Improving water removal efficiency in a PEM fuel cell: Microstructured surfaces for controlling instability-driven pinching Open Access

The injection of a liquid into a crossflow has immense practical relevance in applications such as fuel injection,¹ mixing,² film cooling,³ and water management in PEM fuel cells.^4,5 Increasing applications of hydrogen as a fuel⁶ and, consequently, wide applicability of PEM fuel cell⁷ necessitate greater understanding and administration of PEM fuel cell⁸ than ever before. Therefore, water droplets interacting with solid surfaces have become an imperative research topic⁹ in achieving a high water collection rate¹⁰ and water removal from a PEM fuel cell. The emergence of a water jet¹¹ in the presence of transverse crossflow is analogous to the typical water management system in the gas flow channel in PEMFC, where the reactant gases are distributed, and the breakup manifests the water removal after generation.^12,13 Hence, a better understanding of the water transport and removal procedure to avoid water flooding within a PEM fuel cell is essential to uphold the efficacy of the fuel cell. If the water generated from the cathode catalyst layer is accumulated excessively within the Gas Diffusion Layer (GDL), it will cover the cathode and anode and may reduce the volume fraction of pores available for oxygen transport failing the PEMFCs.^14–16 In PEMFC, water is generated either due to vapor in humidified reaction gas or by electrochemical reaction in the cathode. Today, in most PEMFCs, graphitized carbon cloths produced by spinning and weaving carbon yarns are used in Gas Diffusion Layer (GDL).¹⁷ To characterize water flooding, some researchers have proposed new parameters, such as wetted area ratio.¹⁸ Droplet breakup phenomena under high shear have been studied experimentally,^19–21 theoretically,^22,23 and numerically^24–27 by many researchers. However, most of these experiments have been conducted considering high capillary number, Weber number, Reynolds number, and velocity ratio values. With PEM fuel cells, the breakup occurs at somewhat lower levels of these pertinent characteristics.^28,29 The water generated within the chamber of the PEM fuel cell must be removed at a constant rate to prevent water flooding within the chamber.^10,30

Previous studies on droplet breakup in uniform air flow have broadly categorized the breakup processes into three primary breakups modes,^31,32 i.e., bag breakup, multimode breakup, and shear-stripping breakup, with some other sub-modes, such as bag-stamen,³³ dual-bag,³⁴ bag-plume,³⁵ and plume shear.³⁶ Most of the previous studies were done for a moderate to high jet-to-free stream velocity ratio (R = U_jet/U_air, where U_jet and U_air are the velocities of jet and air, respectively, which are in the orthogonal direction). However, in PEMFC, the jet evolution in transverse airflow may take place at a comparatively much lower Reynolds Number (Re, the ratio of inertia to viscous force), Weber number (We, the ratio of drag force to capillary force), and velocity ratio (R). In this context, where the effects of viscosity are trifling, with filament thinning and pinch-off phenomena manifesting in an inertial regime. Here, the minimum filament radius (h_min) varies with dimensionless time (τ_c) to break up as³⁷  $hminrd∼τc2/3$⁠, where r_d is the radius of the initial filament, τ_c = (t_b − t)/t_I, $tI≡ρrd3/σ$ (Scaling Theories of Pinch-off). Thus, for low-viscosity liquids and in the inertial regime, Oh ≪ 1 holds. In the context of PEM fuel cells, water droplet breakup and water removal investigation necessitate theoretical understanding assisted by numerical and experimental works. Although considerable work has gone into the quantitative study of removing water drops from the gas flow channel, the dynamical effects of various governing parameters on droplet disintegration are still unnoticed, and a qualitative study focusing on the thinning dynamics while breakup is absent. Furthermore, the influence of the surface microstructure leading to instability-driven breakup in the process of the droplet fragmentation after generation within the GDL entails further intervention for sustainable design of PEM fuel cells.

The deformation and breakup phenomena of water jets, subject to the shearing flow of air, are profoundly influenced by a combination of inertial, viscous, aerodynamic, and surface tension forces. While the literature presents numerous notable studies on water droplet removal,³⁸ an extensive investigation aiming to comprehend the behavior of these droplets and their interactions with uneven Gas Diffusion Layers (GDLs) is yet to be undertaken with consideration of breakup, which is still unclear. In this letter, the system under consideration resembles the water chamber of a PEMFC considering all essential features. Here, we consider a microstructured surface and a plane surface for better water removal from the chamber. This hydrophobic microstructured surface helps maintain the spherical shape of the droplets with minimum wetting of the surface as observed in nature like the lotus leaf.³⁹ Although the contact angle may have distinct implications for molecular and continuum scale,⁴⁰ which has been recognized from atomistic simulations, the Young’s contact angle equation remains fairly valid. Additionally, the fragmentation of water into satellite droplets is not desired as they may get trapped within the surface microstructures leading to possible flooding of the chamber. The primary objective is to understand the influence of microstructures and consequently enhance the disintegration of the water generation under the effect of air shear and to achieve a minimum number of satellite water droplets that can be carried away quickly after the breakup. Hence, we study the breakup processes qualitatively and quantitatively for a range of velocity ratios and systematically correlate them with representative relevant non-dimensional numbers. The consequences of Rayleigh–Taylor instability, Kelvin–Helmholtz (KH) instability, and Rayleigh–Plateau (RP) instability are observed to be responsible for the deformation and shapes of the water jet along with the breakup. The RP and the Rayleigh–Taylor (RT) instability play the dominant role in the breakup process. We use the RP instability theory to predict the number of nodes while breakup, and our numerical framework shows excellent agreement with the theory. We observe that the end-pinching takes place when Oh is less than 0.01, and there is rarely any discrepancy in this condition. Furthermore, we study quantitatively the frequency of breakup for a single water generation spot and the water removal rate from the gas flow channel.

The low-velocity water jet is issued from a generation spot into a cross-flow chamber where the velocity ratio, R = U_jet/U_air varies from 0.033 to 0.014, and a fully developed flow of air persists. The water generation spot is located within the microstructures. The distance of the water generation spot from the air inlet is kept constant for both the plain and microstructured GDL surface. The geometrical details of the confinement are given in Fig. 1(a). The length of the confinement is 2000 µm, and the height of the confinement is 400 µm. The details of the numerical methodology and meshing are provided in the supplementary material, Sec. S3 (Figs. S3 and S4).

Starting from the high velocity ratio, we gradually decrease R to thoroughly understand the influence of velocity ratio in droplet deformation and the subsequent breakup process within the confinement under the influence of the shear flow of air. Typically for a PEMFC, based on the dimensions and ratings, we obtain the range of air jet velocity, confinement dimensions, and water generation rate depending on which we calculate We and Re. The corresponding Weber Number is observed to vary in the range of 0.03–0.3, and the Reynolds Number is in the range of 60–200. The breakup process is a complex phenomenon where inertia, viscous, surface tension, and aerodynamic forces are present altogether [as shown in Fig. 1(b)], necessitating detailed study to understand the influence of interplay of surface microstructures, shear airflow, and the confinement. To ensure that the thinning and disintegration occurring within these ranges of We and Re fall into the inertial regime, we have closely observed the evolution of the minimum thickness in the filament thinning process, and the relationship h_min ∼ τ^2/3 remains fairly valid, as illustrated in the supplementary material, Fig. S1. Although the pinch-off process occurs at an inertia-viscous regime (Scaling Theory of Pinch-Off),³⁷ the thinning process mostly takes place in an inertial regime.

First, in an attempt to delve deeper into the effects of velocity ratio on the droplet breakup, we study the breakup phenomena after generation for various velocity ratios over smooth surface. At a comparatively high-velocity ratio (R = 0.033), the water jet does not deform much until it grows into a comparably large spherical (semi-spherical) drop followed by elongation and some considerable deformation [Figs. 2(a) and 2(b)]. However, the time of breakup has been non-dimensionalized with a characteristic time defined as the time taken by the generated water to reach the opposite surface of the confinement (⁠$t*=ttc$⁠, and characteristic time, $tc=ycUjet$⁠, where y_c = characteristic length scale in transverse direction, which is the height of the confinement, t_c ≠ τ_c, τ_c is the dimensionless time mentioned previously, and t_c is the characteristics time). This non-dimensionalization has been performed to include the relative importance of water generation and the consecutive breakup of the water body after generation. As soon as the droplet goes through the deformation stage, it gradually takes the shape of a bell without any filament attached to it (as shown in the supplementary material, Fig. S5) for a smooth surface. In Figs. 2(a) and 2(b), we show the evolution of such bell-shaped water jets with streamlines and velocity vectors at three places within the confinement for R = 0.033. When the water droplet size increases, there remains little space for air to flow through the confinement, increasing its velocity and resulting in a rapid shape change of the water droplet. Flow separation can be observed at the leeward side of the jet, and we would not go into the details of this as this is not the focus of our study. However, the breakup is not observed at an early stage in this case, and it does not detach easily, rather, it elongates from the generation spot, which is not anticipated in a water chamber of PEMFC. However, for R = 0.02, a substantial transformation takes place, leading to a significant alteration in the previously observed bell shape, which manifests as a bulbous structure accompanied by the presence of an appended ligament as shown in Figs. 2(c) and 2(d) (the velocity and pressure contours are provided in the supplementary material, Fig. S6). This can be attributed to the influence of elevated shear resulting from increased air velocity. Under such circumstances, at the given velocity ratio, we can discern the incidence of breakup for a previously even surface, wherein the Rayleigh instability turns out to be instrumental in initiating the breakup process, followed by the potential occurrence of end-pinching⁴¹ as depicted in Figs. 2(c) and 2(d). The escalation in air velocity results in the accelerated formation of blobs and the detachment of the filament. No other undulation in the filament is observed, and this is also predicted by the theoretical formulation taking Rayleigh–Plateau instability theory into account, which is discussed later. For a velocity ratio (R) of 0.014, under the influence of the high velocity of the air, the flow circulation downstream becomes more dominant. A unique evolution of the jet shape stems from this flow circulation. In this case, the blob formation is absent, and the jet is way more elongated, resulting in a comparably thicker filament and a small blob at the top of it. We think this is due to prevailing Kelvin–Helmholtz instability at the upstream side of the filament, which causes the elongation due to high shear resulting from the flow circulation within the air medium. Afterward, though the blob is supplied with a considerable amount of water, the satellite droplet may form in this case. The breakup instances for R = 0.014 is shown in Figs. 2(e) and 2(f). The velocity and pressure contours for R = 0.014 are provided in the supplementary material, Fig. S7 (for the smooth surface) and Fig. S10 (for the microstructured surface). Contrary to expectations, the substantial thickness of the rim does not induce a breakup; instead, a severe reduction in pressure at the generation spot precipitates the manifestation of a breakup. The end pinch-off is not witnessed in this case, as Oh exceeds the value of 0.01. Elevated pressure is detected in the proximity of the breakup node, whereas the minimum pressure manifests precisely at the site of the breakup spot. When R is further decreased to 0.01, the elongation process of the water jet is much faster due to the presence of high shear within the chamber, and the occurrence of Kelvin–Helmholtz instability results in a rapid shape change of the filament. Hence, the disintegration into an almost spherical-shaped droplet is rarely observed in these cases. Disintegrated droplets are fragmented and either slide or roll over the GDL surface again, making the water flooding more probable. Similarly, the elongated structures become evident for microstructured surfaces at a reduced velocity ratio with a subsequent decrease in droplet size, which we will discuss in Sec. IV.

On the contrary, when we introduce the surface microstructures, it acts as an external noise source resulting in an increased number of nodes in the filament during the necking process and, at the same time, making pinch-off more probable even at a higher velocity ratio (e.g., R = 0.033) as depicted in Fig. 3(b). As a consequence of the propulsion of airflow, the droplet makes contact with the microstructured Gas Diffusion Layer (GDL) surface, exhibiting Cassie's wetting state that strengthens disintegration and facilitates the formation of an almost spherical satellite droplet. The interplay between surface texturing, droplet interfacial dynamics, wettability, and confinement has been explored previously by taking pairwise coupling between wettability and structure,^42,43 wettability and confinement,^44,45 and wettability and droplet dynamics.^46,47 We identify this breakup as vibrational breakup, which is observed for a low Weber number (0.0360–0.0980) under the influence of microstructures. We do not observe butterfly or bag breakup, which is generally observed for higher Weber numbers.⁴⁸ The adjacent microstructure makes the liquid filament more susceptible to breakup. However, the breakup occurs at the filament blob connecting point, which can be attributed to the decrease in local Oh. As observed in the supplementary material, Fig. S8, when the filament starts to thin, the fluid accelerates from the neck region, where the pressure is comparably high. Figures 3(c) and 3(d) shows the blob-filament structure with end-pinching, leading to a subsequent breakup for a velocity ratio of 0.02 for the smooth surface and in the presence of microstructures. The velocity and pressure contour plots for the microstructured case are provided in the supplementary material, Fig. S9. The filament has no observable undulations other than the necking at the junction with the blob. Curvature and capillary pressure, which are inversely related to filament radius, rise as filament radius decreases, and fluid is expelled from the neck at an increasing rate. For slightly viscous liquids like water in our case, the filament starts thinning from an inertial regime and breakup occurs at the inertial-viscous regime.³⁷ For the microstructured surface, the necking takes place at a lesser height than for the higher value of the velocity ratio, and the droplet size decreases compared to the smooth surface case, as displayed in Figs. 3(c) and 3(d). The flattened water body, in conjunction with the sack, undergoes a more rapid displacement toward the Gas Diffusion Layer (GDL) surface due to the high air velocity within the chamber. The observed Oh for this case is 0.007, well below 0.01, and we can observe end-pinching. For such small Ohnesorge numbers, the pressure is determined significantly by dynamic effects, and as a consequence, the pressure minimum moves toward the neck. The neck region looks to behave more and more like a Venturi tube as the Ohnesorge number is decreased; high flow rates in the neck reduce the pressure in the neck. The question now arises is whether the numerical difficulties experienced in the case of Oh ≈ 0.01 are in any way related to the onset of end-pinching. As the radius of the neck decreases, the normal stress jumps across the free surface. This fact, together with our observation that the pressure decreases in the neck owing to dynamic effects, ensures that the mechanisms for end-pinching to occur are in place. This observation is also in agreement with the work of Schulkes.⁴¹ Here, we observe that exactly when Oh goes below 0.01, the end-pinching is observed for all cases. Moreover, as observed in Figs. 3(e) and 3(f), for R = 0.014, the breakup is accompanied by a satellite droplet for both the smooth and microstructured surfaces (here, Oh ≥ 0.01).

For R = 0.014, though the breakup time decreases, the elongated filament attached to the drop after a breakup may lead to the formation of undesirable satellite droplets afterward, velocity an pressure contours shown in the supplementary material, Figs. S7 and S10. Furthermore, generated satellite droplets, while moving along, may get trapped in confinement, making their removal process more difficult and increasing the chances of water flooding. Therefore, the reduced breakup time for this case loses its relevance due to the generation of undesirable satellites in applications such as PEM fuel cells. Under decreased velocity ratios, extended ligaments can be discerned, which, in turn, have the potential to undergo fragmentation, leading to diminished droplets.

In the context of Proton Exchange Membrane Fuel Cells (PEMFCs), the microstructured surface may have benefits in terms of droplet generation and quick removal from the water chamber. This is of great interest due to the potential impact on fuel cell durability and efficiency. Inspired by natural surfaces, such as lotus leaves and animal skins,⁴⁹ these features have been widely fabricated at the laboratory scale and studied for handling droplets efficiently. Our study reveals that the stretching process during detachment occurs over a more extended period at higher velocity ratios, resulting in a distinct deformation pattern probably instigated by Kelvin–Helmholtz instability. More details on the theoretical framework for instability adopted here are provided in the supplementary material, Secs. S2 and S5.

Additionally, with a decrease in R, we observe a decrease in droplet size and stretching in ligament due to higher drag force, as depicted in Figs. 4(a)–4(c). Therefore, on the one hand, the droplet volume decreases when either the velocity ratio is decreased or microstructures are introduced.

The Weber number (We) and Ohnesorge number (Oh) are the two most fundamental dimensionless parameters that put forward invaluable insights into the relative significance of various forces in a situation like this. Figure 5 depicts the variation of Weber number (We) with the velocity ratio (R) for various instances of the breakup of droplets for smooth and microstructured cases. Figure 5(a) demonstrates the impact of velocity ratio on the differences between the smooth and microstructured surfaces during liquid stream detachment. Figure 5(a) depicts that at a high-velocity ratio, the difference between the We for the two surfaces is small, but increases as the velocity ratio decreases. The correlation between an increase in Weber number (We) and a decrease in Reynolds number (Re) is intuitively understandable, as heightened air velocity induces greater drag force, particularly at smaller R values. At a reduced velocity ratio, the resulting Weber number for the microstructured surface is much lower than the smooth surface, which establishes that the relative dominance of drag force over capillary force is better mitigated by the microstructures on the surface. This clearly shows that the capillary force is the deciding factor in the breakup at smaller velocity ratios. Therefore, the microstructured surface becomes more efficient regarding droplet detachment as the relative importance of the capillary effects increases. Additionally, the viscosity of droplets plays a significant role in hindering the droplet breakup inducing energy dissipation. This can be attributed to the Ohnesorge number (Oh) as it indicates the relative importance of viscous force to capillary inertia forces. As depicted in Fig. 5(b), we can observe that, with a decrease in R, We increases and Oh also increases. This means that with a decrease in R, viscous effect is becoming prevailing over the capillary effects. Astonishingly, for microstructured surfaces, the increase in Oh is not as rapid as smooth surface cases. Therefore, the microstructures at the surface suppress the dominance of viscous effects at low-velocity ratios. However, end-pinching is hardly observed at a low-velocity ratio, and detachment occurs closer to the generation spot for smooth and microstructured surfaces.

Next, to characterize the Rayleigh instability in the breakup process, we perform theoretical analysis to predict the number of nodes using the Rayleigh instability theory. Previously we observed that the thinning process occurs at an inertial regime, and at this regime, Rayleigh instability dominates the breaking process. Figure 6 relates the theoretically predicted number of nodes using the Rayleigh–Plateau instability theory and the observed number of nodes from our numerical study. While the primary responsibility for the observed instability lies with the Rayleigh–Plateau (RP) mechanism, which renders predictions based on this theory reasonably accurate, it is worth noting that the presence of Rayleigh–Taylor instability may also exert some influence in that context. However, that needs further intervention, and we will not go into any further details in this work. The details of the theoretical analysis are provided in the supplementary material, Secs. S2 and S5 (Fig. S2). Our findings demonstrate a close resemblance between the numerical and the theoretical predictions highlighting the validity of the theoretical framework of RP instability in explaining the observed breakup phenomena in PEM fuel cell confinement.

Afterward, we focus on end-pinching phenomena and attempt to characterize this in terms of the critical Ohnesorge number (Oh). Our results demonstrate that the critical Ohnesorge number (Oh*) for end-pinch-off is 0.01. Furthermore, we find that the breakup process is unaffected by end-pinch-off when the velocity ratio is 0.033 in the presence of a smooth surface. Yet, our study also reveals that microstructures acting as an external noise source may enhance the breakup process and may result in end-pinch-off resulting in lower local Oh (Oh = 0.0078) and R = 0.033. Figure 7(a) shows the variation of the breakup time (t*) with velocity ratio (R) for the smooth and microstructured surfaces. We investigate the interplay of velocity ratio and the microstructures on the breakup time of liquid streams and their dependency on surface microstructures. Our results indicate that the breakup time decreases with a decrease in velocity ratio and is significantly reduced when microstructures are present. Nevertheless, despite continued variations in the velocity ratio, the breakup time does not reduce further after the velocity ratio exceeds the value of 0.0167. Figure 7(b) exhibits the variation of non-dimensionalized breakup frequency and droplet removal rate with velocity ratio for smooth and microstructured surfaces. Although the breakup frequency always increases with a decrease in velocity ratio for smooth and microstructured surfaces, the volumetric removal rate decreases because of the trade-off between the breakup time and the droplet volume. We have already observed that the droplet size decreases with a decrease in velocity ratio, and the presence of microstructures also shows similar variation.

Furthermore, it is observed that the maximum removal rate is obtained for the velocity ratio of 0.033 for the microstructured surface. The non-dimensionalization of volume removal has been done by dividing the values by the confinement volume. The frequency has been non-dimensionalized by the maximum frequency obtained in the microstructured surface with a velocity ratio of 0.014. As we decrease the velocity ratio, despite the frequency of breakup increases, the removal rate decreases due to the decrease in droplet size. The formation of unwanted satellite droplets may occur at very low-velocity ratios in contrast to greater velocity ratios where the breakup is less likely. However, to what extent the interplay of hydrophobicity and the microstructure can affect droplet breakup, and removal is not yet clear. Future studies may reveal several non-intuitive understandings of the same.

In summary, we achieve quantitative prediction of the enhancement of efficiency of water removal with microstructures. The microstructures reduce the droplet's residence time over the surface and drastically reduce the probability of end-pinch-off, leading to the further formation of stray water droplets. To implement microstructures toward preventing water flooding of narrow PEMFC confinements, we observe a decrease in breakup time as we decrease the velocity ratio. However, beyond a threshold of decreasing velocity rations leads to uncontrollable fragmentation of the jet into many droplets, increasing the propensity of flooding as the smaller water droplets trapped within the confinement decrease the efficiency of overall water removal. Our results show that the breakup time is minimum when the velocity ratio is 0.014. However, with microstructures, there is no end-pinching for a velocity ratio of 0.014, and the blob has an extended tiny node, which may lead to a further breakup, generating satellite droplets afterward. We observe that a velocity ratio of 0.02 offers an optimization of reduced breakup time and generation of a single almost spherical droplet after a breakup. In our study, we delineate the effects induced by surface structuration and gain a deeper understanding of its impact on droplet disintegration and removal over the surface. These findings have implications for designing and optimizing interfaces in fuel cells and batteries where product removal is crucial. The results presented here have the potential to inform the development of new materials and technologies that harness the power of microstructures to optimize the performance of a wide range of systems.

Detailed information regarding the identification of the numerical framework with validation, thinning process, a theoretical framework for instability, velocity, pressure contours, and linear stability theory are shown in the supplementary material.

The authors acknowledge the support from Indian Institute of Technology, Kharagpur, through the project Energy Harvesting Using Droplets on Slippery Surfaces. C.B. and D.B. acknowledge the support from Intensification of Research in High Priority Areas (IRHPA) (Grant No. IPA/2021/000081) of the Science and Engineering Research Board (SERB), India.

The authors have no conflict of interest to disclose.

Nilanjan Mondal: Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Diptesh Biswas: Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Chirodeep Bakli: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available within the article and its supplementary material.