nickel hexacyanoferrate (NiHCF) cation intercalation electrode varying both width and depth. Segment 1 corresponds to the first segment at the high-pressure channel inlet, and segment 11 corresponds to the last segment at the channel tip.Segment Segment depth Segment width Segment length no. (mm) (mm) (pm)1 0.200 0.350 111232 0.200 0.300 114543 0.150 0.345 80844 0.150 0.290 40765 0.150 0.225 33986 0.100 0.270 21597 0.100 0.215 13778 0.075 0.325 9379 0.075 0.275 34210 0.075 0.225 37011 0.075 0.180 707Table S3: Geometric parameters of segments used to design tapered-channel IDFFs on the PBA electrode with fixed depth.Segment Segment depth Segment width Segment length no. (mm) (mm) (pm)1 0.200 0.450 18552 0.200 0.400 40953 0.200 0.350 49134 0.200 0.300 37135 0.200 0.250 62476 0.200 0.200 90267 0.200 0.150 10700Table S4: Geometric parameters of segments used to design tapered-channel IDFFs on the PBA electrode with fixed width.Segment Segment depth Segment width Segment length no. (mm) (mm) (pm)1 0.200 0.350 215932 0.150 0.350 141113 0.100 0.350 61334 0.075 0.350 2819Table S5: Geometric parameters of four electrode samples with embedded with parallel channels, each having different inter-channel distance while maintaining constant channel length and depth.Parameters Electrode 1 Electrode 2 Electrode 3 Electrode 4 Channel length, L (mm) 45 45 45 45 Channel depth, h (mm) 0.200 0.200 0.200 0.200 Channel width, w (mm) 0.225 0.225 0.225 0.350 Inter-channel spacing, s (mm) 0.338 1.181 1.506 2.463 No of channel, N (— ) 80 32 26 16

[0165] Analysis of the Gap Distance Between Channel Tips and Electrode Edges

[0166] The gap distance between channel tips and electrode edges g may affect the functionality of IDFFs as a result of a competition between (i) the short-circuiting of flow through channel tips when g is small and (ii) the formation of dead zones around channel tips when g is large. An analytical model may be used to enable the rational selection of gap distances that balance these competing effects. To do this, a single high-pressure channel within an IDFF may be considered. The transverse flow rate V±produced into porous electrode towards adjacent low-pressure channels is expressed in terms of the thickness-averaged transverse superficial velocity us ±, channel length L, and electrode thickness h:V±= 2us lLh

[0167] Using Darcy’s law, us ±at the high-pressure channel’s tip may be expressed in terms of the high-pressure channel’s pressure p"pand the low-pressure channel’s pressure PoUtatthat position:kpe Ptip Pout)us ± = - p S

[0168] For IDFFs using channels that are tapered with linear conductance scaling to produce uniform transverse flow:• 2 / CpgL / l,.v± =~iir~ ^ip - Pout)

[0169] At the same time the flow rate through the tip of such a high-pressure channel is driven by the same pressure difference:• > Ptip ~ PoutVtip ~ n-tip

[0170] where Rtipis the hydraulic resistance associated with such a tip flow. Owing to the similarity between the governing equations for heat conduction and Darcy flow (i.e., the homogeneous form of Laplace’s equation), Rtipmay be expressed in terms of a conduction shape factor a having units of length:1 kpe,\\- = - < JRtip P

[0171] Here, kpe^ is hydraulic permeability within the porous electrode for the associated tip flow. To find a conservative estimate for a, flow between the tip of the channel and the edge of the electrode may be modeled by assuming a planar tip with cross-section that is identical to the maximal cross-section of the channel. For one-dimensional Darcy flow under such conditions, the corresponding shape factor is found to be:woha = - 9

[0172] where w0is the maximal width of the channel.

[0173] To ensure that flow is not short-circuited through channel tips, Vtipmay be small (e.g., < 1%) in comparison with V±:Vfip > ^pe,\\^^ ^pe,||W()S QV, 2knpLh 2knpLq

[0174] Thus, a condition for the gap distance needed to assure tip flow smaller than 1% may be obtained:Wn S k'jjp11S > 50-2-- -p-L K.pe

[0175] Assuming kpe / kpe~ 1, a 350-pm-wide channel with an inter-channel distance of 550 pm requires g > 200 pm for an electrode that is 45 mm long.

[0176] In one experimental example, a 500 pm gap distance to produce a target tapered channel length of 44.5 mm in order to ensure that flow passes through porous-electrode regions between channels rather than through channel tips. However, a region of targeted design near the channel’s tip remained unresolved in practice due to limitations in the microengraving process where the smallest achievable channel width was approximately 150 pm due to the finite width of the engraving tool used.

[0177] Air Permeability Measurement of Patterned Electrodes with Thru-Channels

[0178] To characterize the permeability of electrodes patterned with thru-channels oriented parallel to the flow direction with air as the working fluid, a custom manometer was built to measure pressure while air’s flow rate was controlled using a syringe pump. The resulting air permeability values are shown in Table S6.Table S6: Geometry and permeability of the parallel thru-channels embedded in electrodes for hydraulic permeability measurement with air.Parameters Value Unit Channel length, L 45 mm Channel depth, h 0.2 mm Channel nominal width, w 0.35 mm Channel Coverage 12.4 % Porous electrode permeability, kpe0.42 pm2Experimental apparent hydraulic permeability, 240 pm2Theoretical apparent hydraulic permeability, k^p270 pm2

[0179] Conductivity Measurement and Valve-Delay Time Calibration in an Experimental Example

[0180] To track the real-time salt-concentration changes during desalination experiments, two custom, 3D-printed probes were used based on commercial micro-USB and a Conduino apparatus. Prior to each experiment, each probe was calibrated for NaCl concentrations ranging between 20 to 600 mM. The probes were then connected to the two inlets of the symmetric FDI flow cell. Calibration was performed by attaching the probes to a peristaltic pump, and a solution of know concentration was fed through them at a controlled flow rate. The conductance versus time traces for each probe were generated using an in-house code.Multiple dilutions were carried out to vary the solution concentration with each probe producing a distinct curve. Sufficient time for mixing during each dilution step was ensured. For example, diluting a 5 mL solution of 500 mM by adding 1.25 mL deionized (DI) water yields a 6.25 mL solution with a concentration of 400 mM. At a flow rate of 1 mL / min, the system required 6.25 minutes to achieve complete mixing. Sample data for calibration is shown in Table S7.

[0181] A concentration versus conductance plot was created for both probes, and a second-order polynomial fit was applied to determine the trend of concentration variation.The fitted trend was subsequently utilized to estimate concentration from conductance measurements during desalination experiments.Table S7: Calibration data for conductance probes with a constant 5 mL / min flow rate.K TFlow rate Concentration Volume Added Volume Conductance (S) No.(mL / min) (mM) (mL) (mL) Probe 1 Probe 2 1 5.0 576 10.00 0.00 0.226 0.272 2 5.0 551 11.43 1.43 0.210 0.250 3 5.0 502 13.33 1.90 0.195 0.228 4 5.0 398 16.00 2.67 0.175 0.202 5 5.0 332 20.00 4.00 0.153 0.172 6 5.0 262 26.67 6.67 0.127 0.139 7 5.0 178 40.00 13.33 0.094 0.099 8 5.0 92 80.00 40.00 0.054 0.055 9 5.0 47 160.00 80.00 0.031 0.031 10 5.0 24 320.00 160.00 0.018 0.017

[0182] A delay time between switching the inlet and outlet valves was used to minimize intermixing of the effluent streams following each half-cycle. If all valves were switched simultaneously, charge efficiency A would decrease due to the mixing of the diluate and brine within the electrodes and connecting tubes, which could then flow into the opposite reservoirs. To determine the optimal delay time for switching the outlet valves, a delay time of 4.0 s was initially set for a pair of electrodes with a thickness of approximately 200 pm. Apparent mixing volume was characterized at different flow rates, ranging from 3 to 7 mL / min (Table S8). Two reservoirs were used: one containing a diluate solution(VDo=5 mL) with an initial concentration of about 90 mM, and the other containing a brine solution (VBo= 5 mL) with an initial concentration of about 650 mM. The apparent mixing volume Vmixwas calculated using the change in salt concentration AC = CBQ— CB1and the mass conservation of salt, according to the following equation:AC x VBnvmix p _ pLBOLD0

[0183] where C and V represent the concentration and volume, with subscripts B, D, and 0 corresponding to brine, diluate and initial, respectively.

[0184] The flow rate that resulted in the lowest mixing volume was then used to determine an optimal delay time of 3.2 seconds for a flow rate of 5 mL / min.Table S8: Data for optimizing delay time between inlet and outlet valve switching.Flow Rate CB CD VB VD Vmix °’ (mL / min) (mM) (mM) (mL) I'mL) I'mL) 1 3 584.31 189.09 5.00 5.00 0.63 2 4 547.56 252.52 5.00 5.00 0.47 3 5 516.39 310.72 5.00 5.00 0.53 4 6 484.79 376.23 5.00 5.00 0.77 5 7 450.77 452.39 5.00 5.00 1.57

[0185] Water Recovery and Charge Efficiency for Tapered-Channel IDFFs

[0186] The desalination performance of electrodes with tapered-channel IDFFs for brackish- salinity feed is shown in FIG. 7.

[0187] Wet-Phase Inversion Conditions and Electrode Properties

[0188] Electrodes were solidified using a wet phase-inversion (WPI) procedure in which electrodes after being cast on a graphite foil current collector were immersed immediately into a heated alkaline coagulation bath with pH 12 at 85°C. After about one minute, theelectrodes were taken out and dried under ambient air inside the fume hood. This method allowed electrodes with high electronic conductivity and high mass-loading to be created without cracks. However, PVdF undergoes dehydrofluorination for pH > 11 where HF molecules are released to produce polymer chains with double bonds that can further react with OH’ groups to release more HF. Elemental composition analysis by SEM-EDS confirmed the existence of oxygen in electrodes immersed in alkaline solution and that are absent from electrodes immersed in neutral solution (Table S9). Further, the phase formed by dehydrofluorinated PVdF may act as an immobile sink / source for salt in electrode pores that manifests as charge-efficiency loss.Table S9: Oxygen content obtained from SEM-EDS for various samples.Immerse duration in Temperature of coagulation bath Oxygen Samplealkaline solution (mm) (°C) (% atm) _ _85OC initia||y2 20 85°C initially 8.5 3 60 85 °C initially 10.7 4 5 85°C constant 14.9 5 0 25°C constant 0 6 0 85°C constant 0

[0189] To this end, the coagulation bath temperature was varied for WPI, the particular non-solvent used, and electrode immersion time to examine their effects on electrodemicro structure and mechanics. When using deionized (DI) water as a non-solvent, it was found that high temperatures are favorable to achieve crack- free electrodes. Specifically, crack-free electrodes were made with ~20 mg / cm2loading by immersing electrodes in DI water bath that was maintained at 85°C for 10 min. When ethanol was used as a non-solvent, the bath was kept at room temperature due to ethanol’s low boiling point, thus requiring extended immersion time from 1 minute to 1 hour to eliminate crack formations. Using coagulation baths with elevated temperatures in the phase inversion of PVdF membranes may facilitate liquid-liquid demixing over gelation / crystallization processes. In addition, ethanol is a weak solvent, meaning that it also delays gelation of the PVdF / NMP solution and favors liquid-liquid demixing. PVdF prepared by liquid-liquid demixing may be less prone to cracking than PVdF prepared by gelation within electrodes.

[0190] Galvanostatic cycling was performed at different C-rates for electrodes solidified by various WPI conditions to examine their rate capabilities. At high C-rates, the electrode made with the alkaline WPI procedure, i.e., using coagulation bath with pH 12 at 85°Cinitially, consistently shows higher specific capacities and indicating that alkaline WPI still produces electrodes with higher electronic conductivity. However, all electrodes exhibited comparable specific capacities at 1C except for the one immersed in pH 12 solution at constant 85°C for 5 minutes. Since a cycling rate of 1C is typically used in desalination experiments and since the electrode immersed in pH 7 solution at 85°C for 10 minutes showed the second highest specific capacity at 1C, such WPI conditions were adopted as an alternative to alkaline WPI to avoid dehydrofluorination. In addition, electrodes made by these alternative WPI conditions may be expected to show even higher specific capacity due to their significant improvement in conductivity expected after calendaring.

[0191] The desalination performance of electrodes prepared by the neutral-pH water bath were compared to those prepared by the alkaline water bath. Charge efficiencies obtained with the electrodes made by neutral-pH WPI were consistently -20% higher compared to the electrodes made by the alkaline WPI, respective of the number of half cycles used. These results further support the understanding that PVdF dehydrofluorination is associated with a certain degree of charge efficiency loss and that such loss can be avoided by using a heated, neutral-pH coagulation bath.

[0192] Channel Characterization of an Experimental Example

[0193] The shape of and flow through channels created by using the various engraving strategies were analyzed.

[0194] First, the laser profilometer was used to create a full-length image of the entire tapered channel. This data was output to a comma separated values file (CSV). The data is loaded into a MATLAB script and the image is then divided into 11 discrete segments with length corresponding to the cross-sections used for the channel tapering (Table S2).Alignment of the channel segment and image was accomplished by first creating an orthonormal basis that relates the position of the channel to the image. The first vector in the basis was the t vector, which was determined by creating a line of best fit through the deepest locations in the channel when plotted on the XY plane. The next vector s was parallel to the surface of the electrode. This vector was created by drawing a line of best fit through an edge of the electrode surface on the YZ plane. The Gram- Schmidt procedure was then used to make the s and t vectors orthonormal. To complete the orthonormal basis, a normal vector n was created by calculating the cross-product of both the s and t vectors.

[0195] Once the orthonormal basis was created with the t, s, and n vectors, a rotation matrix was determined using the inverse of the matrix comprised of these vectors, and theimage was then aligned with the channel segment of interest by application of this rotation matrix to the original image coordinates (%, y, z) to obtain aligned image coordinates (%', y', z'). The resulting aligned image was then used to create a representative cross-section for that segment by taking an arithmetic average among all y' positions of channel depth z' for a certain transverse location x'. Sampling of the associated points was done using interpolation of the associated x' and z' points on the aligned surface.

[0196] A representative cross-section was then used with a numerical model of fully developed Poiseuille flow based on the finite- volume method to determine an axial velocity distribution for each representative cross-section. Contour plots of the mean-normalized axial velocity were created to show the velocity distribution at various longitudinal positions along the channel. These can be seen in FIG. 8. These contours show that as the normalized channel position increases, the contour region corresponding to the highest axial velocity decreases in size. This is consistent with the theoretical model, which suggests that as the normalized channel position increases, the axial velocity should decrease. The axial velocity should decrease due to fluid exiting the channel and being pushed into the porous electrode.

[0197] After simulating flow through a given cross-section, hydraulic conductance was determined as the quotient of flow rate and pressure gradient and multiplied by dynamic viscosity. Because fully developed flow was assumed, hydrodynamic entrance-region effects were therefore neglected. Consequently, such conductance values are valid in the inertia-free limit corresponding to small Reynolds number, where hydrodynamic entrance length vanishes. A no-slip boundary condition was assumed on the sides of the representative crosssection and on a surface possessing a common tangent line for the two opposing sides of the cross-section.

[0198] This numerical model was then used to determine the hydraulic conductance Gh= —Vp / dp / dx and cross-sectional flow area for each of these cross-sections. The representative widths were measured as the average of the top and bottom widths of each segment.

[0199] Laser Micromachining: An Alternative Approach

[0200] Aside from using a computer numerical control (CNC) machine, laser micromachining may be used to form flow channels. For example, laser micromachining may be used to engrave channels into porous electrode materials. In one experimental example, to engrave IDFFs into electrodes, a Trotec Speedy 400 Flexx laser was used. This instrument is a dual head laser engraver with a CO2 laser and a 50W Yb- doped fiber laser,but only the fiber laser head was used. The micro structure and composition of electrode material and the morphology of microchannels was analyzed by engraving separate microchannels at various laser powers. 50% of the maximum laser power (50W) was used with the speed of the laser being 1 m s1and the repetition rate being 100 kHz. These settings result in a fluence of 2.25 J cm-2(+ 0.53 J cm-2), which was calculated as Pi: / Aspot, where P is the laser power, Aspotis the area of the laser spot, and T is the laser pulse width. Since the specific pulse width of the Trotec laser could not be easily found, the pulse width was assumed to be 100 ns, which falls within the range reported in the manual (1-1000 ns). The laser spot’s diameter was estimated at around 11.9 + 1.2 pm.

[0201] To generate the tapered channels, a MATLAB script was used to create the outline of the profile based on the equation for linear channel conductance variation. Once the outline was created, the toolpath needed to be shifted inward by half the representative width of the laser or kerf (40 pm) to generate the desired channel profile.

[0202] After creating the channel outline, the material in the center was removed. Various removal strategies were employed. The first two clearing procedures used vector cutting lines while the last two used raster lines. Vector cutting occurs when a laser is controlled to move along a certain curve that engraves with specified laser settings. Raster cutting occurs when a laser is controlled to move along a certain axis over an areal section with specific laser settings.

[0203] Each line in the nested vector engraving case was shifted inward by half the laser kerf (40 pm). In the horizontal vector case, there are horizontal lines that are 20 pm apart. In the section closest to the inlet, there are nine horizontal lines. The number of horizontal lines decreases by one longitudinally along the length of the channel. The horizontal lines from one section overlap with the adjacent sections by 0.25 mm. The specific dimensions of the lines are shown below in Table S10.Table S10: Dimensions of horizontal vector path clearing procedure.No. of lines Line coordinates (mm) Line Dimensions (mm) 9 0 - 1.308 1.3088 1.058 - 4.49 3.4327 4.24 - 7.71 3.476 7.46 - 10.2 2.745 9.95 - 14.24 4.294 13.99 - 17.69 3.73 17.44 - 21.36 3.922 21.11 - 25.12 4.011 24.87 - 35.65 10.78

[0204] INTERDIGITATED CHANNEL TAPERING STRATEGIES TO PRODUCE UNIFORM FLOW WITH CONSTANT FLOW-PATH LENGTH WITH FLUX AND VISCOSITY CONTRAST

[0205] Uniform flow may be produced within interdigitated channels when they are tapered in a particular manner. The associated conditions can be derived to ensure uniform flow for devices having the same fluid density p and viscosity p within high- and low-pressure channels. Such flow channels may be used in electrochemical devices with flowing liquids (e.g., electrochemical desalination and flow batteries for energy storage).

[0206] Some applications (e.g., fuel cells, electrolysis cells, etc.), may possess different fluid properties in the respective channels, owing to the presence of gaseous reactants and / or products. Variations of flow-path length (e.g., inter-channel distance) may be considered to determine appropriate tapering of flow channels. The conservation of volumetric flow rate within each channel may be used:d0dxd— 2 / i|ii5±| = 0dx I pLdx

[0207] The viscosity pland transverse superficial velocity usl±are specific to channel i.us,± may be forced to be invariant with x, allowing the above equations to be integrated directly to yield explicit functions of x for the longitudinal flow rateswithin channel i GHdpH r— 2 / l|14±|% + Kt11pldx

[0208] By neglecting flow through the tip of both channels (i.e., (x = L) = 0 and V|j'(% = 0) = 0), each integration constant Ktmay be found:. GHdpH..GLdpLyi1l1W = - = 2 / iL|u^j_|(x / L)p LLJdx

[0209] Mass balance on a control volume of differential length along x and y but of finite length in z equal to the electrode’s thickness h shows that the mass flux Jmeminto an abuttingmembrane or other planar surface causes the advection mass flux Jadvthrough intervening porous electrode material to change as function of position:Jadv > Jmemdy h

[0210] Assuming that Jmemis uniform, it can be determined from Faraday’s law, together with the associated equation for the stoichiometry of a given half-cell reaction. Under such conditions Jadvis found to vary linearly with y:T>TH JmemJ adv ~ J adv T

[0211] Thus, Jdv= Jadv(y = s) = j”dv- Jmems / h. In turn, w expressions for superficial velocities can be found in terms of J^dv and Jmem'- PH|<±|=JadvP ±| ~ Jadv ~ Jadv ~ Jmem /

[0212] Use of this relationship produces the following high- and low-pressure channel volumetric flow rate distributions:GHdpH2hLJadv^11 W = ~ HM' dxGLdpL2hL(JadvJmem. S / hK / , >yii W = — T~r~ - ~L -11pLdx

[0213] The preceding analysis including dissimilar mass flux and viscosity within highland low-pressure channels enables the adaptation of the theory to arbitrary conditions. While doing so, variations of flow-path length (e.g., inter-channel distance) may be sought to be nullified along the length of each channel, thus establishing consistency with the quasi- ID model formulation where such variations were neglected. Here, the pressure gradient within each channel may be identical at any given position along the channel:d(Ap) dpHdpLdx dx dx

[0214] The above equation does not require the pressure gradient in either channel type to be constant. Instead, it only requires that the local pressure gradient in the high-pressure channel be equal to the local pressure in the low-pressure channel. Using the equations for longitudinal volumetric flow rate, after cancelation of common factors this equation may be written in terms of hydraulic conductance profiles within the respective channels:Jadv^HJ adv^H(1 - (x / L) = 0G GL(Eq. 5)

[0215] Thus, the same strategies for uniformizing flow among working fluids having constant density and viscosity can be applied in contexts with variable viscosity, density, and mass flow rate by varying a modified conductance function Fl= Gl / (ylJaldv) instead of the usual conductance Glthat is varied for fluids having constant density and viscosity. Here, either J^ / dvor j / ldvcan be set arbitrarily and the other parameters can be determined subject to the following constraint based on membrane mass flux: Jmem= (Jadv ~Jadv) / s-

[0216] Anti-symmetric channels may be considered to uniformize flow. Here, the effects of contrasting flow rate, viscosity, and density among high- and low-pressure channels are considered. Anti-symmetry requires FL(x) = FHL — x), thus producing the following constraint equation based on Eq. 5, which is valid when inter-channel distance variations are negligible:x • GH(x) = (L — x) • GHL — x)(Eq. 5 a)

[0217] Inspection reveals that this functional equation which governs GH(x) is identical to the functional equation which governs GH(x) when there is no contrast in flow rate, viscosity, and density. As a result, the same conclusions regarding conductance GH(x) within the high-pressure channel apply, including the use of a trial profile, the use of a cosine series (Eq. 3), and the use of linear scaling (Eq. 3). The difference, however, regards the conductance distribution GL(x) of the associated low-pressure channel. The general form of the anti-symmetry condition is used to infer the low-pressure channel’s conductance distribution:GL(x) = ^^GH(L - x) = |G"(L - x)Jadv J(Eq. 5b)

[0218] Here, f = vHJadv / vHJadv is the flux-viscosity product ratio. Thus, for applications where kinematic viscosity and / or mass flow rate are lower in the low-pressure channel ( < 1), the low-pressure channel is made to have a proportionally smaller conductance than in the high-pressure channel. Eq. 5b reveals an important implication for the utility of tapered-channel IDFFs - when the ratio Jadv / Jadv is either infinity or zero tapered-channel IDFFs cannot produce uniform flow because either a trivial or singular low-pressure channel conductance distribution is produced. Such a scenario is relevant to the anode side of H2 fuel cells. If H2 fuel is fed at an ultra-lean flow rate, such that it is consumed entirely before exiting the electrode, then Jadv / Jadv = 0 is produced as a result.

[0219] In some examples, the flow-path length variations are nullified. To do so, an equation for the associated modified conductance function profile FH(x) may be expressed by also requiring the inter-channel distance to be invariant with a fixed value s. In general high-pressure channel conductance GHis a parametric function of x according to the variations of its width wH(x) and depth hH(x) with x. With fixed s, low-pressure channel conductance GLis a parametric function of that channel’s depth hL(%) and the high-pressure channel’s width wL(%) by virtue of the following geometric relationship: s = scc— wH— wL, where sccis the center-to-center spacing between channels. Using this relationship the above constraint equation may be expressed as follows while also using the relation scc— s = wH+ wL= wHx = 0):_ JadvvL(jx / L) _GH(wH(x), hH(xy) GL(wH(x = 0) — wH(x),hL(x))(Eq. 6)

[0220] In some examples, the flow channels have rectangular cross-sectional area.Additionally, in some such examples, the depth is fixed and width is varied. If depth is fixed and width is varied, then the above formula is found to be a non-linear algebraic equation that can be solved for width wHof the high-pressure channel at certain position x. The variations of wH(x) and hH(x) are determined analytically by solving Eq. 6 for the inverse function x(wH, hH) that specifies the particular x position corresponding to a certain width wHand depth hHof the high-pressure channel:]"dvvHx > VFH> G^~L ~X~ 1 / FL+ 1 / Ftf~, J%dvv"GLGH

[0221] with GH= GH(WH, hHand GL= GL(wH(x = 0) — wH,hL>). A ratio = JadvvH / (JadvvL°f flux- viscosity products may be used to simplify this equation as follows:x * fGLfGL(wH{x = G') — wH,hL>)LXGH+ fGLGH(wH,hH} + fGL(wH(x = 0) — wH,hL>)(Eq. 7)

[0222] After evaluating Eq. 7, the width of the low-pressure channel can be expressed as wL= wH(x = 0) — wH.

[0223] This formula may be used to determine x(wH) for channels of rectangular crosssection having fixed depth h = hH= hL. For channels with wH / h « 1 and wL / h « 1, conductance can be modeled using a local cubic law in the width when the typical no- slip boundary condition is employed at channel walls: G = w3 / i / 12. In such a limit, the following implicit formula relating x* to wH / wH(x = 0) is obtained:y(i — wH / wH(x = o))33(Eq. 9)

[0224] Here, the position corresponding to a certain high-pressure channel width is shown to be a rational function defined as the quotient of two third-degree polynomials. Because Eq. 9 is a rational function, partial fraction expansion of it can be used to show that the corresponding profile shape, given by x* as a function of wH, includes a linear combination of at most three shifted hyperbolas having certain poles that may be complex numbers. Any such complex hyperbolas may superimpose to produce a “witch” of Agnesi. While all non-unitary f values ( 1) produce three hyperbolas superimposed with a constant, unitary f values ( = 1) produce only two hyperbolas superimposed with a line. Such an equation is analytically solvable for wHin terms of x using factorization.

[0225] In some examples of a porous electrode, the porous nature of the walls of a channel embedded within an electrode produces a finite slip velocity usUpthat depends on the electrode’s hydraulic permeability kpeand its Brinkmann-type effective viscosity e / / :1 kpepw2dpUslip2 p Peffkpe dx

[0226] Using the superposition principle, it is readily shown that such slip acts to increase channel hydraulic conductance by an amount Gsiipwhich follows a local “square” law in channel width:pusuphw hw2pkpehw2I kpeGSUP= = — I— = —dxN N

[0227] Here, p*ff is the dimensionless effective viscosity of the porous electrode: pe*ff = eff / k- Superimposing the local-cubic law for conductance and the conductance arising from slip, the following implicit formula for high-pressure channel width is obtained by applying Eq. 7:(wH(x = 0) — wH)3(wH(x = 0) — wH)21 kpef12 2(wHV (wHY l kpe(wH(x = 0) - wH)3, (wH(x = 0) - wH)2\ kpe12 2+ f12 2(Eq. 10)

[0228] From Eq. 10 the high-pressure channel’s shape may be defined by a third-degree rational function, but additional quadratic contributions to the polynomials are introduced in proportion to the square root of the abutting electrode’s hydraulic permeability. The effect of / \(ivH(x=0))2ueffslip on the resulting optimal channel profile is insignificant for H - » 1. A typical 36 / tpgPrussian blue analog porous electrode having kpe~ 1 pm2and a maximal channel width wH(x = 0) of 100 pm this ratio is 300 for= 1, suggesting that the effects of slip are insignificant. However, the impacts of slip could be significant if much smaller channels, as in a hierarchical interdigitated network, were embedded in high permeability electrodes. In that event Eq. 10 provides guidance concerning how to design channels to compensate for slip. Otherwise, slip can be neglected.

[0229] Alternatively, for wH / h » 1 and wL / h » 1, conductance can be modeled using a local linear law in the width w in the absence of wall slip: G = wh3 / 12. In such a limit, the following explicit formula relating wH / wH(x = 0) to x* is obtained:wH / (I — x*)wH(x = 0) x* + / (I - x*)(Eq. 11)

[0230] Equation 11 shows that linear tapering of width only produces uniform flow and a constant flow-path length for wH / h » 1 and wL / h » 1 when the flux-viscosity product ratio is unity. For scenarios with flux-viscosity product contrast between high- and low-pressure channels, Eq. 11 requires an offset, shifted hyperbolic variation of width with position x* in such a limit. The low-pressure channel may exhibit a hyperbolic variation of width wLin this limit:wL_ wH_ x*wH(x = ty wH(x = 0) x* + / (I — x*)

[0231] Having examined the width profiles that are produced in the extreme limits of wH / h and wL / h, optimal channel shape may be determined for arbitrary magnitudes ofwH / h and wL / h when constant flow-path length s is enforced and f is arbitrary. Here, conductance may be modeled using the Poiseuille-flow relation of Boussinesq for a rectangular duct to determine hydraulic conductance as an infinite Fourier series in the absence of wall slip:192w coth(7r(2n — l) / i / w) — csch(n(2n — l) / i / w) G (w, h) n5h Z-i (2n — l)5n=l

[0232] The implicit formula for x* in terms of wHis thus given as follows:fGL(wH(x = 0) — wH,hXGH(wH, h) + fGL(wH(x = 0) — wH, h)(Eq. 12)

[0233] FIG. 9A shows the resulting profiles that are produced by Eq. 12 when no contrast in the flux- viscosity product is present ( = 1). All such high-pressure channel profiles exhibit a sigmoidal shape with two-fold symmetry, resulting in wH(%) — wH(x* = 0.5) being an odd function about x* = 0.5. In the limit of large wH(x = 0) / / i, linear scaling of wH(%) is shown to produce uniform flow with constant flow-path length s, as expected. Even for values of wH(x = 0) / h as large as 10, significant deviations from the former simplistic linear variation of width are observed at the respective ends of the high-pressure channel. Further, values of wH(x = 0) / h as large as unity are approximated well by the third-degree rational-function solution in the limit of wH(x = 0) / h « 1 that is predicted by the formula of Eq. 9. Since f = 1 for such cases, low-pressure channels profiles are also shown to be anti-symmetric to their corresponding high-pressure channel profiles, as depicted in FIG. 9B

[0234] FIG. 10A shows the resulting profiles that are produced by Eq. 12 when the high-pressure channel has two-fold higher flux- viscosity product than the corresponding low-pressure channel, resulting in f = 2. Irrespective of the magnitude of wH(x = O / h, all width profiles for higher-pressure channels vary in a non-linear fashion with position. For wH(x = 0) / / i » 1, the corresponding profile is an offset, shifted hyperbola with downward concavity. As wH(x = 0) / / i is decreased, the profile transitions toward a sigmoidal shape. Such profiles may not possess the two-fold symmetry shown for cases with f = 1 in FIGS.9A-9B. The lack of such symmetry is accompanied by a downstream shift in the axial location x* at which each such profile has half of its maximal width. FIG. 10B shows the boundaries of the corresponding high- and low-pressure channels. Here, high-pressurechannels are shown to have a larger maximal width than low-pressure channels because their greater flux- viscosity product requires a larger hydraulic conductance to match the pressure gradients produced within low-pressure channels.

[0235] FIG. 11 A shows similar but more pronounced effects when f = 5 is considered. The axial location at which each profile has half its maximum value may be shifted even further downstream than the case with f = 2, the net result of which is that all high-pressure channel profiles are concave-down over most of their axial extent. FIG. 1 IB shows that the opposite is true of low-pressure channels, which possess upward concavity over most of the axial extent. Further, low-pressure channels with f = 5 produce even more rapid narrowing than for = 2.

[0236] To simplify the understanding of the channel shapes produced for arbitrary values of w / h, a 2nd-order Pade approximant of the infinite- series solution of hydraulic conductance was derived to match its asymptotic scaling in accordance with the local cubic and the local linear laws in width that are produced in the respective limits of w / h 0 and w / h -> oo;G (w / h)2h3w / 12 1 + 0.3712(w / h) + (w / h)2(Eq. 13)

[0237] By also matching the exact scaling of conductance for w / h = 1 (i.e., G = 0.4217h3w / 12), this Pade approximant produces deviations of less than 3% from the exact conductance formula among all possible values of w / h.

[0238] The accuracy of such a low- order approximation given by Eq. 13, and its representation as a rational function, permits the application of this approximation to channel design for arbitrary w / h and / values. Substitution of Eq. 13 into Eq. 7 produces an implicit channel-shape formula for x* in terms of wHthat is a fifth-degree rational function: / (wH(x = 0) — wH)3(l + 0.3712(wH / h) + (wH / h)2) X ~ / / 2 X \ / (wH)3^1 + 0.3712((wH(x = 0) — wH) / h) + ((wH(x = 0) — wH) / h)\ +f(wH(x = 0) — wH)3(l + 0.3712(wH / h) + (wH / h)2) / (Eq. 14)

[0239] Since Eq. 14 is a rational function, partial fraction expansion can further be used to show that for arbitrary values of w / h and / optimal channel shapes that produce uniform flow and that have constant flow-path length include at superposition of at least one hyperbola and at most five hyperbolas.

[0240] Partial Fraction Expansion of Equation 9

[0241] Partial fraction expansion (PFE) may be applied to Eq. 9 to show that the resulting shape of the channel is a linear combination of a constant, a straight line, and hyperbolas in the limit where wH / h « 1 is satisfied. Numerical PFE with five significant digits may be used to do this for certain values of f. The resulting profiles may be expressed in terms of the dimensionless width w* = wH / wH(x = 0) of the high-pressure channel. Using f = 1 a linear combination of a constant, a straight line, and two complex hyperbolas may be obtained:2 iv* 0.11111 0.11111X * — - 3 3 w* — 0.5 — 0.288681 iv* — 0.5 + 0.288681

[0242] The two hyperbolas have poles equal to 0.5 ± 0.288681 that are complex conjugates, where i = V-l is the imaginary unit. Consequently, when these two complex hyperbolas are superimposed, they produce a real rational function of degree two that includes a “witch” of Agnesi:0.11111 0.11111 (2iv* — l) / 9” i * - 0.5 - 0.288681 ” iv* - 0.5 + 0.288681 “ ” (iv*)2- iv* - 1 / 3

[0243] Using f = 2 PFE produces a linear combination of a constant, one real hyperbola, and two complex hyperbolas:6.2164 —0.10820 + 0.0144331 -0.10820 - 0.0144331 X* — 2 d - 1 - 1 - IV* - 4.8473 IV* — 0.57634 — 0.283611 iv* - 0.57634 + 0.283611

[0244] As with the case using f = 1, the two complex hyperbolas superimpose to produce a real rational function of degree two that includes a witch of Agnesi since the associated poles are complex conjugates:—0.10820 + 0.0144331 -0.10820 - 0.0144331 iv* - 0.57634 - 0.283611+iv* - 0.57634 + 0.283611 0.21640iv* - 0.11653“ ” (iv*)2- 1.1527iv* + 0.41260

[0245] Using f = 5 a linear combination of a constant, one real hyperbola, and two complex hyperbolas is obtained:5, 1.1308 —0.096644 + 0.0299211 -0.096644 - 0.0299211X~ 4+iv* - 2.4085+iv* - 0.67075 - 0.262851+iv* - 0.67075 + 0.262851

[0246] As with the preceding cases, the two complex hyperbolas superimpose to produce a real rational function of degree two that includes a witch of Agnesi since the associated poles are complex conjugates.-0.096644 + 0.0299211 -0.096644 - 0.0299211iv* - 0.67075 - 0.26285i+i * - 0.67075 + 0.26285i0.19329w* — 0.11392“” (iv*)2- 1.3415w* + 0.51900

[0247] For the sake of completeness, PFE for cases with f = 0.5 and f = 0.2 were tested, and the resulting profile was shown to be a linear combination of one constant, one real hyperbola, and two complex hyperbolas that are real when superimposed to include a witch of Agnesi. On this basis, such shapes are produced in the limit of wH / h « 1 for all non-unitary, positive f values. Otherwise with f = 1, this limit produces a profile shape given as a linear combination of a line and two complex hyperbolas that are real when superimposed to produce a witch of Agnesi.

[0248] UNIFORMIZING PROFILES THAT INCORPORATE EFFECTS OF INTER-CHANNEL DISTANCE VARIATIONS

[0249] As described, flow channels can be tapered to achieve uniform flow. Such tapering may be performed to account for the distance between interdigitated channels being invariant along each channel’s length. However, in some examples, rather than the distance between the outer boundary profile of the channel lengths being invariant as described previously, the distance between the outer boundary profile may vary. For example, when the center-to-center distance sccbetween channels is of similar order to the maximal width i0of each channel, the flow-path length s between channels, also called inter-channel distance, may vary significantly as a function of position x along each channel. This may compensate for such variations in the tapering of channels that is used. Here, the type of flow uniformity sought for the associated IDFF may be given by variable s(x). In some examples, the channels may be tapered in such a way to produce uniform velocity transverse velocity magnitude |us ±| (Case I). In other embodiments, the channels may be tapered in such a way to produce uniform residence timer= s / |us ±| (Case II). In either instance, Darcy’s law may be used to relate the associated pressure difference to the corresponding function to be uniformized by cross-sectional tapering:.H LK±| / zsAp = ph— pL= — - -(Eq. 15)

[0250] The variables pH(x) and pL(x) respectively capture the pressure variations with longitudinal position x within high- and low-pressure channels. kpeand p respectively arethe hydraulic permeability of the porous electrode and the dynamic viscosity of the working fluid. Flow uniformity according to either Case I or Case II can thus be enforced by requiring the gradient of the quotient of pressure difference and an appropriate power n of s to be zero:d / Ap\ 1 d(Ap) nAp dsdx \sn) sndx sn+1dx(Eq. 16)

[0251] Here, n = 1 and 2 respectively for Cases I and II. Equation 16 reveals that when inter-channel distance variations are insignificant (ds / dx ~ 0).

[0252] The variations of pressure within each channel may be related to the hydraulic conductance variations to optimize: GH(x) and GL(x). In general, these effects are captured using the following differential equations that enforce the conservation of volumetric flow rate within each channel:d GHdpH+ 2 / i|us ±| = 0dx p dxd GLdpL— 2 / i|us ±| = 0dx p dx

[0253] For Case I, |us ±| may be invariant with x, allowing the above equations to be integrated directly to yield explicit functions of x for the longitudinal flow rates ^(x) andGHdpH— 2h\us l\x + KHp dxGLdpL= 2h\us l\x + KL^11 (*) = - p dx

[0254] By neglecting flow through the tip of both channels (i.e., (x = L) = 0 andI / ,f(x = 0) = 0), the integration constants KHand KLmay be determined:GHdpH..(*) = - - — = 2hL |us ±| (1 — x / L)ClXGLdpL^11 (*) = - p dx= 2 / lLl^.±K% / L)

[0255] The high- and low-pressure channels may be tapered in the same manner such that their conductance distributions are anti-symmetric: GL(L - x) = GH(x).

[0256] For Case I the pressure-difference gradient appearing in Eq. 16 depends on the high-pressure channel’s conductance profile GH(x) as follows:d(Ap) _ 2 / IL|US ±| / Z (Q 27IL|USJ-L| / Z (1 - £) _ / x / L 1 - x / L\ dx ~ GL(x) GH(x) ~2hL\us>1,\ii [GH^L_X>)~GH^ )

[0257] Substitution of the above into Eq. 16, yields the following constraint equation for Case I when Eq. 15 is jointly used to express Ap:2 / iL|us ±|p / x / L l — x / L\ 1 |us.j_|ps dss \GH(L — x) GH(x) / s2kpedxx / L 1 — x / L GodsGH(L — x) / G0GH(X) / GQ2hLkpedx

[0258] Here, Gois the high-pressure channel’s conductance at x = 0. Further, s may be normalized by a representative inter-channel distance srs* = s / sr. Additionally, x may be normalized by channel length L: x* = x / L. Together, the resulting constraint equation becomes:x* 1 — x* Gosrds* 1 ds*GH(1 — x*) / G0GH(x*} 2hL2kpedx* 23 dx*G0(Eq. 17)

[0259] Here E = hL2kpe / G0sris the dimensionless hydraulic resistance within channels.

[0260] By selecting a cross-section type having channel width w that is much smaller than its depth h, a local cubic-law can be used for channel conductance:

[0261] Also, the associated inter-channel distance s can be expressed in terms of the local width of high- and low-pressure channels:s(x) = scc— wH(x) — wL(x) = scc— wH(x) — WH(L — x)(Eq. 18)

[0262] Using Eq. 18 together with Eq. 17, a differential functional equation that governs the dimensionless channel width w* = wH / w0, where w0is the channel’s maximal width may be obtained:■2x’(w’(x’)) - (l - x’)(w’(l -x’))sr[dw’(x’) dw*(l — x*)(w*(x*>*(l - x*))I dx* ' dx*(Eq. 19)

[0263] From this equation the corresponding set of uniformizing profiles may depend on two dimensionless parameters, E and w0 / sr. While Eq. 19 assures that transverse velocity is uniform, it may not be sufficient by itself to determine a unique profile for w*. Given that Eq. 19 is a functional equation that relates w*(x*) to w‘(l — x*), a certain profile for x* E [0.5,1] may be adopted ad hoc and then use Eq. 19 to determine the particular variation of w* for x* E [0,0.5], Alternatively, Eq. 19 may be solved subject to an optimality condition (e.g., minimal pressure drop or minimal mean-square slope). Using the approach with linear conductance scaling, w*(x*) = (1 — x*)1 / 3for x* E [0.5,1] may be used. Substitution into Eq. 19 yields a non-linear differential equation that governs w*:„ wn / sr[dw* 1 ] - (1 - x*)x* = -4^ — x*(w*)3tdx* 3{x*y / i\w0 / srrdw* 1(w’)3— (w’)3+ (1 — x*) = 02E [dx*+3(x*)2 / 3.(Eq. 20)

[0264] In the limit of large w0 / (srE), the corresponding profile accounting for interchannel distance variations is given by solving the following equation because only terms including the factor w0 / (srE) are significant:dw* 1dx* ~ 3(x*)2 / 3

[0265] The following solution is obtained by integration for x* G [0.5,1]:fO-5 (7x(0.5)1 / 3- w* = - = -((O.5)1 / 3- (x*)1 / 3)3(x*)2 / 3 v 7

[0266] Combining this result with the profile assumed for x* G [0,0.5], the complete profile for w0 / (srE) > 10 is given as a piecewise function:, ( Vl — x* for x* E [0.5,1]w = -j[V4 — Vx* for x* G [0,0.5](Eq. 21)

[0267] For arbitrary w0 / (srE) values, Eq. 20 may be solved numerically as it becomes inseparable when all terms are included in the equation. For this purpose, a finite difference discretization may be used that is first-order accurate in the distance Ax* between successive nodes i and i + 1:* * -1 ’wi+l ~wi1«+1)3- «+1) (l - x?+1) =+3+ 0Ax* 3(xt*1)2 / 3

[0268] Here, w* and x* may be approximated by their values at the i + 1thnode to enable solution for w- by marching in space from the channel’s center (i = IV at x* = 0.5) toward that channel’s inlet (i = 1 at x* = 0) by using the following update equation:(Eq. 22)

[0269] Alternatively, the direction may be opposite, approximating w* and x* by their values at the ithnode to obtain the following update equation:(Eq. 23)

[0270] Eq. 22 may be used to generate different channel- width profiles that compensate for inter-channel distance variations in the uniformization of velocity, the results of which are shown in FIG. 12. When the representative flow-path length sris large relative to channel width w0so as to produce large E / (w0 / sr), the resulting profile converges toward the cubic-root profile that neglects inter-channel distance variations. Intermediate E / (w0 / sr) values are shown to require an enlarged inlet region so as to reduce inter-channel distance locally. In the opposite limit where E / (w0 / sr) is small, the uniformized channel profile approaches Eq. 21, thus serving as an upper bound for the width among all uniformizing profiles that satisfy the local-cubic law for conductance.

[0271] MINIMIZING INTER-CHANNEL DISTANCE VARIATIONS IN EXAMPLE EMBODIMENTS

[0272] As described, there is a subspace containing infinitely many profiles of high-pressure channel conductance GHthat uniformize flow:1-'00GH(x) = — > bn[cos(n7r(% — L / 2) / L) — cos(n?r / 2)]% —ln=^(Eq. 24)

[0273] Here, x is distance along the channel, L is the channel’s length, and bnis a cosineseries coefficient for mode n. Such profiles of the form of Eq. 24 produce uniform flow when variations are significant of the flow-path length through porous electrode material that is otherwise known as inter-channel distance s. However, inter-channel distance variations can become significant if the center-to-center spacing between channels sccis smallcompared to the inlet-side width w0of the high-pressure channel. While flow can be uniformized by explicitly compensating for inter-channel distance variations, here it is shown that Eq. 24 can be used to find a particular profile that minimizes inter-channel distance variations, such that they become insignificant and inconsequential to the resulting flow distribution.

[0274] The minimization of inter-channel distance may be posed as a non-linear optimization problem. In some examples, channel width w is varied while keeping channel depth h fixed. Here, variations of the inter-channel distance s(x) = scc— w(GH(x)) / 2 —may be minimized, where w depends either on the conductance of the high- or low-pressure channel. High- and low-pressure channels may be chosen to be anti-symmetric (e.g., GL(x) = GHL — %)) to produce s(x) = scc~ w(GH(xy) — w(GHL — %)). Mapping between GHand the width w is used to implement any conductance profile embodied by Eq. 24. To minimize inter-channel distance variations, profiles that produce the least squared slope of the inter-channel distance may be used. That is, an objective functionthat depends on the infinite set cosine-series coefficients {bn} may be minimized:fL / ds\2= (— ) dxJo\ax / (Eq. 25)

[0275] where the slope profile ds / dx is determined using the chain rule as follows:ds 1 dw dGH1 dw dGHdx 2 dGHdx 2 dGHdxx L-x

[0276] The slope of conductance may be determined using Eq. 24:dGHGH(x) lv-1” nn:— — = - y bn— sin(n7r(% - L / 2) / L) dx x x£_in=1L

[0277] The sensitivity of width w to conductance GHis also determined by the relevant mapping function determined from Poiseuille flow. Though such an approach may apply to generic cross-sections and their variations, a local cubic law in the width (GH= hw3 / 12) may be assumed for this example optimization problem. Assuming that depth is fixed, this mapping becomes w = 12GH / h)1 / 3that produces the following sensitivity:dw 4 „ >

[0278] Thus, the objective function to be minimized is: / -- cL4 / 12GHfx)\ / v — 1°° mi f(.{bn}') = \ 7- - 7 - GH(x) + > bn— Sin(n7r(x - L / 2) / L) Jo\ bx \ h I X 2_in=1L4 / 12GH(L - %) (GH(L - x) h(L — x) \ hnn \ + / bn — sin(n7r(x + L / 2) / L) ) \ dx >n=l b ) /

[0279] ANALYSIS OF GAP DISTANCE BETWEEN CHANNEL TIP AND ELECTRODE EDGE

[0280] The gap distance g between channel tips and electrode edges affects the functionality of embedded, micro-interdigitated flow fields (ep-IDFFs) as a result of a competition between (1) the short-circuiting of flow through channel tips when g is small and (2) the formation of dead zones around channel tips when g is large. Here, an analytical model to enable the rational selection of gap distance is derived that balances these competing effects. To do this a single high-pressure channel within an ep-IDFF is considered. The transverse flow rate V±produced into porous electrode towards adjacent low-pressure channels is expressed in terms of the thickness-averaged transverse superficial velocity {us ±), channel length Lch, and electrode thickness heLcflhe

[0281] Using Darcy’s law, {usat the high-pressure channel’s tip is expressed in terms of the high-pressure channel’s pressure Pfipand the low-pressure channel’s pressure PoUtatthat position:,. kpej_ (Ptip—Pout)< WS,±) = - p s

[0282] Here, kpe l_ is the porous electrode’s hydraulic permeability in the transverse direction, p is dynamic viscosity, and s is inter-channel distance. For ep-IDFFs using channels that are tapered to produce uniform transverse flow:■ _ 2kpe,j. LChhe,H_Lxv± \Ptip Pout)

[0283] At the same time, a flow rate through the tip of such a high-pressure channel is driven by the same pressure difference:Vtip \Ptip Poutj / Rfip

[0284] where Rtipis the hydraulic resistance associated with such a tip flow. Owing to the similarity between the governing equations for heat conduction and Darcy flow (i.e., the homogeneous form of Laplace’s equation), Rtipis expressed in terms of a conduction shape factor a having units of length:1 ^pe,||D—.. °‘‘tip r-

[0285] Here, kpe^ is hydraulic permeability within the porous electrode for the associated tip flow. To find a conservative estimate for a, flow between the tip of the channel and the edge of the electrode is modeled by assuming a planar tip with cross-section that is identical to the maximal cross-section of the channel. For one-dimensional Darcy flow under such conditions, the corresponding shape factor is found to be:w0 / iecr = -9

[0286] where w0is the maximal width of the channel.

[0287] To ensure that flow is not short-circuited through channel tips, Vtipis determined to be small (e.g., < 1%) in comparison with V±:^tip kpe>llas kpe^WQS < Q QiKL 2kpe±Lchhe2kpe LLcflg

[0288] Thus, a condition for the gap distance needed to assure tip flow smaller than 1% is found:WnS k^fj 11a>50_2-. ->iai‘-‘ch *pe,l

[0289] Assuming kpe^ / kpe~ 1, a 350-pm-wide channel with an inter-channel distance of 650 pm requires g > 250 pm for an electrode that is 45 mm long.

[0290] DEAD ZONE AND HOT SPOT ELIMINATION USING TAPERED INTERDIGITATED FLOW FIELDS

[0291] Channels are tapered according to quasi- ID theory to produce uniform transverse flow through the porous electrode region between adjacent channels. The resulting 3D velocity fields and apparent permeability are consistent with those expected from the quasi-1D theory for uniform flow, provided that electrode hydraulic resistance is small compared to that of channels. Based on the associated 3D velocity fields produced by such tapered-channel IDFFs, 3D convective mass transport simulations were conducted with apparent first-order reaction kinetics that map to operation of electrodes used either for intercalation-based desalination or for flow batteries. Such simulations reveal that reaction-rate uniformity increases as flow rate is increased, showing variations only along the transverse coordinate between channels above a critical Peclet number. Using quasi- ID modeling of mass transfer that shows excellent agreement with 3D simulations, transverse back-diffusion of reactive solute from porous electrode regions into high-pressure channels is shown to be the mechanism responsible for such behavior. Thus, the 3D simulations validate the use of quasi- ID theory to inform the design and operation of IDFFs using optimally tapered channels that eliminate both dead zones and hot spots.

[0292] Simulation Methodology

[0293] 3D flow simulation of IDFFs using optimally tapered channels was performed using a porous media flow module implemented previously in OpenFOAM. Specifically, a Darcy-Brinkman formulation is used at steady state to account for the balance of fluid momentum, shear stress, and pressure p to determine a superficial velocity field us(x, y, z):> > r - \ M - V • (pus® us) = -Vp + V • - -usK.(Eq. 25)

[0294] where p and p respectively are the dynamic viscosity and density of the working fluid. Hydraulic permeability k and effective viscosity peff respectively account for the effects of pore-scale and macro-scale shear stress in the fluid. Equation 25 enables the determination of velocity and pressure within porous regions and within channel regions of the simulation domain by setting appropriate values of k and peff. Porous regions use k = kpefor a certain porous electrode permeability kpe. Following from measurements of intercalation electrode permeability, kpeequal to 0.1 pm2was used unless stated otherwise. To recover the Navier-Stokes equations, channel regions use peff = p and k = kcwith channel permeability kcchosen to approach infinity. In practice kc= kpex 106was used to ensure convergence of the iterative numerical solver used. This channel permeability used in 3D simulations is distinct from the channel permeability used in the analytical velocity formula used in Sec. 2 of the SI to verify 3D simulations. The former is made arbitrarily large strictly to effect numerical-model stability since an infinite value is not tractable, whereas the latter one is chosen to account for thru-plane hydraulic confinement that is not locally resolved by the analytical velocity formula. Due to uncertainty concerning thespecific value of peff within electrodes, peff is approximated as p to produce uniform effective viscosity through the entire domain. While this model incorporates the effects of solution inertia by including momentum flux in Eq. 25, inertial effects are negligible. For that reason results are primarily presented in terms of Peclet number, rather than Reynolds number.

[0295] 3D flow model verification was performed for fully developed flow through a straight channel of width w and depth h. abutted by a porous electrode. Simulation results were compared with an exact formula for longitudinal velocity us yas a function of transverse position x that was obtained by solving Eq. 25 analytically: / x-i 1 dp x \ I pkpe( w \ / w M'S y W, — — - — sinh — — + cosh — —yp dy ylkJ \Jveffkc \2-yJk^J \2. / kc

[0296] The effect of hydraulic confinement due to the channel’s finite depth was incorporated into the analytical solution using a local-cubic law for the channel’s permeability: kc= h2 / 12. While it is presently assumed that peff = p, simplification of the above expression in the limit of h / w -> oo shows that the rise in fluid velocity at channel walls above the velocity in the porous electrode - a type of slip velocity usUp- decreases with increasing peff.1 kpedp pw2-slipSy x w / 2) USy x — * oo) — — I - Z / I dy I P-eff^pe

[0297] Thus, experimental measurement of slip velocity can be used to infer peff with an independent measurement of kpe.

[0298] Mass transfer is simulated by modeling steady advection, diffusion, and Faradaic reactions within porous electrodes, having equal applicability to both electrochemical separations devices and to flow batteries. For both such applications it can be shown that active species concentration c is governed by mass conservation with apparent first-order reaction kinetics under certain conditions (see Table B 1):V■ (usc — DeffVc) + k'(c — ceq) = 0(Eq. 26)

[0299] where k' is the apparent rate constant that determines the apparent homogeneous reaction rate: n'" = k'(c — ceq), where ceqis an equilibrium concentration. The relationshipbetween the apparent rate constant k' and electrochemical kinetics parameters are shown in Table B 1 for electrodes and active species that are relevant to Faradaic deionization and flow batteries. Def is active species effective diffusivity that depends on its bulk diffusivity Doand MacMullin number Me: Deff = D0 / Mc. A MacMullin number of 10 was used for electrode regions, consistent with earlier measurements for intercalation electrodes, while a MacMullin number of unity was used for channel regions. To map the Reynolds numbers corresponding to a certain Peclet number, a bulk diffusivity of 1.6x1 O’9m2 / s was assumed for aqueous NaCl solution at infinite dilution, corresponding to a Schmidt number Sc = Pe / Re = ^ / pDo) equal to 556 for pure water at 25°C.

[0300] Table Bl. Formulas for the apparent first-order rate constant and electrochemical kinetic parameters for Faradaic deionization (FDI) using cation intercalation electrodes and for flow batteries using dissolved redox-active molecules. F is Faraday’s constant, R is the gas constant, T is the absolute temperature, a is the active surface area per unit solid volume, v is solid volume fraction, and e is porosity of the electrode.FDI using Cation Intercalation41RFB using Dissolved Redox-Active Moleculesbh"' = k'(ce— ce eq) W-red (Sred ^red,eq^ io inav(l - t+) / 0.5Friin\k = - exp - k = avk01 exp 1RT1Fce,inHV RT / / aFr]0\\+ exp - — —\RT / / (Fl]in\ ^red. O T ^ox,0Ce,eq ^e,in ^Xp JCred’eq~ ( Fp°\\ k'1 1 + exp 1 --^- I IR° = (ps- (pi - E°Rin ~ (jPs 0e—

[0301] aThe following were assumed to derive the volumetric salt depletion rate n"' for steady-state intercalation-based FDI with salt concentration ceButler-Volmer kinetics with overpotential T]in, exchange-current density i0,in, and salt concentration ce inat the electrode’s inlet, a binary electrolyte that possesses equal diffusivity for anions and cations to produce a cation transference number t+~ 0.5, an intercalation material with high specific capacity and facile solid-state diffusion, and electrodes that are thin enough to neglect Ohmic polarization.

[0302] bThe following were assumed to derive the volumetric consumption rate n"e'dof the reduced species in RFBs with a reduced- species concentration credthat deviates in timefrom an initial value cred Qwith an initial oxidized- species concentration cox QButler-Volmer kinetics with standard overpotential i, standard reduction potential E°, and certain anodic (1 — a) and cathodic (a) kinetic transfer coefficients, a sufficient supporting electrolyte having equal diffusivity for reduced and oxidized species, and pseudo- steady mass transfer having a rate of concentration change / 3 with time.

[0303] 3D concentration distributions were simulated in OpenFOAM by including direct coupling of the reaction rate n'" to concentration c. Verification of the mass transport model’s implementation was done by comparing its results for ID flow through a reacting porous electrode with flow-path length s to an exact formula obtained by solving Eq. 26 analytically.c(x* = x / s) - Ceq r_*eO‘+^-) -r+*e(r‘x‘+r;)cin ~ceq r^er+ — r*erl(Eq. 27)

[0304] The preceding formula was derived subject to a known concentration cinat the inlet (x = 0) and a null-diffusion condition at the outlet (x = s). Here, the dimensionless roots r+ and r* are shown to be functions of the dimensionless electrode-wise Peclet number PeSr= ussr / Deff and Damkohler number D aSr= k's^ / Def.Pes+ Pe? + 4Dn.* Nr±“ 2(Eq. 28)

[0305] There is agreement between simulated and analytical results among the various PeSrand DaSrcases, demonstrating that mass transfer has been implemented correctly. These equations are used to develop a quasi- ID model of mass transport within IDFFs that yields good agreement with 3D simulations. The simulated and analytical results both produce dimensionless concentration distributions that are invariant with the inflowing concentration of reactant when defined as (c — ceq) / (cin— ceq). Consequently, the dimensionless reaction rate distribution is shown to be determined solely by Peclet and Damkohler number.

[0306] High- and low-pressure channels using the power-law fitted optimal width profile were incorporated into the simulation domain at a center-to-center distance scc= 700 pm to comprise a repeat unit that defines an infinite array of interdigitated channels embedded in porous electrode material. Further, a gap between the tip of each channel and the electrode’s edge was chosen as 750 pm to suppress flow directly from channel tips through the electrode.Open passages of 5 mm length were appended to the inlet and outlet faces of the L = 45 mm-long repeat unit to allow for flow development into and out of the primary domain. For flow simulations uniform pressure boundary conditions were used respectively across the inlet and outlet faces of the domain, so as to produce a controlled pressure difference from inlet to outlet. For a given geometry, a certain flow rate results from the imposed pressure difference. Additionally, zero normal strain rate was imposed at inlet and outlet boundaries. For mass transfer simulations the inlet face was subjected to a fixed concentration cin, while diffusive flux was neglected at the outlet face to produce zero concentration gradient.Periodic boundary conditions were applied to velocity, pressure, and concentration fields on the sides of the domain to eliminate edge effects. The top and bottom surfaces of the domain that are normal to the z direction were assumed to act as no-slip walls that are also impermeable to fluid flow and active species flux.

[0307] Results of Uniform Flow Using Optimally Tapered IDFF Channels

[0308] 3D flow simulations are used to predict the velocity fields produced by IDFFs that use channels that are tapered to produce uniform flow based on quasi- ID analysis, comparing them to IDFFs using channels with straight and linearly tapered cross-sections, and introducing theory to predict the apparent permeability. The optimal tapering of interdigitated channel widths is determined to make the transverse velocity between high- and low-pressure channels uniform. According to quasi- ID theory, such flow uniformity is assured when the hydraulic conductance GHof each high-pressure channel with length Lchis designed to vary linearly with the longitudinal y coordinate:G«(y- = y / LcJ,- T - = i - y(Eq. 29)

[0309] Here, hydraulic conductance is determined based on appropriate Poiseuille-flow relations for laminar, fully developed flow in terms of volumetric flow rate V, viscosity / z, and the imposed longitudinal pressure gradient dp / dy: G = —V / (dp / dy). Gois the maximum conductance that occurs at y = 0 for a high-pressure channel. It is assumed that the low-pressure channel has the same shape as the high-pressure channel. While certain channel cross-section types have expressions for channel dimensions that can be derived analytically (e.g., a cubic-root profile or a linear profile of channel width w with fixed depth h respectively satisfies Eq. 29 when w « h or w » h is satisfied), a piecewise-linear variation of channel width w was determined by solving for the particular y positionscorresponding to a set of pre- specified widths ranging between a maximal width w0and zero width. Boussinesq’s solution was then used for Poiseuille flow through a rectangular duct for a certain channel depth h to determine conductance for each width w belonging to the set of widths:w3h ( 192 / W\ 1 cosh(7r(2n — — 1 G(w, K) = — I 17T5VhJ 2-i (2n — l)5sinh(7r(2n —n=l (Eq. 30)

[0310] The associated position corresponding to each width w was then found by inverting Eq. 29 as y!ch= 1 — GH(w, h) / G0.

[0311] The exact optimal width profile was then fitted to obtain a power-law width profile of the following form when using h = 200 pm and w0= 266 pm: w / w0=(1 — y / Lch)0A63. This power of 0.463, which produces deviations from the exact profile less than 5 %, is between the one-third and unity powers that are associated respectively with cubic -root and linear profiles that produce uniform flow only under certain asymptotic conditions. The biasing of the present profile away from such ideal profiles results from the channel’s maximal cross-section having width w0that is of similar order to its depth h. and as such neither asymptotic condition (w0« h or w0» h) is satisfied for such a channel. Such a channel corresponds to a dimensionless E value of 0.84 when using a center-to-center distance of scc= 700 pm corresponding to a representative flow-path length sr= 517 pm and when used with the L = 45 mm-long electrodes having 0.1 pm2permeability that are simulated here. Here, a given center-to-center spacing and maximal width produce a certain representative flow-path length sr= (s-1)-1that is estimated as a reciprocal average of the local flow-path length s: (s-1) = f (Lch / s)dy. For such conditions that produce order-unity E, the hydraulic resistance of channels is thus similar to that of the porous electrode material intervening between channels.

[0312] Mid-plane streamlines and pressure field produced for a channel- wise Peclet number PeWoof 360, defined by PeWo= (us n)owo / Dowith (usn)0being the mean streamwise superficial velocity at the high-pressure channel’s inlet were determined. Unless otherwise stated, the subsequent simulations use tapered interdigitated channels whose width is varied according to a power-law fit of the exact optimally tapered width profile given by Eq. 29, also known as a fitted optimal taper. Using an ambipolar diffusivity for dilute, aqueous NaCl, this Peclet number corresponds to a flow rate of 0.26 mL / min through a 45mm x 45 mm electrode, where flow rate is given as Vtotc= Nch{us n)ohwo / 2 = NchPeWoDoh / 2 for an IDFF using Ncflchannels in total. Fluid parcels that enter from the inlet side of the domain preferentially contract into high-pressure channels and then turn at a certain position to flow transversely through porous electrode material. Fluid is then collected by low-pressure channels from porous electrode material, then expanding out of such channels toward the outlet side of the domain. There is a roughly linear gradation of pressure inside each channel, producing a roughly constant difference in pressure between adjacent channels, consistent with the design constraints derived to produce uniform flow.

[0313] The longitudinal pressure variations produced by optimally tapered channels produce the most uniform difference in pressure between high- and low-pressure channels, as compared with IDFFs using channels with either straight cross-section or with a linearly tapered width. Quasi- ID pressure profiles for optimally tapered and straight channels were also determined using analytical formulas derived. While the longitudinal variations of pressure within optimally tapered channels are shown to be roughly linear as expected from the quasi- ID theory upon which they were designed, high- and low-pressure channels with straight cross-section respectively exhibit concave-up and concave-down pressure profiles that are also consistent with their quasi- ID theory. Oppositely, linearly tapered channels exhibit high- and low-pressure channels that respectively produce concave down and concave up pressure profiles, thus demonstrating the unique ability of optimally tapered channels to produce linear pressure variation along their lengths. The pressure distributions at the midplane of each electrode show that all three channel types produce a transverse pressure gradient within porous electrode material that is roughly invariant with the transverse position coordinate, resulting in negligible acceleration of fluid parcels in the Lagrangian reference frame occurring along streamlines through porous electrode material.

[0314] The transverse velocity distributions within porous electrode material are directly related to the associated pressure fields. Despite the present model using the Brinkman correction to Darcy’s law, the difference in pressure between adjacent channels is shown to correlate directly with the transverse superficial velocity. This finding indicates that porescale shear stresses are dominant within the porous electrode region, supporting the use of Darcy’s law within porous electrode domains. Optimally tapered channels produce transverse velocity within 10% of perfect uniformity as expected from quasi-lD theory, while straight channels produce suppressed velocity in the central porous electrode region between channels consistent with their quasi- ID theory, showing large deviations from uniformity.Linearly tapered channels in turn produce suppressed velocity near channel ends with similarly large deviations relative to straight channels. The corresponding streamlines also confirm that flow is unidirectional in the intervening porous electrode region between channels, further justifying the use of quasi- ID theory to model flow through IDFFs using each channel type.

[0315] For fitted optimally tapered channels, the sensitivities of flow uniformity to center-to-center spacing, electrode length, and porous electrode permeability were investigated. As center-to-center spacing is doubled, deviations from flow uniformity are halved, ultimately showing only ~1% deviation from uniformity when using 2.2 mm spacing. However, the weak suppression of velocity in the central region between channels for all cases suggests that factors other than variations of flow-path length are responsible for deviations from uniformity, including the case using a power-law fitted approximation to the exact optimal profile. These deviations from flow uniformity increase monotonically with increasing dimensionless hydraulic resistance parameter E = kpehL2 / Gosr, given that increased representative flow-path length srdecreases E. Correlation between deviation magnitude and E is additionally shown when electrode length L is changed, but its impact is shown to be more significant than changing srincreasing electrode length to 100 mm produces six-fold the deviation produced by 45 mm-long electrodes. However, further increasing electrode length to 200 mm merely doubles the deviation produced by 100 mm-long electrodes. The impacts of porous electrode permeability on the deviation from flow uniformity are also consistent with that expected from the corresponding changes in E.Decreased E results in a decrease to ~1% when porous electrode permeability decreases to 0.001 pm2.

[0316] Quasi- ID theory was also used to quantify the impact on flow uniformity of errors (1) due to the power-law approximation of exact optimal tapering profiles and (2) due to the variations of flow-path length that were assumed negligible. Power-law fitted optimal tapering profiles show transverse velocity distributions that are inconsistent with the present 3D simulations, whereas exact optimal tapering profiles show transverse velocity distributions with similar shape but smaller magnitude relative to the present 3D simulations. These discrepancies indicate that quasi- ID analysis should be improved to accurately predict deviations in velocity that are produced by using optimally tapered channels. Nonetheless, quasi- ID theory was used in addition to predict the root-mean square (RMS) deviation of velocity from uniformity for the various tapering strategies in question. For a given ratiowQ / scc, exact-optimal and fitted- optimal tapering produce velocity deviations that increase sigmoidally with increasing E, resulting in the saturation of maximal deviation for sufficiently large E in a manner that is consistent with 3D simulations. In contrast, straight channels produce unbounded deviations in the large E limit, as a result of them being especially prone to dead zones. Linear tapering generally produces large velocity deviations, though maximal deviations saturate for large E. Among all tapering profiles, exact optimal tapering exhibits lowest velocity deviations and greatest sensitivity toward w0 / scc.

[0317] Velocity and pressure fields were also simulated to calculate the apparent permeability of optimally tapered channels in various IDFF layouts. The apparent permeability kappis inversely defined by considering a superficially homogeneous Darcian medium comprised of an electrode with length L and inlet / outlet face area A that is supplied with a flow rate Vtotaiby an IDFF that is either in contact with or embedded within the electrode. Thus, flow rate scales in direct proportion to the apparent permeability and to the difference in pressure between the inlet of the IDFF’s high-pressure channel (p ) and the outlet of its low-pressure channel (Pout)'- > kapp Pin ~ Pout.'' total,p L(Eq. 31)

[0318] Apparent permeabilities predicted using 3D simulations are shown to exceed the permeability of the porous electrodes in which they are embedded (0.1 pm2) by up to 3xl03times, demonstrating the ability of micro-interdigitated flow fields to enhance flow through electrodes. According to Eq. 31, apparent permeability values reported here may be used to inversely determine the pressure difference needed to produce a certain flow rate. Longer electrodes and shorter channel spacings are shown to produce increased permeability, while decreasing electrode permeability increases normalized apparent permeability. Quasi- ID theory can be used to explain why this effect arises. There, a decrease in electrode permeability decreases the dimensionless parameter E, causing characteristic channel hydraulic resistance to become smaller than that of porous electrode material. This effect causes apparent permeability to be maximized because channel pressure approaches uniformity in that limit.

[0319] Quasi- ID theory was used in tandem to predict apparent permeability. Here, the conservation of volumetric flow was enforced on a differential control volume for high- and low-pressure channels while exploiting the constant pressure gradient that is unique tooptimally tapered channels. This analysis yields a formula for normalized apparent permeability that scales hyperbolically with the dimensionless parameter E:,, max,*^app _ ^appkve1 + 2^(Eq. 32)

[0320] Here, k™pp’* is the upper bound for apparent hydraulic permeability that is obtained for vanishing E, which corresponds to channels possessing negligible pressure variations. Accordingly, k™pp’* is shown to depend on channel length Lch, electrode length L, the center-to-center spacing sccbetween adjacent channels, and the representative flowpath length between channels sri max,» _ LcftLKapp —(Eq. 33)

[0321] The predictions of quasi- ID apparent permeability show least deviation from 3D simulations for small electrode permeabilities. With the exception of the case having large electrode length, the results show that the quasi- ID apparent permeability model deviates from simulated values by less than 10%, validating its use.

[0322] In addition, the present 3D simulations were used to determine the minor loss coefficients Kuand Kdthat are respectively associated with the upstream contraction of flow to enter high-pressure channels and the downstream expansion of flow to exit low-pressure channels. A formula was derived for the modified apparent permeability ka'ppthat incorporates their hydraulic resistances in series with that of a porous electrode having apparent permeability kapp:^ ka' pp _ 1 -1- kapp -i i n $cc(J^u + Kd)kapp1 + Rew°

[0323] Here, ka'ppdecreases with increasing channel- wise Reynolds number ReWo= PewJSc as a result of increased flow inertia, where Sc is the Schmidt number defined as p / pD0. The pre-factor multiplying against Reynolds number (K= scc( / <u+ K^k^ / (W0)2L) is a dimensionless measure of minor-loss coefficients relative to the hydraulic resistance of the electrode fed by a certain IDFF. The preceding formula was fitted to the 3D simulations to infer the following loss-coefficients for channels spaced at 700 pm: Ku= 0.65and Kd= 0.04. There was a negligible drop in permeability realized for sufficiently small Reynolds number, corresponding to vanishing minor losses.

[0324] Results of Reaction Polarization Using Optimally Tapered IDFF Channels

[0325] Mass transfer simulations were performed that incorporate advection and diffusion flux to predict the distribution of active species concentration c subject to apparently homogeneous reactions with a volumetric consumption rate n'" = k'(c — ceq) that obeys apparent first-order kinetics. Unless stated otherwise, 45 mm-long porous electrodes were paired with IDFFs that use the optimally tapered channels for which 3D flow simulations with center-to-center spacing equal to 700 pm. Table B 1 shows that the apparent rate constant k' and the equilibrium concentration ceqused to predict n'" map under certain conditions to the non-linear kinetics of either (1) electrochemical separations devices that absorb / release cations from solution upon red / ox (i.e., intercalative Faradaic deionization) or (2) flow batteries that use soluble redox-active molecules for energy storage. Thus, the use of a simplified kinetics formalism enables applicability across a range of electrochemical technologies, while additionally easing numerical solution to the associated governing equations by eliminating non-linearity and the number of unknown fields to be solved over ~106finite-volume cells. In addition to geometric variations, results of the associated simulations are shown to depend principally on (1) an electrode- wise Damkohler number Da of the first kind that quantifies kinetic rates relative to diffusion rates as DaSr= k's^ / Deff and (2) a channel- wise Peclet number that quantifies advection rates relative to diffusion rates as PeWo= {uSin)owo / Do.

[0326] Simulated concentration distributions that result from varying channel-wise Peclet number when electrode- wise Damkohler number is fixed to a large value (DaSr= 25), corresponding to a diffusion-limited regime where kinetics are relatively facile due to a large apparent rate constant k'. While flow batteries with high k' are generally achievable by using either a sufficiently large cathodic or anodic overpotential (i.e., to approach the Tafel regime as shown by Table B 1), high k' is realized in Faradaic deionization primarily by a large cathodic overpotential due to the asymmetric nature of cation intercalation kinetics (Table B 1), corresponding to intercalation caused by a reducing current. Inspection at low flow rates reveals that low Peclet numbers cause concentration variations and reactions to become focused at the inlet of the high-pressure channel. Though not simulated explicitly here, such a condition would produce large Ohmic polarization because increased liquid- andsolid-phase current density is commensurate with the focusing of reactions near the inlet region to form reaction “hot spots.” Hence, the spatial deviation of n'" from uniformity was used as a surrogate measure of polarization. As Peclet number increases beyond -100, concentration within the high-pressure channel uniformizes toward the imposed inlet concentration cin, corresponding to the concentration ratio (c— Cin) / (ceq— cin) approaching zero. As channel concentration becomes more uniform, the concentration distribution, and hence the reaction rate distribution, assumes a one-dimensional variation along transverse flow paths. Such a reaction distribution possesses lowest spatial deviation, eliminating reaction hot spots.

[0327] 3D simulations of mass transfer were also executed using a 100-fold smaller electrode- wise Damkohler number DaSr. Such a Damkohler number indicates conditions under which the apparent kinetic rate constant k' is sluggish in comparison with activespecies diffusion. Low rate constant occurs in a flow battery by either sufficiently small anodic or cathodic overpotentials (Table Bl), whereas a low rate constant occurs in Faradaic deionization with a sufficiently large anodic overpotential (Table Bl) when cations deintercalate from host material due to oxidizing current. Concentration fields take on the desired ID, transverse variation sought at a much smaller Peclet number than for large Damkohler number. In addition, the spatial distribution of concentration in hot spot regions is markedly different at low Damkohler number than at high Damkohler number, where concentration shows a ID longitudinal variation at low Damkohler number. In any case, the associated reaction hot spots occurring at low Damkohler number are nonetheless located near the electrode’s inlet.

[0328] A quasi- ID theory for mass transfer within electrodes fed by IDFFs that use optimally tapered channels is used, predicated on the hypothesis that the back-diffusion of active species from electrode regions into channels causes reaction hot spots. This quasi- ID theory enforces the conservation of active-species mass on a differential control arranged along a representative high-pressure channel, subject to a back-diffusion flux jbackgoverned by the film law of mass transfer: jback= hm^c^ —cch), where is an associated masstransfer coefficient due to back-diffusion. Predicated on the above hypothesis, the model neglects longitudinal diffusion and hydrodynamic dispersion within channels, yielding a power-law formula for the channel concentration cchShSr^ch ^eq _ A T \eeSr^in Ceq ' ^ch'(Eq. 34)

[0329] Here, the power-law exponent depends on an electrode- wise Peclet number PeSrdefined as PeSr= us lsr / Deff and a back-diffusion Sherwood number ShSrdefined as ShSr= hmsr / Deff. As such, Eq. 34 suggests that hmcan be determined experimentally by fitting measured channel concentration profiles to a power law. Here, us ±is the transverse velocity through porous electrode material that is uniform along each channel due to use of optimally tapered channels. The simulated concentration contours shown in channels agree qualitatively with the trends expected from this simplistic analytical model, irrespective of Damkohler number: as Peclet number increases the power-law exponent decreases, resulting in the focusing of channel concentration gradients toward channel tips. A transition point occurs where this channel concentration distribution becomes linear corresponding to ShSr / PeSr= hm / usequal to unity. Consequently, a design or operating condition is derived based on this quasi- ID model to eliminate reaction hot spots by ensuring uniform channel concentrations: us ±»m, which ensures that ShSr / PeSrfollows a sub-linear power law.

[0330] The 3D simulations quantitatively validate the present quasi- ID model incorporating back-diffusion and the design / operating condition that is derived from it. To perform such validation, the ID, analytical mass transfer model that verified the implementation of 3D simulations was used to find that the back-diffusion Sherwood number ShSrdepends on the electrode-wise Peclet number PeSrand the electrode-wise Damkohler number DaSrin the following manner according to the dimensionless roots r+' and r*S / is= Da,.Pe? + 4Da2(Eq. 35)

[0331] The formula for quasi- ID mass transfer (Eq. 34) was combined with the back-diffusion Sherwood number (Eq. 35) to validate the present quasi- ID mass-transfer model against the 3D simulations. Conservation of volumetric flow through a given high-pressurechannel relates transverse us ±and longitudinal (usy) velocity components as us ±=(usu)w0 / 2Lc / l, enabling electrode-wise Peclet number PeSrto be expressed in terms of channel- wise Peclet number as follows:Zlq | SrSrpesr = ^un L-L = pee fwo Mc— oLj!—f ^ ch

[0332] where Me is the electrode’s MacMullin number equal to D0 / Deff. The resulting distributions of channel concentration cchpredicted by the quasi- ID mass-transfer model agree with 3D simulations within 5% among all Peclet numbers simulated. In addition to providing an ability to validate the present quasi- ID mass-transfer model, this Sherwood number formula (Eq. 35) also provides a basis for design and operational guidance when using IDFFs with optimally tapered channels. The back-diffusion Sherwood number is largest at high Damkohler number and low Peclet number. Conversely, a significant region of the contour shows a back-diffusion Sherwood number is less than unity, corresponding to weak back-diffusion that tends to suppress reaction hot spots by promoting concentration uniformity even at low Peclet number. Beyond this, the close agreement between the quasi-1D model and 3D simulations suggests that hydrodynamic dispersion has an insignificant effect on the concentration distribution within channels, given that the quasi- ID model neglects such effects while 3D simulations implicitly include them due to the local resolution of velocity gradients within channels that is provided by the Darcy-Brinkman formulation. Given that the dispersion coefficient Dafor a channel with circular cross-section scales as the square of Peclet number58(Da / D0= Pe2 / 48), dispersion could cause smearing of channel concentration distributions at Peclet numbers higher than those simulated here.

[0333] The 3D simulated channel concentration distributions were fitted to obtain an apparent power-law exponent that was also compared to analytical results for it based on ID theory. This agreement is not unique to the particular channel spacing, electrode length, and Damkohler number with which the two models were compared. However, for the cases with DaSr= 0.25 theory and simulation deviate substantially at low Peclet number (PeWo~l), but they converge to excellent agreement at sufficiently high Peclet number. This effect is most likely caused by the extreme degree of reaction localization near the inlet of the high-pressure channel in such a circumstance, resulting in 2D diffusion near the electrode edge that is not capture by ID theory. From these results, there are two primary conclusions. Firstly, a sufficiently large Peclet number PeWowill result in a uniform channel concentration.Secondly, a sufficiently large Peclet number PeWowill also result in a one-dimensionalvariation of concentration distribution from each high-pressure channel to its neighboring low-pressure channel. To this end, the criterion that to eliminate hot spots (ShSr / PeSr« 1) can equivalently be expressed in terms of channel- wise Peclet number PeWothat satisfies this condition:pew° Me sr(Eq. 36)

[0334] This result indicates that hot spots are moderated not only by the back-diffusion Sherwood number but also by IDFF layout. Specifically, the critical channel-wise Peclet number to eliminate hot spots increases with the ratio of channel length Lchto the flow-path length srbetween channels. This trend is significant because it suggests that up- scaling of electrodes using such IDFFs promotes hot spots caused by back-diffusion, rather than suppressing them.

[0335] To rigorously characterize hot spots, the degree of reaction uniformity among all simulated cases was quantified by calculating the root-mean-square (RMS) deviation a of the local reaction rate field n'"(x, y, z).o ■ in1 f n \ \T7777 - 1 dVJvpe(Eq. 37)

[0336] Here, Vpeis the volume of porous electrode material over which integration is performed. Small values of a relative to the volume- averaged reaction rate (n"') correspond to reaction-rate distributions that are relatively uniform (cr / (h''' « 1), whereas arof the same order as (n"') is produced by reaction hot spots caused by back-diffusion (cr / (h"' ~1). For a given electrode length and spacing between channels, FIG. 13A shows that reactionrate deviation decreases with increasing Peclet number, consistent with observations of the associated concentration distributions. FIG. 13A also shows that the longest electrode simulated experienced the largest deviation among all cases, while the electrode with the largest center-to-center channel spacing experiences the smallest deviation. These parameters respectively correspond to the cases that produced largest and smallest power-law exponents, where a smaller power-law exponent indicates less impact of back-diffusion. Hence, the reaction polarization produced by a given electrode length with a given channel spacing is minimized by the use of a sufficiently large Peclet number.

[0337] Theoretical results were also obtained for a using quasi- ID theory. Here, local reaction rate was first calculated by combining the formula for the ID transverse concentration distribution within electrodes (Eq. 28) with the quasi- ID formula for the concentration distribution within channels (Eq. 34):r*e(r^x*+rl) _r*e(rlx*+r^) ShSr~ + z-i.

[0338] The roots r± are determined as functions of Peclet and Damkohler number according to Eq. 28. The resulting theoretical a values in FIG. 13 A, determined by numerical integration using Eq. 37, show good agreement between quasi-lD theory and 3D simulations, further validating the use of such simplified models to design optimally tapered IDFFs that eliminate hot spots.

[0339] Having validated this quasi- ID theory using 3D simulations, quasi- ID theory was also used to quantify the spatial extent of hot spots by taking the second moment of the reaction rate distribution according to obtain an RMS reaction moment:7<(Ay)2)aL / 2V3(Eq. 38)

[0340] Here, the subscript of the average operator... ) indicates weighting with respect to the volumetric reaction-rate distribution n”', and the RMS moment is normalized by the maximum possible value of L / 2V3 that is obtained when reactions vary only along the transverse direction between channels. The resulting RMS moment is small for a sufficiently small electrode-scale Peclet number. For example, PeSr= 0.1 produces an RMS moment that is only 5% of the electrode’s length for Da = 20, corresponding to the focusing of reactions into a hot spot that is l / 20ththe size of the electrode. In the opposite limit for the same Damkohler number, above a Peclet number of 10 the RMS moment approaches its maximal value, eliminating hot spots altogether. Accordingly, FIG. 13B shows that the critical electrode-scale Peclet number PeSr Cneeded to eliminate hot spots increases monotonically with increasing Damkohler number. Pes cshows linear scaling with DaSrfor DaSr< 0.1, while it scaling slows to square-root scaling with DaSrfor DaSr> 1.

[0341] Finally, the critical Peclet numbers shown in FIG. 13B were used to determine device- specific residence-time criteria to eliminate hot spots during electrochemical cycling. To determine the Damkohler number for a certain operating condition, certain kinetic and transport parameters were used:

[0342] For intercalative Faradaic deionization (FDI), a volumetric exchange current density of avi0~ 1 A / cm3for Na+intercalation into electrodes with 50 vol.% nickel hexacyanoferrate from 500 mM aqueous NaCl is shown to produce k' at zero overpotential equal to 0.01 1 / s. A bulk diffusivity for NaCl of 1.61X10’9m2 / s, a MacMullin number of 10, and a porosity of 50% were additionally used.

[0343] For a vanadium redox flow battery (RFB), a standard rate constant of 8.5 x 10’6m / s with equal anodic and cathodic transfer coefficients for VO2+ / VO2+redox is shown to produce k' at zero overpotential equal to 0.34 1 / s. A bulk diffusivity for each species was taken as 3.9x 10’10m2 / s, while assuming a porosity of 90% and a MacMullin number of unity.

[0344] The maximum allowable residence time to prevent hot spots within porous electrode material, also referred to as the critical residence time rc. TCwas determined by using each value of DaSrto determine a corresponding PeSr Cvalue based on the asymptotic formulas for it that are shown in FIG. 13B. A corresponding critical superficial velocity was then determined as± c= DeffPeSr C / sr, based upon which TCwas calculated: TC= srE / usj_c. TCis shown to depend on inlet-side overpotential in different ways for FDI and RFBs by virtue of its dependence on k': RFBs shows symmetric criteria, while FDI shows a more demanding criterion when electrodes undergo reduction / intercalation than when they undergo oxidation / deintercalation. RFBs generally require shorter residence times than does FDI, owing to the relatively fast kinetics of RFBs. The use of micro -interdigitated flow fields (ep-IDFFs) also requires shorter residence times than does conventional millimetric IDFFs.

[0345] For RFBs using conventional IDFFs and for FDI using ep -IDFFs, these results show that residence time must be shorter than 1 min to avoid hot spot formation, and these criteria can be even more demanding depending at non-zero overpotential. While such a time scale is short in comparison with 4 hr discharge for an RFB using in grid management and a 6 hr batch-wise desalination process, the ratio of residence time T to either such “batch” timebis determined by electrode pore volume Vp, reservoir volume Vres, and the number of recirculating passes NpassT / b= Vp / (yresNpass. For example, a minimum of three passes,which is generally feasible, must be used to eliminate hot spots for Tc / rb= 0.003 with Fres / Fp = 100.

[0346] Without judicious design, the 3D simulations and quasi- ID theory show that IDFFs are generally susceptible to the formation of dead-zone regions with low transverse superficial velocity, and to the spatial focusing of reaction rate distributions to form hot spots. Dead zone formation is mitigated through the optimal tapering of interdigitated channel cross-sections by using specific conductance functionals that uniformize flow. In contrast, channels that use linear tapering of the presently tested channel cross-sections show dead zones at channel ends, while channels without tapering of any sort show dead zones in the central region between channels. Based on quasi- ID theory, formulas for the apparent permeability of electrodes that are paired with IDFFs using optimally tapered channels were determined, including minor losses that stem from upstream expansion into and downstream contraction from interdigitated channels. These formulas can readily be used for the design of IDFFs by including the effects of maximal cross-section dimensions for the optimally tapered channels that they use, the center-to-center spacing between channels, and the length and thickness of the electrodes with which they are paired. Both the 3D simulations and the quasi- ID theory make use of Darcy’s law for flow through porous media, albeit including macro-scale shear stress by way of the Brinkman correction.

[0347] Hot spot formation is in turn mitigated by using an electrode- wise Peclet number Pesthat exceeds a critical value which scales with electrode-wise Damkohler number Dasin a linear fashion for DaSr« 1 and in square-root fashion for DaSr» 1. The apparent first-order rate constant k' with which DaSrscales linearly is shown to vary in a manner with applied overpotential that is specific to the type of electrode used. While the kinetics of soluble redox-active molecules produce k' that is symmetric with respect to reduction and oxidation if both processes have equal transfer coefficients, cation intercalation processes produce k' that takes a smaller value for oxidation (i.e., deintercalation) than for reduction (i.e., intercalation). Given that Peclet number scales directly with flow velocity, these findings suggest that for both types of electrodes flow rate could be made to vary with time and that dissimilar flow rates could be used in the respective electrodes of a two-electrode flow cell to eliminate hot spots. The strong agreement between the formulas derived using quasi- ID theory also indicate with confidence that back-diffusion is the cause of hot spots. While the present first-order kinetics formulation is applicable to electrochemical processesin certain limits, the present results motivate modeling using multi-component mass transport with process-specific kinetics to capture transient, non-linear phenomena.

[0348] Apparent Permeability Theory

[0349] To derive the apparent permeability of IDFF using optimally tapered channels it is first recognized that flow into and out of each channel satisfies volume conservation. That is, the volumetric flow rate into the high-pressure channel Vy balances with the net volumetric flow rate out of its sides due to a uniform transverse velocity us ±into porous electrode material: Fy = 2usLchh. The channel’s maximal hydraulic conductance Godetermines the supply pressure pnneeded to produce the flow rate Fy:.y _ GQ (ptn~ Ptip')M iPLch

[0350] where Pfipis pressure at the high-pressure channel’s tip that determines the constant pressure gradient that is produced when optimally tapered channels are using with a linear conductance profile (Eq. 29). The local pressure difference Ap±between high- and low-pressure channels additionally drives transverse superficial velocity us ±according to Darcy’s law over a representative flow path of length sr:_ k£e(p£p- pLut)ns± —

[0351] where PoUtis the outlet-side pressure across the domain. Volume conservation that couples Vy and us ±produces a relation for the high-pressure channel’s tip pressure Pfipas a weighted average of Pinand PoUtH _ Go$rPin "h ^‘kpehLCfipouttiv Gosr+ 2kvehL2rt.

[0352] Substitution into the expression for us ±produces an expression for it in terms of the pressure difference p"n— p°utapplied from the electrode’s inlet to its outlet:kpeGo(. Pin ~ Pout)J_ — r 7 \" I- ^kpghLch)

[0353] Using Darcy’s law (Eq. 31) an expression for apparent permeability kappis found:7 C I I j max,*‘'■app _uG0oLLlchLLl_ ^appkpe G0SccSr+ 2kpehLchscc1 + 2I_I

[0354] where k™pp’* is determined from Eq. 33.

[0355] The effects of minor losses to determine a modified apparent permeability ka'ppthat depends on Reynolds number were included. Using Eq. 31, the pressure drop associated with flow through an IDFF using channels with a center-to-center spacing sccand a mean longitudinal velocity < Uy )0and width w0at its channels’ maximal cross-section that feeds an electrode of length L to produce its apparent permeability kappis expressed:, tWo<“n)oMAp'=" ^2T^c7c^ —app

[0356] In turn, the minor head losses due to upstream flow contraction and downstream flow expansion are expressed in terms of each loss coefficient Ktfor the relevant process in terms of the flow’s mean velocity < Uy )0at the maximal cross-section of its channels:kApt - - K K-ioP^°2

[0357] The total pressure difference includes serial contributions for upstream contraction, downstream contraction, and permeation through the electrode via the IDFF:_ Lwo{u]})ou p2Aptotal y 7. T y T£. SCCf^app

[0358] Eq. 31 is then used to determine the modified apparent permeability ka'ppthat incorporates minor losses:k-app _ 1 Lwo< U||)o / z 1kapp kapp2scc^ptotaiscc(ffu+ Kd)kapp1 + R w°

[0359] where the electrode- wise Reynolds number ReWois given as ReWo{un)owop / p = PewJSc.

[0360] Back Diffusion Theory

[0361] Quasi- ID theory is used to model the effects of active-species back-diffusion from electrode material into high-pressure channels that are arranged in an IDFF using optimally tapered channels. Here, back-diffusion flux jbackis governed by the film law of mass transfer: jback= hm(ceq— cch), where hmis a back-diffusion mass transfer coefficient and cchis channel concentration. Back-diffusion occurs in addition to advection into and out of a representative high-pressure channel along both the length of each channel and through its sides. To couple these mass transfer rates, a quasi- ID control volume is invoked along adifferential segment of a representative high-pressure channel. Assuming steady-state conditions, conservation of active species requires the following equation to be satisfied: ^11 (y)cch(y) + 2hm(ceq- cch(y))hdy - VN(y + dy)cch(y + dy) - 2uSi±cch(y)hdy = 0

[0362] After invoking volume conservation (V||(y) — V||(y + dy) = 2us lccffy)hdy). mass conservation produces the following ordinary differential equation that governs channel concentration cch1 dcch2hh-m(Seq ~ ^ch) dy F||

[0363] The preceding equation is made integrable by recognizing that F||(y) scales linearly with distance y for an IDFF using optimally taped channels, due to uniform transverse flow:^11 = ''il.o (1 -T-)

[0364] Integration ultimately yields a power-law profile for channel concentration:ceq ~ceq _ / _ 7 \ V||,oC-in Ceq ' ^ch'

[0365] By introducing the back-diffusion Sherwood number as ShSr= hmsr / De^ and after invoking volume conservation (F||0= 2uS J_Lh), the power-law exponent 2hmhL / V\\0is found to simplify to ShSr / PeSrto produce Eq. 34.

[0366] ID Concentration Distribution and Back-Diffusion Sherwood Number

[0367] For unidirectional transverse flow with negligible longitudinal diffusion, the advection-diffusion-reaction equation (Eq. 27) reduces to an ordinary differential equation at steady-state that governs the variations of active- species concentration c with transverse position x:d2c us ±de k'dx2Deff dx Deff ^Ceq) 0

[0368] One can readily show that the re-scaled function 0 = c — ceqis a linear combination of exponential functions of dimensionless position x* = x / srand two distinct, dimensionless roots rff.0= c — ceq= A+er+x* + A_er~x*

[0369] These roots are shown by Eq. 28 to depend on electrode- wise Peclet number PeSr= ussr / Deff and Damkohler number DaSr= k's2 / Deff. Using a known inletconcentration cinand neglecting diffusive flux at the outlet boundary (dc / dx\x=s= 0), the constants A±are readily determined to yield Eq. 28 after non-dimensionalization.

[0370] Eq. 28 is then used to determine back-diffusion flux jbackas the diffusive flux occurring away from the domain at x = 0:* *def• * * er2 - erijback ~ Dgff, — De^jceq — Cinjr1r2~r~ *;r* ” dxx=Qv 7Sj-fr-L eri — r2er2)

[0371] By recognizing that the product rjr2simplifies to rjr2= —DaSr, the back-diffusion Sherwood number simplifies to the form shown by Eq. 35 after using the film law of mass transfer to determine the mass transfer coefficient as= jbaCk / (ceq ~ctn -

[0372] What has been described above includes mere examples of various embodiments. It is, of course, not possible to describe every conceivable combination of components or methodologies for purposes of describing these examples, but one of ordinary skill in the art can recognize that many further combinations and permutations of the present embodiments are possible. Accordingly, the embodiments disclosed and / or claimed herein are intended to embrace all such alterations, modifications and variations that fall within the spirit and scope of the appended claims. Furthermore, to the extent that the term “includes” is used in either the detailed description or the claims, such term is intended to be inclusive in a manner similar to the term “comprising” as “comprising” is interpreted when employed as a transitional word in a claim.

[0373] Although specific embodiments have been illustrated and described herein, it should be appreciated that any arrangement which achieves the same or similar purpose may be substituted for the embodiments described or shown. This disclosure is intended to cover any and all adaptations or variations of various embodiments. Combinations of the above embodiments, and other embodiments not specifically described herein, can be used. For instance, one or more features from one or more embodiments can be combined with one or more features of one or more other embodiments. In one or more embodiments, features that are positively recited can also be negatively recited and excluded from the embodiment with or without replacement by another structural and / or functional feature. The steps or functions described with respect to the embodiments of the disclosure can be performed in any order. The steps or functions described with respect to the embodiments of the disclosure can be performed alone or in combination with other steps or functions of the disclosure, as well as from other embodiments or from other steps that have not been described in the disclosure.Further, more than or less than all of the features described with respect to an embodiment can also be utilized.

[0374] The illustrations of embodiments described herein are intended to provide a general understanding of the structure of various embodiments, and they are not intended to serve as a complete description of all the elements and features of apparatus and systems that might make use of the structures described herein. Many other embodiments will be apparent to those of skill in the art upon reviewing the above description. Other embodiments may be utilized and derived therefrom, such that structural and logical substitutions and changes may be made without departing from the scope of this disclosure. Figures are also merely representational and may not be drawn to scale. Certain proportions thereof may be exaggerated, while others may be minimized. Accordingly, the specification and drawings are to be regarded in an illustrative (rather than in a restrictive) sense.

[0375] This disclosure relates to the following aspects:

[0376] A first aspect relates to a fluid-flow device comprising a porous medium, a plurality of inlet channels, and a plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium, the plurality of outlet channels being interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel, wherein at least one of the inlet channels and one of the adjacent outlet channels are each defined by an outer boundary profile having a cross-sectional geometry that varies non-linearly along a length thereof, and wherein a shortest flow path distance through the porous medium between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is constant along the length of the inlet channel.

[0377] A second aspect relates to the fluid-flow device of any aspect, wherein at least one dimension of the outer boundary profile of the inlet channel is two-fold symmetric about a point along the outer boundary profile.

[0378] A third aspect relates to the fluid-flow device of any aspect, wherein the device is configured to provide a uniform flow transverse to the length of the plurality of inlet channels.

[0379] A fourth aspect relates to the fluid-flow device of any aspect, wherein the width of plurality of inlet channels at a position x along the length L is determined by the relationship - = — „,, wherein GHis a hydraulic conductance of one of the L GH(wH,hH)+fGL(wHx=O)-wH,hLyJplurality of inlet channels, GLis a hydraulic conductance of one of the plurality of outlet channels, wHis a width of one of the plurality of inlet channels at position x, wH(x = 0) is a maximum width of one of the plurality of inlet channels at an inlet, hHis a depth of one of the plurality of inlet channels, hLis a depth of one of the plurality of outlet channels, and f is a ratio of a flux-viscosity product between one of the plurality of the inlet channels and one of the outlet channels.

[0380] A fifth aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional geometry of the inlet channel and the cross-sectional geometry of the outlet channel are varied such that a flow velocity of a fluid flowing between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is substantially uniform along the length of the inlet channel.

[0381] A sixth aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional geometry of the inlet channel and the cross-sectional geometry of the outlet channel are varied such that a residence time of a fluid flowing between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is substantially uniform along the length of the inlet channel.

[0382] A seventh aspect relates to the fluid-flow device of any aspect, wherein the plurality of inlet channels and the plurality of outlet channels are formed by a process comprising at least one of milling, engraving, stamping, or embossing.

[0383] An eighth aspect relates to the fluid-flow device of any aspect, wherein a pressure gradient of each of the plurality of inlet channels is substantially equal to a pressure gradient of each of the plurality of outlet channels at any given position along a length of each of the plurality of inlet channels.

[0384] A ninth aspect relates to the fluid-flow device of any aspect, wherein the device is included in a fuel cell, reduction-oxidation (redox) flow battery, electrochemical energy system, electrochemical desalination system, electrochemical separation system, electrodialysis, water purification, or thermal system.

[0385] A tenth aspect relates to the fluid-flow device of any aspect, wherein a hydraulic conductance of each of the plurality of inlet channels varies according to the function G(x) = (1 / x) S“=i bn[cos(n7r(x — L / 2) / L) — cos(n?r / 2)], where bnis a real number that produces a real, positive value of G, x is a position along the length of the inlet channel, and L is the length of the inlet channel.

[0386] An eleventh aspect relates to the fluid-flow device of any aspect, wherein a width dimension of the cross-sectional geometry of the inlet channel varies and a depth dimension of the cross-sectional geometry of the inlet channel is fixed along the length of the inlet channel.

[0387] A twelfth aspect relates to the fluid-flow device of any aspect, wherein the outer boundary profile of the outlet channel exhibits anti-symmetry relative to the outer boundary profile of the adjacent inlet channel.

[0388] A thirteenth aspect relates to the fluid-flow device of any aspect, wherein the outer boundary profile of the inlet channel exhibits an offset, shifted hyperbolic variation along the length of the inlet channel.

[0389] A fourteenth aspect relates to the fluid-flow device of the thirteenth aspect, wherein a flux- viscosity product of the plurality of inlet channels is different from a fluxviscosity product of the plurality of outlet channels.

[0390] A fifteenth aspect relates to the fluid-flow device of any aspect, wherein the outer boundary profile of the inlet channels exhibits a sigmoidal variation along the length of the inlet channel.

[0391] A sixteenth aspect relates to the fluid-flow device of the fifteenth aspect, wherein a flux- viscosity product of the plurality of inlet channels is equal to a flux-viscosity product of the plurality of outlet channels.

[0392] A seventeenth aspect relates to the fluid-flow device of the fifteenth aspect, wherein a flux- viscosity product of the plurality of inlet channels is higher than a fluxviscosity product of the plurality of outlet channels, and the cross-sectional geometry of the inlet channel varies nonlinearly along the length of the inlet channel.

[0393] An eighteenth aspect relates to the fluid-flow device of any aspect, wherein a maximum width of the cross-sectional geometry of the inlet channel is larger than a maximum width of the cross-sectional geometry of the outlet channel.

[0394] A nineteenth aspect relates to the fluid-flow device of any aspect, wherein a width of the cross-sectional geometry of the inlet channel at any position along its length is larger than a maximum width of the cross-sectional geometry of the outlet channel along the length of the inlet channel.

[0395] A twentieth aspect relates to the fluid-flow device of any aspect, wherein the device is formed in an electrode body.

[0396] A twenty-first aspect relates to the fluid-flow device of the twentieth aspect, wherein the electrode body is porous.

[0397] A twenty- second aspect relates to the fluid-flow device of the twenty-first aspect, wherein the electrode body comprises electrode pores configured to receive a fluid flow.

[0398] A twenty-third aspect relates to the fluid-flow device of the twenty-first aspect, wherein the electrode body is included in a system further comprising a second electrode body and a separator disposed between the electrode body and the second electrode body, wherein the system is included as part of a reduction-oxidation (redox) flow battery, a fuel cell, an electrolysis cell, or an apparatus configured to facilitate enzymatic reactions, electrochemical separation processing, metal recovery processing, purification processing, or heat transfer.

[0399] A twenty-fourth aspect relates to the fluid-flow device of any aspect, wherein the plurality of inlet channels and the plurality of outlet channels are formed in an impervious substrate.

[0400] A twenty-fifth aspect relates to the fluid-flow device of any aspect, wherein each of the plurality of inlet channels and the plurality of outlet channels comprise a plurality of stepwise segments, each stepwise segment having a constant cross-sectional geometry along a segment length, and the cross-sectional geometry is different between adjacent stepwise segments.

[0401] A twenty- sixth aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional geometry of each of the plurality of inlet channels varies non-linearly and continuously along its respective length.

[0402] A twenty- seventh aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional geometry of each of the plurality of outlet channels varies non-linearly and continuously along its respective length.

[0403] A twenty-eighth aspect relates to the fluid-flow device of any aspect, wherein a hydraulic conductance of each of the plurality of inlet channels varies linearly along its respective length.

[0404] A twenty-ninth aspect relates to the fluid-flow device of any aspect, wherein a hydraulic conductance of each of the plurality of outlet channels varies linearly along its respective length.

[0405] A thirtieth aspect relates to the fluid-flow device of any aspect, wherein each of the plurality of inlet channels comprises a radius at a position where the cross-sectional geometry is at a maximum width.

[0406] A thirty-first aspect relates to the fluid-flow device of any of the preceding aspects, wherein the outer boundary profile of the inlet channel exhibits a first-degree rational-function variation along the length of the inlet channel.

[0407] A thirty-second aspect relates to the fluid-flow device of any of the preceding aspects, wherein the outer boundary profile of the inlet channel exhibits a third-degree rational-function variation along the length of the inlet channel.

[0408] A thirty-third aspect relates to the fluid-flow device of any of the preceding aspects, wherein the outer boundary profile of the inlet channel exhibits a fifth-degree rational-function variation along the length of the inlet channel.

[0409] A thirty-fourth aspect relates to the fluid-flow device of any of the preceding aspects, wherein the outer boundary profile of the outlet channel exhibits a first-degree rational-function variation along the length of the outlet channel.

[0410] A thirty-fifth aspect relates to the fluid-flow device of any of the preceding aspects, wherein the outer boundary profile of the outlet channel exhibits a third-degree rational-function variation along the length of the outlet channel.

[0411] A thirty-sixth aspect relates to the fluid-flow device of any of the preceding aspects, wherein the outer boundary profile of the outlet channel exhibits a fifth-degree rational-function variation along the length of the outlet channel.

[0412] A thirty-seventh aspect relates to a fluid-flow device comprising a porous medium, a plurality of inlet channels, and a plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium, the plurality of outlet channels being interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel, wherein a shortest flow path distance through the porous medium between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is constant along the length of the inlet channel, and wherein at least one of the inlet channels is defined by an outer boundary profile having two-fold symmetry about an inlet intermediate point and at least one of the adjacent outlet channels is defined by an outer boundary profile having two-fold symmetry about an outlet intermediate point.

[0413] A thirty-eighth aspect relates to the fluid-flow device of any aspect, wherein the inlet intermediate point is at a midpoint of the outer boundary profile of the inlet channel and wherein the outlet intermediate point is at a midpoint of the outer boundary profile of the outlet channel.

[0414] A thirty-ninth aspect relates to a fluid-flow device comprising a porous medium, a plurality of inlet channels, and a plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium, the plurality of outlet channels being interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel, wherein at least one of the inlet channels and the adjacent outlet channels are each defined by an outer boundary profile having a cross-sectional geometry that varies non-linearly along a length thereof, and wherein the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel each comprise a plurality of stepwise segments, each stepwise segment having a constant cross-sectional area along a segment length, and wherein the cross-sectional area varies between adjacent stepwise segments.

[0415] A fortieth aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional area of the plurality of segments of the inlet channel are sized to result in an approximate linear variation of hydraulic conductance along a length of the inlet channel.

[0416] A forty-first aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional area of the plurality of segments of the outlet channel are sized to result in an approximate linear variation of hydraulic conductance along a length of the outlet channel.

[0417] A forty- second aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional geometry of the inlet channel and the cross-sectional geometry of the outlet channel are varied such that a flow velocity of a fluid flowing between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is substantially uniform along the length of the inlet channel.

[0418] A forty-third aspect relates to the fluid-flow device of any of the preceding aspects, wherein a shortest flow path distance through the porous medium between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is constant along the length of the inlet channel.

[0419] A forty-fourth aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional area of a first segment of the inlet channel is larger than the cross-sectionalarea of a second segment of the inlet channel, and the second segment is downstream of the first segment.

[0420] A forty-fifth aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional area of a first segment of the outlet channel is smaller than the cross-sectional area of a second segment of the outlet channel, wherein the second segment is downstream of the first segment.

[0421] A forty- sixth aspect relates to the fluid-flow device of any aspect, wherein the plurality of inlet channels and the plurality of outlet channels are formed by a process comprising at least one of milling, engraving, stamping, or embossing.

[0422] A forty- seventh aspect relates to the fluid-flow device of any aspect, wherein the device is included in a fuel cell, reduction-oxidation (redox) flow battery, electrochemical energy system, electrochemical desalination system, electrochemical separation system, electrodialysis, water purification, or thermal system.

[0423] A forty-eighth aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional area of each of the plurality of segments of each of the inlet channels and the outlet channels are rectangular.

[0424] A forty-ninth aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional area of each of the plurality of segments of each of the inlet channels and the outlet channels are circular.

[0425] A fiftieth aspect relates to the fluid-flow device of any aspect, wherein the cross-sectional area of each of the plurality of segments of each of the inlet channels and the outlet channels are trapezoidal.

[0426] A fifty-first aspect relates to a fluid-flow device comprising a porous medium, a plurality of inlet channels, and a plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium, wherein the plurality of outlet channels are interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel, wherein at least one of the inlet channels and one of the adjacent outlet channels are each defined by an outer boundary profile having a cross-sectional geometry that varies non-linearly along a length thereof, wherein a hydraulic conductance of at least one of the inlet channels varies according to the function GH(x) = (1 / x) S“=i bn[cos(n7r(% — L / 2) / L) — cos(n?r / 2)], where bnis a real number that produces a real, positive value of GH, x is a position along the length of the inlet channel, and L is the length of the inlet channel, and wherein a hydraulic conductance of at least one of theadjacent outlet channels varies according to the function GL(x) = -GH(L — x), wherein f is a ratio of a flux-viscosity product between the at least one inlet channel and outlet channel.

[0427] A fifty- second aspect relates to an electrode for an electrochemical cell, comprising a porous electrode body having a first surface and a second surface, a plurality of high-pressure channels embedded within the porous electrode body, each high-pressure channel extending from a high-pressure inlet at the first surface toward the second surface and having a width that varies along a length of said high-pressure channel, and a plurality of low-pressure channels embedded within the porous electrode body and interdigitated with the plurality of high-pressure channels, each low-pressure channel extending from a low-pressure inlet at the first surface toward the second surface and having a width that varies along a length of said low-pressure channel, wherein an inter-channel spacing between adjacent high-pressure and low-pressure channels is substantially constant along the length of said channels, and wherein the width variations of the high-pressure channels and the low-pressure channels are configured to produce a uniform transverse flow velocity through the porous electrode body between adjacent high-pressure and low-pressure channels.

[0428] A fifty-third aspect relates to the electrode of any of the preceding aspects, wherein a pressure gradient within each high-pressure channel is substantially equal to a pressure gradient within an adjacent low-pressure channel at any given position along the length of the channels.

[0429] A fifty-fourth aspect relates to the electrode of any of the preceding aspects, wherein the high-pressure channels and the low-pressure channels are anti-symmetric in their width profiles.

[0430] A fifty-fifth aspect relates to the electrode of any of the preceding aspects, wherein each of the high-pressure channels and each of the low-pressure channels has a rectangular cross-section with a fixed depth h, and wherein the width u / Hof each high-pressure channel at a position x is related to position by:■2x f(l — wH / wH(x = 0))= 0))3+ / (1 — wH / wH(x = 0))3where / is a ratio of flux- viscosity products between the high-pressure and low-pressure channels, and wherein wH / h < 1.

[0431] A fifty-sixth aspect relates to the electrode of any of the preceding aspects, wherein the width profile of each high-pressure channel exhibits a sigmoidal shape with twofold symmetry when the flux- viscosity product ratio / equals 1.

[0432] A fifty- seventh aspect relates to the electrode of any of the preceding aspects, wherein the width profile of each high-pressure channel exhibits a non-linear variation with downward concavity when the flux-viscosity product ratio / is greater than 1.

[0433] Many other modifications of the embodiments above may be made to adapt a particular situation or material to the teachings without departing from the scope of the current disclosure. Therefore, it is intended that the present devices and systems not be limited to the particular embodiments disclosed, but that the disclosed devices and systems include all embodiments falling within the scope of the appended claims. Moreover, the advantages described herein are not necessarily the only advantages of the present disclosure and it is not necessarily expected that every embodiment of the present disclosure will achieve all of the advantages described.

Claims

CLAIMS1. A fluid-flow device comprising:a porous medium;a plurality of inlet channels; anda plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium, wherein the plurality of outlet channels are interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel;wherein at least one of the inlet channels and one of the adjacent outlet channels are each defined by an outer boundary profile having a cross-sectional geometry that varies non-linearly along a length thereof; andwherein a shortest flow path distance through the porous medium between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is constant along the length of the inlet channel.

2. The fluid-flow device of claim 1, wherein at least one dimension of the outer boundary profile of the inlet channel is two-fold symmetric about a point along the outer boundary profile.

3. The fluid-flow device of claim 1, wherein the device is configured to provide a uniform flow transverse to the length of the plurality of inlet channels.

4. The fluid-flow device of claim 1, wherein the width of plurality of inlet channels at a position x along the length L is determined by the relationship =— GH—(wH,hH)+fG<\L~(w°H(Wx=O)-)wH,hL) wherein GHis a hyJdraulic conductance of one of the p Flurality J of inlet channels, GLis a hydraulic conductance of one of the plurality of outlet channels, wHis a width of one of the plurality of inlet channels at position x, wH(x = 0) is a maximum width of one of the plurality of inlet channels at an inlet, hHis a depth of one of the plurality of inlet channels, hLis a depth of one of the plurality of outlet channels, and f is a ratio of a flux- viscosity product between one of the plurality of the inlet channels and one of the outlet channels.

5. The fluid-flow device of claim 1, wherein the cross-sectional geometry of the inlet channel and the cross-sectional geometry of the outlet channel are varied such that a flow velocity of a fluid flowing between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is substantially uniform along the length of the inlet channel.

6. The fluid-flow device of claim 1, wherein the cross-sectional geometry of the inlet channel and the cross-sectional geometry of the outlet channel are varied such that a residence time of a fluid flowing between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is substantially uniform along the length of the inlet channel.

7. The fluid-flow device of claim 1, wherein the plurality of inlet channels and the plurality of outlet channels are formed by a process of at least one of milling, engraving, stamping, or embossing.

8. The fluid-flow device of claim 1, wherein a pressure gradient of each of the plurality of inlet channels is substantially equal to a pressure gradient of each of the plurality of outlet channels at any given position along a length of each of the plurality of inlet channels.

9. The fluid-flow device of claim 1, wherein the device is included in a fuel cell, reduction- oxidation (redox) flow battery, electrochemical energy system, electrochemical desalination system, electrochemical separation system, electrodialysis, water purification, or thermal system.

10. The fluid-flow device of claim 1, wherein a hydraulic conductance of each of the plurality of inlet channels varies according to the function G(x) = (1 / x) S“=i bn[cos(n7r(x — L / 2) / L) — cos(n?r / 2)], wherein bnis a real number that produces a real, positive value of G, x is a position along the length of the inlet channel, and L is the length of the inlet channel.

11. The fluid-flow device of claim 1, wherein a width dimension of the cross-sectional geometry of the inlet channel varies and a depth dimension of the cross-sectional geometry of the inlet channel is fixed along the length of the inlet channel.

12. The fluid-flow device of claim 1, wherein the outer boundary profile of the outlet channel exhibits anti-symmetry relative to the outer boundary profile of the adjacent inlet channel.

13. The fluid-flow device of claim 1, wherein the outer boundary profile of the inlet channel exhibits an offset, shifted hyperbolic variation along the length of the inlet channel.

14. The fluid-flow device of claim 13, wherein a flux-viscosity product of the plurality of inlet channels is different from a flux-viscosity product of the plurality of outlet channels.

15. The fluid-flow device of claim 1, wherein the outer boundary profile of the inlet channels exhibits a sigmoidal variation along the length of the inlet channel.

16. The fluid-flow device of claim 15, wherein a flux- viscosity product of the plurality of inlet channels is equal to a flux-viscosity product of the plurality of outlet channels.

17. The fluid-flow device of claim 15, wherein a flux-viscosity product of the plurality of inlet channels is higher than a flux- viscosity product of the plurality of outlet channels, and wherein the cross-sectional geometry of the inlet channel varies nonlinearly along the length of the inlet channel.

18. The fluid-flow device of claim 1, wherein a maximum width of the cross-sectional geometry of the inlet channel is larger than a maximum width of the cross-sectional geometry of the outlet channel.

19. The fluid-flow device of claim 1, wherein a width of the cross-sectional geometry of the inlet channel at any position along its length is larger than a maximum width of the cross-sectional geometry of the outlet channel along the length of the inlet channel.

20. The fluid-flow device of claim 1, wherein the device is formed in an electrode body.

21. The fluid-flow device of claim 20, wherein the electrode body is porous.

22. The fluid-flow device of claim 21, wherein the electrode body comprises electrode pores configured to receive a fluid flow.

23. The fluid-flow device of claim 21, wherein the electrode body is included in a system further comprising a second electrode body and a separator disposed between the electrode body and the second electrode body, wherein the system is included as part of a reductionoxidation (redox) flow battery, a fuel cell, an electrolysis cell, or an apparatus configured to facilitate enzymatic reactions, electrochemical separation processing, metal recovery processing, purification processing, or heat transfer.

24. The fluid-flow device of claim 1, wherein the plurality of inlet channels and the plurality of outlet channels are formed in an impervious substrate.

25. The fluid-flow device of claim 1, wherein each of the plurality of inlet channels and the plurality of outlet channels comprise a plurality of stepwise segments, each stepwise segment having a constant cross-sectional geometry along a segment length, and wherein the cross-sectional geometry is different between adjacent stepwise segments.

26. The fluid-flow device of claim 1, wherein the cross-sectional geometry of each of the plurality of inlet channels varies non-linearly and continuously along its respective length thereof.

27. The fluid-flow device of claim 1, wherein the cross-sectional geometry of each of the plurality of the outlet channels varies non-linearly and continuously along its respective length thereof.

28. The fluid-flow device of claim 1, wherein a hydraulic conductance of each of the plurality of inlet channels varies linearly along its respective length thereof.

29. The fluid-flow device of claim 1, wherein a hydraulic conductance of each of the plurality of outlet channels varies linearly along its respective length thereof.

30. The fluid-flow device of claim 1, wherein each of the plurality of inlet channels comprises a radius at a position where the cross-sectional geometry is at a maximum width.

31. The fluid-flow device of claim 1, wherein the outer boundary profile of the inlet channel exhibits a first-degree rational-function variation along the length of the inlet channel.

32. The fluid-flow device of claim 1, wherein the outer boundary profile of the inlet channel exhibits a third-degree or lower rational-function variation along the length of the inlet channel.

33. The fluid-flow device of claim 1, wherein the outer boundary profile of the inlet channel exhibits a fifth-degree or lower rational-function variation along the length of the inlet channel.

34. The fluid-flow device of claim 1, wherein the outer boundary profile of the outlet channel exhibits a first-degree rational-function variation along the length of the outlet channel.

35. The fluid-flow device of claim 1, wherein the outer boundary profile of the outlet channel exhibits a third-degree or lower rational-function variation along the length of the outlet channel.

36. The fluid-flow device of claim 1, wherein the outer boundary profile of the outlet channel exhibits a fifth-degree or lower rational-function variation along the length of the outlet channel.

37. A fluid-flow device comprising:a porous medium;a plurality of inlet channels; anda plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium, wherein the plurality of outlet channels are interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel;wherein a shortest flow path distance through the porous medium between an outer boundary profile of at least one of the inlet channels and an outer boundary profile of an adjacent outlet channel is constant along a length of the inlet channel; andwherein at least one of the inlet channels is defined by an outer boundary profile having two-fold symmetry about an inlet intermediate point and at least one of the adjacent outlet channels is defined by an outer boundary profile having two-fold symmetry about an outlet intermediate point.

38. The fluid-flow device of claim 37, wherein the inlet intermediate point is at a midpoint of the outer boundary profile of the inlet channel and wherein the outlet intermediate point is at a midpoint of the outer boundary profile of the outlet channel.

39. A fluid-flow device comprising:a porous medium;a plurality of inlet channels; anda plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium, wherein the plurality of outlet channels are interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel;wherein at least one of the inlet channels and the adjacent outlet channels are each defined by an outer boundary profile having a cross-sectional geometry that varies non-linearly along a length thereof; andwherein the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel each comprise a plurality of stepwise segments, each stepwise segment having a constant cross-sectional area along a segment length, and wherein the cross-sectional area varies between adjacent stepwise segments.

40. The fluid-flow device of claim 39, wherein the cross-sectional area of the plurality of stepwise segments of the inlet channel are sized to result in an approximate linear variation of hydraulic conductance along a length of the inlet channel.

41. The fluid-flow device of claim 39, wherein the cross-sectional area of the plurality of stepwise segments of the outlet channel are sized to result in an approximate linear variation of hydraulic conductance along a length of the outlet channel.

42. The fluid-flow device of claim 39, wherein the cross-sectional geometry of the inlet channel and the cross-sectional geometry of the outlet channel are varied such that a flow velocity of a fluid flowing between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is substantially uniform along the length of the inlet channel.

43. The fluid-flow device of claim 39, wherein a shortest flow path distance through the porous medium between the outer boundary profile of the inlet channel and the outer boundary profile of the outlet channel is constant along the length of the inlet channel.

44. The fluid-flow device of claim 39, wherein the cross-sectional area of a first segment of the inlet channel is larger than the cross-sectional area of a second segment of the inlet channel, and wherein the second segment is downstream of the first segment.

45. The fluid-flow device of claim 39, wherein the cross-sectional area of a first segment of the outlet channel is smaller than the cross-sectional area of a second segment of the outlet channel, wherein the second segment is downstream of the first segment.

46. The fluid-flow device of claim 39, wherein the plurality of inlet channels and the plurality of outlet channels are formed by a process of at least one of milling, engraving, stamping, or embossing.

47. The fluid-flow device of claim 39, wherein the device is included in a fuel cell, reduction- oxidation (redox) flow battery, electrochemical energy system, electrochemical desalination system, electrochemical separation system, electrodialysis, water purification, or thermal system.

48. The fluid-flow device of claim 39, wherein the cross-sectional area of each of the plurality of stepwise segments of each of the inlet channels and the outlet channels are rectangular.

49. The fluid-flow device of claim 39, wherein the cross-sectional area of each of the plurality of stepwise segments of each of the inlet channels and the outlet channels are circular.

50. The fluid-flow device of claim 39, wherein the cross-sectional area of each of the plurality of stepwise segments of each of the inlet channels and the outlet channels are trapezoidal.

51. A fluid-flow device comprising:a porous medium;a plurality of inlet channels; anda plurality of outlet channels in fluid communication with the plurality of inlet channels via the porous medium, wherein the plurality of outlet channels are interdigitated with the plurality of inlet channels such that each inlet channel is positioned adjacent to at least one outlet channel;wherein at least one of the inlet channels and one of the adjacent outlet channels are each defined by an outer boundary profile having a cross-sectional geometry that varies non-linearly along a length thereof; andwherein a hydraulic conductance of at least one of the inlet channels varies according to a function= (1 / x) £”=1bn[cos(nπ(x — L / 2) / L) — cos(nπ / 2)], wherein bnis a real number that produces a real, positive value of GH, x is a position along the length of the inlet channel, and L is the length of the inlet channel, and wherein a hydraulic conductance of at least one of the adjacent outlet channels varies according to a function GL(x) =- GHL — x), wherein f is a ratio of a flux- viscosity product between the at least one inlet channel and outlet channel.

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