ROBUST, HIGH-THERMAL CONDUCTANCE, CAPILLARITY-ENABLED THIN-FILM

DRY CONDENSING SURFACES

PRIORITY CLAIM

[0001] This application claims priority to U.S. Provisional Patent Application No. 63/340,799, filed May 11, 2022, U.S. Provisional Patent Application No. 63/189,555, filed May 17, 2021, each of which is incorporated by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR

DEVELOPMENT

[0002] This invention was made with government support under grant number DE- FE0031677 awarded by the Department of Energy. The government has certain rights in this invention.

FIELD OF THE INVENTION [0003] This invention relates to condensation heat transfer.

BACKGROUND

[0004] Typically, condensation happens on solid surfaces that are wetted by the condensing fluid. In this configuration termed filmwise condensation (FIG. 1 A), the condensing fluid forms a thick liquid film on the surface. The latent heat of condensation must be conducted through this thick, low thermal conductivity liquid film, which presents a significant thermal resistance. For the past eight decades, much work has gone into developing ultra-thin water- repellent coatings to promote dropwise condensation of steam which can enhance heat transfer up to one order of magnitude (FIG. IB) (see, references 1-6). However, these ultra- thin coatings usually degrade are therefore not robust to endure industrial level operations which have a lifetime on the order of tens of years. Achieving robust coatings capable of meeting such lifetime requirements can be extremely difficult and has not been demonstrated.

SUMMARY

[0005] This Summary introduces a selection of concepts in simplified form that are described further below in the Detailed Description. This Summary neither identifies key or essential features, nor limits the scope, of the claimed subject matter. [0006] The invention relates to enhancing the condensation heat transfer performance in applications including power generation, thermal management of high-performance electronics, water purification, distillation, natural gas processing, and air conditioning.

[0007] Condensation heat transfer can be enhanced via a different mechanism. Instead of utilizing ultra-thin hydrophobic coatings, a hierarchical structure is attached on the condenser surface. This novel hierarchical structure is composed of a thin, highly permeable, thermally conductive porous wick and a highly porous, robust, intrinsically hydrophobic membrane bonded or attached on top of the wick.

[0008] In general, a device for providing condensation heat transfer can include a hierarchical structure attached on a condenser surface.

[0009] In one aspect, a capillary-driven condensation surface for a condenser surface can include a thermally conductive porous wick and a porous hydrophobic membrane on the wick.

[0010] In another aspect, a device having hierarchical structure for attachment to a condenser surface can include a thin, highly permeable, thermally conductive porous wick and a highly porous, robust, intrinsically hydrophobic membrane bonded or attached on top of the wick. [0011] In another aspect, a method of improving a heat transfer coefficient of a thermal system can include placing a capillary-driven condensation surface including a thermally conductive porous wick and a porous hydrophobic membrane on the wick on a surface of a condenser element of the thermal system.

[0012] In another aspect, a method of manufacturing a capillary-driven condensation surface for a condenser surface can include placing a porous hydrophobic membrane on a thermally conductive porous wick.

[0013] In certain circumstances, the thermally conductive porous wick can be configured to be in thermal contact with the condenser surface.

[0014] In certain circumstances, the thermally conductive porous wick can include a sintered metal powder, an electrodeposited porous metal, a metal foam, a metal mesh, a laser-etched metal, a 3D printed metal, a molded surface structure, or a patterned substrate.

[0015] In certain circumstances, the thermally conductive porous wick can include a copper foam, a copper mesh, a nickel foam, a stainless steel mesh, or an etched silicon structure. [0016] In certain circumstances, the thermally conductive porous wick can have a porosity of at least 30%. [0017] In certain circumstances, the thermally conductive porous wick can have an average porosity of greater than 30%. In certain circumstances, the thermally conductive porous wick can have an average porosity of less than 98%.

[0018] In certain circumstances, the thermally conductive porous wick can have an average pore size of at least 1 micron.

[0019] In certain circumstances, the porous hydrophobic membrane can be bonded to or mechanically secured to a surface of the porous wick. For example, the bond or securing can be by physical attachment, such as clamps or ties, thermal attachment, such as by diffusion bonding, or localized melting or solidification, or stress-based attachment, such as by pre forming the wick and membrane.

[0020] In certain circumstances, the porous hydrophobic membrane can have an average pore size of less than 20 microns.

[0021] In certain circumstances, the porous hydrophobic membrane can have an average pore size of greater than 10 nanometers.

[0022] In certain circumstances, the porous hydrophobic membrane can include an organic polymer or an inorganic material. For example, the organic polymer can be a fluorinated polyolefin or a polyolefin, such as polyethylene, polystyrene, polypropylene (PP), polytetrafluoroethylene, poly (vinyli dene fluoride) (PVDF), poly(vinylidene fluoride)-co- hexafluoropropylene (PVDF-HFP), or sulfonated polytetrafluoroethylene. The inorganic material can be a rare earth oxide, silicon oxide, or silicon nitride.

[0023] In certain circumstances, the porous hydrophobic membrane can further include a hydrophobic coating. The hydrophobic coating can be a fluorinated alkyl film former, for example, a perfluorinated alkyl silicone alkoxide.

[0024] In certain circumstances, the organic polymer can be an electrospun fiber.

[0025] In certain circumstances, the electrospun fiber can have an average diameter of between 0.5 microns and 4 microns.

[0026] In certain circumstances, the electrospun fiber can have an average diameter of between 0.1 microns and 2 microns.

[0027] In certain circumstances, the thermally conductive porous wick can include microchannels.

[0028] In certain circumstances, the microchannels can be arranged in rows or bands. The rows or bands can have a spacing between the rows or bands of between 0.25 cm and 10 cm. [0029] In certain circumstances, the rows or bands can be arranged substantially perpendicular to a lengthwise axis of the condenser surface.

[0030] In certain circumstances, the porous hydrophobic membrane can include one or more drain ports along a length of the membrane. For example, the drain ports can be adjacent to each of the rows or bands, if present.

[0031] In certain circumstances, the method can include securing the capillary-driven condensation surface to the surface of the condenser element.

[0032] In certain circumstances, the method can include cleaning a surface of the thermally conductive porous wick prior to placing the porous hydrophobic membrane.

[0033] In certain circumstances, the method can include heat treating a surface of the thermally conductive porous wick prior to placing the porous hydrophobic membrane.

[0034] The following Detailed Description references the accompanying drawings which form a part this application, and which show, by way of illustration, specific example implementations. Other implementations may be made without departing from the scope of the disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

[0035] FIGS. 1A-1B are photographs showing condensation. FIG. 1A shows filmwise condensation of steam on a bare copper surface. FIG. IB shows dropwise condensation of steam on a copper surface coated with a monolayer hydrophobic coating. See, reference 7. [0036] FIGS. 2A-2B illustrate schematic of embodiments of a condenser system.

[0037] FIGS. 3A-3E illustrate (FIG. 3A) schematic detailing the working principle of the approach described herein. The schematic depicts condensation at a steady state. Vapor enters through the hydrophobic membrane pores (light sections) and condenses at the convex liquid meniscus, which pins at the wick-membrane interface. The curvature from the bent meniscus yields a capillary pressure proportional to the inverse of the membrane pore diameter. The pressure imbalance between the largest capillary pressure at any membrane pore to the ambient pressure at the exit-port drives fluid flow through the porous wick towards the exit port. FIGS. 3B-3C show the membrane can be made of an inexpensive polymer. FIGS. 3D-3E show the wick can be made from a robust wicking structure such as a porous copper foam.

[0038] FIGS. 4A-4D illustrates a proof-of-concept of the approach micromachined on a silicon wafer with a thin low-stress silicon nitride thin film on top. FIG. 4A is a schematic of the structure and the location of the hydrophobic coating. The top layer is a hydrophobic membrane. FIG. 4B shows a plane-view SEM of membrane top showing the micropillars holding the membrane. FIG. 4C shows a cross-section SEM of wicking structure showing a pillar holding the membrane. FIG. 4D shows condensation in ambient-air environment showing that the top-condensing surface looks dry over a large area during active condensation. From this image it can be seen that the concept is utilizing capillary pressure to drive out fluid to the sides or exit port of the sample.

[0039] FIG. 5 is a schematic of capillary-driven condenser composed of a substrate, a micro- channeled porous metal wick later, and a top-layer hydrophobic membrane. The blue arrow shows the route through which the condensed water can exit out from the microchannel. The red arrow shows the route through which the condensed water can exit out directly from the metal wick. The key dimensions involved in this structure are width of the wick later IF _WiCk . width of the microchannel IF _diannei , length of the microchannel L, and the thickness of the metal wick layer

[0040] FIGS. 6A-6B are schematics of capillary condenser tube in two different designs.

FIG. 6A shows a capillary-driven condenser composed of a bottom layer of porous metal wick and a top layer of hydrophobic membrane with condensate drainage channel across the wick and the membrane at the bottom of the tube. FIG. 6B shows a capillary-driven condenser tube with microchannels for fast drainage and with spotted drainage pores located at the bottom of the each microchannel.

[0041] FIGS. 7A-7B depict electrospinning setups for fabricating hydrophobic membranes. [0042] FIGS. 8A-8D show SEM images of electrospun membrane samples. Fiber diameter and morphology can be characterized accurately using SEM. However, pore size distribution is hard to define due to the random alignment of the fibers.

[0043] FIG. 9 is a graph depicting wet curve and dry curve of an electrospun membrane sample given by the capillary flow porometer (Porolux 1000).

[0044] FIG. 10 is a graph depicting pore size distribution of an electrospun membrane sample given by the capillary flow porometer (Porolux 1000).

[0045] FIGS. 11 A-l IB are schematics showing an example of the reduced number of combinations in a fractional factorial design.

[0046] FIGS. 12A-12F depict SEM images of two electrospun PVDF-HFP samples (top, bottom) and their corresponding fiber diameter distribution and pore size distribution.

[0047] FIG. 13 is a series of graphs depicting fiber diameter as a function of needle distance, voltage supply, and solution feeding rate predicted with experimental data obtained by the FFD. Least squares method was used to find the best fit. Confidence intervals are shown in grey.

[0048] FIG. 14 is a graph depicting effects of electrospinning time on the fiber diameter of the electrospun PVDF-HFP membrane. Four sets of parameters were tested as shown in A,

B, C, and D. Fiber diameters after 20 minutes and 60 minutes of electrospinning were compared. The effect of electrospinning time on the fiber diameter is negligible.

[0049] FIG. 15 is a graph depicting effects of aging of the PVDF-HFP solution on the fiber diameter of the electrospun membrane.

[0050] FIGS. 16A-16B depict SEM images of electrospun PVDF-HFP membranes on the top of a plasma-cleaned copper pillar substrate. FIG. 16A shows electrospun fibers showed larger preference to attach to the tip of the copper pillars than to the fibers themselves. FIG. 16B shows a thick layer of electrospun fibers detached from the pillar substrate.

[0051] FIG. 17 shows electrospun PVDF-HFP membranes on two different substrates with the same heat treatment. Heating the substrate to slightly above the melting point of the polymer during the first few minute of electrospinning was found to be helpful for improving the bonding between the electrospun membrane and the metal wick, although the bonding is also affected by the geometry of the metal wick.

[0052] FIG. 18 shows a schematic of configuration of the exit port in the original design. A high thermal conductivity high porosity micro structured wick is overlaid with a porous hydrophobic membrane on its surface and an exit port or break from the structure is at the bottom to allow the exit of condensate. The fluid transport length is constrained to be half the circumference of the pipe in consideration.

[0053] FIG. 19 depicts an alternative design with flexibility in the design of the fluid transport length. Membrane sections of a rationally designed length can be wrapped around the condensation tube. Fluid may exit at the sides of the bands rather than the bottom. In this scheme, bonding of the membrane and the wick can be that as an elastic band, or by mechanical fasting or clamping to the tube or by a clamp which is not part of the tube but can tie the membrane around the tube.

[0054] FIG. 20 is a graph depicting heat transfer measurements during condensation in a pure vapor ambient. CDC stands for “Capillary-Driven Condensation” and the numbers after CDC represent the micropillar diameter to pitch d/l ratio (first two with a decimal in between the numbers) and the pillar diameter (last two digits in microns). [0055] FIGS. 21A-21B are schematic cross-sections of a condenser tube. FIG. 21 A shows a traditional filmwise condensation on a condenser tube. FIG. 21B shows a capillary-driven condensation on tube with flow driven by the Laplace pressure gradient formed due to the porous membrane, which can improve heat transfer coefficient by over lOx as compared to filmwise condensation, decreasing turbine back pressure by over 50% (see FIGS. 22A-22B). [0056] FIGS. 22A-22B are graphs depicting simulation of capillary-driven condensation performance for water. Both 1” and ^x h ” outer diameter condenser tubes are considered here. The copper foam wick has properties of permeability K _{i k} = 1.25xl0 ⁹ m ² , porosity = 0.87, and thermal conductivity Kick = 15.3 W/mT . The maximum capillary pressure supported by the hydrophobic porous membrane is P _cap -max = 14.2 kPa. The heat transfer coefficient (HTC) enhancement ratio in (FIG. 22A) is compared to traditional filmwise condensation (equation (1)) over the range of heat fluxes shown, and the corresponding percentage reduction in the turbine back pressure as compared to filmwise condensation is plotted in (FIG. 22B).

[0057] FIGS. 23A-23B are graphs depicting parametric sweeps using the model with a condensation heat flux of 150 kW/m ² , with the temperature drops across the structured wick and the porous membrane plotted separately as well as combined (“total”), and compared with the temperature drop predicted by Nusselt’s model for filmwise condensation under the same heat flux. FIG. 23A shows the membrane pore radius was found to produce an enhancement in condensation heat transfer between approximately 2 to 15 microns, holding all other parameters constant. FIG. 23B shows taking a membrane pore radius of 5 microns, the permeability of the structured wick was varied, where an increase in permeability always resulted in improved heat transfer performance (atypical value for metal foams is 1.25xl0 ⁹ m ² ).

[0058] FIG. 24 A is a schematic of typical experiment with round tubes and cooling provided by an external chiller loop. FIG. 24B is an image of environmental chamber (operating pressure up to 300 psig) to be used for condensation studies.

[0059] FIG. 25A shows an image of wick attached to heat transfer characterization-enabled condensation block to be used for condensation studies. FIG. 25B is a graph depicting enhanced condensation heat transfer of low surface tension, low latent heat fluids such as pentane using only the hydrostatic pressure (no porous membrane), with a demonstrated improved heat transfer coefficient of 3x. Water, which has a higher surface tension and higher latent heat, requires the hydrophobic porous membrane to take advantage of capillary pressure driven flow for removal of the condensate. [0060] FIG. 26A shows a schematic of capillary-driven condensation from hydrophilic wick structures with a hydrophobic top surface. Liquid is pinned at the top surface and generates capillary pressure. FIG. 26B shows an example of the liquid within four micropillars with curved liquid-vapor interface, and the fluid velocity profile. FIG. 26C is an SEM image of a representative, fabricated micropillar array wick structure with diameter d, height h and pitch /.

[0061] FIG. 27 is a graph depicting liquid velocity along a 45 mm copper micropillar array wicking surface when the applied condensation heat flux is 187 kW/m ² .

[0062] FIG. 28 is a graph depicting contact angle along a 45 mm copper micropillar array wicking surface when the applied condensation heat flux is 187 kW/m ² .

[0063] FIGS. 29A-29C are graphs depicting membrane pore mean permeabilities as a function pore diameter in the three regimes. FIG. 29A shows the free-molecule flow or Knudsen regime. FIG. 29B shows the transition regime. FIG. 29C shows the viscous flow or Poiseuille regime.

[0064] FIG. 30 is a graph depicting mean permeability of membrane as a function of pore diameter from 0-15 pm in the transition regime for T _sat ~ 35°C. In this diameter range, there is a crossover point for each contribution at around 6 pm. Neglecting the Knudsen flow contribution by assuming purely viscous flow underestimates the permeability by almost ~2x.

[0065] FIGS. 31A-31B depict micro-pillar unit cell utilized for (FIG. 31A) the HTC simulation, which includes solid pillars and (FIG. 3 IB) the permeability simulation, which contains the fluid body with no-slip condition at the wall boundaries. Half of the unit cell was used for the permeability calculation due to symmetry. A quarter of the unit cell shown in (FIG. 31 A) was used for the HTC measurement due to symmetry (not shown).

[0066] FIGS. 32A-32B are graphs depicting the impact of wick size on the heat transfer coefficient by varying the pillar wick diameter and the d/l ratio for a 3 pm thick membrane with (FIG. 32A) 1pm pore diameter and (FIG. 32B) 3pm pore diameter. The pillar height is 30 pm.

[0067] FIGS. 33A-33B are graphs depicting the impact of membrane pore diameter on the heat transfer coefficient for a membrane thickness of 3 pm and two different wick sizes represented by pillar wick diameters of (FIG. 33A) 5pm and (FIG. 33B) 10pm. The pillar height is 30 pm. [0068] FIGS. 34A-34B are graphs depicting the impact of membrane thickness on heat transfer coefficient as the wick d/l ratio is varied for two different wick geometries (FIG. 34A) for a pillar wick diameter of 5 pm and a membrane pore size of lpm and (FIG. 34B) for a larger pillar wick diameter of 15 pm and a membrane pore size of 3 pm. The pillar height is 30 pm.

[0069] FIGS. 35A-35B are graphs depicting the impact of the wicking length on available designs and attainable heat transfer coefficients. Longer wicking lengths require higher capillary-driving pressures, so fewer designs are available due to flooding. The pillar height is 30 pm.

[0070] FIGS. 36A-36F show an approach to fabrication of high performance capillary driven condensation structure.

[0071] FIG. 37 is a photograph depicting a double side polished <100> silicon wafer is coated with 500 nm of low stress silicon nitride, as designed by the fabrication approach described herein.

[0072] FIGS. 38A-38B are micrographs depicting successful large-scale fabrication of sample. FIG. 38A is a plane view SEM image of best sample, showing uniformity in structure at larger areas than previously. The top left inset shows a photograph of the sample, while the bottom right inset shows plane view SEM of a unit cell, showing open pores and pillar locations. FIG. 38B is a cross section SEM image of best sample showing that the membrane hangs on top of the micropillars for large distances. The lower right inset shows a close up of the structures.

[0073] FIGS. 39A-39G depict photoresist-wicking approach to selectively coat the sample as hydrophobic. In FIG. 39A, photoresist is wicked into the structure and rises to a height determined by the photoresist properties. In FIG. 39B, the large sample is coated with a two- step method where photoresist is wicked in from two sample extremes. In FIG. 39C, post- wicking bake shows that distribution of photoresist inside the sample is not uniform. In FIGS. 39D-39F, the photoresist is shown to coat all inside surface of the wick on to the top surface of the membrane. In FIG. 39G, schematic of the top-most surface selective hydrophobic coating given by the approach.

[0074] FIG. 40 is a photograph showing an experimental setup to observe horizontal condensation on structure in ambient conditions under a light microscope under various magnifications. The sample is mounted on top of the cold stage. [0075] FIGS. 41A-41B show schematics and images of coated structure and condensation visualization study under air-ambient conditions for, in FIG. 41 A, a structure coated fully hydrophobic and, in FIG. 41B, a structure selectively coated such that only the top surface of the membrane is hydrophobic. The red outline in the schematic represents the hydrophobic FAS coating. The pillar pitch in each image is 150 pm.

[0076] FIGS. 42A-42B show a visualization study of condensation in ambient air showing that droplets that grow on pillar tops can be continually absorbed into the wick through membrane pores and driven to an exit port.

[0077] FIGS. 43A-43B are images showing the experimental configuration for the acquired data. Namely, the experiment is conducted at two saturated vapor conditions for each sample (~ 40 C and ~60 C). FIG. 43A is an image showing the experimental configuration for the filmwise experiment. FIG. 43B is an image showing condensation on the engineered sample. Notably, a big puddle is formed to promote ensuring a more temperature-stable saturated vapor ambient.

[0078] FIGS. 44A-44B are graphs depicting bar plots of the heat flux (FIG. 44 A) and heat transfer coefficient (FIG. 44B). Taller bars represent experimentally measured values. Gray bars indicate theoretical predictions of the Nusselt filmwise condensation model at the operating condition. Notably, the data for the experimental filmwise experiments are shown to agree with the theoretical Nusselt model.

[0079] FIGS. 45A-45D are images showing how the exit port works in this context. FIG.

45A is an image of a dry-condensation surface at low magnifications. Notably, there are no visible droplets on the surface, or bursting droplets. Rather, vapor condenses on pores in the membrane and is driven to the exit port (edge of the sample). FIGS. 45B-45D are time-lapse images of fluid exiting from the exit port.

[0080] FIG. 46 is a schematic showing comparison of effective thermal conductivities among different metal structures (assuming at the same porosity).

[0081] FIGS. 47A-47C show a diffusion bonding assembly and the high temperature furnace for diffusion bonding. FIG. 47 A is an image of the diffusion assembly. FIG. 47B is a schematic of the diffusion assembly. FIG. 47C is an image of the high temperature furnace. [0082] FIGS. 48A-48C are SEM images of the three hierarchical copper surfaces made by diffusion bonding a thin copper mesh to 200 pm thick copper foam. FIG. 48A shows a 200- mesh-size copper mesh. FIG. 48B shows a 500-mesh-size copper mesh. FIG. 48C is a 1500- mesh-size copper mesh. [0083] FIGS. 49A-49D are graphs depicting permeability characterization of the wick layer under various processing conditions.

[0084] FIG. 50 is a schematic showing vapor deposition of FAS on porous copper sample. [0085] FIG. 51 A is an image and FIG. 5 IB is a schematic showing a copper foam sample machined with microchannels for condensate drainage. Channel width: 100 pm; channel depth: 100 pm; distance between neighboring channels: 1.27mm.

[0086] FIGS. 52A-52C are graphs depicting heat transfer enhancement given by the three hierarchical copper surfaces predicted by the analytical model.

[0087] FIGS. 53A-53C are images taken inside (FIG. 53A) and outside (FIGS. 53B-53C) the environmental chamber.

[0088] FIG. 54A shows filmwise condensation on a flat copper sample and FIG. 54B the corresponding heat transfer data.

[0089] FIGS. 55A-55D show condensation experiments on fully hydrophobic hierarchical copper sample and the resulting heat transfer performance. FIGS. 55A-55C are images of the sample inside the chamber before condensation occurred (FIG. 55A); at the very beginning of the condensation experiment (FIG. 55B); and at later stage of the condensation experiment (FIG. 55C). FIG. 55D is a graph depicting heat flux measured on the sample under various subcools (data points) in comparison with Nusselt model prediction for filmwise condensation heat transfer (curve).

[0090] FIGS. 56A-56D show condensation experiments on biphilic hierarchical copper sample and the resulting heat transfer performance. FIGS. 56A-56C are images of the sample inside the chamber before condensation occurred (FIG. 56A); at the very beginning of the condensation experiment (FIG. 56B); at later stage of the condensation experiment (FIG. 56C). FIG. 56D is a graph depicting heat flux measured on the sample under various subcools (data points) in comparison with Nusselt model prediction for filmwise condensation heat transfer (curve).

[0091] FIGS. 57A-57D show condensation experiments on biphilic, micro-channeled hierarchical copper sample and the resulting heat transfer performance. FIGS. 57A-57C are images of the sample inside the chamber before condensation occurred (FIG. 57A); at the very beginning of the condensation experiment (FIG. 57B); at later stage of the condensation experiment (FIG. 57C). FIG. 57D is a graph depicting heat flux measured on the sample under various subcools (data points) in comparison with Nusselt model prediction for filmwise condensation heat transfer (curve). [0092] FIGS. 58A-58C are SEM images of sintered copper powders. FIG. 58A shows spherical powders with diameter < 10 pm. FIG. 58B shows spherical powders with diameter < 50 pm. FIG. 58C shows dendritic powders with size < 45 pm.

[0093] FIGS. 59A-59C are graphs depicting condensation heat transfer performance of different types of porous copper covered by electrospun fibers, as predicted by heat and mass transfer model. FIG. 59A shows electrospun fibers in combination with a copper foam with 200 pm thickness, 65% porosity, and IE-11 m ² permeability. FIG. 59B shows electrospun fibers in combination with inverse opal copper with 5 pm thickness, 65% porosity, and 5E-11 m ² permeability. FIG. 59C shows electrospun fibers in combination with sintered copper spheres with sphere diameter 50 pm, thickness 200 pm, 50% porosity, and 1.7E-llm ² permeability.

[0094] FIG. 60 is a schematic showing a capillary-driven condensation surface.

DETAILED DESCRIPTION

[0095] Enhancing condensation heat transfer can have important implications to reduce local water consumption, reduce CO2 emissions, while simultaneously enhancing the overall cycle efficiency of steam power plants. This is important since most of the electricity produced in the U.S. comes from steam power plants. The state-of-the-art technology, dropwise condensation (DWC), has demonstrated its ability to achieve condensation heat transfer enhancements of up to an order of magnitude. However, it comes short of industrial implementation due to the requirement for thin coatings, which cannot adhere to the necessary industrial timescales required for operation. Therefore, various condensation approaches have been investigated to enhance the durability of condensation enhancing engineered surfaces.

[0096] Condensation system, structures and methods described herein have the potential to address concerns of durability by combining thick porous hydrophobic membranes with porous wicking structures on condenser tubes. This concept, termed capillary-driven condensation (CDC), can enhance condensation heat transfer to a level comparable to dropwise condensation according to the preliminary models. Herein, this approach is discussed in depth, and investigate the physics during condensation on the CDC surfaces to understand, design, and develop superior condensation surfaces. Two approaches to developing the surface are described. [0097] The fundamental understanding of the physics behavior of the surface during condensation by fabricating a CDC sample on silicon with highly-defined geometry by modeling, optimization, and rational design of a surface, fabrication of that surface, visualization studies to elucidate the physics, and conduct experimental heat transfer measurements. Vapor condensation of a fabricated structure achieved a heat transfer coefficient ~ 240% higher than the theoretical filmwise value at the same operating conditions.

[0098] Hierarchical copper surfaces were fabricated with commercially available copper- based foams and meshes, and hydrophobized a copper mesh layer to realize capillary-driven condensation. Capillary-driven condensation was observed on these hierarchical copper surfaces, and measured a 50% heat transfer enhancement over filmwise condensation on a micro-channeled, biphilic, hierarchical copper sample. Modelling predicts a much higher heat transfer enhancement.

[0099] Scalable fabrication of a hydrophobic membrane and a porous metal wick was also explored. A parametric study has been conducted to optimize the fabrication recipe of the electrospun fibrous membrane. Sintering and electrodeposition have been studied for fabricating porous copper wick. The heat and mass transfer model predicts > 5x heat transfer enhancement over filmwise condensation on an electrospun fiber covered sintered copper powder surface.

[00100] Condensation heat transfer can be enhanced via a different mechanism.

Instead of utilizing ultra-thin hydrophobic coatings, a hierarchical structure is attached on the condenser surface. This novel hierarchical structure is composed of a thin, highly permeable, thermally conductive porous wick and a highly porous, robust, intrinsically hydrophobic membrane bonded or attached on top of the wick. See, for example, FIGS. 2A-2B.

[00101] Condensation heat transfer, whereby a vapor changes phase to liquid by releasing the latent heat, is a critical process in various applications such as steam cycles for power generation, refrigeration cycles for heating, ventilation and air conditioning systems, and heat pipes and vapor chambers for thermal management. Despite decades of research on condenser surface design, state-of-the-art condenser surfaces suffer from low scalability, costly fabrication, and flooding (due to high subcooling). As described herein, condenser surfaces based on the concept of capillary-driven condensation have been designed, fabricated, and tested. By bonding a hydrophobized mesh on top of a highly porous copper foam, the condensate was constrained within the foam layer to form a thin, continuous liquid film. The control of the thickness along with the improved effective thermal conductivity of the liquid film enables us to reduce the thermal resistance; and hence higher heat transfer coefficient. The heat and mass transfer model, which guides surface design, predicts a significant enhancement in heat transfer coefficient compared to the classical filmwise condensation. Experiments are conducted in a custom-built environmental chamber to validate the model prediction. Unlike condenser surfaces designs that rely on sophisticated micro/nanostructuring, the capillary-driven condenser surfaces are easy to fabricate, highly scalable, and can withstand higher subcooling. The insights gained from this work pave a way for enhancing condensation heat transfer in large scale applications.

[00102] As shown in FIGS. 2A-2B, the invention regards the external condensation on the outer surface of the tube. A typical condenser tube made of a metal or metal-alloy is coated with a high-thermal-conductivity, high-porosity wicking structure, and overlayed with a robust intrinsically hydrophobic porous membrane. The wick and membrane are physically or chemically bonded in a robust way. Vapor travels through the membrane pores and condenses in the wick. Liquid-vapor interface is expected to pin at the wick-membrane interface. The capillary pressure generated at the curved liquid vapor interface drives condensate flow to the designated exit ports, as shown at the bottom of the tube in this image in FIGS. 2A-2B.

[00103] Referring to FIG. 2B, which shows a device that exhibits filmwise condensation with enhanced thermal conductivity & controlled condensate film thickness. The device includes a hierarchical surface consisting of a robust hydrophobic membrane and high thermal conductivity wick. Vapor transports through membrane pores and condenses at the wick-membrane interface. Capillary pressure at the membrane-wick interface provides additional driving-force to push condensate from the wick to an exit port for condensate removal.

[00104] The working principle, illustrated in FIG. 3A, is as follows. Pure vapor travels through the membrane pores to the wick-membrane interface where it condenses at the pinned liquid-vapor interface. The condensing fluid must be transported to the exit-port by overcoming the viscous pressure drop of fluid flow along the way from the liquid-vapor interface to the exit-port. The viscous pressure drop is overcome by the capillary pressure that generated by the bent liquid menisci, which can be tailored significantly by the pore size of the membrane. A condition where the maximum viscous pressure drop (calculated with the longest travel distance to an exit port) is larger than the largest capillary pressure generated by the membrane pores would lead to liquid “bursting” from the top of the membrane. This condition is referred to as “flooding” herein, and this condition can be avoided by the surfaces, methods and designs described herein.

[00105] FIGS. 3A-3E illustrate (FIG. 3 A) schematic detailing the working principle of the proposed approach. The schematic depicts condensation at a steady state. Vapor enters through the hydrophobic membrane pores (light sections) and condenses at the convex liquid meniscus which pins at the wick-membrane interface. The curvature from the bent meniscus yields a capillary pressure proportional to the inverse of the membrane pore diameter. The pressure imbalance between the largest capillary pressure at any membrane pore to the ambient pressure at the exit-port drives fluid flow through the porous wick towards the exit port. FIGS. 3B-3C show the membrane can be made of an inexpensive polymer. FIGS. 3D- 3E the wick can be made from a robust wicking structure such as a porous copper foam. [00106] The benefits of a thin, high-permeability, high-thermal conductivity, porous structured wick to enhancing condensation heat transfer are manifold. The wick-condensate composite has a higher effective thermal conductivity which reduces the thermal resistance of the condensate film. Minimizing the height of this layer further reduces its thermal resistance as the heat transport distance decreases. Similarly, the benefits of an intrinsically hydrophobic membrane are manifold. Unlike those highly degradable, self-assembled monolayer hydrophobic coatings, intrinsically hydrophobic membranes are robust and presently used in the water treatment/desalination industries. Most importantly, the micrometer-sized pores of the hydrophobic membrane could generate a significant capillary pressure, which acts as the driving force for the condensate to overcome viscous pressure loss along the wicking structures and flow towards the designated exit ports. Several parameters can be carefully designed in order to optimize the condenser’s heat transfer performance, such as: membrane pore size, membrane thickness, porosity of the membrane, wick thickness, permeability of the wick, porosity of the wick.

[00107] FIGS. 4A-4D illustrate a proof-of-concept of the approach micromachined on a silicon wafer with a thin low-stress silicon nitride thin film on top. FIG. 4A is a schematic of the structure and the location of the hydrophobic coating. FIG. 4B shows a plane-view SEM of membrane top showing the micropillars holding the membrane. FIG. 4C shows a cross-section SEM of wicking structure showing a pillar holding the membrane. FIG. 4D shows condensation in ambient-air environment showing that the top-condensing surface looks dry over a large area during active condensation. From this image it can be seen that the concept is utilizing capillary pressure to drive out fluid to the sides or exit port of the sample.

[00108] FIGS. 4A-4D show a proof of concept of the capillary-driven surface. A dry etching technique was developed to produce a membrane attached to the micropillar wick. A method to selectively coat the top surface of this membrane to make it hydrophobic and condensed on this sample has been demonstrated. As seen in FIG. 4D the top surface of the membrane looks visibly dry during active condensation, showing that condensate is driven out through the wick to the exit port, validating the operation of the concept.

[00109] The present approach can achieve much larger condensation heat transfer performance than conventional extended surfaces, which are commercially available.

[00110] The present approach can overcome the issue of robustness and durability of ultra-thin hydrophobic coatings in dropwise condensation by utilizing components, which are individually robust. The wicking structure can be made from a robust metal foam, and the porous hydrophobic membrane can be a commercially available polymer membrane thick enough to be robust. The performance can be comparable or higher than dropwise condensation depending on the surface design.

[00111] The driving force for condensate flow in the approach, i.e., the capillary pressure generated by the membrane pores, is completely passive and can be enormous (>10 MPa) by utilizing a small membrane pore (on the order of 1 pm), which makes the approach superior than those previously reported wicking condensation surfaces that either use external pumping force (see reference 8) or gravitational force (see reference 9) to drive the condensate flow.

[00112] Some recent studies have applied hydrophobic coatings on the top portion of a wicking structure in order to retain the condensed liquid inside the wicking structure with the capillary pressure generated at the hydrophobic top. These partially -hydrophobically coated (i.e., amphiphilic) wicking structures include nanowires (see reference 10), inverse opals (see reference 10), anodic aluminum oxide (AAO) membranes (see reference 9), micropillar arrays (see reference 9). However, these amphiphilic wicking structures couple the capillary driving pressure generated by the hydrophobic top layer and the viscous pressure loss generated by the hydrophilic bottom layer since the two layers share the same pore geometry, limiting the design space and potential for enhancing heat transfer performance. In addition, many of these wicking structures (e.g., nanowires and micropillar arrays) require expensive, lab-scale fabrication, limiting scale-up applications. Moreover, the ultra-thin hydrophobic coatings utilized by the above amphiphilic structures are non-robust and would not endure industrial-level operations even for weeks. The approach fundamentally surpasses the existing approaches mentioned above by decoupling the capillary driving force and the viscous pressure loss with a two-layer hierarchical surface design. By adding an intrinsically hydrophobic porous membrane on top of the wicking structure, the pore sizes of the wicking structures can be independently tailored from those of the membrane layer to achieve a lower viscous pressure drop inside the wicking structures to form a highly permeable wick while maximizing the capillary driving pressure by reducing the size of the membrane pores. These physics can be modeled, and rational designs can be made. Furthermore, low cost, robust, and scalable fabrication techniques can be used, such as electrospinning fiber membranes and electroless deposition of porous metals.

[00113] The proposed method, systems and surfaces have applicability in steam power generation. Steam power plants account for -60% (see reference 10) of the total electricity generated in the US and -80% in the world. Retrofitting existing power plants, is a huge market for technologies that promise enhancements in condensation heat transfer that can last on the order of 15 years or more.

[00114] Moreover, the thermal management of high-performance electronics sector utilizes heat pipes and vapor chamber technologies, which can be found in conventional laptops, cell phones, and desktops. Heat pipes and vapor chambers both contain evaporators and condensers. A proof-of-concept condenser utilizing this approach is described herein. It is fabricated on silicon utilizing micromachining technology which is common to the industry (see FIG. 4A and 4B).

[00115] As described herein, a device for providing condensation heat transfer can include a hierarchical structure attached on a condenser surface.

[00116] Referring to FIG. 60, capillary-driven condensation surface 10 can include a porous hydrophobic membrane 20 adjacent to a wicking structure 30 on a condensing surface 40 can drive liquid transport and removal from the surface via a capillary pressure gradient along the wicking surface towards an exit port 50. In certain embodiments, optional microchannel 60 in structure 30 can assist with liquid removal. The surface can include a plurality of microchannels and exit ports (not shown).

[00117] Capillary-driven condensation, condensation occurs on the hydrophilic micro/nanostructured wick, and the condensate is then forced out due to the capillary pressure buildup at the menisci formed in the porous hydrophobic membrane. The presence of the structures and the resulting capillarity helps maintain a stable liquid film while driving liquid flow. By tailoring the size of the pores in the membrane and the geometry of the wicking structure, the capillary pressure generated can be maximized and the flow rate of the condensate can be optimized to increase the rate of condensation that the wicking structure can support.

[00118] In one aspect, a capillary-driven condensation surface for a condenser surface can include a thermally conductive porous wick and a porous hydrophobic membrane on the wick.

[00119] In another aspect, a device having hierarchical structure for attachment to a condenser surface can include a thin, highly permeable, thermally conductive porous wick and a highly porous, robust, intrinsically hydrophobic membrane bonded or attached on top of the wick.

[00120] For example, a robust hydrophobic membrane can be combined with a porous, high-thermal-conductivity metal wick and wrapped around the external surface of the condenser tube. During the condensation process, water vapor transports through the membrane pores and condense inside the porous metal wick, forming a thin film with a thickness constrained by the capillary pressure generated at the base of the membrane pore. Condensed water drains out through designated exit port to avoid a flooding issue. The drain can be by gravity or by pump or other pressure differential. [00121] In another aspect, a method of improving a heat transfer coefficient of a thermal system can include placing a capillary-driven condensation surface including a thermally conductive porous wick and a porous hydrophobic membrane on the wick on a surface of a condenser element of the thermal system.

[00122] In another aspect, a method of manufacturing a capillary-driven condensation surface for a condenser surface can include placing a porous hydrophobic membrane on a thermally conductive porous wick.

[00123] Surprisingly, vapor condensation of a fabricated structure can achieve a heat transfer coefficient 50% to 500% higher than the theoretical filmwise value at the same operating conditions. Moreover, capillary-driven condensation was observed on these hierarchical copper surfaces, and measured a 50% heat transfer enhancement over filmwise condensation on a micro-channeled, biphilic, hierarchical copper sample.

[00124] In certain circumstances, the thermally conductive porous wick can be configured to be in thermal contact the condenser surface. [00125] In certain circumstances, the wick can have thickness of between 5 microns and 100 microns, for example, 5 microns, 10 microns, 15 microns, 20 microns, 25 microns,

30 microns, 35 microns, 40 microns, 45 microns, 50 microns, 55 microns, 60 microns, 65 microns, 70 microns, 75 microns, 80 microns, 85 microns, 90 microns, 95 microns, or 100 microns.

[00126] In certain circumstances, the thermally conductive porous wick can include metal wire cloth, perforated sheets, and metal foams, for example, a sintered metal powder, an electrodeposited porous metal, a metal foam, a metal mesh, or a patterned substrate. [00127] In certain circumstances, the thermally conductive porous wick can include a copper foam, a copper mesh, a nickel foam, a stainless steel mesh, or an etched silicon structure, for example, silicon nitride. The wick can include partially-hydrophobically coated (i.e., amphiphilic) wi eking structures include nanowires, inverse opals, anodic aluminum oxide (AAO) membranes, or micropillar arrays. The wick can include perforated stainless steel, perforated brass, perforated steel, perforated aluminum, perforated copper, stainless steel mesh, brass mesh, steel mesh, aluminum mesh, copper mesh, copper foam, or nickel foam, sintered spherical metal powders with diameter < 10 pm, spherical metal powders with diameter < 50 pm, or dendritic metal powders with size < 45 pm, or combinations thereof. [00128] As described in more detail below, copper-based foams and meshes, silicon and silicon nitride structures, and a hydrophobized copper mesh layer have been shown experimentally to achieve capillary-driven condensation. Sintering and electrodeposition have been studied for fabricating porous copper wick. The heat and mass transfer model predicts > 5x heat transfer enhancement over filmwise condensation on an electrospun fiber covered sintered copper powder surface.

[00129] In certain circumstances, the thermally conductive porous wick can have a porosity of at least 30%, at least 35%, at least 40%, at least 45%, at least 50%, at least 55%, at least 60%, at least 65%, at least 70%, at least 75%, at least 80%, at least 85%, at least 90%, or at least 95%. The thermally conductive porous wick can have a porosity of no greater than 98%.

[00130] In certain circumstances, the thermally conductive porous wick can have a thickness of no more than 500 microns, for example, no more than 1 micron, 5 microns, 10 microns, 20 microns, 30 microns, 50 microns, 70 microns, 90 microns, 100 microns, 150 microns, 200 microns, 250 microns, 300 microns, 350 microns, 400 microns, 450 microns, or 500 microns. [00131] In certain circumstances, the thermally conductive porous wick can have an effective thermal conductivity between 0.6 W/mK and 400 W/mK. The effective thermal conductivity can be selected based on the thickness of the wick. For example, if the wick has thickness below 20 microns, then an effective thermal conductivity of above 0.6 W/mK should give some enhancement and if the wick has a thickness above 100 microns, then an effective thermal conductivity of above 10 W/mK can be needed to get some heat transfer enhancement. The higher the effective thermal conductivity the better.

[00132] In certain circumstances, the thermally conductive porous wick can have an average pore size of less than 30 microns, less than 25 microns, less than 20 microns, less than 18 microns, less than 16 microns, less than 15 microns, less than 14 microns, less than 13 microns, less than 12 microns, less than 11 microns, or less than 10 microns.

[00133] In certain circumstances, the thermally conductive porous wick can have an average pore size of at least 1 micron, at least 2 microns, at least 3 microns, at least 4 microns, at least 5 microns, at least 6 microns, at least 7 microns, at least 8 microns, at least 9 microns, or at least 10 microns.

[00134] In certain circumstances, the porous hydrophobic membrane can be bonded to or mechanically secured to a surface of the porous wick. For example, the bond or securing can be by physical attachment, such as clamps or ties, thermal attachment, such as by diffusion bonding, or localized melting or solidification, or stress-based attachment, such as by pre-forming the wick and membrane.

[00135] In certain circumstances, the porous hydrophobic membrane can have an average pore size of less than 20 microns, less than 18 microns, less than 16 microns, less than 15 microns, less than 14 microns, less than 13 microns, less than 10 microns, less than 8 microns, less than 6 microns, less than 5 microns, less than 2 microns, less than 1 micron, less than 0.5 microns, less than 0.25 microns, or less than 0.1 microns.

[00136] In certain circumstances, the porous hydrophobic membrane can have an average pore size of greater than 10 nanometers, greater than 50 nanometers, greater than 100 nanometers, greater than 200 nanometers, or greater than 500 nanometers.

[00137] In certain circumstances, the porous hydrophobic membrane can have a thickness of about 0.1 microns, 0.25 microns, 0.5 microns, 1.0 microns, 1.5 microns, 2.0 microns, 2.5 microns, 3.0 microns, 3.5 microns, 4.0 microns, 4.5 microns, 5.0 microns, 6.0 microns, 7.0 microns, 8.0 microns, 9.0 microns, or 10 microns. In certain embodiments, the porous hydrophobic membrane can have a thickness of 10 mm or less. [00138] In certain circumstances, the porous hydrophobic membrane can include an organic polymer or an inorganic material. For example, the organic polymer can be a fluorinated polyolefin or a polyolefin, such as polyethylene, polystyrene, polypropylene (PP), polytetrafluoroethylene, poly (vinyli dene fluoride) (PVDF), poly(vinylidene fluoride)-co- hexafluoropropylene (PVDF-HFP), or sulfonated polytetrafluoroethylene. The inorganic material can be a rare earth oxide, silicon oxide, or silicon nitride.

[00139] In certain circumstances, the porous hydrophobic membrane can further include a hydrophobic coating. The hydrophobic coating can be a fluorinated alkyl film former, for example, a perfluorinated alkyl silicon alkoxide.

[00140] In certain circumstances, the organic polymer can be an electrospun fiber. The fiber can be a nanofiber or a microfiber. For example, an electrospun fibrous membrane can be deposited on a wick at the time of formation. In certain circumstances, the wick can receive a solvent cleaning treatment or plasma cleaning treatment, or both, to enhance the attachment between the electrospun fibers and to the wick.

[00141] In certain circumstances, the electrospun fiber can have an average diameter of between 0.5 microns and 4 microns. For example, the electrospun fiber can have an average diameter of about 0.1 microns, 0.2 microns, 0.4 microns, 0.6 microns, 0.8 microns, 1.0 microns, 1.2 microns, 1.3 microns, 1.4 microns, 1.5 microns, 1.6 microns, 1.7 microns,

1.8 microns, 1.9 microns, or 2.0 microns.

[00142] In certain circumstances, the thermally conductive porous wick can include microchannels. The microchannels can have a width of about 0.5 microns, 0.6 microns, 0.7 microns, 0.8 microns, 0.9 microns, 1.0 microns, 1.2 microns, 1.3 microns, 1.4 microns, 1.5 microns, 1.6 microns, 1.7 microns, 1.8 microns, 1.9 microns, or 2.0 microns. The microchannels can be spaced by about 1.00 micron, 1.25 microns, 1.50 microns, 1.75 microns, 2.00 microns, 2.25 microns, 2.50 microns, 2.75 microns, 3.00 microns, 3.5 microns, 4.0 microns, 4.5 microns, 5.0 microns, 5.5 microns, 6.0 microns, 6.5 microns, 7.0 microns, 7.5 microns, 8.0 microns, 8.5 microns, 9.0 microns, 9.5 microns, or 10.0 microns. In other circumstances, spacing between microchannels can be about 0.2 cm, 0.4 cm, 0.6 cm, 0.8 cm, 1.0 cm, 1.2 cm, 1.4 cm, 1.6 cm, 1.8 cm, 2.0 cm, 2.2 cm, or 2.4 cm.

[00143] In certain circumstances, wicking lengths can be about half the circumference of the condenser tubes.

[00144] In certain circumstances, the microchannels can be arranged in rows or bands. The rows or bands can have a spacing between the rows or bands of between 0.25 cm and 10 cm. A drainage gap can be formed between each row or band. The row or band can be secured to the condenser surface by shrinkage, elastic behavior, by stitching, by physical clamp, such as using a shrink wrap screen. For example, the rows or bands can have a spacing between the rows or bands of about 0.25 cm, 0.50 cm, 0.75 cm, 1.00 cm, 1.25 cm, 1.50 cm, 1.75 cm, 2.00 cm, 2.25 cm, 2.50 cm , 2.75 cm, 3.00 cm, 3.5 cm, 4.0 cm, 4.5 cm, 5.0 cm, 5.5 cm, 6.0 cm, 6.5 cm, 7.0 cm, 7.5 cm, 8.0 cm, 8.5 cm, 9.0 cm, 9.5 cm, or 10.0 cm. [00145] In certain circumstances, the rows or bands can be arranged substantially perpendicular to a lengthwise axis of the condenser surface. In certain circumstances, the rows or bands can be at an angle relative to the perpendicular, for example, 5, 10, 15, 20, 25, 30, 35, 40, or 45 degrees from perpendicular.

[00146] In certain circumstances, the porous hydrophobic membrane can include one or more drain ports along a length of the membrane. For example, the drain ports can be adjacent to each of the rows or bands, if present. The drain ports can be spaced apart by about 0.25 cm, 0.50 cm, 0.75 cm, 1.00 cm, 1.25 cm, 1.50 cm, 1.75 cm, 2.00 cm, 2.25 cm, 2.50 cm, 2.75 cm, 3.00 cm, 3.5 cm, 4.0 cm, 4.5 cm, 5.0 cm, 5.5 cm, 6.0 cm, 6.5 cm, 7.0 cm, 7.5 cm,

8.0 cm, 8.5 cm, 9.0 cm, 9.5 cm, or 10.0 cm in a row along a bottom edge of the surface. [00147] In certain circumstances, the method can include securing the capillary-driven condensation surface to the surface of the condenser element. Securing can be achieved by physical attachment, such as clamps or ties, thermal attachment, such as by diffusion bonding, using adhesive, or localized melting or solidification, or stress-based attachment, such as by pre-forming the wick and membrane on the substrate.

[00148] Without being bound to any particular theory, there can be vapor transport in membrane pores during operation of a CDC surface. Vapor transport in membrane pores can be modelled. As described herein, CDC operates better when the vapor transport resistance is minimized or eliminated, such as with thinner membranes or with liquid (rather than vapor) transport in the pores. Given that fluid is expected to condense inside of the wicking structure during the initial nucleation stage, vapor transport is expected to occur in the membrane pores. Furthermore, vapor transport is expected when the meniscus of condensate is pinned at the bottom of the membrane pore at the wick-membrane interface. Therefore, it is important to understand and model the physics of vapor transport in the membrane. Nevertheless, the type of transport in the pore can be a function of the wettability of the membrane and its geometry. Here, cylindrical geometry for the membrane pores can be assumed as it simplifies the model. Moreover, the geometry can be made with the fabrication method. Therefore, it is possible to both model and fabricate with the same geometry. The model is developed below along with discussion on the effect of saturated vapor operating conditions on the modelling approach.

[00149] To begin, the operating conditions of a power plant condenser, which are established by the saturation temperature and pressure of the steam, determine the type of flow that will occur through membrane pores. The vapor can flow in two major regimes.

First, the vapor molecules can travel through a pore with minimal collisions between molecules, and most collisions with the membrane pore walls. This type of flow is termed free-molecule flow or Knudsen flow. In the other regime, molecule-molecule collisions dominate over wall-molecule collisions. This is called viscous flow, and the equations for this type of flow are well-developed. There is a regime called the transition regime which considers molecule behaviour from both the Knudsen and the Poiseuille flow regime. To determine which regime is applicable, the mean free path is compared against the pore diameter. The mean free path is the average distance that a molecule travels before it collides with another molecule. The mean-free path is given by the following relation. where k _B is the Boltzmann constant (1.3806485X10 ²³ J/K), T is the absolute temperature, P the average pressure within the membrane pores, and d _c is the collision diameter for water molecules (2.641 xlO ¹⁰ m). The ratio between the mean free path and the pore diameter is called the Knudsen number, and it gives a measure of the flow regime, K _n = /d _p . If K _n > 10 there is free-molecule or Knudsen flow. If K _n < 0.01, Poiseuille flow ensues. If conditions are in between, 0.01 < K _n < 10, a transition regime is applicable where descriptions from both types of flow should be included.

[00150] Many models have been developed to describe fluid flow through porous media. Among them is the well-known dusty-gas model. This model can capture free- molecule flow, viscous flow, and continuum diffusion as part of its development. The system consists of a single, pure vapor of water molecules. Equations for this type of flow simplify significantly for this case.

[00151] In this analysis, a steam saturation temperature of 35 °C was assumed which is not uncommon near the condenser in steam power plants. The resultant mean free path is around 2.4 pm. Thus, different flow types can occur in membrane pores if the pore size is much larger, or much smaller, than 2.4 mhi. In this analysis, pore diameters up to 30 pm were considered. For this operating condition, pore diameters below 250 nm can be said to be in the Knudsen flow regime and those above in the transition regime. For the transition regime, both the Knudsen and the Poiseuille flow contribution were considered. FIGS. 29A-29C show plots of the mean permeability through a pore as a function of pore diameter for the case of pure steam. The operating saturation temperature of steam in power plant condensers can vary with outside-plant environmental conditions from about ~ 40° C to ~ 60° C. See, for example, Berman, A. S. Free molecule transmission probabilities. J. Appl. Phys. (1965), which is incorporated by reference in its entirety. FIG. 29A shows a plot of the mean permeability as a function of pore diameter in the Knudsen regime, where the permeability is much smaller and only free-molecule flow is important. As can be seen, the permeability is much lower than the other two regimes. In contrast, FIG. 29C shows the Poiseuille regime and much larger permeability values, which are desirable, but require larger pore diameters, which are undesirable. FIG. 29B shows the transition regime, where it is necessary to account for both free-molecule and viscous effects through the pore. It is seen that the contribution of free-molecule flow becomes less significant for larger pore diameters, and eventually Poiseuille type flow dominates. FIG. 30 shows a close-up of the transition regime for diameters from 0.1 - 15 pm, which is in the range of interest for CDC. It is seen that in this range, for the given operating condition, it is important to include the effects of both free-molecule and viscous flow. Neglecting the free-molecule flow contribution underestimates the vapor permeability by around a factor of ~2x. Thus, utilizing the Dusty - Gas model is appropriate versus utilizing Hagen-Poiseuille flow. The permeabilities in the Knudsen region are orders of magnitude lower than those in the transition and Poiseuille region. Thus, in that regime the driving force for condensation seen in the total temperature from membrane surface to condenser must be much larger to increase the flow. Alternatively, the transition regime is orders of magnitude higher. However, as the pore diameter increases the available maximum local capillary pressure decreases which is undesirable.

[00152] When condensation occurs, a low-pressure region is created and flow is directed toward the condensing surface. The pure vapor pressure-driven flux in the transition region can be described by ( 2 ) where R is the gas constant (8.3144598 J/mol/K), T is the absolute temperature, Ap is the pressure drop across the membrane, t _m is the thickness of the membrane, and K is the mean flow permeability for a pure gas given by, where D _iK is the Knudsen diffusion coefficient, p is the average pressure in the pore, B ₀ is the viscous flow parameter, and Pa is the viscosity coefficient for pure vapor. The Knudsen diffusion coefficient can be shown to be proportional to the mean molecular speed from the kinetic theory of gasses, where K ₀ is the Knudsen flow parameter or permeability coefficient in free molecule or Knudsen flow regime, and v is the mean molecular speed and is given by,

[00153] It is worth noting the transport coefficients and proportionality constants depend on the structure of the porous medium. For the simple case of a porous membrane consisting of cylindrical pores of length t _m and diameter d _p the Knudsen and Poiseuille flow parameters are simply related as K ₀ — d _p / 4 and B ₀ — d _p /32 respectively. Thus, the mean flow permeability becomes, where t _m is the membrane thickness, R is the gas constant, T is the absolute temperature, is the average pressure in the pore, M is the molar mass, Pi is the liquid viscosity, and d _p is the pore diameter. The first term inside the brackets is the Knudsen mean permeability, which is directly proportional to the pore radius. The second term is the viscous or Poiseuille flow permeability, which is proportional to the pore diameter squared. As such, with increasing pore radius, the permeability increases faster in the Poiseuille regime than in the Knudsen regime. Equation 6 above is for the transition region, and through the dusty-gas model simple additivity of both effects is implemented to account for both, since the contribution for higher order non linear logarithmic terms is assumed to be small. Usually, it is up to individual experiments to determine how good this approximation is, though previous work is stated to have shown excellent agreement. Critiques of this assumption can be found in By on, C. & Kim, S. J. The effect of meniscus on the permeability of micro-post arrays. J. Micromechanics Microengineering 21, (2011), which is incorporated by reference in its entirety. In addition, the quantities inside the pore are approximated, such as the temperature and average pressure.

[00154] The flux through a pore can now be calculated by multiplying the mean permeability through a pore by the pressure across the membrane

J — KAp ( 7 ) where P _vap is the vapor pressure, and P _sat (T _waU + AT _wick ) is the liquid saturation pressure at the wick-membrane interface, here used as an approximation to the saturation pressure right outside the liquid-vapor interface at the bottom of the pore. The units of equation 6 are moles/m ² /s. To convert this to a mass flow rate, one can multiply by the molar mass M in kg/mol. This quantity refers to the mass flux in the area of one pore, so one can multiply by the total frontal surface area A _T and the membrane porosity f _pi to obtain the total mass flow rate through the membrane. The mass flow rate through the pore is then given by, rh _c = K pMA _T( f> _m ( 9 )

[00155] Assuming a square shape to the frontal condensing surface of length L, A _T = L ² , and if L/4t _wick » 1, or if the wick sides are covered by an impermeable material, then condensation on the sides of the wick can be neglected. Maximizing the membrane mass flux is crucial. Since the membrane determines both the maximum possible driving capillary pressure available, and the mass flow capacity through the frontal area, which are both limited by the pore diameter, optimizing the pore diameter and maximizing the porosity is crucial.

[00156] An enthalpy balance yields the following equation, accounting for condensation in each pore through the frontal surface area, q"A _T( l _m — m _c hf _g ( 10 )

[00157] Heat flux through the wick — assuming this is constant through the frontal area of the wick, which is not necessarily true since the membrane is on the frontal surface and the open pore area transfers heat by phase- change but the solid area is assumed to insulate — is given by

[00158] Here, k _e ff is the effective thermal conductivity of the wick, and AT _wick is the temperature drop across the wick. Finally, one can describe flow through the structured wick by first utilizing Darcy’s law, where k is the structured wick permeability, Pi is the liquid dynamic viscosity, p _t is the liquid density, g the gravitational constant, and the additional driving pressure gradient from capillarity.

[00159] Adding a membrane atop the wi eking structure can promote additional fluid driving pressure by capillarity as well as a vapor transport resistance through the thickness of the membrane. With careful design of the membrane, one can find a trade-off where it may be optimized to maximize the additional capillary driving pressure, while minimizing resistance to flow. The additional capillary-driving pressure produced can be approximated by the following expression for cylindrical pores, which represents the maximum capillary pressure that can be sustained by a pinned meniscus, or the burst pressure to enter a pore, where P _ca p,max is the pore-entry burst pressure, P ₀ is the pressure of the vapor above the interface, P _L is the pressure of the liquid just below the interface, s is the liquid surface tension, CA is the contact angle of the hydrophobic membrane, and d _p is the radius of curvature of the liquid-vapor interface emanating from the pore, assumed to be the pore diameter at the burst pressure. A review of key variables is shown below in Table 1. Table 1

Variables Description Units

J Mass Flux moles/s/m ²

K Mean pore permeability moles/s/m ² /Pa

Ap Pressure drop across Pa membrane t

R Heat flux W/m ² tyvick Wick thickness m tra Membrane thickness m d _p Membrane pore diameter m k _e ff Structured wick effective W/m/K thermal conductivity

K Structured wick intrinsic permeability

[00160] Equations 1 through 13 form a complete picture of the preliminary model which considers vapor flow in the transition regime. This model describes the expected behaviour from CDC surfaces when condensation happens with vapor transport in the pores. Despite of the type of transport in the pores, whether liquid or vapor, the transport in the wicking structure of the model continues to apply. To describe liquid transport in the pores, the fluid mechanics in the pore can be coupled with the Schrage equation (or the moment- method) for condensation. The overall trends remain similar. Reasonable values for multiple variables can be prescribed in the model and sweep others to understand trade-offs in design and the expected heat transfer during condensation. The goal was to obtain the maximum possible heat transfer performance while avoiding surface flooding, which degrades heat transfer. Based on modelling, it was found that wicking structures can have high permeability as well as effective thermal conductivity. However, it was found that thermal conductivity becomes less important for higher permeabilities. Thus, designing for higher permeabilities becomes a priority. The model highlights bounding values for the design. Moreover, to optimize heat transfer performance, pore diameters can be below 20pm. Smaller optimized pore diameters yield higher capillary pressure and heat transfer performance. Note that the hydrophobic membrane was here assumed to be a cylindrical through-pore membrane, but this geometry is not necessarily required for operation.

[00161] The preliminary model showed us that it was advantageous to design for a thin wicking structure with high thermal conductivity and high porosity, utilize the thinnest membrane possible with the smallest pores that could sustain the liquid thin film below designed level and avoid flooding at a given operating conditions. For brevity, those results are not presented here, but rather a finite element heat transfer coefficient model is presented below. This model is built utilizing geometry which is directly related to the structure fabricated in the examples described below, one characterized by highly-defined geometry. [00162] The previous section introduced an analytical model which was helpful to get initial estimates of the characteristic length scales required for the hierarchical structured CDC surface and their expected performance. Moreover, that model revealed general trends on how such surfaces operate. The geometry of either the wick or the membrane determine the magnitude of their associated heat and mass transfer resistances, which can be independently designed and optimized to enhance heat transfer. In this section, a model is utilized to rationally design the CDC surfaces utilizing highly defined geometry. This will allows for the tailoring of designs to attain the highest HTC possible at a given operating condition, while simultaneously avoiding top-surface flooding. An explanation of how flooding occurs on these surfaces in the visualization studies is described below. COMSOL is utilized to calculate both the permeability and the heat transfer coefficient (HTC) for micro- pillared wicks as a function of the geometry of the hydrophobic membrane and the wick. Later versions of this model utilize analytical permeability results for micropillar wicks from Byon and Kim (cited above). The results from the HTC model are combined with analytical porous media models, and mechanical stability calculations, to find the flooding criteria, and mechanically stable surfaces, respectively. From the results, designs can be developed. [00163] For the HTC, a conduction model was utilized neglecting convection effects due to a small Jakob and Peclet numbers in the condensate flow direction of the wick. The HTC is calculated on a micro-pillar unit cell (FIGS. 31 A-31B) and later by a quarter-pillar unit cell described by the diameter d, pitch l. and the diameter to pitch ratio d/l, and the pillar height h, which is the wick thickness. The condensation heat flux at the surface of the unit cell can be made a function of vapor transport through a cylindrical pore and scale this over the frontal condensation surface area using a prescribed membrane porosity. Other variations of the present model (not shown here for brevity) also calculate the HTC with the condensate meniscus pinned at the top of the membrane, and liquid transport in the pores. [00164] For vapor transport through the membrane pores, the dusty-gas model from the previous section can be used and the previous equation updated. Previously, the equation was limited to large ratios of the thickness of the membrane t _m to the pore radius r _p (or very thick membranes). In this case, the transmission probability — the probability that a molecule entering the pore from the top surface will make it all the way across the membrane — simplified to h * 2.67(r _p /t _m ). However, as the membrane thickness decreases and the ratio is close to unity or a little less, this assumption is no longer valid. Thus, the transmission probability was modified after the equations provided by Berman, A. S. Free molecule transmission probabilities. J. Appl. Phys. (1965), which is incorporated by reference in its entirety. First, a reduced length, L = t _m /r _p , is introduced for a cylindrical pore. Then, the transmission probability across the pore is approximated by equation 14 below. This equation is then utilized in the dusty-gas model for vapor transport in the transition regime.

((8- ^Z )VL ² +4+L ³ -16) ²

- - - ( 14 ) 72LV ² +4— 288 ln(L+VL ² +4)+288 ln 2

[00165] Next, the HTC was calculated, viscous pressure-drop across the structure for a given subcool was calculated, and the stressed induced on the membrane to predict possible membrane failure was estimated. FIGS. 32A-32B show the effect of wick size on the HTC taking flooding into account. The wick sizes represented include pillar wick diameters from 3 pm to 30 pm, and a sweep of d/l ratios from 0.05 to 0.98. The pillar height is 30 pm. The subcool in these plots was 2 K and so the heat transfer coefficient (HTC) was around ~ 20 kW/m ² , whereas the HTC enhancement was from 100kW/m ² to -220 kW/m ² in this plot, which represents a ~5x - 1 lx enhancement in the HTC. Over this wide range, in both designs the HTC is higher for smaller wick geometries. The HTC also varies as a function of d/l ratio. At low d/l ratios, the HTC is smaller due to a lower solid fraction. As the solid fraction increases, where f = 0.25 n(d/l) ² , the HTC goes up until it reaches a maximum. At higher d/l ratios, the HTC begins to decrease due to less membrane area being available for condensation due to a larger wick solid fraction and the fact that pillar tops are insulated in the model as a conservative estimate (in practice, droplets can nucleate on the top surface of pillars). Moreover, each data point is checked for flooding. Flooding occurs when the viscous pressure drop in the wick exceeds the highest capillary pressure in pinned menisci on membrane pores. Flooded data points are plotted with black circle markers to show they are off limits. Flooded points occur at high solid fractions, where the permeability is low. FIG. 32A represents a structure with a pore diameter of by 1 pm, and FIG. 32B a structure with a pore diameter of 3 pm. The smaller pore diameter produces a larger capillary pressure to drive fluid through the wick. Notably, for the small pore diameter, more points are accessible along the curves before flooding occurs. However, the smaller pore diameter poses a larger vapor transport resistance, and thus the HTC is generally lower (compare the y-axis of FIGS. 32A and 32B). However, the combined effect is such that a similar performance is expected from both designs. The system can operate in a regime where no vapor transport is expected, and in fact find that vapor transport in the ports is not an ideal configuration as droplet absorption is the mechanism by which the surface stays dry. Nevertheless, the present results are directly relevant to the case with no vapor transport, as the general trends are upheld and are shifted by a value. As shown in the visualization experiments, it is possible to add to the model the heat transfer from droplets that grow on pillar tops and are absorbed with some frequency t. Here, a conservative estimate can be taken and this value neglected, since the absorption frequency at the given operating configuration in a pure vapor environment is not known. The results from FIGS. 32A-32B show that multiple designs which can attain to the desired performance may be selected. Thus, a design with the simplest geometry among those with similar performance may ultimately be selected. The black dashed line in both plots represents the expected performance with dropwise condensation at the same vapor temperature and subcool calculated using a simple model by Rose, J. W. Some aspects of condensation heat transfer theory. Int. Commun. Heat Mass Transf. 15, 449-473 (1988), which is incorporated by reference in its entirety. The wall and vapor temperatures were chosen close to commercial operating conditions of condensers as 30°C and 32°C, respectively. Based on the initial model, this lower subcool is a reasonable value to choose since the surfaces are expected to yield a lower the subcool at the same operating conditions than filmwise condensation. This value also allows us to determine the expected HTC for this particular operating configuration with vapor transport in the pores. As discussed below, this configuration — with vapor transport in the pores due to a completely hydrophobic membrane — actually yields many parasitic top-surface droplets. However, visualization studies were conducted to determine the physics and find a membrane configuration which avoids such droplets. [00166] FIGS. 33A-33B show the effect of micro-pillar wick solid fraction and membrane pore diameter on the HTC. The general trend shows that larger pore diameters can produce larger HTC’s due to reduced vapor transport and a higher heat flux at a fixed subcool. However, considering flooding, many of the points which produced enhancement became flooded at larger wick-solid fractions since the capillary pressure in larger pores could not sustain the viscous pressure drop across the wick ( L _w = 1.27 cm). The net result is that larger pore diameters resulted in more flooded points, and the corresponding available designs for a given pore diameter were only accessible at smaller HTC enhancements. In contrast, smaller pores have less flooded points due to higher capillary driving pressures, and designs at higher wick-solid fractions — which have higher HTCs — are accessible. The fabrication capabilities allow pore diameters no smaller than 1 pm in diameter, since this is the resolution limit of the UV and laser exposure tool in the photolithography step.

[00167] FIGS. 34A-34B show the effect of membrane thickness on the HTC. Thinner membranes mean shorter pores and reduced vapor transport resistance, which results in higher possible HTCs. However, as the membrane gets thinner, it can become too weak to sustain the local capillary pressure generated in the pores of the membrane. Here, the mechanical strength of the membrane was considered. The membrane can be made from deposited materials such as SiC or SiN _x on Si wafers. Tsuchiya, Inoue, and Sakata measured the tensile strength and fracture toughness of plasma enhanced chemical vapor deposited (PECVD) SiCh thin films. See, Tsuchiya, T., et al. Sensors Actuators, A Phys. 82, 286-290 (2000), which is incorporated by reference in its entirety. For 650nm thick S1O2 thin films, they found the mean tensile strength to be 1.2- 1.9 GPa in vacuum and 0.6- 1.0 GPa in air, while the mean fracture toughness is 1.3-2.0 MPa-m ^1/2 in vacuum and 0.6-0.9 MPa-m ^1/2 in air. Similarly, Yoshioka, Ando, Shikida and Sato measured the fracture strain and the Young’s modulus of thermally oxidized S1O2 on single crystal silicon and low-pressure chemical vapor deposited (LPCVD) S13N4 thin films. See, Yoshioka, T., et al., Sensors Actuators, A Phys. 82, 291-296 (2000), which is incorporated by reference in its entirety. They measured Young’s Moduli of 74 GPa and 370GPa and fracture strains of 2.5% and 3.8% for S1O2 and S13N4, respectively. The fracture strength can be estimated from these values as Of — Eef, where O is the stress at fracture, E is the Young’s Modulus, and €f is the strain at fracture. Thus, the stress at fracture for S1O2 and S13N4 can be estimated as 1.85 GPa and 14.1 GPa, respectively. As seen, the experimentally measured tensile strength values for these two materials vary significantly. In the fabrication, a low pressure chemical vapor deposited (LPCVD) low-stress SiN _x thin-film (membrane) with a thickness of - 500 nm was utilized. A simple analytical model was set up to compare the lower values of the tensile strength to an estimated bending stress assuming a rectangular beam geometry with the beam base representing the pillar diameter and the beam thickness the membrane thickness. Assuming a beam pinned at both ends, the maximum bending moment at the center of the beam is given by equation 15,

O _L L*

M _c ( 15 ) 384 El where s _; is the distributed load, L is the longest length connecting two pillars in a unit cell, E is the Young’s Modulus, and / is the moment of inertia. The moment of inertia for the assumed geometry is given by equation 16,

( 16 ) where d is the pillar diameter representing the beam width, and t _m is the membrane thickness. The maximum bending stress at the center is then given by equation 17. [00168] The distributed load is overestimated and assumed to be the highest local

Laplace pressure generated in a superhydrophobic pore which is 4 a/d _p . In reality, the pore is not superhydrophobic and this value is scaled down due to a lower contact angle. The smallest pore in this study (1 pm) yields a local pressure of 0.28 MPa for the overestimated value. From this load one can calculate the bending stress for different unit cell geometries and compare it to the available measurements of the tensile strength for the materials and geometries utilized. A quick calculation utilizing a wick geometry with d = 7pm, d/l = 0.5, and membrane made of S13N4 with geometry with d _p — 1pm and t _m — 8pm, reveals that the maximum bending stress is 3.79 GPa, which is close to the 14.1 GPa tensile strength estimated from Yoshioka, T., et ak, Sensors Actuators, A Phys. 82, 291-296 (2000), which is incorporated by reference in its entirety. Smaller thicknesses for the membrane utilizing this model yield higher bending stresses. However, values for the principal stresses which are representative of the geometry here modelled are required to obtain better estimates. Finite element simulations of the first principle stress were computed using the fabricated geometry with COMSOL Multiphysics. The results show that that the first principle stress is much lower (~10 x) than the expected tensile strength for a 500 nm membrane. Therefore, the membrane was expected to uphold the required capillary pressures from the constrained fluid layer in the wick. Despite this result, each designed geometry should be checked for failure, especially as the membrane thickness decreases and the pillar pitch increases. Nevertheless, the main failure mechanism can be pinpointed. Notably, the largest bending moment varies as ~ L ⁴ which is a strong dependence on the pillar spacing. Thus, to reduce the bending stress and achieve thinner membranes, one needs to consider smaller pillar to pillar spacing.

[00169] FIGS. 35A-35B show the effect of wicking length L on the HTC and available designs. For the same design, increasing the wicking length has the effect of reducing the number of available design geometries due to the increase of viscous pressure drop with length, which yields flooding of the structure. This means that the capillary pressure in the pore cannot sustain the viscous pressure drop, and liquid is ejected from the front surface of the membrane. One way to combat this is to utilize smaller pores. However, smaller geometries than the present utilized are significantly more challenging to fabricate. Notwithstanding, significant heat transfer enhancement utilizing the present geometries which are designed for wicking lengths on the order of half the circumference of ½” condenser tubes can be expected. Moreover, thicker wicks can be made to reduce the viscous pressure drop and maintain the membrane size, all while still maintaining HTC enhancement. The exit port can also be engineered to reduce the wicking length and increase the HTC. Engineering the location and number of exit ports also allows us to introduce new design geometries not accessible. Even for larger wicking lengths (FIG. 35B), there are still available designs close to the level of dropwise condensation that will not flood. This means that significant heat transfer enhancement can still be achieved even with a wicking length of 1”. As already mentioned, the wicking lengths modelled — 1” and ½” inch — are industrially relevant length scales for condensers in steam power plants.

[00170] The modelling in this section helps us understand the relationship between the geometry in the wicking structure and that of the membrane. Specifically, the model shows how the geometries affect the HTC, flooding criteria, and mechanical stability of the membrane. The next section details the fabrication for the structures which were rationally selected utilizing a model with the framework shown in this section.

[00171] An approach to fabrication for the capillary-driven condensation structure is shown in FIGS. 36A-36F. A double-side polished silicon wafer (150 mm diameter, ~ 675 pm thick, <001> orientation) was utilized and a 500 nm layer of low-stress silicon nitride was coated utilizing LPCVD (FIG. 36A). Next, photolithography was utilized to pattern a pillar- pore array on the silicon nitride coated side of the wafer (FIG. 36B and FIG. 36C). In FIG. 36D, deep reactive ion etching (DRIE) is utilized to etch the pores in the nitride layer and form the foundation of the wicking structure. In FIG. 36E, an isotropic etch is utilized to remove excess silicon between the pore pattern in the silicon to create the micropillar wicking structure, while maintaining the pore geometry of the silicon nitride membrane. Finally, the remaining photoresist is removed utilizing plasma etching. A self-assembled monolayer (SAM) coating is then utilized to render the surface hydrophobic. Various ways of coating the membrane were attempted and a method detailed in a latter section was used for the experiments. This condenser surface facilitated demonstration of the first successful proof-of-concept on silicon. FIG. 37 shows a 6” wafer coated with ~ 500 nm of low stress silicon nitride which was utilized for one round of fabrication.

[00172] Initial fabrications were done at a millimeter scale sample in order to test the feasibility of the proposed fabrication approach. It was found that it was possible to generally achieve the desired structure and thus showed the feasibility of the approach. The next phase was scaling up the design to the centimetre scale to demonstrate the fabrication at industrially relevant length scales. In this attempt, it was difficult to achieve a large-scale printing of the pattern onto the photoresist utilizing direct-write lithography due to the large size of the drawing. The design was ordered in a photomask from a company for used with a UV light exposure tool. However, this approach did not yield good results due to spatial variations in the exposure dose at these critical dimensions (~ 1 pm feature size). In this section, the MLA-150 direct write lithography tool was used and a feature workaround was utilized to print a large drawing successfully. Previously, it was not possible to print large patterns due to the high density of the drawing caused by the many circles that needed to be converted into the format of the tool. However, a feature on the tool was found that could be used as a workaround. Briefly, a smaller millimetre-sized unit cell of the structure was copied in space and “stitched” together to achieve centimetre scale structures. Smaller drawings could successfully be converted to the tool’s format, whereas larger drawings could not. Thus, a pattern “stitching” approach was utilized to fabricate the centimetre-scale sample.

Fabricating one design at a time then became a feasible approach rather than printing multiple designs on large areas of a 6-inch wafer using either direct-write lithography or a photomask with UV exposure. [00173] FIGS. 38A-38B show scanning electron microscope (SEM) images of the first successful large-scale sample after fabrication. As shown, the surface is uniform and well defined at a low magnification (-1000 x). Note that imaging at even lower magnifications is not ideal since the pattern is not discernible. Nonetheless, the inset at the top left comer shows the successfully fabricated sample at large scale. The characteristic length scale is - 1” inch. Note that diffraction from the sample surface looks visibly uniform indicating structure uniformity (and an open wicking layer) across the entire sample. The lower right inset shows a unit cell image with well-defined and open pores. FIG. 38B shows a cross-section SEM image of the sample. The magnification was sufficiently high to resolve that the membrane hangs on top of the micropillar structure and contains an open space at large-scale. The inset on the lower right shows a close-up of the successful structure. The main components for the hierarchical structure for capillary condensation are achieved, namely, an open wicking structure, and a porous membrane attached on top. Thus, the structure is viable to test during condensation. However, it was found that the bottom of the wicking layer contains undesirable residue material which reduces the permeability of the wick. Therefore, it was important to develop a process to achieve opening the wicking structure completely to more closely match the design. Notwithstanding, such a structure can be utilized for test and study. However, incomplete isotropic etching has been addressed by developing an appropriate recipe in the tool. The samples tested for heat transfer measurements are without this residue layer.

[00174] In order to achieve capillary-driven condensation with the fabricated hydrophilic structure, simple and effective methods to coat the membrane as hydrophobic for condensation tests were investigated. For the greatest simplicity, simply coating the entire sample hydrophobic in a one step method can be used. After all, nucleation is expected to occur everywhere on a subcooled surface. Therefore, it was believed that nucleation would occur inside of the wicking structure, and that nucleates would grow in the structure and connect as a continuous condensate film. However, this was not the case. Although it is the easiest coating method, this configuration turns out to be neither advantageous nor desired. Thus, instead of coating the entire structure hydrophobic, a method was developed to coat only the top surface of the membrane as hydrophobic. This way, a bi-philic structure would be formed where part of the membrane was hydrophobic while the rest of the structure remained hydrophilic. In this configuration, the wicking layer can fill with condensate. [00175] To selectively coat the structure, areas that should not be coated hydrophobic were protected. FIGS. 39A-39C show a process developed for protecting such regions. First, a long slender sample was tested the capillary rise height for various viscosities of photoresist. The SPR-220 photoresist series was tried and wicked in 220-1, 220-3, and 220-7. It was found that the capillary rise height was not sufficient in either of the SPR-220 series photoresists for these purposes (at least ½” inch rise was required). Therefore, the viscosity of the photoresist was decreased by diluting it with isopropanol. FIG. 39A shows that the capillary rise height utilizing a 1:1: ratio of SPR220-1 to isopropanol meets the criteria of a ½” inch rise. Therefore, the strategy was to wick in fluid from two extremes of the sample so that an entire 1” by 1” sample can be protected (FIG. 39B). A post- wi eking bake of the sample at 115 °C for 90 seconds was performed to remove the isopropanol from the structure and leave behind a photoresist layer (FIG. 39C). Notably, the top surface of the baked sample does not look uniform, but rather has lines indicating areas with more photoresist coverage. SEM imaging was performed of the membrane, the bottom of the wick, and the micropillars to find out how the structure was coated with photoresist. FIG. 39D shows that the photoresist is able to wick into the membrane pores up to the top surface of the membrane. FIGS. 39E-39F show that photoresist covers the bottom of the wick and the sidewalls of the pillars with photoresist. To ensure that the top surface of the membrane is not covered with photoresist, an oxygen plasma etch was performed for 20 minutes before applying the hydrophobic coating. Notably, the wick space is not completely filled with photoresist, but all sidewalls inside the structure are covered with a layer of photoresist. This is because the wicked-in solution evaporated in large part, leaving behind a photoresist layer on all walls of the structure. FIG. 39G shows a schematic of the structure after coating. Notably, only the topmost membrane-surface is coated hydrophobic. (Heptadecafluoro-1,1,2,2- tetrahydrodecyl) trimethoxy silane (FAS) was utilized to form a hydrophobic coating, which has shown longer durability on silicon based flat surfaces.

[00176] A horizontal test rig was built on an optical microscope and perform visualization studies of condensation in atmospheric-air conditions at low condensation intensities. The purpose was to study the interaction of condensate with the biphilic surfaces. FIG. 40 shows this setup, whose main components include a chiller loop, a cold stage, a vapor source, and an optical microscope with a CCD camera. The chiller loop consists of a mixture of ice and deionized water that is pumped to the cold stage with a peristaltic pump. The cold stage is a copper plate with welded copper tubing for chilled water flow. A Nikon Eclipse LV100 optical microscope is used to image condensation. In order to increase the intensity of condensation, a water bath was prepared with deionized water which was heated before the onset of boiling to provide enough vapor at around ~ 70° C. The cold stage was maintained at a temperature of ~ 0° C. A dry compressed air line was inserted into this bath to generate bubbles and act as a carrier gas for the vapor. A PixeLINK PL-B742U CCD camera was utilized to record all images and videos. The condensation in this setup is horizontal, meaning that the direction of gravity points towards the front surface of the sample. As a result of this horizontal setup, any droplets on the surface will not shed. The structure visualized during condensation has a pillar diameter of ~15 pm, a pillar pitch of —150 pm, and a membrane pore size of ~ 1 pm.

[00177] Many visualization studies were carried out with various coating techniques. Below, results utilizing samples coated with the FAS coating recipe are shown. To begin the visualization study, the structures were coated in two ways. First, and for ease, the first large- area samples were coated to be entirely hydrophobic. This means that both the membrane as well as the wick were fully hydrophobic (FIG. 41 A). Typically, such a configuration is discouraged since it ruins the water-wicking capabilities of the structure. However, given that the structure does not wick in fluid but rather fills the structure with condensate, condensate nucleation was expected to be imminent on hydrophobic surfaces with a sufficiently high subcool and thus naturally fill the wi eking structure. However, FIG. 41 A shows that this was not the case. In fact, it was observed that such a surface does not fill naturally but forms condensate islands in the wick, rather than a continuous film.

[00178] First, it was observed that some nucleation appears on the top surface of the sample (left image). As more condensation occurs, it was observed that nucleation and coalescence of liquid droplets occurs inside the wi eking structure (center image). Here, top- surface droplets were absorbed through the membrane pores via liquid bridging between the meniscus pinned at the pore and nearby droplets that can make contact and bridge with the meniscus. Moreover, some droplets remain on the top surface of the membrane and grow.

The right-side image shows the formation of small condensate islands inside the wick rather than a continuous film. Further, the incipience of flooding was observed. This occurs in the form of ejected or ‘bursting’ of droplets from the top surface of the membrane. To explain this phenomenon, it was noted that the rough hydrophobic wick increases the contact angle of liquid inside the wick, which increases the viscous pressure drop and causes condensate islands to release their pressure by flooding or “bursting” liquid through nearby membrane pores. One down side of these “bursting” droplets is that they strongly pin on the membrane surface. From these observations, one can expect that the heat transfer would be poor since the pinned droplets increase the overall thermal resistance during condensation. Moreover, below the membrane there are large air-filled sections without condensate islands which lower the effective thermal conductivity of the wick. In addition, these regions show no condensate nucleation meaning that heat transfer is not by phase change. Thus, this form of condensation is not advantageous for heat transfer.

[00179] It is worthy to note that naturally filling the wick with a hydrophobic sample may be possible with taller wicking structures that are less prone to “bursting” droplets. However, such a configuration may lower the HTC. Nevertheless, it was concluded that coating the entire structure hydrophobic is not a preferred configuration in capillary-driven condensation with the present sample geometry.

[00180] Having seen these results, methods to selectively coat the structure were sought, such that the micropillar structure at the bottom of the membrane would remain hydrophilic after fabrication and fill naturally during condensation. It was found that the selectively coated surface not only gets rid of top surface droplets on the membrane by continually absorbing them into the wicking structure through the membrane pores, but it also avoids flooding (FIG. 41B). Moreover, it was also found that the new surface can absorb droplets that grow on the top of pillars (FIGS. 42A-42B). The droplet base diameter on pillars may only grow to the diameter of the pillar itself before being absorbed through one or more of the surrounding membrane pores. This surface gives the impression of a “dry” condensing surface when viewed at low magnifications. A completely hydrophobic configuration is not advantageous. In order to naturally fill the wick with condensate, the wicking structure is preferably hydrophilic to naturally fill the wick, which requires that the wick is not hydrophobic. In preferred embodiments, a hydrophobic membrane is needed for best performance.

[00181] Visualization experiments comparing the fully hydrophobic samples with the selectively coated samples reveal superior performance and understanding of the possible operating modes of CDC. FIG. 41B represents the first successful proof of concept of CDC. Whereas it was postulated that there would be vapor transport in the pores in FIG. 3A, the present results suggest that the mechanism for a dry top-surface is likely direct nucleation of vapor on the top-surface-pinned meniscus, as well as droplet absorption into the wicking structure. Thus, there is liquid rather than vapor transport in the pores. Further, the fluid is removed by capillary pumping from the pinned meniscus at the top of the membrane pore. Thus, it was found that CDC may require either i) a meniscus that is pinned at the top surface of the membrane, or ii) a thin enough membrane such that nucleating droplets on the top surface can be absorbed into the structure through liquid-bridging with the pinned meniscus on the pore. Thus, based on these results, naturally filling the wick is necessary to operate the designed sample in the proposed CDC configuration (FIG. 3A), whether the sample is fully or selectively hydrophobic. Further, by coating the membrane hydrophobic on the structure with the geometry shown, it was possible to avoid flooding and the problem of “bursting” droplets and parasitic top surface droplets on the membrane, which decrease the heat transfer performance.

[00182] In this section, successful heat transfer measurements of condensation in both a bare copper surface and the engineered CDC sample attached to the copper block are reported. A saturated vapor environment at ~ 60 °C was created. To measure the heat flux, five thermocouple readings were utilized along the copper rod at known distances. When the profile is linear at steady state, a data point was taken for ~ 10 mins or longer. FIGS. 43 A- 43B shows the configuration of the experiments conducted. Notably, there are 5 thermocouples measuring the flux, and 5 measuring the vapor temperature in three different ways. Specifically, two thermocouples measure the vapor temperature directly (dry-bulb), two more measure the temperature of the saturated liquid pool in front of the sample (wet- bulb — small pool), and one measures the temperature of the big water bath filled with condensate. Note that the small pool temperature was always lower than the dry-bulb and liquid puddle temperatures. A steady state point is considered desirable when the agreement between all of the vapor temperatures is close.

[00183] FIGS. 44A-44B show data which was collected using the condition mentioned above. FIGS. 44A-44B plot the heat transfer coefficient and heat flux in bar plots for the experimental condition. Moreover, a Nusselt model theoretical prediction is given for each bar to indicate that the subcool values were not the same for the filmwise and CDC samples at the same experimental condition. FIG. 44A shows the results for the heat flux. FIG. 44B shows the heat transfer coefficient results. Notably, there is a 238 % enhancement from ~ 18.84 kW/m ² /K to ~ 44.87 kW/m ² /K in the HTC over filmwise. Note that the error in the HTC for this data point changes significantly over the error of the temperature measurement. Moreover, the measurement is sensitive to the thickness of the bonding layer used to mount the sample, in this case silver epoxy. Minimizing this error will yield more robust measurements since at present the accuracy by which one measures the thickness of this bonding layer presently has a large uncertainty. Future measurements will obtain data points at new conditions, aiming to sweep the subcool range from 0.5 to 6 °C, to more clearly understand the changes in the heat flux and HTC for the samples. In addition, two more samples with denser micropillar wicks for which to measure the heat transfer performance and expect higher HTCs were fabricated.

[00184] FIGS. 45A-45D shows that the surface can achieve a “dry” operating condition where droplets are not observed coming out from the front of the surface or nucleating and growing on top of the membrane. Rather, there are droplets coming out of the edge of the sample, as designed. Thus, the physics of the proposed concept are further elucidated and confirmed by explicitly showing condensate exiting though the exit port at the edges of the sample, without flooding of droplets through the front surface.

[00185] In conclusion, the silicon-based CDC surfaces with highly defined geometry during condensation in an air-ambient were modelled, designed, fabricated, functionalized, and tested. Most importantly, measurements of the heat transfer in pure-vapor were taken. It was found that the surfaces can achieve enhancements in the HTC of up to ~ 240 % (in a 60 °C saturated vapor environment) compared to the theoretical filmwise condensation prediction for the sample with a sparse micropillar spacing of 150 pm. Further, higher HTCs were expected for the samples with closer micropillar spacing. Heat transfer can also be studied with smaller micropillar pitch values. Overall, the samples were seen to lower the temperature drop at the surface or the subcool, which indicated they are lowering the thermal resistance. These results highlight the promise of CDC surfaces in achieving high HTCs. In order to show their promise in a more scalable fashion, the next section concerns the development of scalable CDC surfaces utilizing copper and other scalable materials.

[00186] For achieving capillary-driven condensation on a large scale, i.e., for application in power plant condensers, the vision is to manufacture a hierarchical condenser tube as shown in FIG. 2A. A robust (ideally intrinsically) hydrophobic membrane will be combined with a porous, high-thermal-conductivity metal wick and wrapped around the external surface of the condenser tube. During the condensation process, water vapor will transport through the membrane pores and condense inside the porous metal wick, forming a thin film with a thickness constrained by the capillary pressure generated at the base of the membrane pore. Condensed water will be drained out through designated exit port to avoid flooding issue. [00187] The hydrophobic membrane is a core component of the capillary-driven condenser design, which provides Laplace pressure from the outward-curving liquid-vapor interfaces to drive the condensate transport into the wick structures below the membrane. There are many hydrophobic membrane materials that are commercially available, such as polytetrafluoroethylene (PTFE), polypropylene (PP) and polyvinylidene difluoride (PVDF). A list of some commercially available choices for hydrophobic membranes and their corresponding properties are shown in Table 2.

Table 2

[00188] PTFE membranes are intrinsically hydrophobic, chemically stable, and applicable to large scale industrial applications. They have a relatively wide range of pore size (0.1-10 pm) and thickness that are commercially available, and their prices vary from several hundred to several thousands of dollars per meter squared depending on the geometry. The unlaminated PTFE membranes do not have a supportive layer, while the laminated ones have a supportive layer of PP or polyester (PE) to enhance the mechanical performance of the membrane. Normally, the supportive layer has ~3x of pore size as that of the selective layer and therefore has better permeability.

[00189] Another intrinsically hydrophobic potential membrane material is PP. PP membranes are strong, flexible, and compatible with a broad range of chemicals. They also have a relatively wide range of pore size and thickness that are commercially available, though for large pore sizes (>0.2 pm) the pores are no longer absolutely cylindrical and the nominal pore size is used as a representation. PP membranes have similar range of costs as PTFE membranes.

[00190] PVDF membranes are also intrinsically hydrophobic, though they have relatively higher prices as compared to other membrane materials. PVDF has been widely used in large scale industries such as membrane distillation, oil-water separation, batteries, and tissue engineering.

[00191] Although variety of hydrophobic membranes are available on the market, few of them fulfill the requirement for membrane pore size and thickness. The PVDF membranes were fabricated using a customized electrospinning setup in order to optimize membrane properties for efficient capillary-driven condensation. By the use of electrospinning, one can control the pore size and the thickness of the PVDF membrane and significantly lower down the cost of the PVDF membrane to several dollars per meter square surface area.

[00192] The wick structure is the other core component of the capillary-driven condenser design, which reduces the thermal resistance through integrating the condensate liquid film with a high-thermal conductivity structured wick of a required thickness.

[00193] Various types of structured metal wicks are available commercially. For example, metal wire cloth, perforated sheets, metal mesh, and metal foams with different thickness and porosity can be easily found on the market, as shown in Table 3. Compared to wire cloth, perforated sheets are more rigid and durable for a longer service life in harsh environments. However, both wire cloth and perforated sheets are closed-cell structures that constitute individual enclosures. On the other hand, metal foams consist of cells that are all interconnected, allowing the condensate fluid to pass through the wicking structure. The thermally conductive porous wick can include a sintered metal powder, an electrodeposited porous metal, a metal foam, a metal mesh, a laser-etched metal, a 3D printed metal, a molded surface structure, or a patterned substrate. See, for example, Wang, et al., Nature 582, pages55-59 (2020) and Richard, Bradley et al. “Loop Heat Pipe Wick Fabrication via Additive Manufacturing.” (2017), each of which is incorporated by reference in its entirety. The metal mesh can be a stack of metal meshes. See, for example, Wang, Langmuir 2021, 37, 7, 2289-2297, which is incorporated by reference in its entirety.

Table 3

[00194] In addition to metal foams, sintered powder wicks are another widely used option for metal wick structures. In fact, approximately 80% of conventional heat pipes use sintered powders as wick structures. See, for example, Tang, H. etal.,Appl. Energy 223, 383-400 (2018), which is incorporated by reference in its entirety. Similar to metal wicks made of metal mesh/wire cloths, sintered metal powder can provide large capillary force but relatively low liquid permeability as compared to metal foams.

[00195] The effective thermal conductivity A _¾ t of the porous metal wick is crucial to the heat transfer performance of the capillary-driven condensation. The thermal conductivity of the metal and its volumetric porosity are critical to A¾ _T . Copper has the best thermal conductivity among all the listed materials and has a mid-range price. Higher porosity of the porous wick would decrease its A¾t. although the wick permeability K, which is desirable for condensate flow, increases with increasing pore size and porosity. When designing the wick structures, careful consideration needs to be made on the porosity and the pore size of the structures to ensure good balance between k and A¾ _T .

[00196] The arrangement of the metal network could also affect k _{e ff} . Highly connected metal structures like copper foams usually have higher k _{e ff} than those with poorer connections even at the same porosity. FIG. 46 shows the comparison among effective thermal conductivities of different metal structures.

[00197] The thickness of the metal wicks determines the thickness of the condensate film and therefore plays an important role in the condensation heat transfer performance. Commercially available metal wicks such as copper foams usually have a thickness above 200 pm, limiting the heat transfer performance of the wick layer. Porous metal wicks can be fabricated through electrodeposition, such as electrodeposited copper foam. See, for example, Kim, J., et ak, Electrochem. commun. 10, 1148-1151 (2008). This technique can tune the thickness of the porous metal wick through controlling the deposition time and can also be applied to large scale. The pores of the wick can have average diameters of between 5 microns and 25 microns, for example, 5 microns, 6 microns 7 microns, 8 microns, 9 microns, 10 microns, 11 microns, 12 microns, 13 microns, 14 microns, 15 microns, 16 microns, 17 microns, 18 microns, 19 microns, 20 microns, 21 microns, 22 microns, 23 microns, 24 microns, or 25 microns.

[00198] As discussed below, commercially available copper foams with the smallest thickness available on the market were used as a porous copper wick for proof of concept. Porous copper substrates were also developed with much smaller thickness based on electrodeposition.

[00199] Two commercially available materials were selected for the scalable proof-of- concept study. Porous copper foams with a thickness of -200 pm (thinnest that could be found on the market) and porosity of -70% were purchased from MTI Co. and used as the wick layer. Copper meshes with different mesh sizes (i.e., 200 mesh size, 500 mesh size, and 1500 mesh size) were purchased from TWP Inc. and Structure Probe Inc., and used as the membrane layer after hydrophobic coating. The advantages of using hydrophobized metal meshes as the membrane layer for the proof-of-concept study are (1) metal meshes are relatively easy to be bonded to the metal wick layer; (2) metal meshes with well-defined pore sizes are commercially available; (3) hydrophobic coating has well-developed deposition procedure and its lifetime (typically 1-2 days) is long enough to experimentally validate heat transfer enhancement. Detailed properties of the two layers of materials are shown in Table 4 below.

Table 4

[00200] Capillary-driven condensers were fabricated from a 1-inch diameter copper block by mechanically and then chemically polishing the end of a copper block followed by solvent and acid cleaning the surface. In order to obtain good thermal contact between the copper condenser block, the copper foam wick, and the copper mesh membrane, the copper foam was diffusion-bonded and the copper mesh to the copper block using a furnace.

[00201] A schematic and an image of the latest diffusion bonding assembly are shown in FIGS. 47A-47B. The copper block condenser substrate, the copper foam, and the copper mesh with the same diameter (1 inch) were aligned one by one in between two thin (0.125 inch-thick) ceramic plates, which have a shallow dent of 1.02 -inch-diameter circle in the middle for the copper parts to fit in. The use of the ceramic plates is to prevent possible diffusion bonding from occurring between the stainless-steel clamp and the copper. The slightly larger diameter of the dents on the ceramic plates is to accommodate the thermal expansion of the copper parts under the high temperature during the diffusion bonding process. The two ceramic plates with the copper parts between them are clamped together by a pair of customized stainless-steel parallel clamps. Using different types of stainless steel can prevent diffusion bond from occurring among the stainless-steel parts under high temperature. The torque applied on the nuts can generate a uniform clamping pressure over the stainless-steel parallel plates, assisting gravity force to hold the three copper parts together during the diffusion bonding process.

[00202] Three different hierarchical copper surfaces were designed and fabricated through diffusion bonding, as shown in the scanning electron microscopy (SEM) images in FIGS. 48A-48C. For the wick layer of all three sample surfaces, the same copper foam with a measured permeability of over IE-1 lm ² and a thickness of -200 pm was used. For the membrane layer, three different types of copper mesh were selected for the three sample surfaces: 200 mesh size copper wire cloth (TWP Inc.), 500 mesh size electrodeposited copper mesh (Structure Probe, Inc.), and 1500 mesh size electrodeposited copper mesh (Structure Probe, Inc.), which correspond to pore sizes of -80 pm, -40 pm, and 10 pm, respectively. More details about the geometrical properties of the selected copper foams and copper meshes are shown in Table 4.

[00203] The permeability of the wick layer of the hierarchical copper samples was measured, which is the 200pm-thick copper foam, under various processing conditions, such as different pressure for the diffusion bonding/hot press procedure and different heating time for the sample to be kept in the high temperature furnace. The permeability measurement was conducted by following an established procedure described in the literature (see, for example, reston, D. J. et al. Gravitationally-Driven Wicking for Enhanced Condensation Heat Transfer. (2018), which is incorporated by reference in its entirety).

[00204] FIGS. 49A-49D show the permeability measured on the (FIG. 49 A) original copper foam, (FIG. 49B) copper foam being diffusion bonded/hot-pressed with 30 in-lbs torque clamping force applied on the diffusion bonding assembly, (FIG. 49C) copper foam being diffusion bonded/hot-pressed with 50 in-lbs torque clamping force applied on the diffusion bonding assembly and with a total processing time of 5 hours in the furnace, and (FIG. 49D) copper foam being diffusion bonded/hot-pressed with 50 in-lbs torque clamping force applied on the diffusion bonding assembly and with a total processing time of over 10 hours in the furnace. By comparing FIG. 49A with the other three figures, it was found that permeability increases after the diffusion bonding/hot press procedure. This is potentially a result of the oxidation of the copper structures during the heating process along with the etching of copper oxides by HCL treatment after the diffusion bonding/hot press step. Furthermore, it was found that keeping the sample under high temperature for a longer time would slightly increase its permeability. On the other hand, increasing the clamping force on the diffusion bonding assembly would decrease the permeability of the sample, as shown by FIGS. 49B and 49C. All samples that were tested have permeabilities on the order of IE-11 m ² . Therefore, IE-11 m ² was chosen as a conservative estimation for the permeability of the copper wick/foam and adopted this value into the heat and mass transfer model.

[00205] For the proof of concept study, the membrane layer was fabricated by hydrophobizing the copper mesh. The hydrophobic coating was achieved vapor deposition using FAS ((Heptadecafluoro-l,l,2,2-tetrahydrodecyl) trimethoxy silane) to form conformal hydrophobic coating on metal surfaces. This specific coating exhibits a consistent water contact angle of -105°.

[00206] The details of the coating procedure are as follows. First, the copper sample was cleaned with acetone, ethanol, isopropanol, and water in sequence to remove potential hydrocarbon contaminants. Then, the sample was dipped in to an HCL solution (0.2 M) for 30 seconds to remove oxides. After the samples are rinsed with water and dried with nitrogen, the sample was oxygen plasma cleaned for 20 minutes. With bombardment of air plasma, the copper surface would be more active and easier to form a conformal coating. Then, the sample was placed into a sealed bottle. Along with the porous copper samples, 800 pL of FAS in toluene solution (5 V%) was also put into a small beaker, which sat beside the samples to be coated. The sealed bottle was placed in an oven at 100°C for 3 hours, during which the FAS was coated onto the porous copper sample via vapor phase deposition.

Finally, the samples were taken out of the furnace and cooled down to room temperature in the fume hood. A schematic of the coating process in illustrated in FIG. 50.

[00207] Ideally, only the copper mesh layer needs to be hydrophobized for the proof- of-concept study, and the copper foam layer should remain hydrophilic to promote condensation nucleation. Biphilic coating can be made by depositing hydrophobic coating when covering the bottom-layer structures with a protective coating which can be washed away easily after the hydrophobic coating. For this study, biphilic coating was applied to the hierarchical copper sample by utilizing a heat curing photoresist (AZ3312) as the protective layer for the copper foam structures. The detailed procedure includes following steps. First, the sample was cleaned with acetone, ethanol, isopropanol, and water, followed by 10 minutes of argon plasma cleaning. Then, the sample was dipped into a beaker of photoresist AZ3312 and allowed the photoresist to climb up into the porous copper under capillary force. Once all the copper foam area was covered by photoresist, the sample was baked in an oven at 100°C for 10 minutes until the photoresist was solidified inside the porous copper sample. Next, 20 minutes of oxygen plasma was applied to remove the top layer photoresist off the hierarchical sample, exposing the hydrophilic copper mesh to be hydrophobized. The photoresist-protected hierarchical copper sample was then deposited with FAS coating using the procedure illustrated in FIG. 50. Finally, the hydrophobized hierarchical copper sample was ultrasonically cleaned with acetone to remove the photoresist residue and recovered the hydrophilic copper foam layer. Condensation on hydrophilic surfaces enables a lower nucleation barrier as compared to hydrophobic ones. See, for example, Varanasi, K. K., et al. Appl. Phys. Lett. 95, 094101 (2009), which is incorporated by reference in its entirety. As a result, the biphilic coating ensures condensation start from the hydrophilic copper foam and then reach the hydrophobic mesh layer. [00208] Although the permeability of the copper foam (IE-1 lm ² ) was high enough to prevent flooding at low surface subcools, flooding could potentially occur at local defects such as broken mesh pores, which was hardly evitable during the fabrication process. In order to further facilitate condensate drainage, microchannels were machined on the copper foam layer, as shown in FIGS. 51A-51B. Only one end of the microchannels was machined all the way done to the edge of the sample to ensure directional condensate flow. The channel width, channel depth, and distance between the microchannels were selected as 100 pm, 100 pm, and 1.27 mm, respectively. These dimensions were chosen based on the tools available and were used for a test study. Further modeling is needed to optimize the design of the microchannels.

[00209] An analytical model was developed based on ID conduction through liquid- filled porous copper to predict the heat transfer performance of the three hierarchical sample surfaces. To calculate the effective thermal conductivity of a copper mesh, the following equation was adopted, which is developed and validated by previous literature (see, for example, Reay, D. A., et ak, Heat Pipes (Sixth Edition) 65-94 (Butterworth Heinemann, 2014), which is incorporated by reference in its entirety): where e is the volume faction of the solid phase. k _s is the thermal conductivity of the solid (copper), and k _\ is the thermal conductivity of the liquid (water).

[00210] The effective thermal conductivity of the wick layer (copper foam) can be calculated using the volumetric average value as been shown in previous literature (see, for example, Preston, D. J., et ctl, Langmuir 2018, 34, 15, 4658^1664, which is incorporated by reference in its entirety):

[00211] Heat flux and fluid flow were analytically solved under an environmental condition with vapor temperature of 35°C and surface subcool up to 5°C. This vapor condition is normally seen in steam power plant condensers and this condition was the goal to achieve during the experiments. Conventional filmwise condensation heat transfer predicted by the Nusselt model is used as the benchmark. FIGS. 52A-52C show the model- predicted heat transfer performance of the three samples: (1) 200-mesh-size copper mesh covered copper foam, (2) 500-mesh-size copper mesh covered copper foam, and (3) 1500- mesh-size copper mesh covered copper foam. The y axis on the left shows heat flux, and the y axis on the right shows the flooding criteria P*, which is the ratio of the viscous pressure loss for condensate to travel through the wick layer and the capillary pressure given by the hydrophobized mesh pores. In order to prevent flooding from happening, the experiments should be operated within the regime 0<P*<1. Otherwise, flooding would occur and greatly deteriorate the heat transfer performance of the condensing surface. FIG. 52A shows that, although the 200-mesh-size sample would not flood throughout the subcool regime was being investigating (0-5°C), it only gives marginal enhancement on the heat transfer as compared to the filmwise condensation on a flat surface. This is due to the large thickness of the 200-mesh-size sample (>20x thicker than the other two samples), which significantly decreases the effective thermal conductivity of the mesh layer and consequently reduces the heat transfer coefficient of the system. FIG. 52B shows that, the 500-mesh-size sample could greatly enhance the heat transfer performance as compared to the conventional filmwise condensation, yet only within a narrow range of subcool (0~1.5°C). Flooding would occur on the 500-mesh-size sample once the subcool is larger than ~1.5°C. This is due to the large mesh pore size of the 500-mesh-size sample (~4x larger than the 1500-mesh-size sample), which limits the capillary pressure that drives the condensate to flow throughout the wick layer. The 1500-mesh-size sample has the smallest mesh pore and the smallest mesh thickness among the three samples, which results in its significant heat transfer enhancement (>10x) within a relatively wide subcool regime (0-4.1 °C), as shown in FIG 52C.

[00212] Heat transfer performance of the fabricated sample surfaces was experimentally characterized in a controlled environmental chamber, as shown in FIGS. 53A-53C. The sample copper block was installed onto a test rig inside the chamber, as shown in FIG. 53 A. An array of five thermocouple probes were inserted into the copper block along its length to monitor the temperature distribution along the copper block. Assuming ID (linear) conduction heat transfer, heat flux flowing through the copper block could be extracted from the temperature measurements by Fourier’s Law. The linear fit for the temperature measurements had an R ² value of over 99% for each measurement throughout the experiments, indicating that the assumption of linear conduction was valid and heat transfer through the block walls was negligible. The back side of the copper block was cooled with chiller water provided by an external chiller (Fisher Scientific Isotemp II), and the front side of the copper block, where the hierarchical copper surface was attached to, was served as the condenser surface. The temperature at the condenser surface (right below the copper wick layer) was determined by extrapolating the linear temperature distribution along the copper block measured by the thermocouple array.

[00213] A Pirani gauge was installed on the chamber to accurately monitor the chamber pressure under pumping process, as shown in FIG. 53B. Before every experiment, the chamber was pumped below 0.5Pato eliminate non-condensable gases (NCGs). Following removal of NCGs, pure, degassed vapor was introduced into the chamber from a heated, temperature-controlled water reservoir (as shown in FIG. 53C), and allowed to condense on the condenser surface. The vapor pressure inside the chamber was manually maintained at a constant (~5.3 kPa) by opening the valve on the reservoir and measured by another pressure transducer installed on the chamber (Omega,

MMA030V5B3MB0T3A5CE). Two thermocouples were installed inside the chamber to measure wetbulb and drybulb temperature, respectively. When at steady state, these two temperature agreed with each other with a discrepancy falling under the uncertainty of the thermocouple readings (0.15 °C). Then, the experimentally measured condensation heat transfer coefficient was determined by dividing the heat flux by the surface subcool. For each data point, acquisition waited for over 30 minutes to make sure steady state was reached, and the data point was collected by averaging measurements over a time span of 5 minutes. Visualization of the condensation development was achieved through a viewing window on the chamber, where a camera was placed towards the viewing window outside the chamber. [00214] Condensation heat transfer on a flat copper surface was measured and compared to the Nusselt model prediction to validate the capability of the environmental chamber. FIGS. 54A-54B shows the experimental results of condensation on a flat copper surface. A flat condensate film was observed throughout the experiment, as shown in FIG. 54A. The condensation heat flux was measured under six different subcool conditions ranging from 0.3-7 °C, as shown in FIG. 54B. All the measured data points (dots) are in good agreement with the Nusselt model prediction for filmwise condensation heat transfer (curve), validating that the chamber and the experimental procedures are capable of taking reliable measurements. [00215] For the proof of concept study, three hierarchical copper samples were tested, all of which were based on 1500-mesh-size copper mesh covered copper foam structures. Samples with different wettability (i.e., fully hydrophobic and biphilic) were tested to investigate the influence of surface wettability on the condensation heat transfer performance.

[00216] FIGS. 55A-55D show the experimental results of condensation on a fully hydrophobic hierarchical copper sample. FIG. 55A shows that the sample was completely dry and kept under vacuum before water vapor was introduced into the chamber. At the beginning of the condensation experiment, the chiller temperature was set to 5 °C, which induced a large subcool on the sample surface and initiated condensation. Nucleation of condensation was observed everywhere on the top of the fully hydrophobic hierarchical copper sample in a pattern similar to dropwise condensation on a hydrophobic surface, as shown in FIG. 55B. As condensation continued (after the first few seconds), water started to fill into the interconnected pores inside the copper foam layer, and drained at the bottom of the surface due to gravity. Capillary force generated at the hydrophobic copper mesh layer was able to keep majority of the surface free of droplets, though flooding was observed on certain locations probably due to local defected mesh pores, as shown in FIG. 55C. FIG. 55D shows the heat flux measured on the fully hydrophobic hierarchical copper surface under different subcooling. Due to the existence of the bursting droplets which could act as resistance to heat transfer, the heat transfer performance of this sample was just as good as filmwise condensation.

[00217] FIGS. 56A-56D show the experimental results of condensation on a biphilic hierarchical copper sample. FIG. 56A shows that the sample was completely dry and kept under vacuum before water vapor was introduced into the chamber. The experimental procedure was repeated as the previous experiment by first setting the chiller temperature to 5 °C. Interestingly, a random pattern of condensation nucleation on the top of this biphilic hierarchical copper was not observed. Instead, droplets started to burst out from several locations, as shown in FIG. 56B. As condensation continued (after the first few seconds), water started to fill the interconnected pores inside the copper foam layer, and condensation on this biphilic sample started to show the similar capillary-driven pattern as the fully hydrophobic sample, which demonstrated that capillary-driven condensation would occur as long as the top layer membrane is hydrophobic. Capillary force generated at the hydrophobic copper mesh layer was able to keep majority of the surface free of droplets, though flooding was observed on certain locations probably due to local defected mesh pores, as shown in FIG. 56C. FIG. 56D shows the heat flux measured on the fully hydrophobic hierarchical copper surface under different subcooling. Again, due to the existence of the bursting droplets, the heat transfer performance of this sample was not better than filmwise condensation.

[00218] FIGS. 57A-57D show the experimental results of condensation on a biphilic, micro-channeled hierarchical copper sample. FIG. 57A shows that the sample was completely dry and kept under vacuum before water vapor was introduced into the chamber. The same experimental procedure as the previous experiments was repeated on this sample. Similar to the biphilic hierarchical copper sample, this biphilic micro-channeled hierarchical copper sample also showed local flooding/bursting droplets at the very beginning of the experiments, as shown in FIG. 57B. As condensation continued (after the first few seconds), condensation on this biphilic sample started to show the similar capillary-driven pattern as the previous two samples, as shown in FIG. 57C. Interestingly, the capillary force generated at the hydrophobic copper mesh layer was able to keep most of the surface free of droplets, enabling a condensing surface free of droplets, as shown in FIG. 57C. This was achieved by having the micro-channels as additional exit ports. The flow resistance through the microchannels was lower than the bursting pressure of most broken mesh pores. Therefore, even when there were local defects on the copper mesh layer, many bursting droplets on the top of the surface were not observed. FIG. 57D shows the heat flux measured on the biphilic, micro-channeled hierarchical copper surface under different subcooling. On this biphilic hierarchical copper sample, the heat transfer performance was measured to be 50% better than filmwise condensation. A roughly 50% heat transfer enhancement was shown on this biphilic, micro-channeled hierarchical copper sample as compared to the conventional filmwise condensation.

[00219] The 50% enhancement measured on the biphilic hierarchical copper sample was promising, although it was lower than what was expected from the model prediction.

The following reasons are the hypothesis for the discrepancy between the modelling results and the experimental results. (1) The micro-channels decreased the effective thermal conductivity of the copper foam layer, which was not captured in the model; (2) The copper foam had a poor connection in its vertical direction as compared to its horizontal direction, but the vertical direction was critical for the heat transfer experiment. Therefore, the effective thermal conductivity of the copper foam was overestimated by calculating a volumetric average value. (3) There were inevitable defects such as broken pores in the copper mesh layer due to the diffusion bonding process. These defects would affect flooding criteria and heat transfer performance. Those defects were not considered in the model. Thermal conductivity of the micro-channelled hierarchical copper sample needs to be measured or estimated using a more accurate model. The design of the microchannels (channel width, channel height, distance between nonboring channels) can also be optimized. Combining a thinner copper wick layer with an optimized microchannel structure is expected to further enhance the heat transfer performance of the capillary-driven condenser.

[00220] As mentioned earlier, porous metal wick can be fabricated in a scalable manner by electrodeposition. A recipe to electrodeposit a single layer of inverse opal copper was developed. See, for example, Carlos, D. D., et al., Langmuir 2021, 37, 43, 12568-12576, which is incorporated by reference in its entirety. By using templating spheres with 10 pm diameter, a monolayer inverse opal copper was fabricated with roughly 65% porosity, 5 pm thickness, and a model-predicted porosity of 5E-llm ² permeability. The remaining challenge in the fabrication of inverse opal copper is sustaining a robust attachment and close to full coverage of the templating spheres during the electrodeposition step, which can be critical to the scale of the surface area to be fabricated.

[00221] Sintering copper powders is another scalable way to fabricate porous copper. Sintering copper powders using three different powders was attempted: spherical powders with diameter < 10 pm, spherical powders with diameter < 50 pm, and dendritic powders with size < 45 pm. The resulting porous copper structures are shown in the SEM images in FIGS. 58A-58C.

[00222] Due to the size of the tube furnace, sintered copper powder surfaces were fabricated on the scale of 1 cm. However, sintering copper powers is a highly scalable technique as can be seen in the heat pipe industry. For the capillary-driven condenser, a small thickness (ideally under 200 pm) is required for the porous wick layer in order to reduce thermal resistance of the wick layer. Microporous copper wick monolayers have been achieved by sintering in the literature (see, Hoenig, S. H. & Iii, R. W. B. Journal of Heat Transfer 140, 1-7 (2018), which is incorporated by reference in its entirety), although depositing a monolayer of copper powders over a large surface area is still challenging. Sintering copper powders with smaller size could keep the wick thickness small while depositing multiple layers of powders. However, the wick permeability would decrease with the size of the powders. The permeability of a sintered powder wick can be estimated by: ( 21 ) 150(1— _W ) ² where k is the permeability of the sintered powder wick, d _powder is the size/diameter of the powders, and 0 _W is the porosity of the sintered powder wick. Sintered copper powders has shown a porosity of -50% in literature. See, for example, Leong, K. C., et ah, Journal of Porous Materials, 4, 303-308 (1997), which is incorporated by reference in its entirety. Sintering copper powders with diameter of - 50 pm would be good to balance both the thickness and the permeability of the wick layer. Assuming a sintered copper wick made by four layers of 50 pm copper spheres with a porosity of 50%, the copper wick’s permeability was expected to be ~1.7E-llm ² , which is comparable to the permeability of the copper foam used for the proof of concept study.

[00223] Electrospun membranes are described below.

[00224] A heat and mass transfer model for electrospun-fiber covered porous copper based on vapor transport through the membrane pores and heat conduction through the condensate fdled porous copper was developed. FIGS. 59A-59C show the model prediction for condensation heat transfer performance of different capillary-driven condensers made of electrospun membrane covered porous copper. Three different porous copper were used as the wick layer in the model:

(a) Copper foam with 200 pm thickness, 65% porosity, and IE-11 m ² permeability.

(b) Inverse opal copper with 5 pm thickness, 65% porosity, and 5E-11 m ² permeability. (c) Sintered copper spheres with 200 pm thickness, 50% porosity, and 1.7E-1 lm ² permeability.

(a) is commercially available (b) is a structure fabricated in lab. (c) is a common structure in heat pipe industries and can be fabricate. All three plots shown in FIGS. 59A-59C were obtained at a given vapor temperature (T _v = 35°C) and a subcool of 5°C. X-axis is pore size of the electrospun membrane and Y-axis is the corresponding heat flux predicted by the model. Red, blue, and green lines correspond to three different membrane porosity: 0.9, 0.8, and 0.7. Different thickness of lines means different membrane thickness. The black dash line denotes filmwise condensation predicted by the Nusselt model. In the model, heat flux would drop to zero when "flooding" occurs; this is not true in reality but is used as an indicator for "flooding" in the modelling results. [00225] FIGS. 59A and 59C are similar: with higher membrane pore size (increasing X-axis), higher porosity (line), and smaller membrane thickness (10 pm thick denoted by the thinnest line), heat transfer gets much enhancement and no flooding occurs throughout the sweep. The similarity between these two plots is because similar properties for the copper foam were prescribed (corresponding to the FIG. 59A and the sintered copper (corresponding to FIG. 59C).

[00226] The lines in FIG. 59B (corresponding to inverse opal copper) are not smooth due to the incremental sweep other than a continuous sweep that was performed. Heat flux given by the same membrane increases faster in FIG. 59B as compared to other two plots, but also induces flooding easier, which is because: the ultra-thin inverse opal (5 pm thick) could greatly reduce the thermal resistance of the water-filled porous copper but on the other hand, these thin structures are easier to flood (condenser liquid cannot be completely drained from the edge of the thin wick).

[00227] Overall, the model guided fabrication of a membrane that is thin (on the order of 10 pm), highly porous (90%), and has a large pore size (on the order of 1 pm); it was also important to make sure that flooding would not occur on this membrane under a sufficient subcool (at least 5°C). Once the parametric optimization for electrospinning fibrous membranes is completed, it will be possible to combine an optimized membrane with a porous structured copper substrate to make a scalable version of the capillary-driven condenser. The structure is expected to show over 5x enhancement on the electrospun membrane covered sintered copper powder without flooding under a subcool of 5°C, as predicted by model.

[00228] Electrospinning is an easily tunable technique that utilizes the force balance between the electrostatic force and the solution surface tension to fabricate nanofibrous membrane, is promising for a wide range of applications. Morphology of the electrospun membrane can be directly related to its performance. During an electrospinning process, many factors can affect the morphology of the products, such as properties of the solution, voltage supply, needle distance, feeding rate of the solution, and the electrospinning time. Effective parametric study can be used to optimize the fabrication of electrospun membrane such that the ideal morphology of the electrospun membrane can be achieved by optimizing the most important parameters. Poly(vinylidene fluoride-co-hexafluoropropylene) was chosen as an example for the current study due to its interesting and versatile functionalities such as superior hydrophobicity, high free volume, and piezoelectricity. [00229] As noted above, thermoelectric power generators, which typically use a Rankine cycle, provide the majority of the electricity produced in the US. See, for example, ICF-Intemational, Catalog of CHP Technologies, in Combined Heat and Power Partnership Program, E.P.A. (EPA), Editor. 2008, U. S. Environmental Protection Agency (CHPPP): Washington, D.C., which is incorporated by reference in its entirety. However, they also withdraw the largest amount of water from US water bodies in order to condense the steam generated from the power plant. There is significant interest in enhancing the efficiency of condenser designs, which will not only improve power production but also decrease the amount for water needed for condensation. Significant efforts have focused on advancing condenser designs with higher performance for steam power plants. Various condenser designs have been proposed and incorporated in existing power plants. See, for example,

Hao, M., et al., Program on Technology Innovation: New Concepts of Water Conservation Cooling and Water Treatment Technologies. 2012, Electric Power Research Institute (EPRI): Palo Alto, which is incorporated by reference in its entirety. These condensers, however, typically rely on filmwise condensation (FIG. 1A), where a thin liquid film on the condenser surface forms due to the high surface energy associated with the majority of industrial heat exchanger materials.

[00230] Filmwise condensation is not desired due to the large thermal resistance to heat transfer. See, for example, Nusselt, W., The surface condensation of water vapour. Zeitschrift Des Vereines Deutscher Ingenieure, 1916. 60: p. 541-546, which is incorporated by reference in its entirety. Over the past eight decades, dropwise condensation, where droplets roll off at sizes approaching the capillary length and clear the surface for re- nucleation (FIG. IB), has been researched and promises heat transfer enhancements of 5-10x compared to the filmwise mode. See, for example, Das, A.K., et al., Journal of Heat Transfer- Transactions of the Asme, 2000. 122(2): p. 278-286; Mario, P.J., et al., International Journal of Heat and Mass Transfer, 1986. 29(8): p. 1109-1117; Vemuri, S. and K.J. Kim,

International Journal of Heat and Mass Transfer, 2006. 49(3-4): p. 649-657; Vemuri, S., et al., Applied Thermal Engineering, 2006. 26(4): p. 421-429; Rose, J.W., Proceedings of the Institution of Mechanical Engineers Part a-Joumal of Power and Energy, 2002. 216(A2): p. 115-128; Enright, R., et al., Nanoscale and Microscale Thermophysical Engineering, 2014. 18(3): p. 223-250; and Schmidt, E., et al., Zeitschrift Des Vereines Deutscher Ingenieure, 1930. 74: p. 544-544, each of which is incorporated by reference in its entirety. Despite extensive development, maintaining robust non-wetting behavior has been an ongoing challenge and has prevented steam power plants from taking advantage of the heat transfer enhancement that dropwise condensation offers.

[00231] As described herein, capillary-driven condensation (CDC) can involve a structure in which a porous hydrophobic membrane atop a wicking structure on a condensing surface drives liquid transport and removal from the surface via a capillary pressure gradient along the wicking surface towards an exit port, as shown in Figure 2. This approach has a two-fold advantage. First, the structured wick increases the effective thermal conductivity of the condensate layer, reducing the thermal resistance. Second, the hydrophobic porous membrane helps generate additional capillary driving pressure to drive fluid out and significantly increase the heat transfer performance.

[00232] The invention disclosed herein relates to a robust new approach to enhance condensation heat transfer for steam power plants via capillary-driven condensation (FIGS. 3A-3E). Here, a porous hydrophobic membrane atop a wicking structure on the condensing surface drives liquid transport and removal from the surface via a capillary pressure gradient along the wicking surface towards an exit port. Herein, the traditional phase-change process of thin-film evaporation in wick structures has been reversed (heat pipes, etc.) (see, for example, Coso, D., et al., Journal of Heat Transfer, 2012. 134(10); Weibel, J.A., et al., International Journal of Heat and Mass Transfer, 2010. 53(19-20): p. 4204-4215; and Weibel, J.A., et al., Journal of Heat Transfer, 2013. 135(2), each of which is incorporated by reference in its entirety) to harness an order of magnitude higher heat transfer coefficients than conventional filmwise condensation.

[00233] The key advantages of this novel mode of condensation are: i) a decrease in the thermal resistance of the condensation process due to the increased effective thermal conductivity of the liquid-wick layer; and ii) capillary-assisted removal of the condensate film which provides orders of magnitude higher driving force than gravity. Equally important, this condensation mode promises to achieve both an enhanced heat transfer coefficient and overall heat flux in a robust manner, i.e., without concerns of degradation of surface coatings typical for promotion of dropwise condensation.

[00234] Detailed and systematic studies have supported that this represents a promising solution for steam power plant condensers. Towards this goal of enhancing the heat transfer coefficient and ensuring superior robustness. Some of these studies described herein include: [00235] Develop porous membranes and wi eking structures for capillary-driven condensation. Various wi eking structures and porous hydrophobic membranes have been designed to reduce the thermal resistance and enhance capillary-driven flow.

[00236] Experimentally investigate capillary-driven condensation on flat substrates and tube substrates. Condensation heat transfer performance has been experimentally studied and compared with traditional filmwise condensation on various samples.

[00237] Optimize the capillary-driven condensation structure with model development. A physics-based model has been developed to predict and optimize condensation heat transfer, and validated by experiments.

[00238] Incorporate capillary-driven condensation structure to demonstrate scaled-up proof-of-concept operation. Experiments performed on tube bundles in industrially relevant conditions can provide additional insights.

[00239] Based on the success of these objectives, a robust condenser design can be implemented for steam power plants with greater than five times enhancement in heat transfer coefficients compared to conventional filmwise condensation. Due to the improved heat transfer coefficient of condensation the steam condensation temperature, and accordingly the turbine back-pressure, can be reduced by up 4 °C and 0.7 kPa, respectively. As a result, the overall heat rate of a typical power plant can be expected to decrease by 1.5%, leading to an additional 13.80 MW of generated power for a 950 MW plant and a commensurate savings in water withdrawal and usage. Implementation of this method has a simple payback period of roughly one year.

[00240] In capillary-driven condensation, condensation occurs on the hydrophilic micro/nanostructured wick, and the condensate is then forced out due to the capillary pressure buildup at the menisci formed in the porous hydrophobic membrane. The presence of the structures and the resulting capillarity helps maintain a stable liquid film while driving liquid flow. By tailoring the size of the pores in the membrane and the geometry of the wicking structure, the capillary pressure generated can be maximized and the flow rate of the condensate can be optimized to increase the rate of condensation that the wicking structure can support. This approach promises to be completely passive, robust, and can harness an order of magnitude higher heat transfer coefficients than conventional filmwise condensation. Such a strategy can be a development breakthrough for developing a condensation strategy for power plants. [00241] The capillary-driven condensation mode can be modelled and compared to traditional filmwise condensation theory. This model and the predicted enhancement over filmwise condensation motivate the need to study and further development of the concept of capillary-driven condensation for applications in steam cycle power plants.

[00242] Filmwise condensation can be described as follows. Typically, condensation on industrial materials such as metals results in a fluid film which sheds due to the gravitational force. This hinders heat transfer by adding a significant conduction thermal resistance. Nusselt modeled this phenomenon using lubrication theory, and the model has been shown to agree very well with experimental data. The heat transfer coefficient determined by this model for a horizontal tube is:

( 22 )

[00243] where g is the gravitational acceleration (g = 9.81 m/s ² ), p _v is the vapor density, p is the liquid density, ku _{q id} is the condensate thermal conductivity, / is the condensate dynamic viscosity, r _tube is the radius of the tube, h/ _g is the latent heat of vaporization, and AT is the temperature difference. See, for example, Incropera, F.P., Introduction to heat transfer. 5th ed. 2007, Hoboken, N.J.: Wiley. 901; and Carey, V.P., Liquid-vapor phase-change phenomena : an introduction to the thermophysics of vaporization and condensation processes in heat transfer equipment. 2nd ed. 2008, New York: Taylor and Francis. 742, each of which is incorporated by reference in its entirety. It is clear that the heat transfer can be limited by the driving force of gravity, which cannot be practically increased in an industrial scale condenser. Therefore, the systems, structures and methods described herein rely on capillary driven transport.

[00244] Capillarity can also be important. On a high energy surface, interaction between the liquid and the solid surface leads to the formation of a liquid-vapor interface called the meniscus. The surface tension of the interface leads to a pressure difference across the meniscus. This pressure difference is the capillary pressure P _cap and can be described by the Young-Laplace equation:

Pm ^m ¾ - ¾ “ 2 fr _{( 23 )}

[00245] where s is the surface tension, r is the radius of curvature of the interface, which is the inverse of curvature, Po is the pressure of the vapor phase and P _L is a pressure of the liquid below the interface. As seen in FIG. 3A, the interfaces of the condensing water with the porous wick can result in an increased local pressure that drives flow of the condensate.

[00246] Enhancing fluid flow with a capillary pressure gradient can be an important feature of the structures described herein. In the design described herein, capillary transport through a thermally conductive wick is driven by capillary pressure generated by the convex interfaces formed with a hydrophobic membrane encasing the wicking materials. This design removes condensate more rapidly than filmwise condensation on a bare (without a wick) surface, thereby enhancing heat transfer. A design is shown in 21B as compared to traditional filmwise condensation in FIG. 21A.

[00247] The performance of this design is determined using Darcy’s law for porous media to model the flow within the wick. The mass flux of condensate into the wick, m", which is proportional to the heat flux by the constant factor of the latent heat of the fluid, is assumed spatially uniform due to the wick’s uniform thickness (t _W ick) and thermal conductivity (k _ej j) as detailed in FIG. 3 A and FIG. 2 IB, in which the left side of the schematic is where pressure is at a maximum, while the right side is at ambient pressure (corresponding to T _vapor ). The right and left boundaries could represent either a physical boundary or a symmetry condition. The pressure drop required to remove all of the condensate coming in to a wick of length L is found by integrating Darcy’s law over the length:

[00248] where pw is the condensate dynamic viscosity, K _{i k} is the wick permeability, and twick is the wick thickness. By considering the wick’s thermal resistance, the mass flux for any given temperature difference, AT = T _vap0r T _wa u, is determined:

( 25 )

[00249] where A¾ _/ is the effective thermal conductivity of the wick { _ck ) and condensate (ki, _{q /} ).

[00250] The previous two equations (equations (24) and (25)) relate the heat flux with the required pressure drop. Therefore, for a given maximum capillary pressure drop, one can determine the maximum achievable heat flux which can occur before the capillary pressure generated by the porous membrane can remove all of the condensate. [00251] The results of the modeling effort are shown in FIGS. 22A-22B, applied to a tube geometry with a single fluid exit port at the base of the tube. When considering multiple tube diameters, one can vary the wicking distance or the length L (FIG. 3A). The hydrophobic porous membrane used in this model is aNafion membrane with a characteristic pore diameter of 5 microns and a thickness of 30 microns, and the condensing fluid is water. The baseline properties for the metal foam wick were taken from prior work (see, for example, Semenic, T., et ah, Applied Thermal Engineering, 2008. 28(4): p. 278-283, which is incorporated by reference in its entirety) and laboratory experiments, where the permeability used here is 1.25xl0 ⁹ m ² . The thickness of the metal foam wick was varied as a function of heat flux in order to take full advantage of the capillary pressure budget determined from Equation (23), which was set equal to the required pressure to drive the flow of condensate found in Equation (24); this system of equations was used to determine the optimal wick thickness, which varied from 1-5 mm in the domain considered in FIGS. 22A-22B.

[00252] These results guide the fabrication and synthesis process of the wicking surface, where high permeability wicks are desirable. A key takeaway from the model results in FIGS. 22A-22B is that the heat transfer coefficient is expected to increase up to 12x compared to filmwise condensation. The results indicate that the performance of capillary- driven condensation is comparable to dropwise condensation, but without coating robustness concerns; the corresponding improvement in turbine back pressure reduction can be over 50%.

[00253] The potential economic benefit of capillary-driven condensation for a typical 950 MW nuclear fired power plant (heat rate of 2.942 kW/kW at 3 in Hg abs) was analyzed following the analysis in Webb, R., Enhanced Condenser Tube Designs Improve Plant Performance. Power Magazine, 2010, which is incorporated by reference in its entirety. Estimated costs of the structured porous wick and hydrophobic porous membrane are shown in Table 5 based off of necessary materials to modify a condenser with 23,150 tubes with an outer diameter of 28.6 mm and 13.4 m long. A porous aluminum wick 5 mm thick and a 30 pm thick Nafion membrane were used for cost estimation.

Table 5

[00254] An alternative system can be based on a porous copper powder wick (0.2mm thick) and PVDF membrane (pore size ~1 pm) (Alternative membrane materials: PTFE, PP) are shown in Table 6. Table 6

[00255] With the analysis shown in FIGS. 22A-22B, an unmodified condenser temperature of 54.6°C and 50.7°C for a condenser with enhanced capillary-driven condensation was used. Relevant power plant information is shown in Table 6, which is based on the Arkansas Nuclear One (ANO-1) Unit 1. The modified condenser increases plant power output by 13.80 MW, which corresponds to a capital value of $27.6 million dollars (based on a value of $2,000/kW). Therefore, with the estimated material costs of $26.8 million dollars the simple payback is 0.97 years. More detailed analysis accounting for yearly variation in condenser temperature, more accurate installation and fabrication costs, as well as information specific to a given power plant would be needed for a more precise economic evaluation. These estimates, however, indicate the significant promise with the systems, structures and methods described herein which address a long-felt need in the industry. See Table 7.

Table 7

[00256] The performance of capillary-driven condensation can depend on a number of factors. One factor includes wicking structures and porous hydrophobic wicks. The wick can be a structured wick. The wick can serve at least two main roles: (1) it can serve as a structural support to hold the hydrophobic membrane away from the surface in order to allow condensate to flow underneath it; and (2) it can enhance the effective thermal conductivity of the condensate layer that is flowing beneath the hydrophobic membrane by providing a path of lower resistance for heat transfer through the wick itself. In certain circumstances, a wick can have high permeability and high thermal conductivity, which are two of the most important parameters for enhancing condensation heat transfer. High permeability is required to reduce viscous flow losses; meanwhile, high thermal conductivity decreases the effective thermal resistance of the condensate-wick layer. The effect of an increased permeability is demonstrated in FIGS. 23A-23B; to promote high permeability while maintaining mechanical robustness, metal foams are proposed in the present work, although other scalable approaches including sintered wicks as well as micro-machined grooves can be used.

[00257] The other critical aspect of the capillary-driven condensation design is the porous hydrophobic wick. This membrane is responsible for holding the condensate inside of the structured wick and also generating the capillary pressure (according to Equation (23)) that causes flow of the condensate through the wick towards the exit ports. Several important parameters of the membrane are the pore radius, the thickness, and the intrinsic contact angle. The effect of one of these parameters, the pore radius, is illustrated in FIGS. 23A-23B, where it is shown that only pore radii within a certain range will promote an enhancement compared with filmwise condensation - this indicates the importance of considering these design parameters when engineering capillary-driven condensers. Based on these considerations, low cost Nafion membranes can be used for capillary-driven condensation; however, other materials, e.g., Teflon, can also be used.

[00258] In addition to the design of both the wick and the hydrophobic membrane separately, another consideration is the interfacing of the two when the capillary-driven condenser is assembled. Both the wick and the membrane can be wrapped around a typical tube condenser during fabrication in a scalable and cost-effective manner as discussed below, the approach described herein can be a robust and long-term solution for improved condensation heat transfer in steam power plants. Approaches can include interfacing or bonding the membrane directly onto the wicking material before application to condenser tubes, such as physical, thermal, or stress-based attachment.

[00259] Wicking enhanced filmwise condensation can be characterized experimentally. To investigate capillary-driven condensation heat transfer performance with water as the working fluid, a custom built multi-purpose environmental chamber can be used as shown in FIGS. 24B. The chamber is capable of maintaining a pure fluid environment with internal pressures ranging from 1 atmosphere to high vacuum (-10-8 atm). The cooling/heat removal from the sample will be provided either by a cooling liquid supply using a large capacity chiller (Neslab System-I or Polar Accel 500 LT, Thermo Fischer Scientific, and K4, Applied Thermal Control) or a thermoelectric Peltier device. Both flat and round tube samples can be studied to characterize the performance of water condensation.

The effects of various vapor pressures from near steam power plant conditions can be characterized.

[00260] By monitoring cooling power, surface temperatures, ambient conditions (including presence of non-condensable gases, pressure and temperature, etc.), the performance of the condensation mode can be analyzed: Qcooimg = U · Asurface · ATLMTD where ATLMTD is the log-mean temperature difference between the surface temperature and the vapor environment temperature and Qcooling is the cooling power which is dependent on the mode of cooling; for a chiller: Qcooling = m · c _p · (Tout - I ^' m) where m is the mass flowrate of the refrigerant, c _p is its specific heat, and T„ and Tout are the inlet and outlet refrigerant temperatures. Qcooling for a Peltier device would be: Qcooling, Peltier = P · I where P is the Peltier coefficient and / is the current drawn by the device. The overall heat transfer coefficient U can be used to determine the condensation heat transfer coefficient as a function of the cooling power Q _cooling and the condensation temperature difference. [00261] Apart from the operating fluid, various combinations of wick materials and geometries will also be tested. In situ visualization of the condensation process (similar to FIGS. 1A-1B) through the chamber viewports will allow imaging of the performance under a variety of conditions.

[00262] Prior work includes a study on condensation enhancement for hydrocarbon liquids using only a gravitational pressure gradient in a wick (no hydrophobic porous membrane). The results included a 3x enhancement in heat transfer performance, as shown in FIGS. 25A-25B. The study of capillary-driven condensation advances this prior work by allowing the same performance enhancement with water (which has a much higher latent heat of vaporization that prohibits enhancement with gravitationally-driven flow only) by applying the hydrophobic porous membrane to generate additional pressure to drive condensate out of the wick.

[00263] An accurate numerical model can be developed which captures the liquid- vapor interface and its effect on the fluid transport to better quantify the capillary-driven condensation heat transfer performance on different wick structure geometries. The numerical model can allow for the design of wick structures to maximize the heat transfer enhancement. A finite volume model based on CFD simulations was developed to predict the condensate liquid flow rate for different wick structure geometries. This model will allow one to understand the maximum heat flux that can be achieved via this mode of condensation (i.e., the maximum heat flux before liquid bursts through the hydrophobic top surface). The model also allows us to predict the heat transfer coefficient for different wick structure geometries for comparison with filmwise condensation.

[00264] A modeling framework that was developed for thin-film evaporation on micropillar array wick surfaces, which is a liquid-to-vapor phase change process based on similar design and physics, was modified. See, for example, Zhu, Y., et ah, Langmuir, 2016. 32(7): p. 1920-1927, which is incorporated by reference in its entirety. The micropillar geometry can beneficial for systematic understanding of experiment where well-defined pillar arrays can easily be fabricated. The modified model for capillary-driven condensation can predict liquid velocity, pressure, and meniscus curvature along the wicking direction which is achieved by conservation of mass, momentum and energy. Specifically, models can accurately predict the three-dimensional meniscus shape, which varies along the wicking direction with the local liquid pressure (FIG. 26A). This model can determine the maximum heat flux achievable for different wick structure geometries, based on the maximum allowable contact angle on the pillar surface. This developed model is a tool to optimize the wick structure geometry to maximize the heat flux and heat transfer coefficient.

[00265] Initial calculations based on the preliminary modeling framework were very promising heat transfer performance in this mode of condensation. FIG. 27A and FIG. 28B show the liquid velocity and contact angle along a 45 mm (1 inch diameter tube) wick surface (micropillars with diameter = 40 pm, pitch = 100 pm and height = 200 pm). A maximum condensation heat flux of 187 kW/m ² is obtained when the maximum contact angle approaches 180°. Based on the porosity and assuming copper is used for the pillars, the effective heat transfer coefficient is 244 kW/m ² K (at a temperature difference of 0.77 K, working fluid is water). These performance values are an order of magnitude higher than the corresponding values for traditional filmwise condensation (heat transfer coefficient of 19.2 kW/m ² K and a temperature difference of 9.7 K) at the same heat flux (187 kW/m ² ) based on the Nusselt model.

[00266] The knowledge gained from capillary-driven structure development, heat transfer characterization experiments, and numerical modeling described above can guide the design of optimized capillary-driven condensation structures. Scaled-up condenser prototypes for the proof-of-concept on large tubes (-10-100 cm length) in tube bundles can be fabricated and tested using a Low Pressure Condensation Unit (LPCU).

[00267] A porous hydrophobic membrane atop a wicking structure on the condensing surface drives liquid transport and removal from the surface via a capillary pressure gradient along the wicking surface towards an exit port. Based on the success of the objectives described, a robust condenser design for steam power plants with >5x enhancement in heat transfer coefficients compared to conventional filmwise condensation was developed. Due to the improved heat transfer coefficient of condensation the steam condensation temperature, and accordingly the turbine back-pressure, can be reduced by up to 4 °C and 0.7 kPa, respectively. As a result, the overall heat rate of a typical power plant can be expected to decrease by 1.5%, leading to an additional 13.80 MW of generated power for a 950 MW plant and a commensurate savings in water withdrawal and usage. Implementation of this method has a simple payback period of roughly one year.

[00268] As observed in the condensation experiment with the hierarchical copper sample, microchannels can help with fast drainage of the condensate especially in the case where the hydrophobic membrane has defective pores which could cause local flooding/bursting out of the membrane. A few considerations should be taken when adding microchannels to the capillary-driven condenser design. First and foremost, microchannels will decrease the effective thermal conductivity of the metal wick layer as it would break the continuous connection of metal network for the efficient thermal transport. Therefore, the addition of microchannels should be just enough to prevent the membrane from flooding. When designing the geometry of the microchannels, the viscous pressure loss associated with the process of the condensed water traveling through the porous wick and finally exiting through the microchannel (route is shown as the arrow in FIG. 5) should be less than the viscous pressure loss the condensed water would experience if it exits directly through the metal wick layer without travelling through the microchannel (route as shown in the red arrow in FIG. 5).

[00269] Taking one microchannel and its neighboring metal wick block as a unit of interest. The viscous pressure loss occurred inside the porous metal wick can be described by ID Darcy’s Law:

V(x) _ kάR ^{( 26 )}

A _c mάc [00270] where x is the direction of condensate flow, k is the permeability of the metal wick, m is the viscosity of the condensed water, V (x) is the volumetric flow rate of the condensate in the direction of x, A _c is the surface area of the cross section of the metal wick layer perpendicular to the x direction. In the case of the condensed water traveling across the wick layer into the microchannel (as shown in the blue arrow), A _c = L<5 _W ; in the case of the condensed water exiting directly from the wick layer (as shown in the red arrow), A _c =

^wick^w-

[00271] According to the mass conservation of the condensed water, V(x) can be expressed as a function of the condensation heat flux q (W/m ² ) and the latent heat of water h _fS as [00272] For laminar flow (Re>2300), the viscous pressure drop occurred inside a rectangular microchannel can be calculated by 2 fpu m ² L ( 28 )

DR =

D H

[00273] where D _H is the hydraulic diameter of the rectangular channel:

4A _C ( 29 )

D _H =

R w

[00274] p is the density of the fluid, u _m is the mean flow velocity, and the Fanning friction factor / for rectangular pipe can be derived from the following equation fRe = 24(1 - 1.3553a _c + 1.9467 a ² - 1.7012a _c ³ ( 30 )

[00275] where a _c = 1/a = W _channel /^ _w is the ratio of channel width to channel width. See, for example, Shah, R.K., London, A.L., 1978. Laminar Flow Forced Convection in Ducts, Supplement 1 to Advances in Heat Transfer. Academic Press, New York, NY, which is incorporated by reference in its entirety.

[00276] For turbulent flow (Re>2300) in a rectangular channel, the Fanning friction factor can be calculated by the Blasius equation: where Re ^* is the corrected Reynolds number for rectangular channel geometries

( 32 )

[00277] See, for example, Adams, T.M., Abdel-Khalik, S.I., Jeter, M., Qureshi, Z.H., 1997. An experimental investigation of single-phase forced convection in microchannels. Int. J. Heat Mass Transfer 41 (6-7), 851-857 and Jones Jr., O.C., 1976. An improvement in the calculation of turbulent friction in rectangular ducts. J. Fluids Eng. 98, 173181, each of which is incorporated by reference in its entirety. In the preferable case where the condensed water exits through the microchannel (as shown in the arrow in FIG. 5), the mean flow velocity u _m inside the microchannel can be calculated by

[00278] Comparing the viscous pressure loss calculated by Equation (26) and Equation (28) would determine if the addition of the microchannels can help with the condensate drainage. An optimized microchannel structure should ensure that it accelerates condensate drainage such that flooding would not occur at the defected membrane pores; meanwhile, the microchannel’ s negative impact on the effective thermal conductivity of the metal wick layer should be minimized.

[00279] FIGS. 6A-6B show the integration of microchannels into a capillary-driven condenser tube. FIG. 6A shows the schematic of a capillary-driven condenser composed of a bottom layer of porous metal wick and a top layer of hydrophobic membrane with condensate drainage channel across the wick and the membrane at the bottom of the tube. With this design, condensed water needs to travel half of the perimeter of the tube to exit from the drainage channel at the bottom. The microchannels can be oriented perpendicular to a major access of the condenser tube. FIG. 6B shows the capillary-driven condenser tube with microchannels for fast drainage and with spotted drainage pores located at the bottom of the each microchannel. The microchannels help direct the condensed water to exit from the designated exit ports on the bottom of the membrane layer. Here, the c. Having the exit ports as independent holes instead of a straight channel at the bottom of the condenser tube would allow the hydrophobic membrane to keep stronger attachment to the condenser tube during the condensation process. The spacing between microchannels can be about 0.2 cm, 0.4 cm, 0.6 cm, 0.8 cm, 1.0 cm, 1.2 cm, 1.4 cm, 1.6 cm, 1.8 cm, 2.0 cm, 2.2 cm, or 2.4 cm.

[00280] A microchannel wick can be designed with microchannels with varying geometries. There can be no foam or other structure, just microchannels. The microchannels can have a height of less than 30 microns, a channel wall of less than 100 microns, and a channel pitch of less than 1 mm.

[00281] In an effort to fabricate a scalable, robust, and cost-effective capillary-driven condenser, intrinsic hydrophobic membranes were fabricated based on electrospinning. Electrospinning is a sophisticated technique for fabricating nano- and microfiber membranes, which has been applied to various industries such as water desalination, oil separation, and water harvesting. Electrospinning to directly deposit fibers atop porous metal wick to fabricate capillary-driven condensers can be used in a scalable way.

[00282] A customized electrospinning set up was built, as shown in FIGS. 7A-7B.

Two major components of the electrospinning setup are the high voltage power supply and the syringe pump. A polymer solution (PVDF-HFP in the described current test) is loaded onto the syringe and extruded from the needle tip where a positive charge is applied. For example, 15 wt% PVDF-HFP solutions in a 3:7 vol ratio of N,N-dimethylacetamide/acetone were prepared and pumped through a syringe pump which connects to the highly charged stainless steel needle via a Teflon tube. An aluminum foil covered copper plate supported by a lab jack was used as the collector and connected to ground. The temperature and relative humidity of the environment was monitored throughout the electrospinning process.

[00283] An aluminium foil covered copper plate is used as the fiber collector and is connected to a ground wire. The high voltage difference across the needle and the collector drives the polymer solution to spin onto the collector in a form of nano- or microfibers, under the counteractions of electrostatic force and surface tension experienced by the solution. The electrospinning process involves many operation parameters, some of which are critical to the fiber/membrane fabricated by the process. Some key parameters include voltage being applied, distance between the needle and the collector, the solution feeding rate, the properties of the polymer solution, and the time the electrospinning process lasts. Without limitation, electrospinning PVDF-HFP was initially studied since this polymer is intrinsically hydrophobic and is known to form uniform membrane pores.

[00284] Three properties of the membrane are critical to vapor transport and heat transfer through the membrane layer: pore size, thickness, and porosity. For electrospun fibrous membranes, the pore size can be proportional to the fiber diameter. In order to characterize these important properties, several techniques were used as detailed below. [00285] Scanning electron microscopy (SEM) images were taken to help us characterize the morphology of the electrospun fibers, as shown in FIGS. 8A-8D. SEM images could clearly show important information of independent fibers (e.g., fiber diameter, presence of beads) and how the fibers are being aligned into a membrane. However, due to the random alignment of the electrospun fibers, it was hard to characterize pore size of the membrane using SEM. However, pore size distribution is important information needed in order to predict the capillary pressure that the membrane can generate and the vapor transport resistance through the membrane. Therefore, in addition to the SEM, a capillary flow porometry (POROLUX™ 1000) was used to characterize pore size distribution of the electrospun membrane. Fiber diameter can be positively correlated to the membrane pore size for the electrospun membrane samples that were fabricated.

[00286] FIG. 9 shows how a capillary flow porosity works to get the pore size distribution. The sample membrane was first wetted with a known fluid (Galpore, surface tension 15.6 mN/m) and put into the porometer for the wet curve measurement. During the measurement, a nitrogen flow was applied to the membrane sample with incremental pressure difference across the membrane. At each pressure, flow rate of nitrogen gas was recorded. At the beginning of the measurement, flow rate stayed at zero since all the pores were blocked by the liquid. At one point, nitrogen started to flow through the membrane, which denoted that the biggest pore inside the membrane (i.e. bobble point pore size) was opened up by the gas flow. After this point, the nitrogen gas flow rate would keep increasing until a point when the smallest pore inside the membrane was open, i.e., the membrane became completely dry. Following the wet curve measurement, another round of gas flow rate measurement was repeated on the dry sample with the same incremental pressure being applied, resulted in a dry curve. A curve with 50% of the dry curve slope was plotted as “half-dry curve”, as shown in FIG. 9. Pore size at which 50 % of the total gas flow can be accounted (half the flow is through pores larger than this diameter) is defined as the mean flow pore (MFP) diameter. Pore size distribution could be extracted from the wet curve and dry curve measurements, as shown in FIG. 10.

[00287] The thickness of the membrane was measured by a micrometer with a resolution of 1 pm. The average of three independent thickness measurements was calculated at the thickest part of the electrospun membrane and used it as the membrane thickness. In addition, the membrane weight and the membrane surface area were measured to calculate membrane porosity. A consistent porosity of - 80% was obtained through all of the PVDF- HFP samples, which is in good agreement with the literature. See, for example, Ahmed, F.

E., et ak, DES 356, 15-30 (2015), which is incorporated by reference in its entirety.

[00288] An optimized parametric study on the electrospinning process based on fractional factorial design (FFD) was completed. Investigating the effects of each independent parameter (full factorial design) tends to be laborious and redundant. In statistics, fractional factorial design (FFD) is a powerful tool for reducing the experimental burden while recognizing the major effects. FFD consists of a carefully chosen subset (fraction) of the experimental runs of a full factorial design, identifying the most important parameter to the response of interest. The effects of multiple fabrication parameters on the morphology of the electrospun membrane can be studied efficiently by FFD. FIGS. 11 A-l IB show an example of the reduced number of combinations in a fractional factorial design. See, for example, T. Berling & Runeson, 2003. P. Efficient evaluation of multifactor dependent system performance using fractional factorial design. IEEE Trans. Softw. Eng. 769-781, which is incorporated by reference in its entirety. The filled circles in FIG. 11A correspond to the eight runs in the calculations in a full factorial design. The filled circles in FIG. 1 IB correspond to the four performed runs of the eight possible in the calculations in a fractional factorial design. FIGS. 11 A-l IB represent a schematic representation of a full 8-run factorial design with three parameters in comparison to a reduced 4-run fractional factorial design. + or -indicates if the parameter is being maximized or minimized.

[00289] The FFD method used in this study varied factors together and fitted the results using a polynomial method. It could determine how factors interact but could have errors due to polynomial assumption. Therefore, FFD was used for screening parameters before moving on to full factorial study on the most important parameter.

[00290] During the screening study, the effects of the three important fabrication parameters were studied, namely the voltage supply, the discharge/needle distance, and the syringe pump feeding rate on the morphology of the electrospun membrane. Table 8 shows the selected key parameters studied and their variation range. The variation range for each parameter was empirically determined to ensure successful electrospinning without bead formation. The duration of the electrospinning was held constant (60 minutes) and the solution concentration unchanged (15% wt PVDF-HFP in 3:7 vol ratio of N,N- dimethylacetamide/acetone) throughout all runs of experiments. The environmental temperature and humidity was controlled at around 23 °C and 30% respectively. The electrospun fiber was left in a well ventilated area overnight for complete evaporation of the solvent before SEM imaging.

Table 8 [00291] Table 9 shows the subset of the experimental runs guided by the fractional factorial design (solution concentration 15 wt% and electrospinning time of 60 minutes). The fractional factorial design simplified a full 8-run parametric study into 4 runs. Each set of parameters were run though at least 2 times under similar environmental conditions and the morphology of the resulted membranes was characterized with SEM and a capillary flow porometer.

Table 9

[00292] FIGS. 12A-12F show the characterization results for two electrospun PVDF- HFP samples. FIGS. 12A-12C show the SEM image, the fiber diameter distribution, and the membrane pore size distribution of one electrospun PVDF-HFP sample, and the FIGS. 12D- 12F show the SEM image, the fiber diameter distribution, and the membrane pore sizedistribusion of another sample electrospun with a different set of parameters. It shows that fiber diameter is positively correlated to the membrane pore size of the electrospun membranes. Therefore, fiber diameter can be used as an indicator of the membrane pore size, which is critical to the capillary pressure the membrane is able to generate and thus is important to the design of the capillary-driven condenser.

[00293] FIG. 13 shows the parametric study on the effects of the needle distance, voltage supply, and solution feeding rate on the fiber diameter of the electrospun PVDF-HFP membrane guided by fractional factorial design (FFD). FFD simplifies the parametric study and guides the optimization of the fabrication. As shown by the results, solution feeding rate has the most significant impact on the fiber diameter of the electrospun PVDFD-HFP membrane. Fiber diameter increases with decreasing solution feeding rate. The membrane pore size was expected to also increase with decreasing solution feeding rate as a result of the positive correlations between the fiber diameter and the membrane pore size. Voltage supply has a moderate impact while needle distance has a negligible impact on the fiber diameter. [00294] To optimize the fabrication of the electrospun PVDF-HFP membrane for the best performance of the capillary-driven condensation, membrane pore size needs to be optimized. A larger membrane pore will allow easier vaport transport, but the pore size should not exceed the maximum limit of the membrane pore size at which the capillary pressure cannot hold the condensed water in place and cause flooding. FFD tells us that the solution feeding rate is the most important parameter among the three key paratemers that were studied (voltage supply, needle-to-collector distance, and feeding rate). Future work entails a detailed full factorial study on the effect of solution feeding rate on the electrospun membrane pore size. The membrane thickness was not studied here because membrane thickness is a very tunable characteristic by simply increasing the duration of electrospinning to increase membrane thickness. Moreover, the effects of the duration of electrospinning on the fiber diameter were studied and demonstrated that there is little correlation between the duration of the electrospinning time and the resulted fiber diameter, as shown in FIG. 14. [00295] Effects of solution aging on fiber diameter was studied. FIG. 15 is a graph depicting effects of aging of the PVDF-HFP solution on the fiber diameter of the electrospun membrane. The solution was kept stirring on an 80 G hot plate. Two sets of parameters were tested on Day 1, 2, and 3 after the solution was prepared. Fiber diameter consistently decreases with time, indicating the aging of the solution. Therefore, it is important to start electrospinning with freshly prepared solution to prevent any aging effect.

[00296] Getting a strong bonding between the electrospun membranes and the porous metal substrates can be challenging. In a preliminary test, electrospun membranes directly deposited on untreated substrates could be peeled off the substrates easily. However, it was found that solvent clean treatment and plasma clean treatment on the substrates could enhance the attachment between the electrospun fibers and the metal substrates. For this preliminary test, copper pillar surfaces were used as the porous metal substrate for easier imaging. These are square-shape copper pillars with lOOpm height, lOOpm neighborhood distance and 100pm side length on a 3mm thick copper substrate. The substrates were cleaned with acetone, ethanol, isopropanol and water, dried the substrates with nitrogen flow, and then plasma cleaned the substrates with oxygen plasma for 10 minutes. These samples were used for electrospinning right after the cleaning process. The resulting electrospun fibers showed larger preference to attach to the metal surfaces than to the fiber themselves, as shown in FIG. 15 A. However, these fibers could detach from the metal substrate as a discrete layer when the membrane became too thick, as shown in FIG. 15B. [00297] Heat treatment was found to be an effective approach to enhance the bonding between the electrospun membrane and the porous metal substrate. During a preliminary study, the porous copper substrate was put on a grounded hot plate and during the first 5 minutes of electrospinning heated the substrate to 160°C, which is slightly higher than the melting point (143°C) of the polymer used for electrospinning (PVDF-HFP). By heating up the substrate, it was expected the electrospun fibers deposited on the substrate during the first few minutes to melt and form better and more uniform bonding with the porous metal substrate, after which the electrospun fiber could form a strong bonding to the substrate driven by the attraction between PVDF-HFP fibers without the need of the heat. It was also found that the geometry of the metal wick has a significant effect on the bonding between the electrospun membrane and the metal wick. Electrospun PVDF-HFP membranes were formed on two different substrates with the same heat treatment (kept of a 160 °C hot plate during the first 5 minutes of electrospinning). Better bonding was formed on the 200 pm pore size copper foam than the membrane spun on the 100x100 size copper mesh (200 pm equivalent pore size).

[00298] Based on the above preliminary experiments, the following strategy for fabricating the capillary-driven condenser tube in a scalable way can be followed:

[00299] Fabricating the porous metal wick layer by template-assisted electrodeposition or sintering. The type of the metal being used in this step is not fixed by should be compatible with the tube material. Conventional materials being used in condensers include copper, copper-nickel alloy, stainless steel, etc.

[00300] Machining of exit ports or channels on the porous copper wick layer. This step can be integrated into the first step or post-processed after the first step. Depending on the geometry for the channels, different tools can be used.

[00301] Adding a layer of hydrophobic membrane on top of the porous metal wick. It is believed electrospinning has a high potential to customize the hydrophobic membrane in the current application. Securing the porous wick covered condenser tube on a rotating system (similar to the electrospinning drum collector) would allow the electrospun fiber to be uniformly deposited onto the tube surface. On a tube geometry, the bonding of the electrospun membrane would be secured by the alignment of the nanofibers. Heating the tube substrate above the melting point of the polymer being electrospun during the first few minutes of electrospinning can further enhance the bonding between the membrane and the porous metal wick layer. [00302] Machining of exit ports or channels on the membrane layer. This can be done by simply locally cutting or melting the membrane.

[00303] Mechanical clamping can be added to further secure the membrane in place. [00304] FIG. 18 shows the configuration of the exit port. In the conventional configuration, the exit port was shown to be located at the lower section of the condensation pipe. In this configuration, the fluid transport length Lf in the wick was given by half the circumference of the tube (due to a line of symmetry — as shown in the figure). For example, a conventional tube size in power plant condensers is 1 inch, and thus Lf is around -1.57 inch or - 4cm. However, depending on the design of the structure, gravity typically does not play a significant role since the bond number (Bo) — which characterizes the ratio of gravitational forces to surface tension forces — is small (Bo « 1). Therefore, the exit port does not necessarily have to be located at the bottom of the tube (FIG. 18). Moreover, the configuration in FIG. 18 has design limits since the fluid transport length L is determined by half of the circumference of the tube. Therefore, a design whereby the length Lf can take on any reasonable value which would be advantageous to heat transfer is proposed. Specifically, the flooding of the structure, which is the escape of fluid inside of the wicking structure through membrane pores to the top surface of the membrane in the form of drops or puddles, which is caused by the inability of the capillary pressure of pinned condensate at membrane pores to sustain the viscous pressure drop that must be surpassed in order to drive fluid flow to the exit port, has to be avoided by design. However, the constrain ton a fixed Lf means less design space for the geometry of the membrane and the wicking structure. Therefore, a configuration which relaxes the tight constraint on Lf can be beneficial. To understand the tradeoffs better, Equation (34) is introduced which is a pressure “budget” term that can predict the incipience of flooding.

[00305] This equation shows the competition between the capillary pressure P _cap which is a function of the pore diameter d _p against the viscous pressure drop in the wick AP _wicfe which is a function of the fluid transport length Lf. and the wick geometry. Note that the viscous pressure drop is proportional to the square of the fluid transport length in a simple model utilizing the Darcy equation for flow in porous media. Therefore, flooding is strongly dependent on this term. In the case of a micropillar wick, the membrane geometry is characterized by h and d/l, which are the thickness of the wick (pillar height) and the pillar diameter to pillar pitch ratio, respectively, whose quantities determine the pore size of the wick, and its porosity.

[00306] FIG. 19 shows a design whereby L can be made independent of the tube geometry and can independently tailor it to relax constraints on the design of the geometry of the wicking structure (thickness of the wick, pore size, porosity etc.) and the membrane (thickness, pore size, porosity). This means the structure can be tailored, for example, to have larger membrane pores that can be designed to still avoid flooding and which are more readily available commercially, and thinner wicking structures to reduce the thermal resistance yet still avoiding flooding. Note that based on equation 1, the constraint on Lf in the viscous pressure drop term can be relaxed and thus have more flexibility on the overall design.

[00307] If elasticity is utilized, a membrane can wrap around the tube and be held by tension without the need to have adhesives for attachment. Moreover, mechanical clamping can be utilized in some configurations to avoid the necessity to utilize bonding methods which can require adhesives or multistep coatings to achieve stronger bonding. For example, the membrane sections can be held at the bottom by stitching, mechanical clamping or other method that does not require adhesives. By eliminating the requirement to bond a membrane to the wicking structure in the conventional way of using adhesives, the membrane only needs enough clamping force to sustain the required capillary pressures in its membrane pores during operation. All materials in this category should be stable in a steam environment.

[00308] FIG. 20 shows heat transfer measurements on capillary-driven condensation samples in a pure vapor ambient. The error bars on the measurement are large due to the large oscillation of the dry bulb vapor temperature in the chamber, and the difference between the dry-bulb temperature, and wet-bulb temperature. Therefore, a specific value for the enhancement as of this point has not been defined at this point, but in general that there is enhancement above the filmwise condensation that can be seen to range ~ 3x to 4x for some values. Moreover, no difference in the performance between the designed samples was captured, possibly due to the large errors involved in the experimental setup and lack of sensitivity to slight differences in measurements. Future experiments will aim to reduce this difference and utilize pressure measurements to reduce the error of the saturated vapor temperature and obtain finer details on the possible heat transfer enhancements.

[00309] Modelling predicts ~ 4x to 12x heat transfer enhancements compared to filmwise condensation. By comparison, dropwise condensation is known to have ~5x - ~7x enhancements measured experimentally. The pore size can range from about 1 - 20 microns. Commercial membranes are typically thicker there will be a larger vapor transport resistance. However, increasing the pore size also allows better vapor transport. This can mean less capillary pressure to sustain fluid flow in the wick.

[00310] Thus, the model can predict enhancements beyond those experimentally measured for dropwise condensation.

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[00312] The following United States patent documents as also incorporated by reference in its entirety:

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[00313] It should be understood that the subj ect matter defined in the appended claims is not necessarily limited to the specific implementations described above. The specific implementations described above are disclosed as examples only.