Heat Transfer Enhancement During Water and Hydrocarbon Condensation on Lubricant Infused Surfaces

Vapor condensation is routinely used as an effective means of transferring heat or separating fluids. Dropwise condensation, where discrete droplets form on the condenser surface, offers a potential improvement in heat transfer of up to an order of magnitude compared to filmwise condensation, where a liquid film covers the surface. Low surface tension fluid condensates such as hydrocarbons pose a unique challenge since typical hydrophobic condenser coatings used to promote dropwise condensation of water often do not repel fluids with lower surface tensions. Recent work has shown that lubricant infused surfaces (LIS) can promote droplet formation of hydrocarbons. In this work, we confirm the effectiveness of LIS in promoting dropwise condensation by providing experimental measurements of heat transfer performance during hydrocarbon condensation on a LIS, which enhances heat transfer by ≈450% compared to an uncoated surface. We also explored improvement through removal of noncondensable gases and highlighted a failure mechanism whereby shedding droplets depleted the lubricant over time. Enhanced condensation heat transfer for low surface tension fluids on LIS presents the opportunity for significant energy savings in natural gas processing as well as improvements in thermal management, heating and cooling, and power generation.


S1. Results from Xiao et al. Compared to Filmwise Condensation from Nusselt's Model
Xiao et al. reported that "the overall heat transfer coefficients on [dropwise condensation] surfaces in this work (h < 2-7 kW/m 2 K) are much lower compared to pure vapor conditions [in previous work by Miljkovic et al.] (h < 12-13 kW/m 2 K) 1 due to the presence of NCGs acting as a diffusion barrier to the transport of water vapor towards the condensing surface." 2 Therefore, the fact that the noncondensables (NCGs) would degrade heat transfer performance was recognized. Unfortunately, the data reported is the overall heat transfer coefficient including the tube wall and the chiller water flow, as opposed to the isolated condensation heat transfer coefficient. We can approximate the condensation heat transfer coefficient (including NCG effects) by assuming that the effective heat transfer coefficient of the non-condensing sections of the resistance network, h fixed , namely, the tube wall and the chiller water flow, remains fixed between experiments (see Figure 1 in the main text for a schematic of the resistance network): we can also determine the vapor temperature, T v , at each vapor pressure for a saturated mixture, Figure S1. Adjusted Figure   S1, where the heat transfer is comparable to that which would be expected from filmwise condensation.

S2. Droplet Departure Size Characterization
The average droplet departure size was calculated using imaging analysis performed on videos of condensation on the different surfaces with at least 20 departure events observed for each sample. The results are shown in Table S1.

S3. Condensation Chamber Setup
The custom environmental chamber used for this work consists of a stainless steel frame with a door (sealed with a rubber gasket), two viewing windows, and apertures for various components.
Resistive heater lines were wrapped around the exterior of the chamber walls to prevent condensation at the inside walls and then insulated on the exterior walls. The output power of the SI-5 resistive heater lines was controlled by a voltage regulator. Two insulated stainless steel water flow lines (Swagelok) were fed into the chamber to supply cooling water to the chamber from a large capacity chiller.
A secondary stainless steel tube line was fed into the chamber that served as the flow line for the incoming vapor supplied from a heated steel reservoir. The vapor line was wrapped with a rope heater (60 W, Omega) and controlled by a power supply. The vapor reservoir was wrapped with another independently-controlled heater (120 W, Omega) and insulated to limit heat losses to the environment. The access tubes were welded to the vapor reservoir, each with independently-

S4. Condensation Procedure
For each experimental trial, a set of strict procedures was followed to ensure consistency throughout the experiments. The first step of the process was to turn on the voltage regulator to heat up the environmental chamber walls, which prevented condensation on the chamber walls.
Simultaneously, the vapor reservoir was filled with approximately 2 liters of either DI water or toluene. After opening the vapor inflow valve and closing the vapor release valve, the rope heater around the vapor reservoir was turned on with the heater controller set to maximum output (1200 W). Then the rope heater connected to the vapor inflow valve was turned on. The temperature of the reservoir was monitored with the installed thermocouples; the temperature at SI-7 the top of the reservoir was higher than that of the middle/bottom of the reservoir due to the water thermal mass present at the middle/bottom section. Hence, we ensured that the regions of the reservoir of higher thermal capacity were brought to a sufficiently high temperature for boiling. During the boiling process, aluminum foil was placed on the bottom surface of the inner chamber to collect any of the liquid leaving the vapor inflow line. Once boiling was achieved and the internal thermocouple on the reservoir was 5˚C above the boiling point for at least 10 minutes, the vapor inflow valve was closed. The excess fluid that spilled inside the chamber during de-gassing of the reservoir was removed.
In order to install the samples onto the rig, the Swagelok female adapters at the ends of the tube samples were connected to the male connecters on the rig. Before installing the entire sample setup in the chamber, all adapters/connecters were tightened to ensure that there were no leaks that could affect vacuum performance. Finally, the bellows tubes (for the chiller water inflow/outflow) were connected to the chiller water lines.
The next step was to begin the vacuum pump-down procedure. Initially, the liquid nitrogen cold trap was filled to about half capacity. The ambient exposed valves connecting the chamber and the vacuum pump were both closed and the valve connected to the liquid nitrogen cold trap was opened. The vacuum pump was then turned on, initiating the pump-down process. The pressure inside the chamber was monitored during the pump-down process. This process took approximately one hour in order to achieve the target vacuum conditions (0.5 Pa < P < 1 Pa).
The experimental operating pressure of non-condensable was set to be a maximum of 0.25% of the operating pressure. Non-condensable gas content of above 0.5% (pressure) has been shown SI-8 to significantly degrade performance during dropwise condensation. 3 In our experiments, extreme care was taken to properly de-gas the vacuum chamber and vapor reservoir prior to experimental testing. In addition, the chamber leak rate was characterized prior to each run in order to estimate the maximum time available for acquiring high fidelity data with noncondensable content of less than 0.25%.
The setup of the chiller water flow-loop is described as follows: the water pump reservoir was filled and turned on to a flow rate of 5 L/min. The flow rate was monitored with the flow meter integrated in the inflow water line. In order to bring the chilled water into the flow loop and to the tube sample, the external chilled water lines were opened.
Prior to beginning experiments, the camera was turned on for visual imaging of the sample during condensation. Afterwards, the rope heater around the water reservoir was turned off and the vapor inflow valve was slowly turned open until the operating pressure was reached. Steady state conditions were typically reached after 2 minutes of full operation.

S5. Heat Transfer Coefficient and Error Propagation
An energy balance was applied to the tube sample to determine the overall condensation heat transfer by calculating the change in enthalpy of the chiller water flowing inside the tube: where Q is the overall condensation heat transfer rate, ̇ is the chiller water mass flow rate, is the chiller water specific heat, and T in and T out are the tube condenser inlet and outlet temperatures, respectively. From the overall heat transfer rate, we calculated the heat flux by dividing by the condenser surface area: where A is the surface area of the outer tube surface (A = 2πrL, where r = 3.175 mm, L = 13.1 cm). The temperature difference between the chiller water and vapor far from the tube sample was also determined, represented here as the log mean temperature difference (LMTD) to account for the change in temperature of the chiller water along the tube length: where T v is the temperature of the surrounding vapor far from the tube sample (T v = T sat (P v )).
From the overall condensation heat transfer and the log mean temperature difference, the overall heat transfer coefficient, ̅ , was determined: where A is the surface area of the outer tube surface (A = 2πrL, where r = 3.175 mm, L = 13.1 cm). Note that the overall heat transfer coefficient is a function of only the product of experimentally measured parameters raised to powers. Therefore, the error associated with ̅ is calculated as follows: The condensation heat transfer coefficient can be extracted from ̅ by considering a series of thermal resistances that sum to ̅ and isolating the resistance associated with condensation: Rearranging to solve explicitly for h c : where A i is the surface area on the inner surface of the tube (A i = 2πr i L), R t is the thermal resistance of the tube (R t = ln(r/r i )/(2πk t ), k t is the tube material thermal conductivity), and the internal heat transfer coefficient, h i , is determined from the Gnielinski correlation for pipe flow: SI-11 ℎ = ( 2 ) ( /8)( − 1000) 1 + 12.7( /8) 1/2 ( 2/3 − 1) (S11) = (0.790 ln − 1.64) −2 (S12) where f is the friction factor, Re is the Reynolds number, Pr is the Prandtl number, ρ is the chiller water density, k i is the chiller water thermal conductivity, and µ is the chiller water dynamic viscosity. Solving for h i and substituting into the above Equation S10 allows for determination of h c . Once h c is known, the condenser surface subcooling is determined as follows: As h c is not a simple function of a product of powers, the error is determined as a function of the first partial derivatives of h c with respect to its components.
The error in ̅ was determined in Equation S8 and the error in h i was estimated as 10% associated with the Gnielinski correlation. 4 Table S2 below summarizes the uncertainty associated with each experimental measurement.
where q" is the steady state dropwise condensation heat transfer rate per unit area of the condensing surface, ΔT is the temperature difference between the saturated vapor and sample outer surface (ΔT = (T sat (P) -T s )), R * is the critical radius for heterogeneous nucleation (R * = r c ), 6 R c is the droplet coalescence radius, q(R) is the individual droplet heat transfer (Equation S19), n(R) is the non-interacting droplet size distribution, 5 N(R) is the coalescence dominated droplet size distribution, 5, 7 R is the droplet radius, σ is the condensate surface tension, h fg is the latent heat of phase change, ρ w is the condensate density (liquid water), θ is the droplet contact angle, h int is the interfacial heat transfer coefficient, 8 k w is the condensate thermal conductivity, k HC is the hydrophobic coating thermal conductivity, φ is the structured surface solid fraction (equal to one for the flat surfaces considered here), h is the structured surface height (equal to zero for flat surfaces), and δ HC is the hydrophobic coating thickness (≈1 nm). 6 The first integral in Equation S14 represents the heat flux component from droplets smaller than the coalescence SI-14 length scale (R < R e ), where direct growth by vapor accommodation at the liquid-vapor interface dominates and neighboring droplet coalescence is absent. The second integral represents the component of the heat flux from droplets growing mainly by coalescence with other droplets (R > R e ). These two components contribute to the total surface heat transfer per unit area (q"). The model results were obtained using experimentally determined droplet departure radii Ȓ (see Table S1, above) and contact angles.
Note that the equation above is modified on the lubricant infused surface because lubricant is between the structures on the surface. The appropriate equation for the LIS is: where the k w has been changed to k l in the right-most section of the denominator. Another where g is the gravitational acceleration (g = 9.81 m/s 2 ), ρ v is the water vapor density, µ w is the condensate dynamic viscosity, h' fg is the modified latent heat of vaporization accounting for the change in specific heat of the condensate, and c p,l is the condensate specific heat. 4,8 An example of the dropwise and filmwise heat transfer coefficients as a function of condenser subcooling yielded from the above models is plotted in Figure S2

SI-17
In the present study, the condenser subcooling during dropwise condensation of water ranged from ≈1.5-6 K. The experimentally determined heat transfer coefficient enhancement of 3-4x for dropwise condensation of water on flat hydrophobic and LIS-coated tubes compared to filmwise condensation is in excellent agreement with the model shown in Figure S2 in the 2-5 K subcooling range where the experimental measurements were taken. Note that, at higher subcooling over ≈10 K, the typically reported heat transfer coefficient enhancement of up to one order of magnitude 3b for dropwise compared to filmwise condensation would be realized.

S7. Lubricant Layer Thickness
The lubricant layer thickness was approximated by dividing the total volume of lubricant added to the surface by the projected surface area of the condenser tube. The total volume of added lubricant was calculated with Equation S25 based on the procedure described in the main text: where n is the number of droplets of lubricant added to the surface (1 droplet in the present work) and r lub is the radius of each droplet of lubricant added to the surface (1 mm in the present work). The surface area of the condenser tube is: where r tube and L tube are the radius and the length of the condenser tube, respectively. Then, the thickness of the lubricant layer is: In the present work, this lubricant layer thickness is 1.6 μm, on the same order as the copper oxide nanostructures, which have a characteristic height of 1 to 2 μm. 12 Therefore, the lubricant completely filled the nanostructured surface without significant excess. The thermal resistance SI-18 of the lubricant layer was also estimated to determine its effect on the overall heat transfer performance. The thermal resistance is: where k lub is the thermal conductivity of the lubricant, ≈ 0.09 W/m-K for Krytox GPL 101.
Therefore, in the present work, the thermal resistance of the lubricant is approximately 1.8x10 -5 m 2 K/W. Comparing this to the thermal resistances for filmwise condensation of water (8.2x10 -5 m 2 K/W) and toluene (5.1x10 -4 m 2 K/W), it is clear that, even with the lubricant layer, we can expect a maximum heat transfer enhancement of up to 4.5x for water and 28x for toluene compared to filmwise condensation based on this simplified analysis. In fact, the actual maximum enhancement is greater due to the relatively high thermal conductivity of the copper oxide nanostructures compared to the lubricant, which essentially form a composite material with the lubricant and significantly lower the thermal resistance.

S8. Transition to Filmwise Condensation of Toluene
Although toluene has a finite positive contact angle on the flat hydrophobic condenser, the low value of contact angle and high contact angle hysteresis result in a transition to filmwise condensation even at very low subcooling of less than 1 K, shown in Figure S3 below. This phenomenon has been reported previously, where low-contact-angle fluids typically condense in the filmwise mode. 13 SI-19 Figure S3. Transition from dropwise to filmwise condensation for toluene condensing on the flat hydrophobic surface. These time-lapse images show that discrete droplets initially form and grow on the surface; however, due to the low contact angle and high contact angle hysteresis, the condensation mode transitions to filmwise even at low (<1 K) subcooling.