WO2016099814A1

WO2016099814A1 - Optimization of configuration of u-type fluidic systems for uniform flow distribution

Info

Publication number: WO2016099814A1
Application number: PCT/US2015/062102
Authority: WO
Inventors: Joshua JACKSON; Mateusz L. Hupert; Steven Soper
Original assignee: The University Of North Carolina At Chapel Hill
Priority date: 2014-12-16
Filing date: 2015-11-23
Publication date: 2016-06-23

Abstract

A fluid array includes: (a) a fluid input header, (b) a fluid output header, and (c) a plurality (N) of adjacent fluidic channels (e.g., parallel channels) connected to both the input header and the output header in a U-array configuration. The header segments are dimensioned to enhance fluid flow therein. In some embodiments, the adjacent fluidic channels are (i) concentrically arranged (ii) decrease sequentially in length, or (iii) are both concentrically arranged and decrease sequentially in length. Such arrays may be used in a variety of applications, including cooling (e.g., in electronic applications), in fuel cells for circulating air and/or fuel, and in diagnostic arrays.

Description

OPTIMIZATION OF CONFIGURATION OF U-TYPE FLUIDIC SYSTEMS

FOR UNIFORM FLOW DISTRIBUTION Joshua Jackson, Mateusz Hupert, and Steven Soper

Related Applications

This application claims the benefit of United States Provisional Patent Application Serial No. 62/092.51 1 , filed December 16, 2014, the disclosure of which is incorporated by reference herein in its entirety.

Field of the Invention

The present invention concerns fluidic and microfiuidic arrays for use in fuel cells, diagnostic devices, cooling devices and the like.

Background of the Invention

Fuel cells with parallel channel configurations require significantly reduced pressure to distribute reagents, thereby minimizing pumping power that parasitically feeds off the fuel cell's power output. However, the practical implementation of parallel configurations have been hampered by intrinsic flow non-uniformity, especially during scale-up, and the difficulty in removing water byproducts.

Flow non-uniformity arises due to non-negligible header channel resistance that perturbs the system from an ideally uniform distribution. These effects have been well documented [1-25], but more recently, geometries optimized for flow uniformity have been developed, specifically for Z-type [1, 19, 24] and pin-type [25] systems. Relatively straightforward geometric ratios for headers in Z-type systems were deduced from a discrete geometry optimization method to correct for flow non-uniformity. The primary advantages of this design strategy were computational and conceptual simplicity as well as preserving the geometry of the electroactive channels.

Summary of the Invention

As noted above, the present invention provides a fluid array, comprising:

(a) a fluid input header. (b) a fluid output header, and

(c) a plurality (N) of adjacent fluidic channels (e.g.. parallel channels) connected to both the input header and the output header in a U-array configuration.

The header segments are dimensioned as described herein, or the architecture or configuration of the array meets the criteria described herein, for enhancing fluid flow therein. Such arrays may be used in a variety of applications, including cooling (e.g., in electronic applications), in fuel cells for circulating air and/or fuel, and in diagnostic arrays.

In some embodiments, the adjacent fluidic channels are (i) concentrically arranged (ii) decrease sequentially in length, or (Hi) are both concentrically arranged and decrease sequentially in length.

In some embodiments, the fluid input header and the fluid output header are (i) arranged in parallel relationship, (ii) arrange in a non-parallel angled relationship, or (Hi) arranged in a back-to-back parallel relationship.

In some embodiments, the adjacent fluidic channels are non-parallel.

Arrays as described herein may be used for a variety of purposes, including but not limited to:

(a) detecting a first member of a binding pair in a liquid sample by (a) passing the fluid through a microarray having a second member of a binding pair immobilized therein, and (b) detecting the binding of the first member to the second member in the array; or

(b) transferring heat to or from a coolant or refrigerant fluid by circulating the fluid through a fluid channel array in a heat exchanger in a heat-transfer effective amount; or

(c) circulating fuel or oxygen through a fuel or oxygen channel array in a fuel cell in an energy-generating effective amount.

Note that the dimensions of header segments given herein are average dimensions for each segment. It will be appreciated that the segments are generally "smoothed" such as by linear or curved functions, to avoid step changes or irregular changes in channel heights and widths between segments.

The present invention is explained in greater detail in the drawings herein and the specification set forth below.

Brief Description of the Drawings

Figure 1. (A) Schematic diagram and (B) discrete representation of a 3 channel U- type configuration. Arrows indicate flow direction. Figure 2. Schematic diagram of trapezoidal U-type configurations.

Figure 3. Plot of Eq. (12) vs M_t for the 8^th, 9^th, and 10^th adjacent fluidic channels in the 11 -channel geometry I (see Table 1).

Figure 4. Effect of Δ/, on the 1^st and 10^th header widths for geometry I in a trapezoidal configuration.

Figure 5. Schematic diagram of square radial U-type configurations.

Figure 6. Schematic diagram of circular radial U-type configurations.

Figure 7. Effect of ΔΖ_έ on the 1^st and 10^th header widths for geometry I in square radial configurations.

Figure 8. Air flow distributions for (A) initial and (B) optimized, trapezoidal geometries I -111 calculated via (green) discrete calculations and (red) CFD simulations.

Figure 9. CFD air flow velocity profiles for (A) initial and (B) optimized, trapezoidal geometries I-III. For initial geometries, the bottom channel is the inlet, and for trapezoidal, the left channel is the inlet. All images are scaled in size and to their respective velocity ranges (v_max is 3 m s^"1 for all geometries except the initial and optimized geometry I, which are scaled to 1.5 and 2 m s^"1 respectively).

Figure 10. CFD air flow velocity profiles and air flow distributions acquired from (green) discrete and (red) CFD results for (A) initial and (B) optimized, square radial geometries.

Figure 11. CFD air flow velocity profiles and air flow distributions acquired from

(green) discrete and (red) CFD results for (A) initial and (B) optimized, circular radial geometries.

Figure 12. Pressure distributions in (A) the optimized, square radial geometry shown in Figure 10B and (B) the optimized, trapezoidal geometry II shown in Figure 9B. P_max is 450 Pa in (A) and 350 Pa in (B). The top header channel is the inlet for both geometries. (C) The absolute pressure halfway along each parallel channel is plotted for the distributions shown with black squares in (A) and with x-marks in (B). Channel numbers are normalized to the total number of channels.

Figure 13. Examples of U-type geometries where the fed channels are ( A) concentric circles, (B) concentric squares, (C) a combination of concentric squares and circles, (D) alternating concentric squares or, generically, concentric polygons, (E) arrayed without an immediately discernable pattern, (F) a combination of concentric circle arrays, (G) a square concentric array with inlet/outlet headers in a back-to-back parallel relationship, and (H) parallel array addressed with angled inlet/outlet headers. Such U-type geometries can also be arrayed as subsets of larger geometries, e.g., (I) a back-to-back coupling of two geometries shown in Fig. 13(H), and (J,K) the geometries shown in Fig. 13(H,I) arrayed as a larger U- type geometries (that is, multi-bed arrays), respectively. All lines represent fluidic channels, and arrows indicate the direction of flow from inlet to outlet.

Detailed Description of Illustrative Embodiments

The present invention is now described more fully hereinafter with reference to the accompanying drawings, in which embodiments of the invention are shown. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather these embodiments are provided so that this disclosure will be thorough and complete and will fully convey the scope of the invention to those skilled in the art.

Like numbers refer to like elements throughout. In the figures, the thickness of certain lines, layers, components, elements or features may be exaggerated for clarity. Where used, broken lines illustrate optional features or operations unless specified otherwise.

The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms "a," "an" and "the" are intended to include plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms "comprises" or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements components and/or groups or combinations thereof, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components and/or groups or combinations thereof.

As used herein, the term "and/or" includes any and all possible combinations or one or more of the associated listed items, as well as the lack of combinations when interpreted in the alternative ("or").

Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the specification and claims and should not be interpreted in an idealized or overly formal sense unless expressly so defined herein. Well- known functions or constructions may not be described in detail for brevity and/or clarity.

It will be understood that when an element is referred to as being "on," "attached" to, "connected" to, "coupled" with, "contacting," etc., another element, it can be directly on, attached to, connected to, coupled with and/or contacting the other element or intervening elements can also be present. In contrast, when an element is referred to as being, for example, "directly on," "directly attached" to, "directly connected" to, "directly coupled" with or "directly contacting" another element, there are no intervening elements present. It will also be appreciated by those of skill in the art that references to a structure or feature that is disposed "adjacent" another feature can have portions that overlap or underlie the adjacent feature.

Spatially relative terms, such as "under," "below," "lower," "over," "upper" and the like, may be used herein for ease of description to describe an element's or feature's relationship to another element(s) or feature(s) as illustrated in the figures. It will be understood that the spatially relative terms are intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. For example, if the device in the figures is inverted, elements described as "under" or "beneath" other elements or features would then be oriented "over" the other elements or features. Thus the exemplary term "under" can encompass both an orientation of over and under. The device may otherwise be oriented (rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein interpreted accordingly. Similarly, the terms "upwardly," "downwardly," "vertical," "horizontal" and the like are used herein for the purpose of explanation only, unless specifically indicated otherwise.

It will be understood that, although the terms first, second, etc., may be used herein to describe various elements, components, regions, layers and/or sections, these elements, components, regions, layers and/or sections should not be limited by these terms. Rather, these terms are only used to distinguish one element, component, region, layer and/or section, from another element, component, region, layer and/or section. Thus, a first element, component, region, layer or section discussed herein could be termed a second element, component, region, layer or section without departing from the teachings of the present invention. The sequence of operations (or steps) is not limited to the order presented in the claims or figures unless specifically indicated otherwise. " Adjacent fluidic channels" as used herein refers to the plurality of channels between the input header and the output header. In some embodiments, and as is most commonly seen in fuel cells technology, the adjacent fluidic channels are parallel channels. However, in other embodiments, made possible as noted herein, the adjacent fluidic channels are non- parallel (e.g., at angled relations to one another, a curved channel or channel segment adjacent a linear channel or channel segment, etc.). All may be referred to as "fed channels" as they are fed by the input header.

As noted above, the present invention provides a fluid array, comprising:

(a) a fluid input header,

(b) a fluid output header, and

(c) a plurality (N) of adjacent fluidic channels (e.g., parallel channels) connected to both the input header and the output header in a U -array configuration

In some embodiments, the adjacent fluidic channels are non-parallel.

As shown in greater detail below, in some embodiments the adjacent fluidic channels are convex or concave curved (e.g., parabolic, arc, etc.) linear, or a combination of any thereof. Thus, the adjacent fluidic channels may be linear, arranged in the shape of curve (e.g., an arc such as a half circle, a parabola, etc.), a polygon (e.g.. a triangle, a quadrilateral such as a rectangle, a pentagon, a hexagon, a star, etc.) or combination thereof.

The plurality (TV) of adjacent fluidic channels may comprise at least 5. 10. 20 or 25 adjacent fluidic channels (e.g., up to 2,500 or 5.000 adjacent fluidic channels or more).

The array may be formed in any suitable material, including inorganic substrates (e.g., silicon, glass, etc. ) and polymer substrate (e.g., fluorocarbons).

The array may be configured to carry any of a variety of fluids, including gases and liquids, and (for liquids) both Newtonian and non-Newtonian fluids. Examples include, but are not limited to, air. 0₂, C0₂, H₂, alcohols, hydrocarbons and hydrocarbon mixtures, dielectric fluids, refrigerants and coolants (e.g., halocarbons such as fluorocarbons), oils, liquid nitrogen, water or aqueous solutions, biological fluids (e.g., blood, blood serum, blood plasma, buffy coat, urine, saliva, cerebral spinal fluid, optionally diluted and/or partially purified), etc.

Diagnostic arrays. In some embodiments, at least some, a major portion, or all of said adjacent fiuidic channels have a binding ligand immobilized therein (e.g., a protein, peptide, nucleic acid, carbohydrate, etc., binding ligand, such as an antibody). The binding ligand may serve as a second member of a binding pair. Such arrays may be used to detect a first member of a binding pair (e.g., cells, proteins, peptides, hormones, drugs, etc. ) in a liquid sample (e.g., a biological fluid such as those described above) by (a) passing the fluid through a microarray having a second member of a binding pair immobilized therein, and (b) detecting the binding of said first member to said second member in said array (with detection being carried out by any of a variety of known techniques, such as sandwich assay with a fluorescent labeled antibody).

Heat exchangers. In some embodiments, the array comprises a fluid channel array for a heat exchanger (e.g., for cooling circuitry in electronic and microelectronic applications). Such arrays may be used to transfer heat to or from a coolant or refrigerant fluid by circulating the fluid through a fluid channel array in such a a heat exchanger in a heat-transfer effective amount (ultimately circulating the fluid to a heat sink in accordance with known techniques).

MEA layers for fuel cells. In some embodiments, the array comprises a fuel channel array or an air channel array for a fuel cell electrode-electrolyte assembly (MEA) layer (examples of which are described further below). In use, such embodiments provide a method of circulating fuel or air through a fuel or air channel array in a fuel cell in an energy- generating effective amount.

Fuel cells. Fuel cells such as is a planar solid-oxide fuel cell (SOFC), generally comprise: (a) a primary fuel inlet header; (b) a primary fuel outlet header; (c) a primary oxygen inlet header; and (d) a primary oxygen outlet header; and (e) a plurality of at least 10. 20, 30, 40 or 50, up to 100, 200, 1.000 or 2,000 or more membrane electrode assembly (MEA) layers, each of said layers comprising: a semipermeable membrane, a plurality of fuel channels on one side of said semipermeable membrane, a plurality of oxygen channels on the opposite side of said semipermeable membrane, a secondary fuel inlet header and a secondary fuel outlet header, each in fluid communication with said plurality of fuel channels; and a secondary oxygen inlet header and a secondary oxygen outlet header, each in fluid communication with said plurality of oxygen channels. In general, fuel channels and oxygen channels are arranged in counter-flow, cross- flow, or co-flow configurations. In some embodiments, the semipermeable membrane comprises an anode layer facing said fuel channels, a cathode layer facing said oxygen channels, and an electrolyte layer separating said anode layer and said cathode layer.

"Oxygen channel" as used herein with respect to fuel cell arrays may carry pure oxygen gas, or any suitable fluid or gas in pure or mixed form containing sufficient oxygen for the fuel cell to generate energy (e.g., air).

As noted above, the individual oxygen or fuel arrays in each MEA layers may be an array as described above. In addition, however, (f) the primary fuel input and primary fuel output headers, and/or (g) the primary oxygen input and primary oxygen output headers, may be dimensioned in like manner as described above, to further enhance fluid flow therein.

In use, the fuel cells provide a method of circulating fuel and/or oxygen through the fuel and/or oxygen channel arrays therein in an energy- generating effective amount, wherein flow of the fuel and/or oxygen is enhanced by imparting the configurations described herein, thereby increasing the efficiency and/or capacity of the fuel cell.

The present invention is explained in greater detail in the following non-limiting Examples.

Experimental

Herein, we apply a similar methodology to that which we have described in Z-type systems to improve flow uniformity in also prevalent U-type systems, ultimately improving flow uniformity according to single-phase dynamics but only after arraying the electroactive channels in either trapezoidal arrangements or preferably concentric radial patterns, which is in itself a significant shift away from typical parallel designs and potentially opens new avenues for water management.

Water management remains a significant obstacle to the implementation of parallel configurations of fuel cells. Simply put, while the parasitic pressure required to drive reagents is reduced in parallel systems, the hydrodynamic forces available to remove water byproducts drop as well. Relative to a single serpentine channel, flow velocities are generally reduced by a factor of N, the number of adjacent fluidic channels. The serpentine design also employs a more subtle method to remove water byproducts. The pressure differential between adjacent turns of the serpentine channel are on the order of the pressure drop along the channel's length, thereby forcing air through the porous electrode and convectively removing water byproducts from the electrode surface. This same effect is negligible in traditional parallel configurations, and together, reduced fluid velocities and negligible convective forces result in relatively poor water management in nearly all parallel systems that hamper their commercial implementation [26].

Discrete geometry optimization methods, such as those utilized previously and herein, lack the complexity necessary to describe and correct for water dynamics. In turn, these methods offer a unique simplicity for designing systems with both uniform reagent distribution and hydrodynamic forces for water removal, a first step in developing practical parallel fuel cell systems. Moreover, the concentric radial patterns of U-type systems presented herein not only correct flow non-uniformity but develop pressure differentials between the adjacent fluidic channels that are no longer negligible, potentially enabling convective water management in parallel fuel cells for the first time. In light of these observations, more complex modeling (three dimensional, multi -phase, and/or time- dependent) and experimental observations are proposed to explore convective water management and assess the complexity imparted to reagent distribution, even in steady-state.

Nomenclature:

A cross-sectional area of channel (m )

O-i cross-sectional area of i^th parallel channel (m²)

a_p cross-sectional area of adjacent fluidic channels (m )

At cross-sectional area of i^th inlet header (m²)

A'i cross-sectional area of i^th outlet header (m²)

A in cross-sectional area of plate inlet (m )

contact area of tee junction (m²)

c? resistance coefficient of i^th inlet header (1/m⁴)

c^p resistance coefficient of adjacent fluidic channels (1/m⁴)

D hydraulic diameter (m)

Di hydraulic diameter of I^th inlet header (m)

dp hydraulic diameter of adjacent fluidic channels (m)

f friction factor

non-uniformity index

H channel height (m) i height of i^th inlet header (m)

h_p height of adjacent fluidic channels (m)

L channel length (m)

Li length of i* inlet header (m)

h length of i^th parallel channel (m)

Mi change in length between i^th and (t + l)^th adjacent fluidic channels

N number of channels

P channel perimeter (m)

Pi perimeter of i^th inlet header (m)

Vv perimeter of adjacent fluidic channels (m)

R resistance (kg/(m s))

n resistance of I^th parallel channel (kg/(m² s))

Ri resistance of I^th inlet header (kg/(m² s))

Ri' resistance of i* outlet header (kg/(m² s))

resistance of tee junction (kg/(m s))

Re Reynolds number

Vi velocity of i^th parallel channel (m/s)

v_v velocity of adjacent fluidic channels (m/s)

Vi velocity of i^th inlet header (m/s)

n velocity of i^th outlet header (m/s)

Vin velocity of plate inlet (m/s)

w width (m)

width of i^lh inlet header (m)

w₀ width of the header opening area in radial configurations (m) w_p width of adjacent fluidic channels (m)

Greek letters:

a aspect ratio

μ viscosity (kg/(m s))

Pi radius of i^th parallel channel (m)

ω angular displacement (rad) 2. Computational Methods

2.1 Assumptions. Both discrete and CFD models make the same assumptions: (1) the fluid's density (p) and viscosity (μ) are constant; (2) temperature is constant at 293.15 K; (3) fluid flow is steady-state and laminar; and (4) mass transfer with the electrode layer is neglected [ 1 , 19. 21 ].

2.2 CFD Validation. We validated the subsequent algorithms by conducting CFD simulations of both air and hydrogen flow using COMSOL Multiphysics® 4.3a. Details regarding these simulations are available elsewhere [1].

2.3 Discrete Model for U-Configuration Fuel Cells. The discrete model for describing a U -configuration fuel cell is illustrated in Figure 1. The geometry is segmented into interconnected fluidic resistors with a resistance (R) given by

1 {Re Πμ-P h

2 D-A (1) where the channel's geometry is defined by its cross-sectional area (A), perimeter (P), length (L), width ( ). and height (H). The hydraulic diameter (0) is given by AW^■ H /2 W + H , and the product of the Reynolds number and friction factor (Re f) is approximated by Kays and Crawford [27] as 13.84 + 10.38e (V-— a )), where a is the channel's aspect ratio that is > 1. Additionally, the merging of the adjacent fluidic channels and header segments in a tee junction implicitly adds a resistance term (R_T) to a header segment due to the contact area of the tee junction (A_T), where all geometric parameters regard the associated header segment [19]:

D A

The generalized pressure and mass balance equations for a U-configuration geometry with N adjacent fluidic channels are:

_Vi = VtRi + V!R[ + v_i+1r_i+1 (3a) ViAi = V_i+1A_i+1 + v_i+1a_i+1 (3b) v_INA_IN = VtAi + V;A[ (3c) where v₍. . and a_t are the average linear velocity, resistance, and cross-sectional area of the parallel channel. Similarly, V,, R and A_t describe the i^x inlet header segment, and V,-'. R_t', and Ai correspond to the /* outlet header segment. Eqs. (3) can be expressed in matrix form:

[M] = [R] [V]

where

Note that the [R] matrix reduces to those previously published [19] if the proper substitutions are made. We constructed an algorithm in FORTRAN to solve Eqs. (4) as previously described [1]. Again, we use a non-uniformity index as a metric for flow maldistribution, where F = 0 if flow is perfectly uniform and F_x→ 1 as maldistribution becomes increasingly severe [6, 11]: 2.4 Discrete Geometry Optimization— Trapezoidal Configurations . To reduce the discrete model into simple equations that may be used to guide design of U-type systems with uniform distribution, we assume the case that flow is perfectly uniform throughout the adjacent fluidic channels (v_{t N} = v_p) and that the adjacent fluidic channels are completely identical (ι _N = r_p). Eq. (3a) then reduces to

VtRi + V!R[ = 0 (6)

However, Eq. (6) is physically impossible as neither the velocity nor resistance terms may be negative or zero if fluid is flowing. We may avoid Eq. (6) by modulating the length of the I^th parallel channel length such that l_t = + Δ/j, where is the change in length between i^th and (t + l)^th adjacent fluidic channels. This simultaneously ensures i\≠ r_i+1 and does not alter the adjacent fluidic channels' height and width, which are typically optimized [28]. We now apply additional assumptions to simplify Eqs. (3a,b). Firstly, the U- conliguration geometry is symmetric (V_t— V, '. R_t = /?_;'. etc.), Eq. (3a) now becomes v_p (r_i - r_i+t) = 2 - V_iR_i (7)

Secondly, we ensure all adjacent fluidic channels' heights and widths are equal (h _N = h_p, w_{1 N} = w_p). Several parameters of the adjacent fluidic channels are then constant, namely cross-sectional area (a_p), hydraulic diameter (d_p), perimeter (p_p), and the Reynolds number and friction factor ((Re f _p), These constants can be compiled into a single resistance coefficient for the adjacent fluidic channels such that C^p = (Re f)_p p_p/(d_pa_p ) and r, = μ(Γ^ρα„ί_ί·/2. Then Eq. (7) reduces to: μν_ρ a_p ( - = 4 - ViRi (8a)

Moreover, Eq. (3b) implies that

V_N_₁A_N__t = v_pa_p (9a)

V_N-₂A_N_2 = VN^A^ + v_pa_p = 2^■ v_pa_p (9b) which can be generalized to

= (N— 0^■ v_pa_p (9c)

Substituting Eq. (9c) into Eq. (8b) and simplifying results in

4(N- _

1 μεΡΑί ¹ '

To simplify further, we define resistance coefficients for the i^th header segment such that

⁽ _'i = (Re f i Pi/iDiAf ) and R_L = μΟ^Α^ + ¾/2, where {Re f P D A and A_Ti are the Reynolds number and friction factor product, perimeter, hydraulic diameter, cross- sectional area, and tee junction contact area for the i^th header, respectively. After substitution and rearrangement, Eq. (10) becomes

M_i - 2(N - i) - ^C (L_i + ^) = 0 (11)

Thus, to perfectly distribute flow through a U-type configuration, we can choose geometric parameters such that the left hand side of Eq. (11) equals zero.

First, regardless of its magnitude, note that must be positive since all other terms yield a negative number. This stipulates that adjacent fluidic channels nearest the stack manifold inlet must be longer. This is not a surprising result since the pressure balance equations of a U-type configuration (Eq. 3 a) require that the pressure drop across the I^th parallel channel must equal the pressure drop across both the i^th inlet headers and the i + 1^th parallel channel. Thus, to ensure v_t = v_i+1, it is reasonable that must be greater than r_i+1, and Eq. (10) quantitates this change in resistance due to

Secondly, an immediately intuitive method of satisfying Eq. (11) is to vary Δ/,_: alone, but this approach is greatly complicated by Lj, the length of the I^th header. As shown in Figure 2, if we lengthen the adjacent fluidic channels by Δ/,·, the corres onding header's length increases by a factor that can be roughly approximated as L_t =

where W_rib is the rib width and equal to L_t if Δ/j = 0. Substitution into Eq. (11) yields

While Mi must be positive to balance Eq. (12), it also imparts a negative contribution to the left hand side of Eq. ( 12) due to the adjusted term, and this negative component is weighted by an (N— t) factor. Consequently, even if all other geometric parameters are chosen such that there is a M_N→ that satisfies Eq. (12) for the N— 1 parallel channel, it may not be possible to balance Eq. (12) for channels closer to the inlet, where i is smaller and (N— i) is larger. This is illustrated for the 1 1 -channel geometry 1 (see Section 3) in Figure 3. Keeping all geometric parameters besides Δ/; constant (see Table 1), we found a suitable ΔΖ; for i— 10 and i = 9, evident as the plot of Eq. ( 12) crosses zero. However, for i— 8 or less, where the (N— i) factor is 3 or more, a feasible solution does not exist. Thus, in addition to increasing parallel channel length, we must also counterbalance the negative (/V— i) factor by simultaneously changing another geometric parameter. As discussed previously [1], we then adjust header widths in order to modulate the C-^{ term in Eq. ( 12). Table 1. Geometric parameters (units: mm) for the initial, non-optimized geometries I-III.

Parallel Channel Properties Rib Inlet Inlet

Geometry Ref.

Number Length Width Height Width Width Height

I 1 1 50.00 1.50 0.60 1.50 3.00 0.60 5

II 21 50.00 1.50 0.60 1.50 3.00 0.60 5

III 26 50.00 2.00 0.72 2.00 4.00 0.72 18

The effect of the magnitude of Δ/j on the optimized header widths and overall fuel cell footprint is illustrated for geometry I in Figure 4. As Δ/, is increased, smaller reductions of the negative terms in Eq. ( 1 1 ) are necessary, and the optimal header width decreases. Furthermore, the i = 1 header widths are larger than the i— 10 header widths due to the (N— i) term. One would then want to increase as much as possible since this leads to the largest electroactive surface area and the smallest header footprint, but for this method, one can minimize the header widths only to a relatively large value due to the inclusion of Mi in the negative terms of Eq. (12). At best, one achieves such trapezoidal geometries as shown in Figure 9. As we will show in Section 2.5, there are more practical designs for these optimized U-type configurations.

Lastly, in terms of geometry design, simultaneously varying header widths and parallel channel lengths to find values that satisfy Eq. ( 12) leads to a multitude of solutions. To yield unique solutions, we chose to first set ΔΖ, the same for all adjacent fluidic channels (Δ/_{j w} = ΔΖ,·) and then find the appropriate header width to satisfy Eq. (12) using a previously published algorithm [ 1 j. Clearly, other constraints are plausible.

2.5 Discrete Geometry Optimization— Square and Circular Radial Configurations. To achieve l^≠ 0 (see Section 2.4), we may adopt radial, U-type configurations, where adjacent fluidic channels concentrically traverse about a center point in, for example, a square or circular pattern. Adjacent fluidic channels farther from the center point intrinsically span a longer distance than those closer to the center, and this change in length is both constant and easily approximated from basic geometric principles. Thus, we set the innermost parallel channel as the N^Ih channel so that Δ/, is always positive.

A schematic diagram of adjacent fluidic channels arranged in a square format is illustrated in Figure 5. A parallel channel, displaced from the device's center, has a length approximated by l_i = 8p_l - W₀ ( 1 3 ) Here, p_t is the average half side length of the i^th parallel channel's square geometry

(taken from the channel's midline), and W₀ is the width of the opening necessary to assemble headers along the ends of the adjacent fluidic channels. Given constant channel properties, it can be shown that the length of the t^lh parallel channel is generally given by /,· = 8 (p_N + ( V - (w_p + W_rib)) - W₀ ( 14)

The di ference in length between any set of adjacent fluidic channels is then Ak = 8(w_p + W_rib) (15)

Thus, to assemble a U-type configuration of adjacent fluidic channels arranged in a square radial format, we need only set the average half side length of the innermost channel ( ½), the width of the header opening (W₀), and the parallel channel properties (h_p, w_p, and W_rib). Then, the widths of the headers can be easily determined using simple algorithms [1] and Eq. ( 1 1). Note that the use of Eq. (1 1) is applicable to this design since the ends of the adjacent fluidic channels are flush (L; = W_rib), and consequently, the header widths necessary to satisfy Eq. (1 1) are smaller than in the trapezoidal configuration because Δ/; does not contribute any negative component to the left hand side of Eq. (1 1) (Note that Al_t alone cannot be used to balance Eq. (1 1 ) for all adjacent fluidic channels since in the outlined geometries, it is a constant set via Eq. (15). Header widths must be adjusted as well.). This fact is illustrated for geometry I in Figure 7, where significantly smaller widths are necessary to optimize the radial geometry. For example, with a Δί, of 15 mm, the first and largest header's optimized width is 13.99 mm in a trapezoidal configuration but only 4.71 mm in a square radial configuration. This decrease in the optimal headers' widths is owed to the flush adjacent fluidic channels only, and we achieve the same exact results in a circular radial configuration as well.

As shown in Figure 6, adjacent fluidic channels patterned in a circular fashion are quite similar to the square radial U-type configurations. In a circular radial arrangement, the length traveled by the i^th channel, which has an average radius of curvature pj (taken from the channel's midline) and revolves an angular displacement ω (in radians) is h = ^ωΡί (16)

The difference in length between the i^th and (i— 1)^Λ adjacent fluidic channels is then

Μ_ί = ω(ρ_ί - ρ_ί→) (17)

Since all parallel channel properties except length are held constant, then h = ω ( _N + (N - + ^wrib)) (18) and

Al_i = _M(w_p + W_rib) (19)

Assembling a circular U-type configuration requires setting ω, then is automatically generated by Eq. (19) and the header widths satisfying Eq. ( 1 1 ) are generated via the search algorithm [1]. As a frame of reference, to give a of 15 mm for geometry I, the adjacent fluidic channels must have an angular displacement of 286°, which is feasible given the small header widths.

3. Validation

We validated the discrete method (Section 2.3) by comparison with air flow CFD simulations (Section 2.2) for the three published U-type geometries [6, 19] that are parameterized in Table 1. For comparability, all inlet velocities (Table 2) were set so that if perfectly distributed, air stoichiometry would be 2 at 1 A cm^" [29].

Air flow distributions through the adjacent fluidic channels of geometries I-III are shown in Figure 8A, and CFD velocity surfaces are shown in Figure 9A. Both sets of F_x parameters, calculated by Eq. (5) and shown in Table 2, are nearly identical between discrete and CFD results. Additionally, these results coincide with previously published results [6, 19], all of which validates the discrete model.

Table 2. Air flow parameters for initial geometries I-III: target average air velocity through parallel channels (v_p, units: m s^~ ), total electroactive surface area (SA, units: cm ), and F parameters from discrete and CFD calculations.

Geometry v_p SA Discrete Fj CFD F,

I 0.74 8.3 0.80 0.81

II 0.74 15.8 0.98 0.98

III 0.62 26.0 1.00 1.00

4. Flow Field Optimization - Trapezoidal Configurations We applied the geometry optimization scheme in Section 2.4 to geometries I-III, generating the trapezoidal configurations with the air flow distributions and velocity profiles shown in Figs. 8B and 9B, respectively. In all cases, the length of the parallel channel farthest from the inlet was set to the initial design's parallel channel length, 50 mm, but note that this assignment bears no effect on the optimal header widths. As seen in Eq. (12), these widths are only dependent on Δ/,·. which was set to 5 mm for all geometries. Inlet channel velocities were set so that the outermost channel, which has the largest surface area, would be fed air at a stoichiometry of 2 at 1 A cm^" [29]. No attempt was made to conserve stoichiometry for all channels because the reduced flow velocities in the shorter, central channels would likely result in water mismanagement and channel flooding [30].

It is clear from the CFD results that the optimization process reduced flow non- uniformity. The F_x parameters decreased by approximately 60% after optimization (see Table 3). However, all F_x parameters from the CFD simulations remained greater than 0.2, increasing from geometries I to III in accordance with increasing F_x parameters prior to optimization. Non-uniformities in the optimized flow distributions were not predicted by the discrete model. These results were also unlike discrepancies between a discrete model and CFD solutions previously observed in Z-type configurations, which were due to Reynolds number dependent flow recirculation in tee junctions [1, 15]. For example, a CFD simulation of hydrogen flow distribution in the trapezoidal geometry III (Target hydrogen flow velocities were also set to provide the outermost channel with a stoichiometry of 2 at 1 A cm^"

².) yielded approximately the same f parameter, 0.40, as air flow, 0.39.

Table 3. Inlet widths (units: mm) and air flow parameters for optimized, trapezoidal geometries I-III: target average air velocity through parallel channels (v_p, units: m s^~

1 2

), total electroactive surface area (SA, units: cm ), and F parameters from discrete and CFD calculations.

Geometry Vp SA Discrete Fj CFD Fi

I 0.74 8.3 0.80 0.81

11 0.74 15.8 0.98 0.98

III 0.62 26.0 1.00 1.00

We suspect the trapezoidal U-type configuration's non-uniformity was due to the rough approximation of header segment lengths. If in Eq. (12) (Section 2.4), the portion of the I^th header segment's length due to the Δ/j term was approximated via its center line rather than the effective rib length (Figure 2), more uniform distributions would likely be obtained. This more accurate approximation would require even wider headers since the header segments would be approximated as more resistive. Considering the significant portion of the fuel cell's surface area already occupied by the enlarged headers, further efforts to improve the accuracy of the discrete model's trapezoidal geometries were not undertaken, and research was focused on developing the radial designs in Section 5.

5. Flow Field Optimization - Radial Configurations

The geometric parameters optimized for hydrogen consumption by Kumar, et al. [28] were used to construct the radial U-type configurations. Parallel channel width and height were 1.5 mm, and rib width was 0.5 mm. Headers also had a constant height of 1 .5 mm. The two radial configurations were designed to have roughly the same electroactive surface area (Table 4). Square radial configurations had 15 adjacent fluidic channels: ₁₅ was 17.5 mm; ΔΖ; was 16 mm; and the header opening W₀ was 20 mm for all channels. For circular radial configurations that had 25 adjacent fluidic channels, ω was 270° (yielding a ΔΖ₍· of 9.42 mm); p_2$ was 9.75 mm; and straight segments (2.5 mm long) were added to each end of the adjacent fluidic channels. For both radial configurations, initial, straight headers were 3 mm wide, and in all cases, target air flow velocities were set to provide the outermost channel with a stoichiomctry of 2 at 1 A cm^" (as discussed in Section 4).

Table 4. Inlet widths (units: mm) and air and hydrogen CFD flow parameters for square and circular radial configuration with initial, straight headers and optimized headers: target average air velocity through parallel channels (v_p, units: m s^'1), total electroactive surface area (SA, units: cm ), and F_t parameters.

The square radial configuration without optimized headers (Figure 10A) had a CFD F₁ parameter for air flow of 0.35. In comparison to geometries I and II, which had 11 and 21 channels and CFD F₁ parameters of 0.81 and 0.98, respectively, it is apparent that modulating the adjacent fluidic channels' lengths immediately improved flow distribution uniformity even without adjusting header widths. This improved uniformity was predicted by the discrete algorithm, albeit not exactly with an F₁ parameter of 0.31. After optimizing header widths, the headers narrowed from only 3.06 mm to 0.73 mm from channel 1 to 15 (Figure 1GB). essentially preserving the device^'s initial footprint. After optimization, the CFD F_l parameters for air flow and hydrogen flow decreased from 0.35 and 0.37 to 0.18 and 0.12, respectively. The relative standard deviation (RSD) of the air flow distribution was 6.5%, which is very close to the target of 5% for acceptable flow non-uniformity [23].

The initial circular radial configuration (Figure 11 A) had CFD air and hydrogen flow j parameters of 0.49 and 0.46 respectively, which reduced to 0.18 and 0.13 after utilizing the optimized headers (Figure 11B) that narrowed from 7.05 mm to 0.91 mm from channel 1 to 25. Here, the optimized air flow RSD was within 5.7%.

6. Pressure Distributions in Radial Configurations

Two representative pressure distributions of air flow in an optimized trapezoidal geometry and an optimized, square radial geometry are shown in Figure 12 along with a plot of the absolute pressures midway along the adjacent fluidic channels for both geometries. The pressure differential between adjacent channels is significantly higher in the radial geometry in Figure 12A (16.4 Pa on average) compared to the trapezoidal geometry in Figure 12B (0.2 Pa on average) or to traditional parallel geometries in general. As discussed in Section 1, such pressure drops between channels are extremely advantageous for water management as air can be forced through the porous electrode to convectively remove water byproducts in a manner akin to the commercialized serpentine geometries [26]. It must be noted that the models utilized within this manuscript neglect such convective forces, and more complex simulations such as multi-phase dynamics are required to assess both the convective forces and their perturbations to the steady-state flow distributions shown in Figs. 10 and 11; thus, we refrain from commenting further other than to say that to our knowledge, these radial geometries represent the first parallel type fuel cell geometries where convective water management is plausible.

7. Conclusions

We have presented three methodologies for constructing U-type configurations of fuel cells that are optimized for steady-state, single phase flow uniformity, trapezoidal and square and circular radial configurations. All of these designs increased the electroactive channels' lengths and header widths according to simple geometric ratios but did not alter dimensions of the electroactive channels critical to fuel cell efficiency. We highlight the radial configurations in particular as they exhibited relatively uniform flow with little sacrifice of surface area to adjusted headers, which was a significant drawback of trapezoidal geometries. Lastly, we conclude that in the radial configurations, non-negligible pressure differences were observed between adjacent adjacent fluidic channels, and we hypothesized that these radial geometries may exhibit convective water removal, which is unique amongst parallel systems and especially important considering the poor water management observed in traditional parallel systems.

Lastly, we note that the concentric geometries chosen herein (i.e., square and circular) were chosen for demonstrative purposes and that any alternative format where the fed channels progressively shorten with respect to the stack manifold inlet/outlet, either technically parallel or not, is a plausible template to actualize Eq. (12) for improved flow uniformity. In Figure 13, we illustrate examples of this fact, which range in complexity from simpler arrays of concentric arcs or polygon to combinations thereof or arrays with non- parallel and irregularly shaped channels, and assembly thereof into multi-bed arrays through linking with master input and output headers. All of these examples can be addressed via Eq. (12) and only differ in advantage by specific application, fabrication or engineering constraints, etc. References

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Claims

THAT WHICH IS CLAIMED IS:

1. A fluid array, comprising:

(a) a fluid input header,

(b) a fluid output header.

(c) a plurality (N) of adjacent fluidic channels (e.g., parallel channels) connected to both said input header and said output header in a I J -array configuration; wherein:

said adjacent fluidic channels are (i) concentrically arranged (ii) decrease sequentially in length, or (Hi) are both concentrically arranged and decrease sequentially in length; and/or said fluid input header and said fluid output header are (i) arranged in parallel relationship, (ii) arrange in a non-parallel angled relationship, or (Hi) arranged in a back-to- back parallel relationship; and/or

said adjacent fluidic channels are non-parallel.

2. The fluid array of claim 1 , wherein said adjacent fluidic channels are convex or concave curved (e.g., parabolic, arc, etc.) linear, or a combination of any thereof.

3. The array of claim 1 or 2, wherein said adjacent fluidic channels are linear, arranged in the shape of curve (e.g., an arc such as a half circle, a parabola, etc.), a polygon (e.g., a triangle, a quadrilateral such as a rectangle, a pentagon, a hexagon, a star, etc.) or combination thereof.

4. The array of claim 1 to 3, wherein said plurality (N) of adjacent fluidic channels comprises at least 5, 10, 20 or 25 adjacent fluidic channels (e.g., up to 2,500 or 5,000 adjacent fluidic channels or more).

5. The array of claim 1 to 4, wherein said array is formed in an inorganic substrate or a polymer substrate.

6. The fluid array of claim 1 to 5, wherein said fluid is a liquid or gas.

7. A multi-bed fluid array comprising:

(a) a plurality of fluid arrays of claim 1 to 6

(b) a master fluid input header connected to all of said fluid input headers; and (c) a master fluid output header connected to all of said fluid output headers.

8. The array of claim 1 to 7, wherein at least some, a major portion, or all of said adjacent fluidic channels have a binding ligand immobilized therein.

9. The array of claim 1 to 7, wherein said array comprises a fluid channel array for a heat exchanger.

10. The array of claim 1 to 7, wherein said array comprises a fuel channel array or an oxygen channel array for a fuel cell electrode-electrolyte assembly (MEA) layer.

1 1. A method, comprising:

(a) detecting a first member of a binding pair in a liquid sample by (a) passing the fluid through a microarray having a second member of a binding pair immobilized therein. and (b) detecting the binding of said first member to said second member in said array, wherein an array of claim 1 to 7 is used as said microarray; or

(b) transferring heat to or from a coolant or refrigerant fluid by circulating the fluid through a fluid channel array in a heat exchanger in a heat-transfer effective amount, wherein an array of claim 1 to 7 is used as said fluid channel array; or

(c) circulating fuel or oxygen through a fuel or oxygen channel array in a fuel cell in an energy-generating effective amount, wherein an array of claim 1 to 7 is used as said array in which said fuel or oxygen is circulated.

12. A fluid array, comprising:

(a) a fluid input header,

(b) a fluid output header.

(c) a plurality (N) of adjacent fluidic channels (e.g., parallel channels) connected to both said input header and said output header in a IJ -array configuration;

each of said fluid input and fluid output headers comprising a terminal channel followed by a plurality (N-l) of segments, with each of said segments forming a junction with a corresponding one of said adjacent fluidic channels, and with each of said segments having a length (L), a width (W), a height (II), and a contact area (A_T) at said junction: with the length of the adjacent fluidic channels (Δ/, ) and geometry of the input and/or output header segments satisfying the relationship:

' P[

where

All is ^me difference in length of the i^th and the adjacent fluidic channels;

Re_t is the Reynolds number in the z* inlet header segment;

fi is the friction factor in the z^'th inlet header segment;

Pi is the perimeter of the /^'* inlet header segment;

Dj is the hydraulic diameter of the ζ^'Λ inlet header segment;

Aj is the cross sectional area of the * inlet header segment;

Lj is the total length of the z^'Ih inlet header segment;

A_{T i} is the junction contact area of the z^"'^h inlet header segment;

Re- is the Reynolds number in the z^'th outlet header segment;

/ is the friction factor in the z^'th outlet header segment;

P/ is the perimeter of the I^th outlet header segment;

D[ is the hydraulic diameter of the i^lh outlet header segment;

A is the cross sectional area of the z^'th outlet header segment;

L'l is the total length of the z^'th outlet header segment;

A_T'_i is the junction contact area of the z^'th outlet header segment;

Rt'_p is the Reynolds number in the adjacent fluidic channels;

f_p is the friction factor in the adjacent fluidic channels;

P_p is the perimeter of the adjacent fluidic channels;

D_p is the hydraulic diameter of the adjacent fluidic channels; and

A_p is the cross sectional area of the adjacent fluidic channels;

to minimize P to less than 0.2 or a relative standard deviation less than 10%.

13. The fluid array of claim 12, wherein said adjacent fluidic channels are convex or concave curved (e.g., parabolic, arc, etc.) linear, or a combination of any thereof.

14. The array of claim 12 or 13, wherein said adjacent fluidic channels are linear, arranged in the shape of curve (e.g., an arc such as a half circle, a parabola, etc.), a polygon (e.g., a triangle, a quadrilateral such as a rectangle, a pentagon, a hexagon, a star, etc.) or combination thereof.

15. The array of claim 12 to 14, wherein said plurality (TV) of parallel fluid channels comprises at least 5, 10, 20 or 25 adjacent fluidic channels (e.g., up to 2,500 or 5,000 adjacent fluidic channels or more).

16. The array of claim 12 to 15, wherein said array is formed in an inorganic substrate or a polymer substrate.

17. The fluid array of claim 12 to 16, wherein said fluid is a liquid or gas.

18. A multi-bed fluid array comprising:

(a) a plurality of fluid arrays of claim 12 to 17;

(b) a master fluid input header connected to all of said fluid input headers; and

(c) a master fluid output header connected to all of said fluid output headers.

19. The array of claim 12 to 18, wherein at least some, a major portion, or all of said adjacent fluidic channels have a binding ligand immobilized therein.

20. The array of claim 12 to 18, wherein said array comprises a fluid channel array for a heat exchanger.

21. The array of claim 12 to 18, wherein said array comprises a fuel channel array or an oxygen channel array for a fuel cell electrode -electrolyte assembly (MEA) layer.

22. A method, comprising:

detecting a first member of a binding pair in a liquid sample by (a) passing the fluid through a microarray having a second member of a binding pair immobilized therein, and (b) detecting the binding of said first member to said second member in said array, wherein an array of claim 12 to 18 is used as said microarray; or

transferring heat to or from a coolant or refrigerant fluid by circulating the fluid through a fluid channel array in a heat exchanger in a heat-transfer effective amount, wherein an array of claim 12 to 18 is used as said fluid channel array; or

circulating fuel or oxygen through a fuel or oxygen channel array in a fuel cell in an energy-generating effective amount, wherein an array of claim 12 to 18 is used as said array in which said fuel or oxygen is circulated.