Self-supervised learning of shedding droplet dynamics during steam condensation Open Access

Characterization of vapor-to-liquid phase change processes can be challenging due to the demand for special consideration of measurement protocols and sensor accuracies.¹ For external condensation on flat surfaces, the conventional approach for heat transfer measurement involves using thermocouples inserted into the sample on which the condensation is happening. Assuming one-dimensional heat conduction, Fourier’s law is applied to estimate the steady-state condensation heat flux.^2,3 On condenser tubes, temperature sensors are inserted at the inlet and outlet of the coolant test section to measure the coolant temperature difference due to the condensation occurring on the outer surface.^4,5 Capturing the droplet dynamics during condensation yields useful information regarding condensation heat and mass transfer.^6,7 Statistical analysis of the droplet dynamics during condensation helps to extract meaningful physical features and gain insights into their relationship with condensation heat flux and heat transfer coefficient.^8,9

Recently, research has been directed toward elucidating the correlation between heat transfer performance and various parameters governing droplet dynamics, such as droplet diameter, shedding velocity, droplet number density, and droplet formation mechanism.^1,8,10 In filmwise condensation (FWC), a thin liquid film is formed and grows in thickness along the gravitational direction.¹¹ The FWC heat transfer efficiency is strongly dependent on removing the liquid film and keeping it as thin as possible.¹² Droplet dynamics are more complex during dropwise condensation (DWC) and jumping droplet condensation.⁵ During DWC, nano-scale droplets are formed at nucleation sites on the surface. These droplets grow, may merge or coalesce, and shed from the surface when they reach the critical size, depending on interfacial energy and surface orientation in the presence of gravity.^13,14 Condensation on superhydrophobic surfaces offers an additional gravity-independent droplet removal mechanism known as jumping droplet condensation.¹⁵ 

Multi-scale droplet dynamics during the nucleation, growth, coalescence, and shedding stages is intricately related to the fluid-surface interfacial tension and surface morphology at different length scales.¹⁶ Within this context, researchers have used numerical simulations to develop drop-size distribution models predicting the possibility of coalescence, showing that coalescence possibility is higher for smaller droplets during jumping droplet condensation, resulting in higher heat fluxes.¹⁷ Others have shown that smaller droplets have a higher jumping probability, which can enhance heat flux.¹⁸ Surface orientation plays a notable role in the performance of jumping droplet condensation.¹⁹ Theoretical analysis of jumping droplet condensation on inclined surfaces revealed up to a 100% enhancement in heat transfer coefficient compared to the horizontal orientation due to a significant decrease in the maximum droplet size.¹⁹ Droplet coalescence, in conjunction with gravity, can remove mid-sized droplets that would otherwise remain trapped on the surface due to surface tension if gravity is acting alone.¹⁹ The heat transfer coefficient is higher for dropwise condensation on a vertical condenser since the departing droplet size decreases, enabling a larger fraction of the dry surface area for the steam to condense.²⁰ Essentially, coalescence aids in detaching larger droplets from their pinned necks, substantially reducing the critical size required for gravitational shedding. Additionally, the heat transfer coefficient decreases as the maximum droplet diameter increases on the surface.²⁰ 

Despite the high number of recent studies on droplet dynamics during condensation,^16–19,21 the majority have focused on steam condensation on flat surfaces. Furthermore, few have examined the relationship between droplet characteristics like size and velocity to heat transfer parameters such as heat flux and heat transfer coefficient.^22,23

The shedding velocity of droplets is another significant parameter that has remained under-explored in external condensation. Multiple coalescence-induced droplets (>2) merging together can lead to higher falling droplet velocity on flat superhydrophobic surfaces.²⁴ Theoretical studies of the relationship between droplet velocity and the droplet merging process unveiled that introducing more droplets into the merging process results in increased velocity for each droplet, consequently leading to a smaller falling droplet radius.²⁵ Similarly, studies of droplet dynamics, including departing size and velocity on biphilic tubes, have shown that droplet dynamics and heat transfer are strongly dependent on the spacing between stripes of the same wettability (e.g., more hydrophilic or hydrophobic regions).¹⁰ 

While providing useful information, studying droplet dynamics during condensation on tube surfaces is a cumbersome and time-consuming task. Obtaining departing droplet size and velocity is of great importance and is the fundamental step to deriving the relationship between these parameters and condensation heat transfer. However, extracting these parameters manually is labor-intensive and subject to human bias and error. In addition, this task is not scalable to many samples. Therefore, it is important to develop an automatic method for extracting departing droplet size and velocity from a variety of tube samples.

Recent advances in computer vision and machine learning have provided a great tool for studying phase change heat transfer phenomena.^26–28 Previous studies have used these methods to extract meaningful physical features such as droplet departure frequency, droplet distribution, and growth time during condensation, as well as bubble statistics such as size and count during boiling.^29,30 One major challenge in developing supervised computer vision deep learning models is the preparation of labeled training datasets. When using these models for the segmentation of the droplets, a large enough annotated dataset (e.g., masked droplets) should be provided to the model to learn from. This step is generally very time-consuming, and not enough annotated data are available in some cases. This could be a real bottleneck for the practical deployment of vision-based deep learning models.³¹ Self-supervised methods aim to alleviate the data annotation bottleneck by removing or limiting the manual data annotation step. Recently developed self-supervised methods for object segmentation rely mostly on representation learning from a huge unlabeled dataset and a final fine-tuning step on a small annotated dataset.^32–34 Other common self-supervised techniques create synthetic data and labels using generative models (e.g., GANs) or data augmentation techniques and use synthetic labels along with the real data to learn the fine segmentation in a supervised fashion.^35,36

Here, we developed a simple approach by taking advantage of a small “high-quality” dataset along with classical image processing and clustering techniques to create masked droplet images automatically. The smaller datasets along with their masks are then subjected to data augmentation techniques to create a larger dataset (>100 images), which is used in training a deep fully convolutional network for droplet segmentation. The trained model is then used to obtain shedding droplet size and velocity during condensation on a variety of tube sizes and surface morphologies. The droplet dynamics results are used to develop a data-driven model to find the relationship between droplet dynamics, and tube size, surface properties, and heat flux.

Pure steam condensation experiments were conducted with the test rig depicted in Fig. 1. High-speed imaging data required for the computer vision model, as well as temperature and pressure sensor signals, were collected during the experiments. Comprehensive details about this experimental facility can be found in previous work.¹ Briefly, the setup comprises a custom-designed stainless-steel vacuum/pressure chamber fabricated by Gladwin Tank with an internal diameter of 30.5 cm. The chamber can accommodate straight tubes up to 30 cm in length and is sealed at both ends with 4.5 cm thick stainless-steel flanges designed to withstand an internal pressure of 5.06 MPa. These flanges are effectively sealed using Neoprene rubber gaskets (Parker O-rings, part No. 2-280) immersed in vacuum grease (Dow Corning part No. 146355D) to minimize the penetration of non-condensable gases (NCGs) into the chamber during the experiments. The chamber is equipped with several feedthroughs, allowing for the installation of thermocouples, resistance temperature detectors (RTDs), fluidic lines, and pressure transducers. Two independent pressure transducers, a Micro-Pirani from MKS Instruments and a Baratron 728A, are installed to monitor the pressure within the chamber. Six optical viewports (five with a 5.08 cm diameter from MPF Products and one with a 6.35 cm diameter from MDC Vacuum) are installed on the chamber for visualization purposes. These viewports are designed to withstand internal pressures up to 2.75 MPa and operate in vacuum conditions. Prior to each experiment, the chamber is evacuated using a vacuum pump (Model Adixen Alcatel 2005) to remove NCGs. A liquid nitrogen cold trap (Model No. TLR4XI100QF, Kurt J. Lesker) is incorporated in the vacuum line between the chamber and the pump to eliminate moisture from the air and improve the vacuum quality.

The steam supply system (standard CF TEE from Kurt J. Lesker with a 20.3 cm outer diameter) is filled with deionized (DI) water. The water is heated using three tape heaters (Part No. AWH-101-040DP, ETS Equipment) installed on the outer surface of the vapor generator. The heating rate of the heaters is controlled by a variable voltage regulator (Model PM-1220B, ETS Equipment). To prevent over-pressurization, the fluid temperature is monitored using a T-type thermocouple (Part No. SCPSS-032, Omega). The vapor generator is equipped with four ports on top, which are connected to the main chamber, a pressure relief valve set at 300 kPa (Part No. SS-RL4S8, Swagelok), a vapor vent line for pressure control, and a fill line used to introduce deionized (DI) water before starting the test.

In the coolant flow loop, tap water is cooled using a large capacity chiller (Part No. 327005091602, System III TU7 Pump, Thermo Fisher Scientific) and directed to the test section inside the chamber. The flow rate of the coolant is measured using an electromagnetic flow meter (Part No. FMG93, Omega), while the inlet and outlet temperatures are monitored using two feedthrough RTDs (Part No. AT-PX1123Y-LR4S1T2T, ReoTemp). To minimize condensate formation on undesired surfaces and limit it to the test section, all auxiliary connections and tubing inside the chamber are insulated. Additionally, tape heaters (Part No. AWH-101-040DP, ETS Equipment) are wrapped around the chamber to heat it and prevent condensation on the internal side of the chamber wall and viewports. The vapor pressure is controlled by adjusting a manual valve (Model No. 6L-LD8-BBXX, Swagelok) connecting the vapor generator to the chamber.

During the chamber pump-down process, the heaters around the vapor generator are activated to boil the DI water. However, at this stage, the valve connecting the vapor generator to the chamber remains closed. Once the chamber pressure drops below 5 Pa and the coolant inlet and outlet reach steady state readings, the condensation experiment commences. At this point, the vapor is introduced into the chamber with the pressure set by adjusting the position of the valve between the vapor generator and the chamber. A high-speed camera (Vision Research AMETEK Phantom VEO 710) is positioned in line with one of the viewports for visualization. The high-speed camera captures the moments of the falling droplet at a rate of 600 frames per second (fps). To optimize lighting conditions, the exposure time is set to 1000 µs, and the video size is adjusted to 1920 × 1080 pixels as the maximum possible resolution at 600 fps. A minimum of ten droplet-falling moments are captured for every test condition (i.e., for every tube and every heat flux). An example of the visual data is shown in Fig. 1(e).

Atmospheric condensation data are collected using a setup consisting of a closed coolant flow loop [Fig. 1(d)]. To maintain a precise working fluid temperature range of 5 ± 0.1 °C, a high-capacity chiller (Model No. 6160T21A130DDNT, PolyScience) is connected to the test tubes. Effective heating of the water was achieved using two ceramic heating plates (Model No. C-MAG HS 7 S001, IKA and Model No. SP88857100, Thermo Scientific) placed underneath the sample. The entire setup is enclosed within a controlled lab space, ensuring the generated vapor remains near the tube surface and expediting the condensation process. To quantify the flow rate and inlet temperature, a magnetic inductive flow meter (Model No. MIM-12 05G N4 C3T 0, KOBOLD) is installed just before the coolant line inlet. Visual observations are facilitated by a high-speed camera (Vision Research AMETEK Phantom VEO 710) positioned in line with the testing tube, capturing falling droplets at a sampling rate of 600 fps. Optimal lighting conditions are achieved by setting the exposure time to 500 µs, and the video size is adjusted to a maximum resolution of 1920 × 1080 pixels. At least twenty droplet-falling moments were captured for each test condition.

Generally, a self-supervised learning task is formulated based on a pre-training step to learn some proxy tasks from a large unlabeled dataset. Examples of proxy tasks are learning to classify the rotation angle, filling a masked area, or distinguishing between augmented views of the same image (a positive sample) or augmented views of other images (negative samples). The latter is known as contrastive learning.^37–39 After the pre-training step, a representation of the data is learned, which is fine-tuned based on limited labeled data for a segmentation task.⁴⁰ While they are strong self-supervised learning schemes, these methods have complexities such as high computational costs and large unlabeled dataset requirements, which are not always available depending on the scientific domain. Another simpler, yet strong, approach is to create synthetic data using data augmentation techniques to eliminate the manual data annotation step. The deep segmentation model is trained on synthetic data in a supervised manner and finally tested on real data.^41,42 Here, we take advantage of a small high-quality dataset and image processing techniques to create a new dataset with corresponding masks. Then, data augmentation techniques are used to create additional synthetic data with corresponding masks that are used to train a supervised U-Net⁴³ semantic segmentation model. In this context, high quality data are images with optimized lighting conditions and a uniform background containing high resolution droplets with boundaries readily visible by naked eye [Fig. 1(e)].

The input images obtained from the condensation videos are first cropped out to an area below the tube. Due to light reflection from the shedding droplet, the brightest pixel in the image is inside the droplet with a high probability. Therefore, we crop out an area of around 200 pixels in width and height around the brightest pixel location for every image in the “high-quality” dataset to create a new dataset. Every image in the new dataset includes a droplet surrounded by a small and uniform background region around itself (Fig. 2). The simplicity of the new dataset allows us to use simple clustering algorithms to filter out the background. Here, we first apply a Gaussian filter and a median filter to smooth the image and help eliminate noise from the image. Then, k-means clustering (k = 3 or 4) is applied to pixel intensity values. Given the uniform background, most background pixels have similar pixel intensities. However, intensities could greatly vary within the droplet area. The pixels inside the droplets are generally clustered into two or three clusters, while the background pixels are clustered separately, forming the biggest cluster around each droplet. A closing morphological transformation is used to fill out any pixels inside the droplet that are wrongly clustered as the background. Closing morphological transformation is a strong tool for filling holes inside the object boundary.⁴⁴ This step ensures that all the pixels inside the droplet are labeled as the foreground. Then, the biggest cluster (background) pixels are turned into 0, and the remaining pixel values are turned into 255 (e.g., maximum pixel intensity) to create masked droplet images. Although this method works on images with uniform, less noisy backgrounds and high-resolution droplets having distinguishable boundaries with the background, it fails in many cases if one or more of these criteria are not met. Therefore, a generalizable method applicable to images with different lighting and background conditions is required.

The U-Net semantic segmentation model has been successfully used for object segmentation in different fields such as biomedical engineering and thermofluidic sciences.^45,46 However, deep segmentation models require a large enough training dataset to learn from. Therefore, we first used image augmentation techniques on the reduced dataset to create a large synthetic dataset for training the segmentation model. Augmentation techniques such as noise addition (e.g., Gaussian noise and multiplicative noise), geometrical transformations (e.g., random rotation, vertical and horizontal flipping), and blur were applied to the initial images. These transformations were chosen not only to create the synthetic dataset and increase the dataset size but also to help with model generalizability so that the model learns to segment the droplets correctly even from noisy images, images with different lighting conditions, and images with non-uniform backgrounds. U-Net with the ResNet34 backbone⁴⁷ was used as the deep learning model, and the encoder weights were initialized using the model weights pre-trained on the ImageNet⁴⁸ dataset. The model was trained with the Adam optimizer and a step learning rate scheduler, decreasing the learning rate gradually. The initial learning rate was set at 0.0002. The model training results on the training and validation datasets are shown in Fig. 2.

The trained deep learning model successfully segmented droplets even from images where the k-means clustering method failed, showing that although k-means clustering was used to create initial masks from high-quality data, the deep learning model is more generalizable to other images with different backgrounds and noise conditions. The average intersection over union (IOU) on the test dataset was more than 0.95, showing good accuracy in estimating shedding droplet size. IOU, also known as the Jaccard index, is one of the most commonly used evaluation metrics in object detection and segmentation tasks and is defined as the ratio of the intersected area between the predicted and ground truth masks to the union area of the predicted and ground truth masks.⁴⁹ In order to eliminate the first image processing step (e.g., brightest pixel location), the predicted masks were mapped into the coordinates of the original full-sized images, and a new dataset, including the original full resolution images with their corresponding masks, was created. Then, a new deep learning model with the same network architecture was initialized with the weights of the previous model and re-trained with the new dataset. The new model achieved an IOU value of 0.9, which is reduced by 0.05 compared to the first model. However, this is expected as the new model makes predictions on full-sized images with more complexity. The second model training results on the training and validation datasets are shown in Fig. 2. The summary of our self-supervised methodology is shown in Fig. 3.

To study the effect of surface morphology on droplet shedding behavior, we collected data from a variety of surfaces with different surface wettability and roughness. The surfaces include hydrophilic smooth copper (Cu), micro-structured copper oxide (CuO), smooth Cu coated with a hydrophobic Parylene C coating, smooth Cu coated with a hydrophobic self-assembled monolayer (SAM) coating, and micro-structured CuO coated with a hydrophobic Parylene C coating having different coating thicknesses resulting in different surface wettability. Parylene C was deposited on the tubes by chemical vapor deposition (CVD) using a Specialty Coating Systemes Labcoater 2 system, resulting in a conformal coating on the surface.⁵⁰ The micro-structured CuO surfaces were fabricated through the oxidation of Cu in a basic solution at elevated temperatures. Fabrication details are available in previous studies.^5,51 Heptadecafluorodecyl-trimethoxy-silane (HTMS) was deposited on the Cu surface as the hydrophobic SAM coating. The deposition was performed using chemical vapor deposition at atmospheric pressure. More details about the HTMS deposition procedure are available in previous work.⁴ Details of all tested surfaces are shown in Table I. All coatings were applied to tubes with outer diameters of d_o = 6.35 mm (0.25 in.) and d_o = 12.7 mm (0.5 in.) and tube lengths of L = 26.7 cm (10.5 in.).

The Parylene C coating provides a uniform coating on the Cu surface as it is characterized by lower contact angle hysteresis and, therefore, weaker droplet adhesion to the surface. However, the surface wettability of CuO is dependent on the thickness of the deposited Parylene C layer. Depositing a 1 µm thick Parylene C film results in advancing contact angles greater than 90° (hydrophobicity); however, the contact angle hysteresis is relatively high, showing strong adhesion forces between the droplet and the surface. This is due to the underlying rough surface, which a thin Parylene C layer is not able to fully fill in. Increasing the coating thickness allows for the filling of all gaps between the structures and the formation of a more uniform layer on top. As depicted in Fig. 4, scanning electron microscopy (SEM) images of the cross-section of the CuO substrate with different thicknesses of Parylene C reveal different surface roughness on each surface. Focused ion milling (FIB) was used to cut through the coating cross-section to observe both the CuO/Parylene interface as well as the surface roughness on each sample.

The shedding droplet size results are shown in Fig. 5. Shedding size is strongly dependent on surface wettability and droplet-surface adhesion forces. Tubes with no hydrophobic Parylene layer have higher average droplet shedding diameters. FWC happens on both bare Cu and CuO surfaces. While the CuO surface is superhydrophilic, resulting in higher adhesion forces, the shedding droplet size is slightly smaller than that observed on the bare Cu surface in atmospheric condensation at low heat fluxes. This contradicts the observations of pure steam condensation at higher heat fluxes. During pure steam condensation in the absence of NCGs and at high heat fluxes, the droplet shedding diameter is slightly higher on the superhydrophilic CuO surface compared to the hydrophilic Cu surface. This is due to the higher droplet adhesion to the CuO surface. During atmospheric condensation and at very low heat flux values (<5 kW/m²), a mixture of filmwise and dropwise condensation is observed on the hydrophilic Cu. In this mode, there are several hanging droplets that grow to near critical gravitational shedding departure size. These droplets sometimes coalesce and form larger shedding droplets. In addition, condensate film thinning might happen on the micro-structured surface at very low heat fluxes observed in atmospheric condensation,^12,53 which might further reduce the shedding droplet size on the CuO surface. Among the hydrophobic surfaces, decreasing the contact angle hysteresis generally decreases the shedding droplet size. The Cu-2P and CuO-4P samples have the most uniform hydrophobic coating, resulting in the lowest droplet shedding size, followed by CuO-2P and CuO-1P. A similar trend is observed in the contact angle hysteresis (Table I). Lower contact angle hysteresis is an indicator of lower droplet-solid adhesion force and higher surface chemical and topographical homogeneity.^54,55 While the Cu surface coated with HTMS is hydrophobic, it only shows DWC at low heat fluxes. At larger heat fluxes, this surface shows a mixture of FWC and DWC with FWC component increasing with heat flux. Therefore, we demonstrated its shedding results along with other FWC results. Furthermore, it should be noted that the error bars are larger for rougher hydrophobic surfaces (e.g., CuO-1P). The error bars show the random uncertainty of the variable rather than measurement errors.

During DWC, the driving forces for shedding can differ. In its simplest form, a droplet grows to a certain size while hanging from the bottom of the tube and sheds from the surface as soon as the gravitational forces become larger than the adhesion forces (gravity-driven). However, droplets could coalesce and shed because of the coalescence momentum (coalescence-driven). The coalescence could happen while droplets are still on the surface (not hanging from the bottom of the tube), and the merged droplet travels down and spontaneously sheds from the surface (without any hanging stage). On the other hand, several hanging droplets could coalesce and create a large enough droplet shedding from the surface. In another mechanism, it is possible that a hanging droplet and a traveling droplet coalesce, and the merged droplet sheds from the surface due to the coalescence momentum and gravitational forces when the merged droplet is large enough. Given the potential variability in the shedding-triggering element, the shedding droplet size also varies for each individual droplet on the same surface. On surfaces having more uniform coatings (e.g., Cu-2P), the probability distribution of shedding happening due to different triggering elements is skewed toward one or two of them. Our visualization results show that on uniform coatings, shedding mainly happens by hanging droplets falling by themselves or by two hanging droplets coalescing, with three droplet coalescence rarely being the driving force for shedding. On the other hand, the probability distribution function for shedding driving forces is more uniform on rough hydrophobic surfaces, and all the discussed triggering elements are observed occasionally during condensation.

Our results also show shedding droplet size dependency on the condensation heat flux during both FWC and DWC. During FWC on Cu and CuO coated tubes, shedding size slightly increases with heat flux and rapidly plateaus (e.g., at q″ ∼ 60 kW/m²). The shedding droplet size dependency on heat flux is more pronounced for DWC. At higher heat fluxes, condensation happens at higher rates, and the chance of smaller droplets coalescing is higher. The coalesced droplet travels down and washes small hanging droplets with itself. Similarly, the resulting merged droplet is also small and sheds from the surface due to the momentum caused by the coalescence or due to the droplet being slightly larger than the critical size. Here, the droplet critical size is defined as the size where the gravitational force and adhesion force on the droplet are equal. At lower heat fluxes, droplet growth is slower, and droplets generally grow to a larger size before coalescence. In most cases, several hanging droplets grow to near the critical size before coalescence and shed at sizes larger than the critical size. At very low heat fluxes (q″ < 40 kW/m²), the average shedding droplet size slightly decreases as more droplets shed only due to gravitational force (no coalescence), resulting in smaller shedding droplets. The heat flux value at which the trend changes is dependent on surface morphology (e.g., wettability and roughness) and the tube size. On smaller tubes, this heat flux value is smaller and may not be observed within the data points [Fig. 5(d)] as opposed to larger tubes [Fig. 5(e)], where the data points clearly indicate an initial increase in droplet size and a decreasing trend afterward.

Following a single shedding droplet trajectory in consecutive frames, starting from the departure frame until the final frame where the droplet is completely visible, the shedding droplet initial speed in the direction of gravity was estimated using the equations for falling objects subject to gravitational force.⁵⁶ For every droplet, the droplet distance traveled is estimated from consecutive frames (∼20 frames) by passing the frame images into the model, obtaining the masked droplet, and specifying the center of the mask (e.g., the center of the droplet) as the droplet location (Fig. 6). The shedding speed results are shown in Fig. 6. In general, shedding droplets have higher speeds on larger tubes. However, the speed differences based on tube size are more pronounced for the atmospheric condensation experiments when compared to the pure steam condensation experiments. During atmospheric water vapor condensation, the speed trend is similar to the shedding droplet size, with more uniform surfaces with lower roughness and contact angle hysteresis (e.g., Cu-2P) having lower speed compared to rougher surfaces with higher contact angle hysteresis (e.g., CuO-1P). The only exceptions are the HTMS coated tubes (Cu-HTMS), which have lower speeds compared to CuO-1P but higher shedding sizes.

As shown in Fig. 6, the shedding speeds have smaller variations with heat flux in pure steam condensation when compared to droplet shedding sizes. However, they generally follow the same trend observed in the shedding droplet size data. For example, increasing the surface roughness (e.g., CuO-1P) increases the shedding speed. Similarly, speeds have a stronger dependency on heat flux during DWC when compared to FWC. On uniform surfaces (e.g., Cu-2P), this dependency is weaker when compared to rougher surfaces and can be neglected. During DWC, the droplet speed varies depending on the cause of shedding. Droplets shed due to direct growth and gravitational forces overcoming the droplet-surface adhesion force. These gravity-driven shedding events exhibit slower droplet departure speeds when compared to coalescence-driven droplet shedding events, which are momentum mediated. Typically, the frequency of coalescence-driven shedding events surpasses that of gravity-driven shedding, especially at higher heat fluxes. While our measurements were taken randomly at each heat flux value, it is possible that the ratio of coalescence-driven shedding to gravity-driven shedding may not represent the true ratio if hundreds of data were taken. In these few instances, fluctuations are observed in the results. Although these fluctuations are generally lower during FWC, it is still possible to observe them as the size of shedding droplets could slightly vary for each measurement, resulting in different speeds.

The models used were ridge regression (RR),⁵⁷ random forest (RF),⁵⁸ extreme gradient boosting (XGB),⁵⁹ and artificial neural networks (ANNs).⁶⁰ All the models were trained on 80% of the data and evaluated on the remaining 20%. The RR, RF, XGB, and ANN models achieved MAEs of 0.302 (6.24%), 0.085 (1.74%), 0.108 (2.29%), and 0.149 (3.03%), respectively.

The RF model showed the best performance in predicting the shedding droplet size. Therefore, we chose RF model to further predict the shedding droplet speed as well. Three different RF models were trained by varying the input variables. In the first model, the experimentally measured shedding droplet size was added to the previously mentioned inputs, resulting in an MAE of 0.0164 (6.48%). In the second model, the experimentally measured droplet sizes were replaced with the shedding sizes predicted from the first model, achieving an MAE of 0.0145 (6.22%). This could be due to the smaller fluctuations in the predicted values compared to the actual experimental values. In the third model, the droplet diameter data were removed from the input, and MAE was reduced to 0.0138 (5.37%), showing that using a dependent variable (droplet size) as an independent variable would degrade the shedding speed prediction accuracy. Therefore, the RF models showed excellent accuracy in predicting the shedding size and speed based on surface properties, tube size, condensation mode, and condensation heat flux. Examples of shedding droplet size and speed prediction results from the RF models during both DWC and FWC modes are shown in Fig. 7.

While showing promising capability for predicting droplet dynamics during FWC and DWC, it should be noted that the current models are trained on a relatively small dataset (∼200 data points), which is created by conducting expensive and time-consuming condensation experiments. The results obtained from this work demonstrate the potential power of data-driven machine learning based models in predicting droplet dynamics. However, more data from a variety of surface conditions and tube geometries are needed for inclusion in the training process to develop a generalized model. The model could help in designing optimized surfaces for enhanced condensation. For example, there is high interest in the thermo-fluids and nanoengineering research communities to design biphilic surfaces with varying wettability.^61–63 Different biphilic surfaces have different droplet dynamics and heat and mass transfer performance, accordingly.^10,64 In addition to droplet shedding size and speed, droplet shedding and removal frequency are other important parameters that are directly related to condensation heat transfer.^1,65 Droplet shedding frequency refers to the number of droplets shed from the surface per unit time at specific heat flux values.¹ Droplet removal frequency denotes the number of removed droplets during a given period of time in a given area.⁶⁵ It is noteworthy that to obtain precise time-averaged values, sufficiently long condensation videos should be captured at each heat flux value. Future research efforts can extend this study to the prediction of droplet shedding and removal frequency, advancing our understanding of droplet dynamics during condensation.

Furthermore, jumping droplet condensation on superhydrophobic surfaces could be added to the FWC and DWC modes studied in the current work. A generalized data-driven droplet dynamics predictive model including condensation on biphilic and superhydrophobic surfaces could greatly ease the design optimization of these surfaces. In addition to collecting real data from complex experiments, synthetic data could be made through generative models.^66,67 Generative models are trained to create images similar to real-world images. After training, the model could be used to create synthetic data and increase the training dataset size, helping with generalizing the model. While promising in many research disciplines, further investigations and studies are required to align the generative models with physical constraints occurring in complex physics-based problems such as phase change heat transfer.

The recent developments in computer vision and machine learning models have created an opportunity to expedite our comprehension of the fundamental understanding and modeling of phase change heat and mass transfer. Object segmentation and tracking models have provided great insight into the statistical analysis of condensation and boiling phenomena by quantifying bubble and droplet statistics with reasonable resolution in time and space. The bottlenecks for training these models are the data annotation or labeling steps. Previous studies using computer vision models relied on manual data annotation for preparing the dataset. Manual data annotation requires intensive labor, especially for training object segmentation models. In addition, the trained model may not generalize to other datasets with different underlying statistical distributions. Self-supervised models aim to solve this drawback by removing or minimizing the manual data annotation step. Most self-supervised models are first trained to learn a representation of the data based on a proxy task through data augmentation techniques. These models are mostly computationally intensive, and they further require a limited training step with labeled datasets. Here, we avoid these computational difficulties and data annotation steps by using a hierarchical model to learn from a small, high-quality, reduced dataset and transfer the learning to a final larger dataset created by data augmentation techniques. In the last step, the droplets and masks are mapped into the original coordinates of the full-sized images, and the segmentation model is re-trained with the full-sized dataset. The initial high-quality dataset allows taking advantage of classical image processing and unsupervised clustering techniques to automatically create masked objects out of the images, which are further used in a data augmentation step to create a larger dataset to train a U-Net segmentation model. The final model can segment shedding droplets from condensation videos with an average IOU of more than 0.9.

The shedding droplet size and speed results collected from the self-supervised computer vision model revealed dependency on tube size, condensation mode (e.g., FWC or DWC), surface properties, and condensation heat flux in pure steam condensation. Shedding droplet size and speed are generally higher on rougher surfaces when compared to more uniform surfaces. The shedding droplet size generally decreases with increasing heat flux during DWC. However, the trend is the opposite for FWC, with a smaller dependency on heat flux. Shedding speeds have a slight variation with heat flux during FWC; however, they follow the same trend as shedding size. The results show great complexity in analyzing the droplet dynamics on a variety of surfaces. We developed a random forest model to predict the shedding droplet size and speed given the droplet advancing and receding contact angle, tube size, condensation mode (FWC or DWC), and condensation heat flux with acceptable accuracy. Although trained on a relatively small dataset containing data from condensation on superhydrophilic, hydrophilic, and hydrophobic tubes, the models achieved mean absolute errors of 1.74% and 5.37% in predicting average shedding droplet size and speed on an unseen test dataset. These results show a promising opportunity for using data-driven machine learning based modeling to predict the droplet dynamics in phase change processes.

The authors gratefully acknowledge funding support from the Office of Naval Research under Grant No. N00014-16-1-2625, the National Science Foundation under Award No. 1554249, and the Air Conditioning and Refrigeration Center. N.M. gratefully acknowledges funding support from the International Institute for Carbon Neutral Energy Research (WPI-I2CNER), sponsored by the Japanese Ministry of Education, Culture, Sports, Science and Technology.

The authors have no conflicts to disclose.

S.K. and P.K. contributed equally to this paper.

Siavash Khodakarami: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (lead); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Pouya Kabirzadeh: Conceptualization (equal); Data curation (supporting); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Nenad Miljkovic: Conceptualization (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (lead); Writing – review & editing (equal).

The initial dataset including samples of high-speed imaging of atmospheric and pure steam condensation and the codes are provided via GitHub at: https://github.com/SiaK4/SSL_DropletDynamics.